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Establishment Probability in Fluctuating Environments: A Branching Process Model
Theoretical Population Biology TP1284 theoretical population biology 50, 254280 (1996) article no. 0031
Establishment Probability in Fluctuating Environments: A Branching Process Model Patsy Haccou Section Theoretical Biology, Institute of Evolutionary and Ecological Sciences, Leiden University, P.O. Box 9516, 2300 RA Leiden, The Netherlands
and Yoh Iwasa Department of Biology, Faculty of Science, Kyushu University, Fukuoka, 812, Japan Received June 9, 1995
We study the establishment probability of invaders in stochastically fluctuating environments and the related issue of extinction probability of small populations in such environments, by means of an inhomogeneous branching process model. In the model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period, individuals have (independent) Poisson distributed numbers of offspring. The expected numbers of offspring per individual are independently identically distributed over the periods. It is shown that the establishment probability of an invader varies over the reproduction periods according to a stable distribution. We give a method for simulating the establishment probabilities and approximations for the expected establishment probability. Furthermore, we show that, due to the stochasticity of the establishment success over different periods, the expected success of sequential invasions is larger then that of simultaneous invasions and we study the effects of environmental fluctuations on the extinction probability of small populations and metapopulations. The results can easily be generalized to other offspring distributions than the Poisson. 1996 Academic Press, Inc.
INTRODUCTION Success of invasion to an empty habitat is an important life history characteristic of many organisms. So-called ``r-selected'' species are considered to have life history traits that lead to a higher success of invasion to a novel habitat (MacArthur and Levins, 1967; MacArthur, 1972). Populations of many species are maintained by recurrent events of extinction of local populations and subsequent invasion from other 254 0040-580996 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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populations (e.g., Hanski, 1992). To understand the dynamics and longterm persistence of a metapopulation, we need to know the establishment success of invaders as well as the extinction probability of small populations. Of course, these two issues are intimately related. Invasion success is also important in an evolutionary context, because any novel trait must arise initially as rare mutations. Thus, the establishment of new traits in the long run may be influenced by the dynamics at small population size. In addition, many evolutionary stability concepts are defined in terms of invasion probability (Maynard Smith, 1982; Charlesworth, 1994). The fate of invaders (or mutants) and small populations can be expected to be strongly affected by demographic stochasticity. This is usually modelled by means of branching processes or birth-and-death processes, in which it is assumed that the environment is constant between generations (Goel and Richter-Dyn, 1972; Eshel, 1981, 1984; Charlesworth, 1994). Several theoretical studies have shown that environmental stochasticity may lead to profoundly different population dynamics than observed in constant environments. For instance, in fluctuating environments so-called ``bet-hedging'' may occur (Slatkin, 1974; Seger and Brockmann, 1988; Philippi and Seger, 1989). Effects of environmental stochasticity, however, are usually studied by considering long-term expected growth rates in large populations, ignoring demographic stochasticity. (See, e.g., Levins, 1968; Lewontin and Cohen, 1969; Schaffer, 1974; Tuljapurkar, 1990; Yoshimura and Clark, 1991; Iwasa and Haccou, 1994; McNamara, 1995; Haccou and Iwasa, 1995; Sasaki and Ellner, 1995.) Establishment success in stochastic environments has been studied for low population densities. Keiding (1975), Ludwig (1976), and Turelli (1980) used diffusion approximations of inhomogeneous branching processes for this purpose. Chesson (1982) and Chesson and Ellner (1989) developed the idea of stochastic boundedness. According to this criterion a population can establish itself in the long run if the asymptotic population density is bounded from below by a positive random variable. Whereas relative densities of mutants can be low in these models, the numbers of individuals are still assumed to be infinitely large. When population size is large, the effect of demographic stochasticity on population growth rate is small compared to that of environmental stochasticity (Keiding and Nielsen, 1973; Yoshimura and Clark, 1991). However, in small populations, effects of demographic stochasticity cannot be ignored. On the other hand, there are indications that environmental fluctuations may have large effects on small populations. For instance, Hanski (1992) found strong empirical evidence of such effects. Iwasa and Mochizuki (1988) showed theoretically that occasional catastrophic events can considerably increase the extinction chance.
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The combined effects of environmental and demographic stochasticity can be studied by means of the inhomogeneous branching process, which was first examined by Smith (1968) and Smith and Wilkinson (1969). Mountford (1971, 1973) was the first to use the model in biological research. In this model both sources of stochasticity are incorporated. First, the expected numbers of offspring per individual, here denoted by m t (t=1, . . .), are independently identically distributed over reproduction periods. Second, within reproduction period t individuals have stochastic numbers of offspring, which are independently identically distributed, conditional on m t . Athreya and Karlin (1971a, b) examined the model in detail and proved that the establishment probability of an initially rare mutant is positive if and only if the geometric mean of m t is larger than one. This conclusion was extended to the case with correlated environmental fluctuation, and with multiple state branching models (Athreya and Karlin, 1971a, b). These results, although very general, do not give information about the magnitude of the establishment success of invaders. It is important, however, to be able to estimate this parameter, e.g., when studying metapopulation dynamics, co-evolution, in the control of foreign organisms, or in life-history studies. Wilkinson (1969) gives an approximation for the establishment success, which we briefly describe in this paper. However, we found that for this approximation to be sufficiently accurate, extensive computations are needed. We give an alternative approximation method, which is easier to calculate and quite accurate. Although it is derived for small environmental variability, it appears to work well for relatively high environmental variances too. Its results are compared with Wilkinson's approximations. Furthermore, we give a method for estimating establishment success from simulations. The simulation method is based on the mathematically derived, distributional properties of the extinction probabilities and avoids having to simulate the actual branching processes themselves. An important difference between constant and fluctuating environments is that, whereas in a constant environment the probability of a successful invasion is fixed and does not vary over time, in a fluctuating environment the fate of a newly arrived individual strongly depends on the timing of invasion. Since there is a large probability of extinction in the first generations, an invasion in a favorable period has a much higher chance of eventual success than an invasion during an unfavorable period. This implication has been noted in the mathematical literature (e.g., Athreya and Karlin, 1971a, b), but the consequences for biological applications have not been fully understood until now. One such consequence is that sequential invasions have a higher probability of success than simultaneous ones. This has large implications for invasion strategies, biological control strategies, and metapopulation dynamics.
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As an example, we consider an inhomogeneous branching process model in which the number of surviving offspring follows a Poisson distribution, with an expectation that fluctuates independently between years. After describing the simulation method and approximations for the establishment probability, we examine the effects of different modes of invasion (e.g., sequential versus simultaneous) on establishment probabilities. The results can be translated directly into consequences for survival probabilities of small populations. Our conclusions are also valid for other offspring distributions than the Poisson, and the simulation method as well as the approximation can easily be generalised in this respect.
A MODEL FOR DEMOGRAPHIC STOCHASTICITY IN FLUCTUATING ENVIRONMENTS We assume a simple life-cycle, where all individuals reproduce asexually during separate reproduction periods. Furthermore, individuals produce offspring independently of one another. Let N t&1 be the population size just before period t. We consider the probability of successful invasion if the population starts from a single individual. Hence N 0 equals one. For ease of notation, we will use extinction probabilities rather than establishment probabilities (which can easily be calculated from these) in our equations. Before we describe the full model, in which both demographic stochasticity and fluctuating environments are included, we will discuss the classical models used for studying evolution in the presence of, respectively, fluctuating environments and demographic stochasticity. Case a: Fluctuating Environment, No Demographic Stochasticity Evolution in stochastic environments is usually studied in infinitely large populations, where effects of demographic stochasticity are ignored. This is the situation considered, e.g., by Schaffer (1974), Yoshimura and Clark (1991), McNamara (1995), Haccou and Iwasa (1995), and Sasaki and Ellner (1995). Let m t denote the expected number of offspring per individual in period t. It is assumed that the m t are independent and identically distributed. According to this model, t
N t =m tN t&1 = ` m j N 0 .
(1)
j=1
This model only has a biological interpretation for large population sizes, when effects of individuals differences in offspring numbers can be ignored. Therefore, it cannot be used for studying extinction probabilities of small populations andor establishment probability of small numbers of invaders.
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However, for comparison with later results, we will consider what happens in this model if N 0 equals one and define the ``extinction probability'' in this case as the chance that N t becomes less than one. Although this is biologically uninterpretable, it provides a way to compare the model with the more realistic (branching process) models considered later. Consider the log-population size: t
log(N t )= : log(m j ).
(2)
j=1
Since the m j are independently identically distributed, the process [log(N t )] is a random walk. Let m~ denote the geometric mean of m t , m~ =exp[E[log(m t )]].
(3)
From the theory of random walks (see, e.g., Feller, 1971, Section XII.7) we can infer that the ``extinction probability'' of the process equals:
_
1 Pr[log(N t )<0] , t t=1
p a =Pr[inf log(N t )<0]=1&exp & : t
&
(4)
and that the following relations hold: m~ 1 p a =1, m~ >1 p a <1.
(5)
Consider the situation where m~ is larger than one. Then, according to (5), the extinction probability is less than one. However, note that, given a realization of the process [m 1 , m 2 , . . .] extinction occurs either with probability one or with probability zero. Thus, p a corresponds to the expectation of the extinction probability over different realizations of [m 1 , m 2 , . . .]. Case b: Constant Environment, Demographic Stochasticity The classical model for studying extinction probability in a constant environment is the GaltonWatson branching process. It is assumed that in each period, individuals have independent identically distributed numbers of offspring, with a fixed expectation. Let Y j, t be the number of offspring of the j th individual in period t. The population size after reproduction is Nt&1
N t = : Y j, t ,
with Y j, t i.d.d. for all j and t.
j=1
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(6)
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Let Y be a random variable with the same distribution as the Y j, t . The extinction probability of the process is implicitly defined by
p b = : Pr[Y=k] p kb ,
(7)
k=0
where the smaller root should be chosen if there are two positive roots. The following relations hold: E[Y] 1 p b =1
(8)
E[Y]>1 p b <1 (see, e.g., Feller, 1968, Chap. XII). Whereas in case a, conditional on a fixed sequence of environments, extinction occurs with either probability one or zero, note that in case b, although there is no environmental fluctuation, extinction does not occur with probability one or zero if E[Y] is larger than one, due to the stochasticity in offspring numbers. p b corresponds to the expected proportion of processes that goes extinct if we consider infinitely many processes starting with one individual. Case c: Fluctuating Environment and Demographic Stochasticity (Full Model ) We will now combine the models of cases a and b. As in case b, let Y j, t be the offspring of the j th individual reproducing in period t. We now assume that the expected number of offspring per individual reproducing in period t equals m t . Similar to case a, we let the m 1 , m 2 , . . . be independent identically distributed variables and, similar to case b, we assume that conditionally on m t , the Y j, t ( j=1, ..., N t&1 ) are independently identically distributed. Thus combining the models of cases a and b gives Nt&1
N t = : Y j, t ,
with Y j, t , ..., Y j, N t&1 i.d.d. for each t, E[Y j, t | m t ]=m t .
j=1
(9) This is the inhomogenous branching process which was first considered by Smith (1968) and Smith and Wilkinson (1969). We will now consider the conditional probability of extinction: p=Pr[extinction | m 1 , m 2 , . . .].
(10)
Athreya and Karlin (1971a) proved that if the conditions |E[log(m t )]| < E[ |log(1&Pr[Y j, t =0 | m t ])| ]< ,
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(11)
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hold, we have the relations m~ 1 Pr[ p=1]=1 m~ >1 Pr[ p<1]=1,
(12)
where m~ is as defined in Eq. (3). (Their result is more general, but here we only need it in this restricted form). We will assume from now on that the conditions specified in Eq. (11) do hold. Note that there is an important difference with cases a and b, namely, whereas the extinction probabilities p a and p b (cf. Eqs. (4) and (7)) are constants, in case c the extinction probability is a random variable, which is nondegenerate if m~ is larger than one. Thus, if m~ is larger than one, p varies over different possible realisations of [m 1 , m 2 , . . .], as p a varies over different realisations. However, as opposed to case a, there is variation in offspring numbers as well. Therefore, as in case b, extinction does not occur with probability one or zero. For a given realisation, p has the same interpretation as p b . The stochasticity of p has additional consequences which we will discuss below. Time Dependence of Extinction Probability in Case c Now we take a slightly different view and consider the fate of a single invader arriving just before period t. To study this, we define the extinction probability, conditional on the future environmental process, p t =Pr[extinction | m t , m t+1 , . . .];
(13)
then p t =Pr[zero offspring at t | m t , m t+1 , . . .] + : Pr[k offspring at t, which have no progeny k 1
in the long run | m t , m t+1 , . . .]
= : Pr[Y=k | m t ] p kt+1 ,
(14)
k=0
where Y denotes the random number of offspring of the individual. Now, consider the different situations discussed above. In case a, the conditional extinction probability, p t , defined in Eq. (13), is either zero or one and thus does not vary with t. In case b, m t equals m for all t, since the environment does not vary, so in Eq. (14) p t = p t+1 = p b for all t. In case c, however, p t varies with t, unless extinction is certain (m~ 1) in which case it equals
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one for every t. If m~ >1, p t is less than one and changes in time. It follows directly from the ergodicity of the process [m 1 , m 2 , . . .] that 1 T : p t =E[ p] T T t=1 lim
a.s.,
(15)
where p is as defined in Eq. (10) (see, e.g., Karlin and Taylor, 1975, Section 9.5). Similar conclusions can be derived for the other moments; for instance, the variance of p t over time corresponds to the variance of p. Equation (14) proves to be very useful for simulating the distribution of p as well as for deriving approximations for its moments. We will illustrate this for the case where the numbers of offspring have a Poisson distribution, with m~ >1.
THE POISSON BRANCHING-PROCESS MODEL IN A FLUCTUATING ENVIRONMENT We assume now that conditionally on m t , the Y j, t are Poisson random variables: Pr[Y j, t =k | m t ]=e &mt
m kt k!
(k=0, 1, . . .).
(16)
Then, using Eq. (14) we get
p t = : e &mt k=0
m kt k p =e &mte mtpt+1 O p t =e &mt(1& pt+1 ). k! t+1
(17)
From the results of Athreya and Karlin (1971a) it follows that p, p t , and p t+1 have equal marginal distributions. Furthermore, since the m t are independently distributed, p t+1 and m t are independent random variables. Let .( p) denote the probability density function of p and h(m t ) that of m t , then, after some calculations this leads to p.( p)=
|
0
1 1 . 1+ log( p) h(m) dm m m
\
+
(18)
(see, e.g., Springer, 1979). Unfortunately, Eq. (18) cannot be solved explicitly (for numerical methods, see, e.g., Kondo, 1991, Chap. 7). However, there are several ways in which Eq. (17) can be used to derive the distribution of p.
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First, note that Eq. (17) simply describes a recurrence relationship for a sequence of random variables p t , with values between zero and one. Based on this equation, sequences of p t values can be simulated, without reference to the underlying branching processes, through backward iteration: given a certain value of p t+1 , p t can be generated by drawing m t from a given distribution and subsequently using Eq. (17). In Appendix A it is shown that as t tends to minus infinity the sequence converges with probability one to a random variable. Since (18) must hold for this random variable as well as p, and furthermore, has a unique solution (Athreya and Karlin, 1971a), they must be equal with probability one. Thus, to simulate p, we start with an array of nonzero p 0 's, simulate an array of values of m &1 according to a certain distribution, calculate the values of p &1 , etc., until the distribution is stable, and estimate the moments from the resulting array of values of p t . Product moments of the sequence of p t values, such as p t p t+1 p t+2 , can be estimated from sequences of such arrays. Alternatively, moments can be calculated from the time averages of a sequence of p t values simulated in the same way (see, e.g., Eq. (15)). However, the first method provides more accurate estimates, since the different p t values within the array are independent, whereas in the second method, autocorrelations between subsequent p t values can inflate the variance of the estimators. Second, approximations for the moments of p can be derived from the fact that similar relationships as given in Eq. (18) hold for these. Define the function f (m, p)=e &m(1& p);
(19)
E[ p]=E[ f (m, p)]
(20)
Var[ p]=Var[ f (m, p)].
(21)
then
and
Below, we will use Eqs. (20) and (21) to derive approximations for the expectation of p and compare these with simulation results.
APPROXIMATIONS FOR THE EXPECTED EXTINCTION PROBABILITY For convenience, we will denote the expectation of m, E[m], by m and the expectation of p, E[ p], by p. Furthermore, we denote the extinction
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probability in a constant environment where the mean number of offspring per individual is equal to m, by p A . This extinction probability is implicitly defined by p A = f (m, p A )
(22)
(see, e.g., Feller, 1968, Chap. XII). For any value of m, p A can easily be calculated by numerically solving (22). From Eqs. (19), (20), and (22), together with Jensen's inequality, it can easily be proved that p is larger than p A (see Appendix B). One way to derive an approximation for p is to expand f (m, p) (defined in Eq. (19)) around p= p A and m=m (see below). However, for values of m~ near one, this approximation may not be so accurate, since m can still be much larger than one and, as a consequence, p A will be much smaller than one. However, p must be near one for m~ near one. We therefore also examined an approximation based on expansion of f (m, p) around m=m~. Let p G be the extinction probability in the branching process with a constant mean number of offspring equal to m~: p G = f (m~, p G ).
(23)
Since p G equals one when m~ is less than or equal to one, we expect that the values of p G lie closer to p than those of p A . This was examined through simulations. Figure 1 shows the establishment chances (1& p ), estimated from simulations where m is uniformly distributed over an interval for several values of m and two values of Var[m]. The estimates were based on an array of 10,000 p t 's. The widths of the 950 confidence intervals of these estimates are at most 0.005 on both sides. (They are not given in any of the figures, since they either do not show up on the scale used, or they obscure the picture.) We used a value of t=500. Previous investigations showed that distributions are stable after this value of t. Simulations were done with the APL 68000 random generator on a Macintosh PowerPC. As can be seen from Fig. 1, environmental variability considerably reduces the establishment chance of invaders. Furthermore, the values of p G are indeed much closer to p than those of p A . We will now describe different approximations of p and compare their accuracy. Approximation Based on the Arithmetic Mean of m We assume that the variance of m is small, so that for every k>1, E[(m&m ) k ] and E[( p& p k ] are both O(= k ). To derive an approximation
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Fig. 1. Establishment probabilities in a stochastic environment with m uniformly distributed, 1& p( v ), in a constant environment with expected number of offspring equal to m, 1& p A(Q), and in a constant environment with expected number of offspring equal to m~, 1& p G ( b ).
.
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.
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for p, we use the Taylor-expansion of f (m, p) around m=m and p= p A , together with Eqs. (20) and (21) (details are given in Appendix C). This leads to p = p A +
1 2f 2f ( f m) 2 Var[m] + +O(= 3 ), 2(1&( f p)) m 2 p 2 1&( f p) 2
\
+
(24)
where the derivatives are evaluated at m=m and p= p A . Substituting the values of the derivatives in Eq. (24) gives p = p A +
= pA +
Var[m] e &m(1& pA ) (1& p A ) 2 +O(= 3 ) 2(1&me &m(1& pA) ) 1&m 2e &2m(1& pA )
\
+
p A(1& p A ) 2 Var[m]+O(= 3 ). 2(1&mp A )[1&(mp A ) 2 ]
(25)
Approximation Based on the Geometric Mean of m Define z=log m
(26)
z =E[log m]; then z
f (m, p)= g(z, p)=e &e (1& p),
(27)
g(z, p)= f (m~, p)=e &m~(1& p).
(28)
We will now assume that E[(z&z ) k ], rather than E[(m&m ) k ], is of order = k. As before, we assume that E[( p& p ) k ] is of this order too. We now expand g(z, p) around z=z and p= p G (see Eq. (23)) and use the relations given in Eqs. (20) and (21) to derive approximations for the expectation of p. The derivation is completely analogous to that given in Appendix C for the approximation given in Eq. (25) and gives a similar result: p = p G +
m~(1& p G ) p G m~(1& p G ) &1 Var[z]+O(= 3 ). 2(1&m~p G ) 1&(m~p G ) 2
{
=
(See Appendix D for details.)
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(29)
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Wilkinson's Approximation Wilkinson (1969) gives a method for calculating approximations for p as well as the higher moments E[ p n ], which we will briefly describe. Let ? be a vector of the moments of p, ?=( p, E[ p 2 ], ..., E[ p n ]),
(30)
and let q also be a vector of length n with j th element equal to q( j)=Pr[N 1 =0 | N 0 = j]=E m[Pr[N 1 =0 | N 0 = j; m 1 =m]],
(31)
where N i is the population size in the ith generation. When the numbers of offspring are Poisson distributed, we have q( j)=E m[e & jm ],
(32)
since the convolution of j independent Poisson distributions with parameter m is again a Poisson distribution, with parameter jm (see, e.g., Manoukian, 1986, Section 4.13). Furthermore, let A be a n_n matrix with elements: A(i, j)=Pr[N 1 = j | N 0 =i]=E m[Pr[N 1 = j | N 0 =i ; m 1 =m]],
(33)
which in our case corresponds to A(i, j)=E m
_
(im) j &im e . j!
&
(34)
Wilkinson (1969) proved that ?=(I&A) &1 q+=,
(35)
where I denotes the identity matrix and = is at most equal to E[ p n+1 ]. Thus, (I&A) &1 q provides an approximation for ?, which increases in accuracy as n increases. Accuracy of the Approximations Figure 2 shows the accuracy of the approximations for p given in Eqs. (25), (29) and Wilkinson's approximation (Eq. (35), with n=100). The approximation given in Eq. (25) is fairly accurate for low variances of m, provided that m is relatively large. However, for values of m near one it does not work so well, especially if Var[m] is large. Wilkinson's approximation (Eq. (35)) works better than Eq. (25), but the approximation in Eq. (29) is by far the most accurate for values of m~ close to one,
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Fig. 2. Differences between p and its approximated values: (Q) based on m (Eq. (25)); ( v ) based on m~ (Eq. (29)); ( b ) Wilkinson's (1969) approximation (Eq. (35), with n=100).
.
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.
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especially when Var[m] is large. Increasing n would make Wilkinson's approximation more accurate, but the computations are already formidable at n=100.
CONSEQUENCES OF ENVIRONMENTAL VARIABILITY FOR INVASION SUCCESS The probability of success of a single invader at a given time, say t, corresponds to p t . The stochasticity of p t implies that the chance of successful invasion depends on the way in which invasions take place. The chances differ for simultaneous or sequential invasion at one site and for invasions at different sites. When there are n sequential invasions into a single site, the chance that at least one invasion is successful is
_
n
&
q seq =E 1& ` p t =1&E t=1
_
n
&
` pt . t=1
(36)
When n invaders arrive simultaneously, the chance of success is q sim =E[1&( p t ) n ]=1&E[ p n ].
(37)
The probability of at least one success if n invaders arrive at different sites which are independent with respect to the expected numbers of offspring equals n
q ind =1& ` E[ p t ]=1& p n.
(38)
t=1
Note that, in nonstochastic environments with expected numbers of offspring equal to m, all these chances would equal q const =1& p nA .
(39)
The differences between q seq , q sim , and q ind can have profound effects on invasion strategies. From Holder's inequality (see, e.g., Manoukian, 1986, Section 1.4) it follows that E
_
n
& \
n
` p t ` E[ p nt ] t=1
t=1
+
1n
=E[ p n ],
(40)
where the rightmost equality results from the fact that the p t are identically distributed. From Eqs. (36), (37), and (40) it can be concluded that, in a
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stochastically fluctuating environment, the probability of success of n sequential invasions is higher than that of a simultaneous arrival of n invaders. As shown in Fig. 3, the difference between the two increases rapidly with n. From Jensen's inequality it follows that q ind is higher than q sim , so invasions into different independent sites have higher chances of success than those into one or, equivalently, into perfectly correlated sites. Our simulation results indicate that q ind is higher than q seq (see Fig. 3). This implies that E
_
n
&
n
` p t >p n = ` E[ p t ] t=1
(41)
t=1
and, thus, in this case there appears to be a positive autocorrelation between the values of p t . It is not a trivial task, however, to examine analytically whether this is true in general. Increasing the times between
Fig. 3. Establishment chances of different modes of invasions: (_) q sim (Eq. (36)), ( v ) q seq (Eq. (37)), ( + ) q ind (Eq. (38)), (Q) q const (Eq. (39)), for m uniformly distributed, m =1.3, Var[m]=0.50.
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successive invasions will decrease the correlation between the p t values at different invasion times and therefore, in the case investigated here, it will increase the chance of success. This is illustrated in Fig. 4. Since q ind is much closer to q const than either q seq or q sim (Fig. 3), spatial heterogeneity apparently reduces the negative effects of temporal variation on establishment chances. To study the effects of spatial heterogeneity more closely, consider the establishment chances of n invaders into k independent sites (which provide identically distributed expected numbers of offspring). The establishment chances of, respectively, sequential and simultaneous invasions then become
\ _
q~ seq =1& E
n
` pt t=1
k
&+ ,
q~ sim =1&(E[ p n ]) k.
(42) (43)
It can easily be seen that as long as E[ p n ]
Fig. 4. Establishment chances of five sequential invasions, for increasing numbers of periods between invasions, for m uniformly distributed, m =1.3, Var[m]=0.50.
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Fig. 5. Establishment chances of 100 invaders divided over different numbers of independent sites: ( v ) sequential invasions (see Eq. (41)); ( b ) simultaneous invasions (Eq. (42)), for m uniformly distributed, m =1.3, Var[m]=0.50.
DISCUSSION The value 1& p corresponds to the chance that a single invader arriving at one site at a random time succeeds to establish itself in the long run. It also corresponds to the expected proportion of successful invasions if large numbers of invaders arrive at different times or at different sites which provide independent identically distributed expected numbers of offspring. Note, however, that we use an unbounded population size model. In practice, density-dependent processes will ultimately limit the size of populations. Therefore, the establishment probabilities that we define should be considered as approximations for the chances that populations will eventually reach a large and stable size and persist for a long time. Being able to estimate p is important for studying r-selection, biological control strategies, and metapopulation dynamics. For many evolutionary considerations, it will matter only whether or not there is a positive establishment chance. This depends on the geometric mean, m~ (Athreya and Karlin, 1971a). However, note that in co-evolutionary processes the relative chances of invasion success can make a difference. In such cases, an estimate of p might be needed. The approximation based on p G (Eq. (29)) is very accurate (Fig. 2). Approximations for offspring distributions other than the Poisson may be
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derived in a similar way, provided that an explicit expression can be found for f (m, p) (i.e., the right-hand side of Eq. (14)). Note that this function corresponds to the probability generating function of the offspring distribution in p t+1 (see Eq. (17)). These functions are listed in, e.g., Abramowitz and Stegun (1972, Chap. 26). An alternative is to use Wilkinson's (1969) approximation method. Although the method was originally derived for discrete m, it can easily be proved to be applicable directly to continuous m (as in our example). This approximation has the advantage that by increasing n, its accuracy can be increased to any value. However, especially for high values of n, the required computations are formidable. From simulation results it appears that for uniformly distributed m, n should be larger than 100 to approximate p with reasonable accuracy, especially if m~ is near one (see Fig. 2). From further investigations we found that the accuracy increases very slowly with n. As shown in Fig. 2, the approximation given in Eq. (29) works much better at low m~ than Wilkinson's approximation for n=100. Branching processes in random environments have also been studied by approximating changes in population densities with diffusion processes (Keiding, 1975; Ludwig, 1976; Kurtz, 1978; Turelli, 1980). However, such approximations can only be expected to be accurate for large population sizes. The k th moment of p, E[ p k ], corresponds to the extinction probability of a population of size k, so it can be useful to be able to estimate the higher moments of p. In Appendix D we give an approximation for the variance of p, based on p G . In principle, our approximation method can be generalized for higher moments, but it becomes very complicated. The method derived by Wilkinson (1969; see also Eq. (35)) can be used to approximate E[ p k ] up to any value of k. However, as shown in Fig. 2, the error in the approximation of the first moment of p can be quite large for low expectations of m. For higher moments, this problem will increase. Another method for estimating higher moments of p is to use simulations. Note that the fact that p t is stochastic has consequences for estimation of p from simulations. A single value of p t is not a good estimate of p, unless environmental variability is very low (compare Turelli, 1980). The simulation method can also be used for different offspring distributions than the Poisson, provided that an explicit expression for f (m, p) can be found. Athreya and Karlin (1971b) also proved their convergence theorem for autocorrelated m t . To apply a similar simulation method in that case, formally it has to be proved that here, too, the sequence of p t 's defined in Eqs. (13) and (14) converges to the random variable p defined in Eq. (10). However, the proof of convergence of p t given in Appendix A is based on the martingale limit theorem, which is very general. We expect that it can easily be adjusted for autocorrelated m t .
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The success of sequential invasions into one habitat is much larger then that of simultaneous invasions (Fig. 3, Eq. (40)). This finding has major consequences for biological control as well as the study of reproduction strategies, such as the production of offspring or phenomena such as dormancy and diapause. Although it has been noted before from theoretical modelling that random fluctuations in the environment can promote dormancy or the proportion of nonbreeding individuals (Cohen, 1966; Mountford, 1971, 1973), the connection with stochasticity of the probability of successful invasion in stochastic environments was not made. Our results show that this may be the major cause of such phenomena. Furthermore, since Holder's inequality, which is the basis for this result (see Eq. (40)) holds very generally, the conclusion that sequential invasion is advantageous does not depend on the offspring distribution, the distribution of the m t , or on the independence of the m t . When groups of invaders arrive sequentially, the relative advantage of sequential invasion will decrease with the group size. The effects of positive autocorrelation in the m t can be predicted by considering that in the limiting case where m t is constant and equal to m, all establishment probabilities given in Eqs. (36) to (38) will converge to q const . Thus, in general the success of invasion, whether it is sequential or not will be higher in positively correlated environments. Furthermore, the relative advantage of sequential when compared to simultaneous invasion will decrease. Our results indicate that, at least for independent homogeneously distributed m t and Poisson distributed numbers of offspring, q seq is less than q ind (Fig. 3), which implies that the p t are positively autocorrelated. Increasing the times between successive invasions will decrease the correlation between the p t values at different invasion times and therefore it will increase the chance of success. This is illustrated in Fig. 4. It remains to be examined however, whether the p t are in general positively autocorrelated for independent m t . If they are, the effect of negative correlation in the sequence of m t values would be to reduce the autocorrelation in p t values, which would make sequential invasion even more advantageous in such environments. Furthermore, it would imply that increasing the lag between successive invasions is advantageous in any nonperiodic temporally varying environment, since it decreases the dependence between the p t values at the different invasions times. The advantage of sequential versus nonsequential invasion also depends on the amount of information about the expected number of offspring in a given reproduction period, m t . If there is information about these values, it is a better strategy to produce large numbers of offspring in favorable years (see, e.g, Cohen, 1967). This might for example explain the occurrence of mast years in certain species versus continuous reproduction in others.
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If invasions occur in different, uncorrelated sites, the difference between success of sequential and simultaneous invasions becomes much less, since environmental heterogeneity induces an element of independence between success probabilities at any given time (see Eqs. (41) and (42) and Fig. 5). So, for instance plant species with seed dispersal over long distances, relative to the scale of spatial variation in the environment, can be expected to show less dormancy than those with seed dispersal over relatively short distances (given that there is temporal environmental variability). Many previous conclusions in the literature, derived from specific models, which are much more complicated than the one considered by us, can be understood from our general results concerning the relation between invasion modes and establishment success (cf. Eqs. (36) to (43)). For instance, Levin et al. (1984) studied optimal dispersal strategies in spatially and temporally heterogeneous environments. They found that environmental fluctuations promote dispersal, even if it is risky. This conclusion can be understood from our results by comparing the establishment success when n seeds are deposited into one site, q sim (Eq. (37)), with the success of dispersing seeds to n different sites, q ind (Eq. (38)). This difference is quite large (Fig. 3) and will increase with increasing variance of m. Cohen and Levin (1991) found that the occurrence of dormancy reduces the optimal dispersal fraction in uncorrelated and negatively correlated environments considerably. In our model, dormancy corresponds to (forced) sequential invasion. The establishment success if both dormancy and dispersal occur equals q~ seq (Eq. (42)), whereas, if no dispersal occurs it equals q seq (Eq. (36), with nk substituted for n). In independent and negatively correlated environments, this difference will be much smaller than the difference between q sim and q ind (compare Figs. 3 and 5), thus reducing the advantage of dispersal. They also found that in positively correlated environments dormancy either has a slight decreasing effect, or it increases optimal dispersal, depending on the probability of finding a new patch (Cohen and Levin, 1991). This might be related to our finding that positive temporal correlation brings both q seq and q sim closer to q ind . However, this will have to be investigated further. In studies of metapopulation dynamics, both colonization and extinction probabilities of small populations are important parameters (e.g., Hanski, 1994). The effects of random environmental fluctuations described in this paper may have important consequences for such dynamics. For instance, the survival chance of a population of size n is equal to q sim (cf. Eq. (37)). As shown in Fig. 3, it is much smaller than the survival chance in a constant environment with the expected numbers of offspring equal to m, q const (Eq. (39)). This illustrates that effects of environmental variability should not be neglected.
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Survival chances of more than one population at different sites can be calculated in similar ways. For instance, the chance that out of k populations of size n at least one will survive equals q~ sim (Eq. (42)). Increasing k will increase q~ sim , even if n is decreasing (see Fig. 5). This indicates that spatial heterogeneity decreases the disadvantages of temporal variation for the persistence of metapopulations.
APPENDIX A: PROOF OF CONVERGENCE OF p t Let m be the expectation of m and let p A be such that p A =e &m(1& pA ),
(A1)
so p A is the long-term extinction probability in the branching process with constant mean number of offspring equal to m. First, we assume that m >1, so that p A <1. Define $ t = p A & p t =e &mt(1& pA ) &e &m(1& pt+1 ) =e &m(1& pA )(1&e &mt(1& pt+1 )+m(1& pA ) ).
(A2)
The conditional expectation of $ t equals E[$ t | m t+1 , m t+2 , . . .] =e &m(1& pA )E[1&e &mt(1& pt+1 )+m(1& pA ) | m t+1 , m t+2 , . . .]. (A3) Applying Jensen's inequality gives E[$ t | m t+1 , m t+2 , . . .] e &m(1& pA )[1&e &m( pA & pt+1 ) ] =e &m(1& pA )[1&e &m$t+1 ].
(A4)
From Eq. (A1) we find that (1& p A )+e &m(1& pA ) =1. For m >1, the function x+e &mx equals 1 for x equal to zero and has a minimum at x=log mm and goes to a value larger than one as x approaches one. Thus, 1& p A is larger then log mm and me &m(1& pA ) <1 for all m >1. Since, furthermore, 1&e &x x: E[$ t | m t+1 , m t+2 , . . .] e &m(1& pA )[m$ t+1 ]<$ t+1 .
(A5)
Now consider the case when m 1. Then p A =1 and $ t as defined in Eq. (A2) becomes equal to 1& p t . Accordingly,
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E[$ t | m t+1 , m t+2 , . . .]=1&E m[e &mt(1& pt+1 ) ] 1&e &m(1& pt+1 ) =1&e &m$t+1 m$ t+1 $ t+1 ,
(A6)
where we used Jensen's inequality and the fact that m 1. From Eqs. (A5) and (A6) we can conclude that [$ t ] is a reverse supermartingale and E[$ t ] is monotonically nonincreasing as t decreases. From the definition of $ t it follows that [ p t ] is a reverse submartingale, since p A does not depend on t. Since p t is restricted to the interval [0, 1], its moments are bounded, and thus it follows from the martingale limit theorem that p t p almost surely as t & , where p is a random variable with bounded expectation and variance (see, e.g., Hall and Heyde, 1980, Section 2.3).
APPENDIX B:
PROOF THAT p IS LARGER THAN p a
Let p A be defined as in Eq. (A1). From Eqs. (19) and (20): log( p )=log(E p, m[e &m(1& p) ]).
(B1)
Jensen's inequality gives log( p ) E p, m[ &m(1& p)]=&m(1& p ).
(B2)
From (B2) and the fact that log( p A ) equals &m(1& p A ) it can be concluded that p is larger than p A .
APPENDIX C: DERIVATION OF EQ. (25) Define x=m&m, r= p& p.
(C1)
Using the definitions in (C1), we can rewrite f (m, p) as f (m, p)= f (m +x, p A +$+r),
(C2)
where $= p & p A . Expansion of the right-hand side around m and p A gives
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f (m, p)=f (m, p A )+x
f f +($+r) m p
1 2f 2f + x2 +x($+r) 2 m 2 m p 1 2f + ($+r) 2 2 +O(($+x+ p) 3 ), 2 p
(C3)
where the partial derivatives are evaluated at m and p A . Note that the variances of x and r equal those of respectively m and p. Furthermore, x and r are independent. Using this, Eq. (20) and Eq. (C3) give: p = p A +$
2f 1 2f 1 2 2f f 1 + Var[m] Var[ p] + + $ +O(($+=) 3 ), p 2 m 2 2 p 2 2 p 2 (C4)
and, thus (using the definition of $)
\
$ 1&
2f 1 2f 1 2f f 1 = Var[m] + Var[ p] 2 + $ 2 2 +O(($+=) 3 ). 2 p 2 m 2 p 2 p
+
(C5) Since Var[m] and Var[ p] are both of order = 2, we can conclude from (C5) that $ is of order = 2 too, so
\
$ 1&
2f 1 2f f 1 Var[ p] = Var[m] + +O(= 3 ), p 2 m 2 2 p 2
+
(C6)
and, thus, p = p A +
1 2f 2f Var[m] +Var[ p] 2 m 2 p 2
\
+\
1&
f p
+
&1
+0(= 3 ).
(C7)
Using Eq. (21) and the expansion in (C3) gives Var[ p]=Var[m]
f m
\ +
2
+Var[ p]
f p
2
\ + +O(= ). 4
(C8)
This leads to Var[ p]=
( f m) 2 Var[m]+O(= 4 ). 1&( f p) 2
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(C9)
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By combining this with Eq. (C7) we get Eq. (24). Substituting the values of the derivatives gives Eq. (25). APPENDIX D:
DERIVATION OF EQ. (29)
An analogous derivation as given in Appendix C gives p = p G +
1 2g 2g Var[z] 2 +Var[ p] 2 2 z p
Var[ p]=
\
+\
1&
g p
+
&1
+O(= 3 ),
( g z) 2 Var[z]+O(= 4 ). 1&( g p) 2
(D1) (D2)
The partial derivatives of g( v ) evaluated at z and p G are g z
}
z
=&e z(1& p G ) e &e (1& pG) =&m~(1& p G ) e &m~(1& pG ) z, pG
=&m~(1& p G ) p G , 2
g z 2
}
z(1&pG )
=&e z(1& p G ) e &e
z(1&pG )
+e 2z(1& p G ) 2 e &e
z, pG
=&m~(1& p G ) p G +m~ 2(1& p G ) 2 p G , g p
} g p }
(D3)
z
=e ze &e (1& pG ) =m~p G , z, pG
2
z
=e 2ze &e (1& pG ) =m~ 2p G .
2
z, pG
Substituting these values in Eqs. (D1) and (D2) gives Eq. (29). ACKNOWLEDGMENTS The research was made possible by a support to Patsy Haccou from the Royal Dutch Academy of Sciences and by a grant-in-aid in scientific research by the ministry of education, science, and culture, Japan, to Yoh Iwasa. We thank Hans Metz, John McNamara, Geza Meszena, Sean Collins, Evert Meelis, and John Val for their useful comments.
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Athreya, K. B., and Karlin, S., 1971b. Branching processes with random environments, II. Limit theorems, Ann. Math. Statist. 42, 18431858. Charlesworth, B., 1994. ``Evolution in Age-Structured Populations,'' 2nd ed., Cambridge Univ. Press, Cambridge. Chesson, P. L., 1982. The stabilizing effects of a random environment, J. Math. Biol. 15, 136. Chesson, P. L., and Ellner, S., 1989. Invasibility and stochastic boundedness in monotonic competition models, J. Math. Biol. 27, 117138. Cohen, D., 1966. Optimizing reproduction in a randomly varying environment, J. Theor. Biol. 12, 119129. Cohen, D., 1967. Optimizing reproduction in a randomly varying environment when a correlation may exist between the conditions at the time a choice has to made and the subsequent outcome, J. Theor. Biol. 16, 114. Cohen, D., and Levin, S. A., 1991. Dispersal in patchy environments: The effects of temporal and spatial structure, Theor. Popul. Biol. 39, 6399. Eshel, I., 1981. On the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size-a branching process model, J. Math. Biol. 12, 355362. Eshel, I., 1984. On the survival probability of a slightly advantageous mutant gene in a multitype population: A multidimensional branching process model, J. Math. Biol. 19, 201209. Feller, W., 1968. ``An Introduction to Probability Theory and Its Applications,'' Vol. I, 3rd ed., Wiley, New York. Feller, W., 1971. ``An Introduction to Probability Theory and Its Applications,'' Vol. II, 2nd ed., Wiley, New York. Goel, N. S., and Richter-Dyn, N., 1972. ``Stochastic Models in Biology,'' Academic Press, New York. Haccou, P., and Iwasa, Y., 1995. Optimal mixed strategies in stochastic environments, Theor. Popul. Biol. 47, 212243. Hall, P., and Heyde, C. C., 1980. ``Martingale Limit Theory and Its Applications,'' Academic Press, New York. Hanski, I., 1992. Inferences from ecological incidence functions, Am. Nat. 139, 657662. Hanski, I., 1994. A practical model of metapopulation dynamics, J. Anim. Ecol. 63, 151162. Iwasa, Y., and Mochizuki, H., 1988. Probability of population extinction accompanying a temporary decrease of population size, Res. Popul. Ecol. 30, 145164. Iwasa, Y., and Haccou, P., 1994. Emergence patterns of male butterflies in stochastic environments, Evol. Ecol. 8, 503523. Karlin, S., and Taylor, H. M., 1975. ``A First Course in Stochastic Processes,'' 2nd ed., Academic Press, New York. Keiding, N., and Nielsen, J. E., 1973. The growth of supercritical branching processes with random environments, Ann. Probab. 1, 10651067. Keiding, N., 1975. Extinction and exponential growth in random environments, Theor. Popul. Biol. 8, 4963. Kondo, J., 1991. ``Integral Equations,'' Clarendon Press, Oxford. Kurtz, T. G., 1978. Diffusion approximations for branching processes, in ``Branching Processes'' (A. Joffe and P. Ney, Eds.), pp. 269292, Dekker, New York. Levins, R., 1968. ``Evolution in Changing Environments,'' Princeton Univ. Press, Princeton, NJ. Levin, S.A., Cohen, D., and Hastings, A., 1984. Dispersal strategies in patchy environments, Theor. Popul. Biol. 26, 165191. Lewontin, R. H., and Cohen, D., 1969. On population growth in a randomly varying environment, Proc. Nat. Acad. Sc. USA 62, 10561060. Ludwig, D., 1976. A singular perturbation problem in the theory of population extinction, Soc. Indus. Appl. Math.Am. Math. Soc. Proc. 10, 87104.
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Maynard Smith, J., 1982. ``Evolution and the Theory of Games,'' Cambridge Univ. Press, Cambridge. Manoukian, E. B., 1986. ``Modern Concepts and Theorems of Mathematical Statistics,'' Springer-Verlag, New York. McArthur, R. H., 1972. ``Geographical Ecology,'' Harper 6 Row, New York. McArthur, R. H., and Levins, R., 1967. ``Theory of Island Biogeography,'' Princeton Univ. Press, Princeton, NJ. McNamara, J., 1995. Implicit frequency dependence and kin selection in fluctuating environments, Evol. Ecol. 9, 185203. Metz, J. A. J., Nisbet, R. M., and Geritz, S. A. H., 1992. How should we define ``fitness'' for general ecological scenarios?, TREE 7, 198202. Mountford, M. D., 1971. Population survival in a variable environment, J. Theor. Biol. 32, 7579. Mountford, M. D., 1973. The significance of clutch-size. in ``The Mathematical Theory of the Dynamics of Biological Populations'' (M. S. Bartlett and R. W. Hiorns, Eds.), pp. 315323, Academic Press, London. Philippi, T., and Seger, J., 1989. Hedging one's evolutionary bets, revisited, TREE 4, 4144. Sasaki, A., and Ellner, S., 1995. The evolutionarily stable phenotype distribution in a random environment, Evolution 49, 337350. Schaffer, W. M., 1974. Optimal reproductive effort in fluctuating environments, Am. Nat. 108, 783790. Seger, J., and Brockmann, H. J., 1988. What is bet-hedging?, in ``Oxford Surveys in Evolutionary Biology'' (P. H. Harvey and L. Partridge, Eds.), Vol. 4, pp. 182211, Oxford Univ. Press, Oxford. Slatkin, M., 1974. Hedging one's evolutionary bets, Nature 250, 704705. Smith, W. L., 1968. Necessary conditions for almost sure extinction of a branching process with random environments, Ann. Math. Statist. 39, 21362140. Smith, W. L., and Wilkinson, W. E., 1969. On branching processes in random environments, Ann. Math. Statist. 40, 814827. Springer, M. D., 1979. ``The Algebra of Random Variables,'' Wiley, New York. Tuljapurkar, S., 1990. ``Population Dynamics in Variable Environments,'' Lecture Notes in Biomathematics, Springer-Verlag, Berlin. Turelli, M.,1980. Niche overlap and invasion of competitors in random environments, II. The effects of demographic stochasticity, in ``Biological Growth and Spread'' (W. Jager, H. Rost, and P. Tautu, Eds.), Lecture Notes in Biomathematics, Vol. 38, pp. 119129, Springer-Verlag, Berlin. Wilkinson, W. E., 1969. On calculating extinction probabilities for branching processes in random environments, J. Appl. Probab. 6, 478492. Yoshimura, J., and Clark, C. W., 1991. Individual adaptations in stochastic environments, Evol. Ecol. 5, 173192.
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Establishment Probability in Fluctuating Environments: A Branching Process Model Patsy Haccou Section Theoretical Biology, Institute of Evolutionary and Ecological Sciences, Leiden University, P.O. Box 9516, 2300 RA Leiden, The Netherlands
and Yoh Iwasa Department of Biology, Faculty of Science, Kyushu University, Fukuoka, 812, Japan Received June 9, 1995
We study the establishment probability of invaders in stochastically fluctuating environments and the related issue of extinction probability of small populations in such environments, by means of an inhomogeneous branching process model. In the model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period, individuals have (independent) Poisson distributed numbers of offspring. The expected numbers of offspring per individual are independently identically distributed over the periods. It is shown that the establishment probability of an invader varies over the reproduction periods according to a stable distribution. We give a method for simulating the establishment probabilities and approximations for the expected establishment probability. Furthermore, we show that, due to the stochasticity of the establishment success over different periods, the expected success of sequential invasions is larger then that of simultaneous invasions and we study the effects of environmental fluctuations on the extinction probability of small populations and metapopulations. The results can easily be generalized to other offspring distributions than the Poisson. 1996 Academic Press, Inc.
INTRODUCTION Success of invasion to an empty habitat is an important life history characteristic of many organisms. So-called ``r-selected'' species are considered to have life history traits that lead to a higher success of invasion to a novel habitat (MacArthur and Levins, 1967; MacArthur, 1972). Populations of many species are maintained by recurrent events of extinction of local populations and subsequent invasion from other 254 0040-580996 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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populations (e.g., Hanski, 1992). To understand the dynamics and longterm persistence of a metapopulation, we need to know the establishment success of invaders as well as the extinction probability of small populations. Of course, these two issues are intimately related. Invasion success is also important in an evolutionary context, because any novel trait must arise initially as rare mutations. Thus, the establishment of new traits in the long run may be influenced by the dynamics at small population size. In addition, many evolutionary stability concepts are defined in terms of invasion probability (Maynard Smith, 1982; Charlesworth, 1994). The fate of invaders (or mutants) and small populations can be expected to be strongly affected by demographic stochasticity. This is usually modelled by means of branching processes or birth-and-death processes, in which it is assumed that the environment is constant between generations (Goel and Richter-Dyn, 1972; Eshel, 1981, 1984; Charlesworth, 1994). Several theoretical studies have shown that environmental stochasticity may lead to profoundly different population dynamics than observed in constant environments. For instance, in fluctuating environments so-called ``bet-hedging'' may occur (Slatkin, 1974; Seger and Brockmann, 1988; Philippi and Seger, 1989). Effects of environmental stochasticity, however, are usually studied by considering long-term expected growth rates in large populations, ignoring demographic stochasticity. (See, e.g., Levins, 1968; Lewontin and Cohen, 1969; Schaffer, 1974; Tuljapurkar, 1990; Yoshimura and Clark, 1991; Iwasa and Haccou, 1994; McNamara, 1995; Haccou and Iwasa, 1995; Sasaki and Ellner, 1995.) Establishment success in stochastic environments has been studied for low population densities. Keiding (1975), Ludwig (1976), and Turelli (1980) used diffusion approximations of inhomogeneous branching processes for this purpose. Chesson (1982) and Chesson and Ellner (1989) developed the idea of stochastic boundedness. According to this criterion a population can establish itself in the long run if the asymptotic population density is bounded from below by a positive random variable. Whereas relative densities of mutants can be low in these models, the numbers of individuals are still assumed to be infinitely large. When population size is large, the effect of demographic stochasticity on population growth rate is small compared to that of environmental stochasticity (Keiding and Nielsen, 1973; Yoshimura and Clark, 1991). However, in small populations, effects of demographic stochasticity cannot be ignored. On the other hand, there are indications that environmental fluctuations may have large effects on small populations. For instance, Hanski (1992) found strong empirical evidence of such effects. Iwasa and Mochizuki (1988) showed theoretically that occasional catastrophic events can considerably increase the extinction chance.
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The combined effects of environmental and demographic stochasticity can be studied by means of the inhomogeneous branching process, which was first examined by Smith (1968) and Smith and Wilkinson (1969). Mountford (1971, 1973) was the first to use the model in biological research. In this model both sources of stochasticity are incorporated. First, the expected numbers of offspring per individual, here denoted by m t (t=1, . . .), are independently identically distributed over reproduction periods. Second, within reproduction period t individuals have stochastic numbers of offspring, which are independently identically distributed, conditional on m t . Athreya and Karlin (1971a, b) examined the model in detail and proved that the establishment probability of an initially rare mutant is positive if and only if the geometric mean of m t is larger than one. This conclusion was extended to the case with correlated environmental fluctuation, and with multiple state branching models (Athreya and Karlin, 1971a, b). These results, although very general, do not give information about the magnitude of the establishment success of invaders. It is important, however, to be able to estimate this parameter, e.g., when studying metapopulation dynamics, co-evolution, in the control of foreign organisms, or in life-history studies. Wilkinson (1969) gives an approximation for the establishment success, which we briefly describe in this paper. However, we found that for this approximation to be sufficiently accurate, extensive computations are needed. We give an alternative approximation method, which is easier to calculate and quite accurate. Although it is derived for small environmental variability, it appears to work well for relatively high environmental variances too. Its results are compared with Wilkinson's approximations. Furthermore, we give a method for estimating establishment success from simulations. The simulation method is based on the mathematically derived, distributional properties of the extinction probabilities and avoids having to simulate the actual branching processes themselves. An important difference between constant and fluctuating environments is that, whereas in a constant environment the probability of a successful invasion is fixed and does not vary over time, in a fluctuating environment the fate of a newly arrived individual strongly depends on the timing of invasion. Since there is a large probability of extinction in the first generations, an invasion in a favorable period has a much higher chance of eventual success than an invasion during an unfavorable period. This implication has been noted in the mathematical literature (e.g., Athreya and Karlin, 1971a, b), but the consequences for biological applications have not been fully understood until now. One such consequence is that sequential invasions have a higher probability of success than simultaneous ones. This has large implications for invasion strategies, biological control strategies, and metapopulation dynamics.
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As an example, we consider an inhomogeneous branching process model in which the number of surviving offspring follows a Poisson distribution, with an expectation that fluctuates independently between years. After describing the simulation method and approximations for the establishment probability, we examine the effects of different modes of invasion (e.g., sequential versus simultaneous) on establishment probabilities. The results can be translated directly into consequences for survival probabilities of small populations. Our conclusions are also valid for other offspring distributions than the Poisson, and the simulation method as well as the approximation can easily be generalised in this respect.
A MODEL FOR DEMOGRAPHIC STOCHASTICITY IN FLUCTUATING ENVIRONMENTS We assume a simple life-cycle, where all individuals reproduce asexually during separate reproduction periods. Furthermore, individuals produce offspring independently of one another. Let N t&1 be the population size just before period t. We consider the probability of successful invasion if the population starts from a single individual. Hence N 0 equals one. For ease of notation, we will use extinction probabilities rather than establishment probabilities (which can easily be calculated from these) in our equations. Before we describe the full model, in which both demographic stochasticity and fluctuating environments are included, we will discuss the classical models used for studying evolution in the presence of, respectively, fluctuating environments and demographic stochasticity. Case a: Fluctuating Environment, No Demographic Stochasticity Evolution in stochastic environments is usually studied in infinitely large populations, where effects of demographic stochasticity are ignored. This is the situation considered, e.g., by Schaffer (1974), Yoshimura and Clark (1991), McNamara (1995), Haccou and Iwasa (1995), and Sasaki and Ellner (1995). Let m t denote the expected number of offspring per individual in period t. It is assumed that the m t are independent and identically distributed. According to this model, t
N t =m tN t&1 = ` m j N 0 .
(1)
j=1
This model only has a biological interpretation for large population sizes, when effects of individuals differences in offspring numbers can be ignored. Therefore, it cannot be used for studying extinction probabilities of small populations andor establishment probability of small numbers of invaders.
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However, for comparison with later results, we will consider what happens in this model if N 0 equals one and define the ``extinction probability'' in this case as the chance that N t becomes less than one. Although this is biologically uninterpretable, it provides a way to compare the model with the more realistic (branching process) models considered later. Consider the log-population size: t
log(N t )= : log(m j ).
(2)
j=1
Since the m j are independently identically distributed, the process [log(N t )] is a random walk. Let m~ denote the geometric mean of m t , m~ =exp[E[log(m t )]].
(3)
From the theory of random walks (see, e.g., Feller, 1971, Section XII.7) we can infer that the ``extinction probability'' of the process equals:
_
1 Pr[log(N t )<0] , t t=1
p a =Pr[inf log(N t )<0]=1&exp & : t
&
(4)
and that the following relations hold: m~ 1 p a =1, m~ >1 p a <1.
(5)
Consider the situation where m~ is larger than one. Then, according to (5), the extinction probability is less than one. However, note that, given a realization of the process [m 1 , m 2 , . . .] extinction occurs either with probability one or with probability zero. Thus, p a corresponds to the expectation of the extinction probability over different realizations of [m 1 , m 2 , . . .]. Case b: Constant Environment, Demographic Stochasticity The classical model for studying extinction probability in a constant environment is the GaltonWatson branching process. It is assumed that in each period, individuals have independent identically distributed numbers of offspring, with a fixed expectation. Let Y j, t be the number of offspring of the j th individual in period t. The population size after reproduction is Nt&1
N t = : Y j, t ,
with Y j, t i.d.d. for all j and t.
j=1
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(6)
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Let Y be a random variable with the same distribution as the Y j, t . The extinction probability of the process is implicitly defined by
p b = : Pr[Y=k] p kb ,
(7)
k=0
where the smaller root should be chosen if there are two positive roots. The following relations hold: E[Y] 1 p b =1
(8)
E[Y]>1 p b <1 (see, e.g., Feller, 1968, Chap. XII). Whereas in case a, conditional on a fixed sequence of environments, extinction occurs with either probability one or zero, note that in case b, although there is no environmental fluctuation, extinction does not occur with probability one or zero if E[Y] is larger than one, due to the stochasticity in offspring numbers. p b corresponds to the expected proportion of processes that goes extinct if we consider infinitely many processes starting with one individual. Case c: Fluctuating Environment and Demographic Stochasticity (Full Model ) We will now combine the models of cases a and b. As in case b, let Y j, t be the offspring of the j th individual reproducing in period t. We now assume that the expected number of offspring per individual reproducing in period t equals m t . Similar to case a, we let the m 1 , m 2 , . . . be independent identically distributed variables and, similar to case b, we assume that conditionally on m t , the Y j, t ( j=1, ..., N t&1 ) are independently identically distributed. Thus combining the models of cases a and b gives Nt&1
N t = : Y j, t ,
with Y j, t , ..., Y j, N t&1 i.d.d. for each t, E[Y j, t | m t ]=m t .
j=1
(9) This is the inhomogenous branching process which was first considered by Smith (1968) and Smith and Wilkinson (1969). We will now consider the conditional probability of extinction: p=Pr[extinction | m 1 , m 2 , . . .].
(10)
Athreya and Karlin (1971a) proved that if the conditions |E[log(m t )]| < E[ |log(1&Pr[Y j, t =0 | m t ])| ]< ,
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(11)
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hold, we have the relations m~ 1 Pr[ p=1]=1 m~ >1 Pr[ p<1]=1,
(12)
where m~ is as defined in Eq. (3). (Their result is more general, but here we only need it in this restricted form). We will assume from now on that the conditions specified in Eq. (11) do hold. Note that there is an important difference with cases a and b, namely, whereas the extinction probabilities p a and p b (cf. Eqs. (4) and (7)) are constants, in case c the extinction probability is a random variable, which is nondegenerate if m~ is larger than one. Thus, if m~ is larger than one, p varies over different possible realisations of [m 1 , m 2 , . . .], as p a varies over different realisations. However, as opposed to case a, there is variation in offspring numbers as well. Therefore, as in case b, extinction does not occur with probability one or zero. For a given realisation, p has the same interpretation as p b . The stochasticity of p has additional consequences which we will discuss below. Time Dependence of Extinction Probability in Case c Now we take a slightly different view and consider the fate of a single invader arriving just before period t. To study this, we define the extinction probability, conditional on the future environmental process, p t =Pr[extinction | m t , m t+1 , . . .];
(13)
then p t =Pr[zero offspring at t | m t , m t+1 , . . .] + : Pr[k offspring at t, which have no progeny k 1
in the long run | m t , m t+1 , . . .]
= : Pr[Y=k | m t ] p kt+1 ,
(14)
k=0
where Y denotes the random number of offspring of the individual. Now, consider the different situations discussed above. In case a, the conditional extinction probability, p t , defined in Eq. (13), is either zero or one and thus does not vary with t. In case b, m t equals m for all t, since the environment does not vary, so in Eq. (14) p t = p t+1 = p b for all t. In case c, however, p t varies with t, unless extinction is certain (m~ 1) in which case it equals
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one for every t. If m~ >1, p t is less than one and changes in time. It follows directly from the ergodicity of the process [m 1 , m 2 , . . .] that 1 T : p t =E[ p] T T t=1 lim
a.s.,
(15)
where p is as defined in Eq. (10) (see, e.g., Karlin and Taylor, 1975, Section 9.5). Similar conclusions can be derived for the other moments; for instance, the variance of p t over time corresponds to the variance of p. Equation (14) proves to be very useful for simulating the distribution of p as well as for deriving approximations for its moments. We will illustrate this for the case where the numbers of offspring have a Poisson distribution, with m~ >1.
THE POISSON BRANCHING-PROCESS MODEL IN A FLUCTUATING ENVIRONMENT We assume now that conditionally on m t , the Y j, t are Poisson random variables: Pr[Y j, t =k | m t ]=e &mt
m kt k!
(k=0, 1, . . .).
(16)
Then, using Eq. (14) we get
p t = : e &mt k=0
m kt k p =e &mte mtpt+1 O p t =e &mt(1& pt+1 ). k! t+1
(17)
From the results of Athreya and Karlin (1971a) it follows that p, p t , and p t+1 have equal marginal distributions. Furthermore, since the m t are independently distributed, p t+1 and m t are independent random variables. Let .( p) denote the probability density function of p and h(m t ) that of m t , then, after some calculations this leads to p.( p)=
|
0
1 1 . 1+ log( p) h(m) dm m m
\
+
(18)
(see, e.g., Springer, 1979). Unfortunately, Eq. (18) cannot be solved explicitly (for numerical methods, see, e.g., Kondo, 1991, Chap. 7). However, there are several ways in which Eq. (17) can be used to derive the distribution of p.
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First, note that Eq. (17) simply describes a recurrence relationship for a sequence of random variables p t , with values between zero and one. Based on this equation, sequences of p t values can be simulated, without reference to the underlying branching processes, through backward iteration: given a certain value of p t+1 , p t can be generated by drawing m t from a given distribution and subsequently using Eq. (17). In Appendix A it is shown that as t tends to minus infinity the sequence converges with probability one to a random variable. Since (18) must hold for this random variable as well as p, and furthermore, has a unique solution (Athreya and Karlin, 1971a), they must be equal with probability one. Thus, to simulate p, we start with an array of nonzero p 0 's, simulate an array of values of m &1 according to a certain distribution, calculate the values of p &1 , etc., until the distribution is stable, and estimate the moments from the resulting array of values of p t . Product moments of the sequence of p t values, such as p t p t+1 p t+2 , can be estimated from sequences of such arrays. Alternatively, moments can be calculated from the time averages of a sequence of p t values simulated in the same way (see, e.g., Eq. (15)). However, the first method provides more accurate estimates, since the different p t values within the array are independent, whereas in the second method, autocorrelations between subsequent p t values can inflate the variance of the estimators. Second, approximations for the moments of p can be derived from the fact that similar relationships as given in Eq. (18) hold for these. Define the function f (m, p)=e &m(1& p);
(19)
E[ p]=E[ f (m, p)]
(20)
Var[ p]=Var[ f (m, p)].
(21)
then
and
Below, we will use Eqs. (20) and (21) to derive approximations for the expectation of p and compare these with simulation results.
APPROXIMATIONS FOR THE EXPECTED EXTINCTION PROBABILITY For convenience, we will denote the expectation of m, E[m], by m and the expectation of p, E[ p], by p. Furthermore, we denote the extinction
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probability in a constant environment where the mean number of offspring per individual is equal to m, by p A . This extinction probability is implicitly defined by p A = f (m, p A )
(22)
(see, e.g., Feller, 1968, Chap. XII). For any value of m, p A can easily be calculated by numerically solving (22). From Eqs. (19), (20), and (22), together with Jensen's inequality, it can easily be proved that p is larger than p A (see Appendix B). One way to derive an approximation for p is to expand f (m, p) (defined in Eq. (19)) around p= p A and m=m (see below). However, for values of m~ near one, this approximation may not be so accurate, since m can still be much larger than one and, as a consequence, p A will be much smaller than one. However, p must be near one for m~ near one. We therefore also examined an approximation based on expansion of f (m, p) around m=m~. Let p G be the extinction probability in the branching process with a constant mean number of offspring equal to m~: p G = f (m~, p G ).
(23)
Since p G equals one when m~ is less than or equal to one, we expect that the values of p G lie closer to p than those of p A . This was examined through simulations. Figure 1 shows the establishment chances (1& p ), estimated from simulations where m is uniformly distributed over an interval for several values of m and two values of Var[m]. The estimates were based on an array of 10,000 p t 's. The widths of the 950 confidence intervals of these estimates are at most 0.005 on both sides. (They are not given in any of the figures, since they either do not show up on the scale used, or they obscure the picture.) We used a value of t=500. Previous investigations showed that distributions are stable after this value of t. Simulations were done with the APL 68000 random generator on a Macintosh PowerPC. As can be seen from Fig. 1, environmental variability considerably reduces the establishment chance of invaders. Furthermore, the values of p G are indeed much closer to p than those of p A . We will now describe different approximations of p and compare their accuracy. Approximation Based on the Arithmetic Mean of m We assume that the variance of m is small, so that for every k>1, E[(m&m ) k ] and E[( p& p k ] are both O(= k ). To derive an approximation
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Fig. 1. Establishment probabilities in a stochastic environment with m uniformly distributed, 1& p( v ), in a constant environment with expected number of offspring equal to m, 1& p A(Q), and in a constant environment with expected number of offspring equal to m~, 1& p G ( b ).
.
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.
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for p, we use the Taylor-expansion of f (m, p) around m=m and p= p A , together with Eqs. (20) and (21) (details are given in Appendix C). This leads to p = p A +
1 2f 2f ( f m) 2 Var[m] + +O(= 3 ), 2(1&( f p)) m 2 p 2 1&( f p) 2
\
+
(24)
where the derivatives are evaluated at m=m and p= p A . Substituting the values of the derivatives in Eq. (24) gives p = p A +
= pA +
Var[m] e &m(1& pA ) (1& p A ) 2 +O(= 3 ) 2(1&me &m(1& pA) ) 1&m 2e &2m(1& pA )
\
+
p A(1& p A ) 2 Var[m]+O(= 3 ). 2(1&mp A )[1&(mp A ) 2 ]
(25)
Approximation Based on the Geometric Mean of m Define z=log m
(26)
z =E[log m]; then z
f (m, p)= g(z, p)=e &e (1& p),
(27)
g(z, p)= f (m~, p)=e &m~(1& p).
(28)
We will now assume that E[(z&z ) k ], rather than E[(m&m ) k ], is of order = k. As before, we assume that E[( p& p ) k ] is of this order too. We now expand g(z, p) around z=z and p= p G (see Eq. (23)) and use the relations given in Eqs. (20) and (21) to derive approximations for the expectation of p. The derivation is completely analogous to that given in Appendix C for the approximation given in Eq. (25) and gives a similar result: p = p G +
m~(1& p G ) p G m~(1& p G ) &1 Var[z]+O(= 3 ). 2(1&m~p G ) 1&(m~p G ) 2
{
=
(See Appendix D for details.)
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(29)
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Wilkinson's Approximation Wilkinson (1969) gives a method for calculating approximations for p as well as the higher moments E[ p n ], which we will briefly describe. Let ? be a vector of the moments of p, ?=( p, E[ p 2 ], ..., E[ p n ]),
(30)
and let q also be a vector of length n with j th element equal to q( j)=Pr[N 1 =0 | N 0 = j]=E m[Pr[N 1 =0 | N 0 = j; m 1 =m]],
(31)
where N i is the population size in the ith generation. When the numbers of offspring are Poisson distributed, we have q( j)=E m[e & jm ],
(32)
since the convolution of j independent Poisson distributions with parameter m is again a Poisson distribution, with parameter jm (see, e.g., Manoukian, 1986, Section 4.13). Furthermore, let A be a n_n matrix with elements: A(i, j)=Pr[N 1 = j | N 0 =i]=E m[Pr[N 1 = j | N 0 =i ; m 1 =m]],
(33)
which in our case corresponds to A(i, j)=E m
_
(im) j &im e . j!
&
(34)
Wilkinson (1969) proved that ?=(I&A) &1 q+=,
(35)
where I denotes the identity matrix and = is at most equal to E[ p n+1 ]. Thus, (I&A) &1 q provides an approximation for ?, which increases in accuracy as n increases. Accuracy of the Approximations Figure 2 shows the accuracy of the approximations for p given in Eqs. (25), (29) and Wilkinson's approximation (Eq. (35), with n=100). The approximation given in Eq. (25) is fairly accurate for low variances of m, provided that m is relatively large. However, for values of m near one it does not work so well, especially if Var[m] is large. Wilkinson's approximation (Eq. (35)) works better than Eq. (25), but the approximation in Eq. (29) is by far the most accurate for values of m~ close to one,
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Fig. 2. Differences between p and its approximated values: (Q) based on m (Eq. (25)); ( v ) based on m~ (Eq. (29)); ( b ) Wilkinson's (1969) approximation (Eq. (35), with n=100).
.
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.
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especially when Var[m] is large. Increasing n would make Wilkinson's approximation more accurate, but the computations are already formidable at n=100.
CONSEQUENCES OF ENVIRONMENTAL VARIABILITY FOR INVASION SUCCESS The probability of success of a single invader at a given time, say t, corresponds to p t . The stochasticity of p t implies that the chance of successful invasion depends on the way in which invasions take place. The chances differ for simultaneous or sequential invasion at one site and for invasions at different sites. When there are n sequential invasions into a single site, the chance that at least one invasion is successful is
_
n
&
q seq =E 1& ` p t =1&E t=1
_
n
&
` pt . t=1
(36)
When n invaders arrive simultaneously, the chance of success is q sim =E[1&( p t ) n ]=1&E[ p n ].
(37)
The probability of at least one success if n invaders arrive at different sites which are independent with respect to the expected numbers of offspring equals n
q ind =1& ` E[ p t ]=1& p n.
(38)
t=1
Note that, in nonstochastic environments with expected numbers of offspring equal to m, all these chances would equal q const =1& p nA .
(39)
The differences between q seq , q sim , and q ind can have profound effects on invasion strategies. From Holder's inequality (see, e.g., Manoukian, 1986, Section 1.4) it follows that E
_
n
& \
n
` p t ` E[ p nt ] t=1
t=1
+
1n
=E[ p n ],
(40)
where the rightmost equality results from the fact that the p t are identically distributed. From Eqs. (36), (37), and (40) it can be concluded that, in a
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stochastically fluctuating environment, the probability of success of n sequential invasions is higher than that of a simultaneous arrival of n invaders. As shown in Fig. 3, the difference between the two increases rapidly with n. From Jensen's inequality it follows that q ind is higher than q sim , so invasions into different independent sites have higher chances of success than those into one or, equivalently, into perfectly correlated sites. Our simulation results indicate that q ind is higher than q seq (see Fig. 3). This implies that E
_
n
&
n
` p t >p n = ` E[ p t ] t=1
(41)
t=1
and, thus, in this case there appears to be a positive autocorrelation between the values of p t . It is not a trivial task, however, to examine analytically whether this is true in general. Increasing the times between
Fig. 3. Establishment chances of different modes of invasions: (_) q sim (Eq. (36)), ( v ) q seq (Eq. (37)), ( + ) q ind (Eq. (38)), (Q) q const (Eq. (39)), for m uniformly distributed, m =1.3, Var[m]=0.50.
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successive invasions will decrease the correlation between the p t values at different invasion times and therefore, in the case investigated here, it will increase the chance of success. This is illustrated in Fig. 4. Since q ind is much closer to q const than either q seq or q sim (Fig. 3), spatial heterogeneity apparently reduces the negative effects of temporal variation on establishment chances. To study the effects of spatial heterogeneity more closely, consider the establishment chances of n invaders into k independent sites (which provide identically distributed expected numbers of offspring). The establishment chances of, respectively, sequential and simultaneous invasions then become
\ _
q~ seq =1& E
n
` pt t=1
k
&+ ,
q~ sim =1&(E[ p n ]) k.
(42) (43)
It can easily be seen that as long as E[ p n ]
Fig. 4. Establishment chances of five sequential invasions, for increasing numbers of periods between invasions, for m uniformly distributed, m =1.3, Var[m]=0.50.
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Fig. 5. Establishment chances of 100 invaders divided over different numbers of independent sites: ( v ) sequential invasions (see Eq. (41)); ( b ) simultaneous invasions (Eq. (42)), for m uniformly distributed, m =1.3, Var[m]=0.50.
DISCUSSION The value 1& p corresponds to the chance that a single invader arriving at one site at a random time succeeds to establish itself in the long run. It also corresponds to the expected proportion of successful invasions if large numbers of invaders arrive at different times or at different sites which provide independent identically distributed expected numbers of offspring. Note, however, that we use an unbounded population size model. In practice, density-dependent processes will ultimately limit the size of populations. Therefore, the establishment probabilities that we define should be considered as approximations for the chances that populations will eventually reach a large and stable size and persist for a long time. Being able to estimate p is important for studying r-selection, biological control strategies, and metapopulation dynamics. For many evolutionary considerations, it will matter only whether or not there is a positive establishment chance. This depends on the geometric mean, m~ (Athreya and Karlin, 1971a). However, note that in co-evolutionary processes the relative chances of invasion success can make a difference. In such cases, an estimate of p might be needed. The approximation based on p G (Eq. (29)) is very accurate (Fig. 2). Approximations for offspring distributions other than the Poisson may be
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derived in a similar way, provided that an explicit expression can be found for f (m, p) (i.e., the right-hand side of Eq. (14)). Note that this function corresponds to the probability generating function of the offspring distribution in p t+1 (see Eq. (17)). These functions are listed in, e.g., Abramowitz and Stegun (1972, Chap. 26). An alternative is to use Wilkinson's (1969) approximation method. Although the method was originally derived for discrete m, it can easily be proved to be applicable directly to continuous m (as in our example). This approximation has the advantage that by increasing n, its accuracy can be increased to any value. However, especially for high values of n, the required computations are formidable. From simulation results it appears that for uniformly distributed m, n should be larger than 100 to approximate p with reasonable accuracy, especially if m~ is near one (see Fig. 2). From further investigations we found that the accuracy increases very slowly with n. As shown in Fig. 2, the approximation given in Eq. (29) works much better at low m~ than Wilkinson's approximation for n=100. Branching processes in random environments have also been studied by approximating changes in population densities with diffusion processes (Keiding, 1975; Ludwig, 1976; Kurtz, 1978; Turelli, 1980). However, such approximations can only be expected to be accurate for large population sizes. The k th moment of p, E[ p k ], corresponds to the extinction probability of a population of size k, so it can be useful to be able to estimate the higher moments of p. In Appendix D we give an approximation for the variance of p, based on p G . In principle, our approximation method can be generalized for higher moments, but it becomes very complicated. The method derived by Wilkinson (1969; see also Eq. (35)) can be used to approximate E[ p k ] up to any value of k. However, as shown in Fig. 2, the error in the approximation of the first moment of p can be quite large for low expectations of m. For higher moments, this problem will increase. Another method for estimating higher moments of p is to use simulations. Note that the fact that p t is stochastic has consequences for estimation of p from simulations. A single value of p t is not a good estimate of p, unless environmental variability is very low (compare Turelli, 1980). The simulation method can also be used for different offspring distributions than the Poisson, provided that an explicit expression for f (m, p) can be found. Athreya and Karlin (1971b) also proved their convergence theorem for autocorrelated m t . To apply a similar simulation method in that case, formally it has to be proved that here, too, the sequence of p t 's defined in Eqs. (13) and (14) converges to the random variable p defined in Eq. (10). However, the proof of convergence of p t given in Appendix A is based on the martingale limit theorem, which is very general. We expect that it can easily be adjusted for autocorrelated m t .
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The success of sequential invasions into one habitat is much larger then that of simultaneous invasions (Fig. 3, Eq. (40)). This finding has major consequences for biological control as well as the study of reproduction strategies, such as the production of offspring or phenomena such as dormancy and diapause. Although it has been noted before from theoretical modelling that random fluctuations in the environment can promote dormancy or the proportion of nonbreeding individuals (Cohen, 1966; Mountford, 1971, 1973), the connection with stochasticity of the probability of successful invasion in stochastic environments was not made. Our results show that this may be the major cause of such phenomena. Furthermore, since Holder's inequality, which is the basis for this result (see Eq. (40)) holds very generally, the conclusion that sequential invasion is advantageous does not depend on the offspring distribution, the distribution of the m t , or on the independence of the m t . When groups of invaders arrive sequentially, the relative advantage of sequential invasion will decrease with the group size. The effects of positive autocorrelation in the m t can be predicted by considering that in the limiting case where m t is constant and equal to m, all establishment probabilities given in Eqs. (36) to (38) will converge to q const . Thus, in general the success of invasion, whether it is sequential or not will be higher in positively correlated environments. Furthermore, the relative advantage of sequential when compared to simultaneous invasion will decrease. Our results indicate that, at least for independent homogeneously distributed m t and Poisson distributed numbers of offspring, q seq is less than q ind (Fig. 3), which implies that the p t are positively autocorrelated. Increasing the times between successive invasions will decrease the correlation between the p t values at different invasion times and therefore it will increase the chance of success. This is illustrated in Fig. 4. It remains to be examined however, whether the p t are in general positively autocorrelated for independent m t . If they are, the effect of negative correlation in the sequence of m t values would be to reduce the autocorrelation in p t values, which would make sequential invasion even more advantageous in such environments. Furthermore, it would imply that increasing the lag between successive invasions is advantageous in any nonperiodic temporally varying environment, since it decreases the dependence between the p t values at the different invasions times. The advantage of sequential versus nonsequential invasion also depends on the amount of information about the expected number of offspring in a given reproduction period, m t . If there is information about these values, it is a better strategy to produce large numbers of offspring in favorable years (see, e.g, Cohen, 1967). This might for example explain the occurrence of mast years in certain species versus continuous reproduction in others.
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If invasions occur in different, uncorrelated sites, the difference between success of sequential and simultaneous invasions becomes much less, since environmental heterogeneity induces an element of independence between success probabilities at any given time (see Eqs. (41) and (42) and Fig. 5). So, for instance plant species with seed dispersal over long distances, relative to the scale of spatial variation in the environment, can be expected to show less dormancy than those with seed dispersal over relatively short distances (given that there is temporal environmental variability). Many previous conclusions in the literature, derived from specific models, which are much more complicated than the one considered by us, can be understood from our general results concerning the relation between invasion modes and establishment success (cf. Eqs. (36) to (43)). For instance, Levin et al. (1984) studied optimal dispersal strategies in spatially and temporally heterogeneous environments. They found that environmental fluctuations promote dispersal, even if it is risky. This conclusion can be understood from our results by comparing the establishment success when n seeds are deposited into one site, q sim (Eq. (37)), with the success of dispersing seeds to n different sites, q ind (Eq. (38)). This difference is quite large (Fig. 3) and will increase with increasing variance of m. Cohen and Levin (1991) found that the occurrence of dormancy reduces the optimal dispersal fraction in uncorrelated and negatively correlated environments considerably. In our model, dormancy corresponds to (forced) sequential invasion. The establishment success if both dormancy and dispersal occur equals q~ seq (Eq. (42)), whereas, if no dispersal occurs it equals q seq (Eq. (36), with nk substituted for n). In independent and negatively correlated environments, this difference will be much smaller than the difference between q sim and q ind (compare Figs. 3 and 5), thus reducing the advantage of dispersal. They also found that in positively correlated environments dormancy either has a slight decreasing effect, or it increases optimal dispersal, depending on the probability of finding a new patch (Cohen and Levin, 1991). This might be related to our finding that positive temporal correlation brings both q seq and q sim closer to q ind . However, this will have to be investigated further. In studies of metapopulation dynamics, both colonization and extinction probabilities of small populations are important parameters (e.g., Hanski, 1994). The effects of random environmental fluctuations described in this paper may have important consequences for such dynamics. For instance, the survival chance of a population of size n is equal to q sim (cf. Eq. (37)). As shown in Fig. 3, it is much smaller than the survival chance in a constant environment with the expected numbers of offspring equal to m, q const (Eq. (39)). This illustrates that effects of environmental variability should not be neglected.
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Survival chances of more than one population at different sites can be calculated in similar ways. For instance, the chance that out of k populations of size n at least one will survive equals q~ sim (Eq. (42)). Increasing k will increase q~ sim , even if n is decreasing (see Fig. 5). This indicates that spatial heterogeneity decreases the disadvantages of temporal variation for the persistence of metapopulations.
APPENDIX A: PROOF OF CONVERGENCE OF p t Let m be the expectation of m and let p A be such that p A =e &m(1& pA ),
(A1)
so p A is the long-term extinction probability in the branching process with constant mean number of offspring equal to m. First, we assume that m >1, so that p A <1. Define $ t = p A & p t =e &mt(1& pA ) &e &m(1& pt+1 ) =e &m(1& pA )(1&e &mt(1& pt+1 )+m(1& pA ) ).
(A2)
The conditional expectation of $ t equals E[$ t | m t+1 , m t+2 , . . .] =e &m(1& pA )E[1&e &mt(1& pt+1 )+m(1& pA ) | m t+1 , m t+2 , . . .]. (A3) Applying Jensen's inequality gives E[$ t | m t+1 , m t+2 , . . .] e &m(1& pA )[1&e &m( pA & pt+1 ) ] =e &m(1& pA )[1&e &m$t+1 ].
(A4)
From Eq. (A1) we find that (1& p A )+e &m(1& pA ) =1. For m >1, the function x+e &mx equals 1 for x equal to zero and has a minimum at x=log mm and goes to a value larger than one as x approaches one. Thus, 1& p A is larger then log mm and me &m(1& pA ) <1 for all m >1. Since, furthermore, 1&e &x x: E[$ t | m t+1 , m t+2 , . . .] e &m(1& pA )[m$ t+1 ]<$ t+1 .
(A5)
Now consider the case when m 1. Then p A =1 and $ t as defined in Eq. (A2) becomes equal to 1& p t . Accordingly,
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E[$ t | m t+1 , m t+2 , . . .]=1&E m[e &mt(1& pt+1 ) ] 1&e &m(1& pt+1 ) =1&e &m$t+1 m$ t+1 $ t+1 ,
(A6)
where we used Jensen's inequality and the fact that m 1. From Eqs. (A5) and (A6) we can conclude that [$ t ] is a reverse supermartingale and E[$ t ] is monotonically nonincreasing as t decreases. From the definition of $ t it follows that [ p t ] is a reverse submartingale, since p A does not depend on t. Since p t is restricted to the interval [0, 1], its moments are bounded, and thus it follows from the martingale limit theorem that p t p almost surely as t & , where p is a random variable with bounded expectation and variance (see, e.g., Hall and Heyde, 1980, Section 2.3).
APPENDIX B:
PROOF THAT p IS LARGER THAN p a
Let p A be defined as in Eq. (A1). From Eqs. (19) and (20): log( p )=log(E p, m[e &m(1& p) ]).
(B1)
Jensen's inequality gives log( p ) E p, m[ &m(1& p)]=&m(1& p ).
(B2)
From (B2) and the fact that log( p A ) equals &m(1& p A ) it can be concluded that p is larger than p A .
APPENDIX C: DERIVATION OF EQ. (25) Define x=m&m, r= p& p.
(C1)
Using the definitions in (C1), we can rewrite f (m, p) as f (m, p)= f (m +x, p A +$+r),
(C2)
where $= p & p A . Expansion of the right-hand side around m and p A gives
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f (m, p)=f (m, p A )+x
f f +($+r) m p
1 2f 2f + x2 +x($+r) 2 m 2 m p 1 2f + ($+r) 2 2 +O(($+x+ p) 3 ), 2 p
(C3)
where the partial derivatives are evaluated at m and p A . Note that the variances of x and r equal those of respectively m and p. Furthermore, x and r are independent. Using this, Eq. (20) and Eq. (C3) give: p = p A +$
2f 1 2f 1 2 2f f 1 + Var[m] Var[ p] + + $ +O(($+=) 3 ), p 2 m 2 2 p 2 2 p 2 (C4)
and, thus (using the definition of $)
\
$ 1&
2f 1 2f 1 2f f 1 = Var[m] + Var[ p] 2 + $ 2 2 +O(($+=) 3 ). 2 p 2 m 2 p 2 p
+
(C5) Since Var[m] and Var[ p] are both of order = 2, we can conclude from (C5) that $ is of order = 2 too, so
\
$ 1&
2f 1 2f f 1 Var[ p] = Var[m] + +O(= 3 ), p 2 m 2 2 p 2
+
(C6)
and, thus, p = p A +
1 2f 2f Var[m] +Var[ p] 2 m 2 p 2
\
+\
1&
f p
+
&1
+0(= 3 ).
(C7)
Using Eq. (21) and the expansion in (C3) gives Var[ p]=Var[m]
f m
\ +
2
+Var[ p]
f p
2
\ + +O(= ). 4
(C8)
This leads to Var[ p]=
( f m) 2 Var[m]+O(= 4 ). 1&( f p) 2
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(C9)
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By combining this with Eq. (C7) we get Eq. (24). Substituting the values of the derivatives gives Eq. (25). APPENDIX D:
DERIVATION OF EQ. (29)
An analogous derivation as given in Appendix C gives p = p G +
1 2g 2g Var[z] 2 +Var[ p] 2 2 z p
Var[ p]=
\
+\
1&
g p
+
&1
+O(= 3 ),
( g z) 2 Var[z]+O(= 4 ). 1&( g p) 2
(D1) (D2)
The partial derivatives of g( v ) evaluated at z and p G are g z
}
z
=&e z(1& p G ) e &e (1& pG) =&m~(1& p G ) e &m~(1& pG ) z, pG
=&m~(1& p G ) p G , 2
g z 2
}
z(1&pG )
=&e z(1& p G ) e &e
z(1&pG )
+e 2z(1& p G ) 2 e &e
z, pG
=&m~(1& p G ) p G +m~ 2(1& p G ) 2 p G , g p
} g p }
(D3)
z
=e ze &e (1& pG ) =m~p G , z, pG
2
z
=e 2ze &e (1& pG ) =m~ 2p G .
2
z, pG
Substituting these values in Eqs. (D1) and (D2) gives Eq. (29). ACKNOWLEDGMENTS The research was made possible by a support to Patsy Haccou from the Royal Dutch Academy of Sciences and by a grant-in-aid in scientific research by the ministry of education, science, and culture, Japan, to Yoh Iwasa. We thank Hans Metz, John McNamara, Geza Meszena, Sean Collins, Evert Meelis, and John Val for their useful comments.
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Athreya, K. B., and Karlin, S., 1971b. Branching processes with random environments, II. Limit theorems, Ann. Math. Statist. 42, 18431858. Charlesworth, B., 1994. ``Evolution in Age-Structured Populations,'' 2nd ed., Cambridge Univ. Press, Cambridge. Chesson, P. L., 1982. The stabilizing effects of a random environment, J. Math. Biol. 15, 136. Chesson, P. L., and Ellner, S., 1989. Invasibility and stochastic boundedness in monotonic competition models, J. Math. Biol. 27, 117138. Cohen, D., 1966. Optimizing reproduction in a randomly varying environment, J. Theor. Biol. 12, 119129. Cohen, D., 1967. Optimizing reproduction in a randomly varying environment when a correlation may exist between the conditions at the time a choice has to made and the subsequent outcome, J. Theor. Biol. 16, 114. Cohen, D., and Levin, S. A., 1991. Dispersal in patchy environments: The effects of temporal and spatial structure, Theor. Popul. Biol. 39, 6399. Eshel, I., 1981. On the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size-a branching process model, J. Math. Biol. 12, 355362. Eshel, I., 1984. On the survival probability of a slightly advantageous mutant gene in a multitype population: A multidimensional branching process model, J. Math. Biol. 19, 201209. Feller, W., 1968. ``An Introduction to Probability Theory and Its Applications,'' Vol. I, 3rd ed., Wiley, New York. Feller, W., 1971. ``An Introduction to Probability Theory and Its Applications,'' Vol. II, 2nd ed., Wiley, New York. Goel, N. S., and Richter-Dyn, N., 1972. ``Stochastic Models in Biology,'' Academic Press, New York. Haccou, P., and Iwasa, Y., 1995. Optimal mixed strategies in stochastic environments, Theor. Popul. Biol. 47, 212243. Hall, P., and Heyde, C. C., 1980. ``Martingale Limit Theory and Its Applications,'' Academic Press, New York. Hanski, I., 1992. Inferences from ecological incidence functions, Am. Nat. 139, 657662. Hanski, I., 1994. A practical model of metapopulation dynamics, J. Anim. Ecol. 63, 151162. Iwasa, Y., and Mochizuki, H., 1988. Probability of population extinction accompanying a temporary decrease of population size, Res. Popul. Ecol. 30, 145164. Iwasa, Y., and Haccou, P., 1994. Emergence patterns of male butterflies in stochastic environments, Evol. Ecol. 8, 503523. Karlin, S., and Taylor, H. M., 1975. ``A First Course in Stochastic Processes,'' 2nd ed., Academic Press, New York. Keiding, N., and Nielsen, J. E., 1973. The growth of supercritical branching processes with random environments, Ann. Probab. 1, 10651067. Keiding, N., 1975. Extinction and exponential growth in random environments, Theor. Popul. Biol. 8, 4963. Kondo, J., 1991. ``Integral Equations,'' Clarendon Press, Oxford. Kurtz, T. G., 1978. Diffusion approximations for branching processes, in ``Branching Processes'' (A. Joffe and P. Ney, Eds.), pp. 269292, Dekker, New York. Levins, R., 1968. ``Evolution in Changing Environments,'' Princeton Univ. Press, Princeton, NJ. Levin, S.A., Cohen, D., and Hastings, A., 1984. Dispersal strategies in patchy environments, Theor. Popul. Biol. 26, 165191. Lewontin, R. H., and Cohen, D., 1969. On population growth in a randomly varying environment, Proc. Nat. Acad. Sc. USA 62, 10561060. Ludwig, D., 1976. A singular perturbation problem in the theory of population extinction, Soc. Indus. Appl. Math.Am. Math. Soc. Proc. 10, 87104.
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