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Evolutionarily stable germination strategies with time-correlated yield
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Theoretical Population Biology 70 (2006) 255–261 www.elsevier.com/locate/tpb
Evolutionarily stable germination strategies with time-correlated yield Angelo Valleriani Max Planck Institute of Colloids and Interfaces, Theory Division D-14424 Potsdam, Germany Received 28 April 2005 Available online 12 August 2006
Abstract We investigate the effect of auto-correlated yield on the evolutionarily stable germination fraction of dormant seeds. By using both analytics and numerics, we first show that in a regime of small fluctuations a positive correlation reduces dormancy and a negative correlation enhances dormancy. By extending the numerical analysis we also show that in the regime of large fluctuations a more complex picture emerges where also negative correlations can reduce dormancy. r 2006 Elsevier Inc. All rights reserved. Keywords: Delayed germination; Environmental stochasticity; Auto-correlation; Dormancy; Seed bank; Evolutionarily stable strategy
1. Introduction Plant species living in deserts and strongly varying environments need to keep a permanent soil seed bank in order to avoid extinction (Cohen, 1966, 1967; Venable and Lawlor, 1980; Bulmer, 1984; Ellner, 1985a, b; Brown and Venable, 1986; Venable and Brown, 1988; Gutterman, 2002; Rees, 1994). This is achieved by means of delayed germination, which is a mechanism that allows seeds to remain dormant in the soil even if optimal conditions for germination are met. In this work, we describe delayed germination as an evolutionary strategy in stochastic environments (Cohen, 1966; Bulmer, 1984; Ellner, 1985a, b). According to this approach, it is the large level of the environmental variation entering through the between year changes of the yield Y, which is the average number of seeds per plant, that determines the degree of dormancy of the seeds. In order to simplify the formulation of models, it is usually assumed that all seeds produced after one given season share the same germination probability g so that, for large seed banks, g is the fraction of germinating seeds. The first study dedicated to the determination of the germination fraction as an evolutionary strategy was presented by Cohen (1966). There, the germination fraction Fax: +49 0 331 567 9612.
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was obtained as optimal strategy in a stochastic environment. The assumption of that work was that the dynamics of the seed bank is described by a multiplicative process without density dependence. Due to this assumption, it was possible to identify the optimal strategy as the value of the germination fraction that maximized the geometric average of the growth rate. Under the simple assumption that only two kinds of season occurs, namely good season with probability p and a season with no seed production with probability 1 p, Cohen (1966) showed that the optimal germination fraction is approximately equal to p. This study showed that under certain simplifying assumptions, there is a simple relationship between the optimal g and the statistical properties of the environment. This strategy was called bet-hedging (Cohen, 1966; Slatkin, 1974; Philippi and Seger, 1989). A further major step in the development of the theory was presented by Bulmer (1984) and Ellner (1985a, b), where density dependence effects have been included in the dynamics of the seed bank. The density dependence was introduced in terms of a decreasing seed production as function of the seedlings density. In these cases, the number of seeds in the seed bank fluctuates around a certain average. The nature of these fluctuations depended on the statistical properties of the environmental fluctuations. Since there was no long-term growth of the size of the seed bank, it was not possible to apply the method used by Cohen (1966) to compute the optimal germination strategy.
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In fact, what had to be done was to study this system by means of the invasibility analysis in order to find the evolutionarily stable strategy (Maynard Smith, 1982) for the system. It was then possible to show that large fluctuations of the environmental variables and of the seed bank size select for small evolutionarily stable germination fractions. Recently, Valleriani (2005) has also shown that the evolutionarily stable value of g depends on the moments of the distribution of the yield in a rather non-trivial way. In particular, it was shown that the second, third and fourth moment act differently in the determination of the evolutionarily stable germination fraction. Of interest here is another property of environmental stochasticity, namely the auto-correlation of its variables. It was stressed already some time ago (Halley, 1996) that environmental noise might be correlated in time and that the correlations should have an effect on the population dynamics, on the risk of extinction, and on community composition (Lawton, 1988; Steel and Henderson, 1984; Ripa et al., 1998; Halley and Kunin, 1999). Recently, Levine and Rees (2004) have shown that the autocorrelation properties of the year quality can influence the survival of rare annuals in a grass-dominated system. Their conclusion was that positive auto-correlation would not enhance coexistence whereas negative correlations would do. The basic mechanism was that positive yearto-year correlations in the stochastic variable would tend to produce long series of seasons with the same quality and favor the grasses. On the other hand, negative correlations would tend to produce short series and a strong year-toyear variability. Another study by Vasseur and Yodzis (2004) showed moreover that most, if not all environmental variables (e.g. precipitation, temperature) are auto-correlated, with properties of the correlation depending on the particular environment. Although Vasseur and Yodzis (2004) did not propose any particular general model on how to take into account their observation, they correctly stressed that the presence or absence of effects due to correlated noise must be taken into account in ecological modeling. In particular, Vasseur and Yodzis (2004) found that going from the inland towards the sea we have to expect an increase in the length of the auto-correlation. Thus, if we want to compare populations located at different geographical positions it is necessary and convenient to have a theory that is able to incorporate this important aspect of the environmental variables. The model system we want to consider here is given by annual plants in strongly stochastic and unpredictable environment like deserts. The most natural stochastic variable that enters those models is given by average yearly precipitation because this, at the end, influences the yield of the plants. Although we do not have a strong argument to say that average yearly precipitation must be autocorrelated, the work of Vasseur and Yodzis (2004) tells us that it is likely that it will be so.
We consider here the seed bank dynamics governed by the equation Sðt þ 1Þ ¼ SðtÞ½gf ðgSðtÞÞY ðtÞ þ ð1 gÞð1 dÞ,
(1)
where SðtÞ is the size of the soil seed bank at the beginning of season t. The stochastic variable Y ðtÞ is the average yield per adult plant in season t. The distribution of Y is given by the between year variation of the average yield per adult plant and in this and the next section we assume that it is a continuous variable. As we shall see later, in the numerical part of this work we will consider a discrete Y. The variable g in Eq. (1) is the fraction of SðtÞ that germinates and d is the fraction of non-germinating seeds, given by ð1 gÞSðtÞ, that become unviable during season t. The function f ðsÞ is responsible for density dependence and gives the fraction of germinating seeds s ¼ gSðtÞ that survive competition. We will assume (Ellner, 1985b; Nilsson et al., 1994; Mathias and Kisdi, 2002) that this function is given by the reciprocal yield model f ðsÞ ¼
K , K þs
(2)
where K is the carrying capacity of the system, namely the maximum density of adult plants supported by the environment. The previous studies on evolutionarily stable germination fraction assumed that the stochastic variable Y ðtÞ is not correlated. In this work instead we consider a correlated function as follows. Let us define yðtÞ ¼ Y ðtÞ Y , where Y is the average value of Y. We will then define M 2 ðnÞ ¼ hyðtÞyðt þ nÞi,
(3)
the two-point correlation function of yðtÞ, where M 2 ð0Þ is the variance of the distribution of Y, and where hfi denotes the long time average of the function f. In the following, we will sometimes also use the short form f to denote this time average. We would then recover the function for non-correlated yield when for n40 we have M 2 ðnÞ ¼ 0. In principle, M 2 ðnÞ can be positive or negative depending on whether the two values of y are positively or negatively correlated. The determination of the evolutionarily stable value of g occurs by exploiting the method of invasibility analysis (Bulmer, 1994). This method, already extensively discussed in the literature (Bulmer, 1984; Ellner, 1985a, b) provides the Evolutionarily Stable Strategy (ESS), which is given as solution of the equation (Bulmer, 1984; Ellner, 1985a) f ðgSðtÞÞY ðtÞ ð1 dÞ ¼ 0. (4) gf ðgSðtÞÞY ðtÞ þ ð1 gÞð1 dÞ To solve Eq. (4) it is necessary to numerically perform the time average over many generations of the dynamics given by Eq. (1) for many different values of g until the value that satisfies Eq. (4) is found (Bulmer, 1984; Ellner, 1985a). For this reason we will hereafter refer to the solution of Eq. (4) as the numerical solution and we will
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denote it with gESS . Nevertheless, the solution of Eq. (4) can be achieved with any degree of accuracy, the only limit being computational time and numerical precision. 2. Analytical approach In a recent work, we have presented an analytical solution that approximates well the numerical one obtained from Eq. (4) in absence of time correlations (Valleriani, 2005). Indeed, we showed that when fluctuations are small and when Y is symmetric around its average, the solution gESS of Eq. (4) can be approximated with the solution g of 1 ð1 dÞ½1 þ T 2 ¼ 0,
(5)
where hx2 ðtÞi hxðtÞxðt þ 1Þi (6) S2 is the second order contribution in the expansion of Eq. (4) in x defined as SðtÞ ¼ S þ xðtÞ and S is the average size of the seed bank. In absence of time correlations in Y we could show that the function T 2 becomes simply
imation is equivalent to state that in the regime x5K5gS the reciprocal yield model is well approximated by the saturated yield model (Ellner, 1985b). As further step in the determination of T 2 , we have to determine the dynamical equation for xðtÞ. This is given again by Eq. (1) after inserting SðtÞ ¼ S þ xðtÞ as xðt þ 1Þ ¼ gSðtÞf ðgSðtÞÞY ðtÞ ð1 vÞS þ vxðtÞ.
ð1 vÞ2 C2, ð1 þ vÞ
(7)
where v ¼ ð1 gÞð1 dÞ and C 2 ¼ M 2 ð0Þ=Y 2 . This allowed to use Eq. (5) to find an approximate solution of Eq. (4) and we could show that this solution is good when the distribution of Y is symmetric around the average and it is not too broad. We must notice here that a result similar to our Eqs. (5) and (7) was obtained also in Ellner (1985b). To obtain T 2 in presence of time correlations we must first look back at Eq. (1) together with Eq. (2) to compute the average size of the seed bank. By using the expansion in SðtÞ ¼ S þ xðtÞ, we first obtain the exact relation ð1 vÞS ¼ hgSðtÞf ðgSðtÞÞY ðtÞi,
(8)
where we now look for an approximate expression of the average in the rhs. To do this, we expand SðtÞ in the rhs around its average S and take into account the effect of the correlation between SðtÞ and Y ðtÞ. In the limit x5K5gS this leads to the first order approximation 2 K K ð1 vÞS ¼ KY 1 hxðtÞY ðtÞi þ , (9) þg gS gS where we see that the corrections to the first term are at most of the order K=ðgSÞ51 and we thus neglect them in this approximation. Therefore, in the regime of small fluctuations, we then discard the smaller term and we take the approximate value KY , (10) 1v which is the same result as it was obtained without timecorrelations (Valleriani, 2005). We notice here that Eq. (10) is equivalent to assume gSðtÞf ðgSðtÞÞ ¼ K. This approx-
S¼
(11)
By using now the approximation gSðtÞf ðgSðtÞÞ ¼ K, we get xðt þ 1Þ ¼ KyðtÞ þ vxðtÞ,
(12)
where we have also used Eq. (10) and the definition of yðtÞ ¼ Y ðtÞ Y . After this preamble, we are now ready to compute the numerator of T 2 . By recursively using Eq. (12), one obtains the identity
T2 ¼
T2 ¼
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hxðtÞxðt þ 1Þi ¼ vhx2 ðtÞi þ
1 K2 X vj M 2 ðjÞ, v j¼1
(13)
where M 2 ðnÞ was defined in Eq. (3). By means of the same technique we obtain also ! 1 X K2 2 j hx ðtÞi ¼ M 2 ð0Þ þ 2 v M 2 ðjÞ , (14) 1 v2 j¼1 P ~ ¼ 1 vj M 2 ðjÞ we have that so that by defining M 1 hx2 i hxðtÞxðt þ 1Þi ¼
~ 1v K 2 M 2 ð0Þ K 2 M . 1þv v 1þv
(15)
By dividing now by S we finally obtain T2 ¼
ð1 vÞ2 0 C , ð1 þ vÞ 2
(16)
where C 02 ¼ C 2
~ 1v M . Y2 v
(17)
We can compare the formal similarity between Eqs. (16) ~ will have an effect similar and (7) and see that a negative M ~ will have an effect to an increased C 2 and that a positive M similar to a decreased C 2 . To proceed further, we consider now the following specific form of the correlation function: M 2 ðnÞ ¼ M 2 ð0ÞLn ,
(18)
where jLjX1 is the parameter that determines the correlation length N ¼ 1= logðjLjÞ and L can be both positive and negative. By taking into account now Eq. (18) and the definition of ~ we can easily see that M ~ ¼ M 2 ð0Þ v , (19) M Lv so that now hx2 i hxðtÞxðt þ 1Þi ¼
K 2 M 2 ð0Þ L 1 , 1þv Lv
(20)
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where we see that the L-dependent factor is larger than unity for Lo 1 and is smaller than unity for L41. Putting it all together, we have that the equation producing the approximate value of the evolutionarily stable germination fraction can be derived from Eq. (5) and is given by ð1 vÞ2 0 1 ð1 dÞ 1 þ C 2 ¼ 0, (21) 1þv
addition, the moments of the distribution of Y do not depend on the specific values of chosen. With this construction, we have that L¼
1 , 2 1
where we see that for ! 0:5 we have a non-correlated series. By using this value of L we can determine the value of T 2 as function of q and :
where
3. Numerical analysis In this section, we describe an analysis where we compare the values of gESS and g . To this purpose, we need to make several simplifications. First of all, we simplify the stochastic variable Y ðtÞ, which was a continuous variable in the previous section, to take only two values: ( Y þ ¼ Y ð1 þ qÞ with probability pþ ; Y ðtÞ ¼ (23) Y ¼ Y ð1 qÞ with probability p ; where Y is the average yield and p ¼ 1 pþ . In the absence of time correlations, we would fix pþ ¼ 0:5 so that q2 Y 2 would give the variance of the distribution of Y. The variable 0pqp1 is therefore the coefficient of variation of the yield Y. When q ¼ 0 there are no fluctuations around the average whereas when q ¼ 1 the variable Y ðtÞ will take the value zero with probability p . We would like now to create an exponentially correlated time series where the probability that we observe Y ðtÞ ¼ Y þ at any arbitrary t is equal to 0:5 and where the values of the moments of the distribution of Y are the same as in the uncorrelated case. In order to create such a time series, we will make pþ dependent on the value of the yield at time t 1. It is easy to create such series by imposing ( if Y ðt 1Þ ¼ Y þ ; pþ ¼ (24) 1 if Y ðt 1Þ ¼ Y ; where 0oo1. With this prescription, 0oo0:5 will generate a negatively correlated series, 0:5oo1 will generate a positively correlated series, and ¼ 0:5 will generate a series without correlations. From the symmetry of the construction, it is possible to realize that the events Y þ and Y will occur with equal frequency. In
T 2 ¼ q2
2 2 ð1 vÞ2 , 1 vð2 1Þ 1 þ v
(26)
which we have checked against the value of T 2 determined from Eq. (6). We have also checked that the higher order terms (see also Valleriani, 2005) in expansion (5) are small compared to T 2 . After this check, Eq. (5) takes the final form 2 2 ð1 vÞ2 1 ð1 dÞ 1 þ q2 ¼ 0, (27) 1 vð2 1Þ 1 þ v from which we can compute g , to be compared with gESS . The results are summarized in Fig. 1. This figure refers to the regime of small q, where we expect that the solution of Eq. (27) is good. As one can see, we have in fact a rather good qualitative and quantitative agreement between gESS and g . Moreover, we see also that positive time correlations enhance germination and that negative time correlations enhance dormancy. For larger values of q, the approximated g computed from Eq. (27) is no more precise. In this regime, therefore, we must rely only on an analysis of gESS . This is shown in Fig. 2. There we clearly see that as q becomes large, the value of gESS as function of becomes flat and nonmonotonous. For q ¼ 1, moreover, it becomes a monotonous decreasing curve. In this particular case, therefore, the role played by the correlations is reversed in 1 0.9 0.8 Germination fraction
M 2 ð0Þ L 1 (22) C 02 ¼ Y2 L v is the squared of the modified coefficient of variation of Y. Eq. (21) is the main analytical result of this paper. In order to verify this result, we must compute the value of g from it and compare it with the value of gESS computed from Eq. (4). This is possible and easy if we consider a specific realization of the stochastic variable Y ðtÞ and its time auto-correlation as we shall see in the following section.
(25)
0.7 0.6 0.5
ESS for q=0.4 g* for q=0.4 ESS for q=0.5
0.4
g* for q=0.5 ESS for q=0.6
0.3
g* for q=0.6 zero correlation
0.2 0.1
negative correlation
positive correlation
0 0.1
0.2
0.3
0.4 0.5 0.6 Value of ε
0.7
0.8
0.9
Fig. 1. Comparison of gESS with g in the regime of small q. Other parameters are: d ¼ 0:1 and Y ¼ 500.
ARTICLE IN PRESS A. Valleriani / Theoretical Population Biology 70 (2006) 255–261 1 0.9
ESS for q=0.8 ESS for q=0.9
0.8
ESS for q=1.0
Germination fraction
zero correlation
0.7 0.6 0.5 0.4 0.3 0.2 0.1
negative correlation
positive correlation
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Value of ε
Fig. 2. Evolutionarily stable strategy gESS for large q. Other parameters are: d ¼ 0:1 and Y ¼ 500. A behavior qualitatively similar to q ¼ 0:9 has been observed also for larger values of q.
comparison to the regime of small q. Here indeed, negative correlations select for higher germination fractions. The flattening of g as a function of for strong negative correlations is expected to happen whenever the variable v is very close to unity. This occurs when q is large (selecting for small g ) and d is small. Appendix A is devoted to a more detailed analysis of this rather technical aspect. 4. Conclusions In this work we have developed an equation to compute the approximate value of the evolutionarily stable germination fraction. The equation took into account a generic form of the auto-correlation of the continuous stochastic variable of the problem, namely of the yield. The result, given in Eq. (16), shows that positive correlations should favor germination and that negative correlations should enhance dormancy. The nature of this result is independent of the form of the correlation. Thus, this result holds both for colored and for exponential correlated noise. To verify our prediction, we have considered exponentially decaying positive and negative correlations. In the numerical part, we have also restricted our stochastic variable to take only two states. The prescription, given in Eqs. (23) and (24), ensures that the two states are visited with equal frequency in the average and nevertheless allows to tune the correlation by using just one parameter. To parameterize the changing broadness of the yield function we have used the parameter q. With this parameter we could therefore study the effects in very stable environments by taking q small. This is the regime were our analytical results were correct. Nevertheless, we could also study, at least numerically, the conditions of
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strongly varying environments by taking larger q and even putting q ¼ 1. These environments would resemble the desert conditions, where seasons with zero yield occur frequently. Despite these simplifications, we could see in Fig. 1 that our prediction is accurate in the range of parameters where our approximate solution is expected to hold. We can pretty well understand why negative correlations enhance dormancy in the small fluctuations regime. This is possible if we compare Eqs. (16) with (7). In fact, we see that negative correlations have the effect of increasing the effective size of the fluctuations, i.e. C 02 4C 2 , whereas positive correlations have the opposite effect. Intuitively, we can rephrase a similar situation studied by Kisdi and Mesze´na(1993) by saying that negative correlations tend to make a good year better and a bad year worse than without any correlation. In fact, the correlations in the yield induce a covariance between yield Y ðtÞ and the size of the seed bank SðtÞ. Since the covariance has the same sign of the correlation, after a good year, where seed production has been large, a bad year is more likely to follow and competition between seedlings will be large due to both the effect of the large number of seeds and to the low yield. After a bad year instead, a good year is likely to follow and competition between seedlings will be small due to both the effect of the smaller number of seeds and of the large yield. This implies that, compared to the case without time correlations, negative correlations tend to increase the fluctuations in year quality. A similar argument can be made also in the case of positive correlations. Furthermore, we have computed, by means of numerical averages, the evolutionarily stable germination fraction also outside the range of validity of our approximation. The results (see in Fig. 2) show that a flattening of the effect of the negative correlation occurs. We could demonstrate that this is expected to occur whenever both gESS and d are small. Since gESS is a decreasing function of q, we expected therefore that at large q and small d the derivative of gESS with respect to becomes zero. We could verify also this prediction in Fig. 3. A quite different behavior occurs whenever one of the two states of the yield takes the value zero. In Fig. 2 this is shown by the curve q ¼ 1. In this particular case, the negative correlation favors germination and positive correlations enhance dormancy. The intuitive explanation for this result is that with negative correlations the occurrence of zero-yield seasons is restricted to few or just one consecutive seasons, depending on the strength of the correlation. With positive correlations instead, long series of zero-yield seasons can occur with increased frequency as the strength of the correlation is increased. This long series of zero-yield seasons can severely reduce the size of the seed bank so that an enhanced dormancy is favored. The work presented here suggests that we should expect a dependence of the germination fraction on the autocorrelations of the main environmental variable of the model. A possible way to verify this prediction would be to
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1 0.9
ESS for q=0.8, d=0.5
Germination fraction
0.8
0.6 0.5 ESS for q=0.8, d=0.1
0.4
where v ¼ ð1 g Þð1 dÞ. To compute this derivative we would have to compute g from Eq. (27). However, even without knowing the exact value of g , we can see that dg lim ¼ 0. (A.3) v !1 d ¼0
0.3 ESS for q=0.8, d=0.01
0.1 0 0.1
0.2
0.3
(A.1)
holds for any v, where we recall that v ¼ ð1 gÞð1 dÞ. Due to the relation between g and v we then have dg 1 ð1 v Þ2 , (A.2) ¼ d ¼0 1 d 4
0.7
0.2
we obviously see that the relation dv qT 2 qT 2 ð1 vÞ2 ¼ ¼ d ¼0 q qv ¼0 4
0.4
0.5
0.6
0.7
0.8
0.9
Value of ε Fig. 3. Effect of mortality on gESS as dependent on . For large fluctuations and small mortality the effect of negative correlations becomes negligible.
consider a species distributed over a gradient of the correlation length of the yield Y. This suggestion has been inspired by the evidence provided in Vasseur and Yodzis (2004) where the correlation length of environmental variables was observed to change in a systematic way when going from the inland, dominated by series with short-range correlations, to the sea, dominated by series with long-range correlations. Yet, the work of Vasseur and Yodzis (2004) considers time series on the time scale of months, whereas the time scale of the population dynamics of the seed bank is one year. It is therefore unclear how much of the correlations seen in Vasseur and Yodzis (2004) are still visible at a larger time scale. Nevertheless, we believe that the empirical verification of the results presented here is a great challenge for field ecologists. Acknowledgments The author would like to thank Monika Schwager and Marcel Ausloos for suggesting several relevant references about correlated time series and their ecological relevance. Appendix A. Negative correlations and large fluctuations In this section we would like to understand the mechanism underlying the reduced effect of negative correlations for large fluctuations. To explain these results, we would again use T 2 and Eq. (27). Although Eq. (27) will never predict a decreasing g with , we can verify the conditions under which the derivative dg =d becomes zero at ¼ 0. Since we cannot compute the exact value of g from Eq. (27), we can use a method to see at least under which conditions this derivative changes from positive to zero. From Eq. (27)
This means that when both g and d are very small, so that v is close to unity, we should expect a small growth of g with . To test this hypothesis, we have computed gESS corresponding to q ¼ 0:8 for different values of d and plotted the result in Fig. 3. In conclusion, due to the fact that large q selects for small g we see that Eq. (A.3) correctly predicts that for large q and small d the effect of the negative correlations become weaker and weaker. References Brown, J.S., Venable, D.L., 1986. Evolutionary ecology of seed bank annuals in temporally varying environments. Am. Nat. 127, 31–47. Bulmer, M.G., 1984. Delayed germination of seeds: Cohen’s model revisited. Theor. Popul. Biol. 26, 367–377. Bulmer, M.G., 1994. Theoretical Evolutionary Ecology. Sinauer Associates Publishers, Sunderland, MA. Cohen, D., 1966. Optimizing reproduction in a randomly varying environment. J. Theor. Biol. 12, 119–129. 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 be made and the subsequent outcome. J. Theor. Biol. 16, 1–14. Ellner, S., 1985a. ESS germination strategies in randomly varying environments. I. Logistic-type models. Theor. Popul. Biol. 28, 50–79. Ellner, S., 1985b. ESS germination strategies in randomly varying environments. II. Reciprocal yield-law models. Theor. Popul. Biol. 28, 80–116. Gutterman, Y., 2002. Survival Strategies of Annual Desert Plants. Springer, Berlin Heidelberg. Halley, J.M., 1996. Ecology, evolution and 1=f -noise. TREE 11, 33–37. Halley, J.M., Kunin, W.E., 1999. Extinction risk and the 1=f family of noise models. J. Theor. Biol. 56, 215–230. Kisdi, E´., Mesze´na, G., 1993. Density dependent life history evolution in fluctuating environments. In: Yoshimura, J., Clark, C.W. (Eds.), Adaptation in Stochastic Environments. Lecture Notes in Biomathemaics, vol. 98. Springer, Berlin. Lawton, J.H., 1988. More time means more variation. Nature 334, 563. Levine, J.M., Rees, M., 2004. Effects of temporal variability on rare plant persistence in annual systems. Am. Nat. 164, 350–363. Mathias, A., Kisdi, E´., 2002. Adaptive diversification of germination strategies. Proc. R. Soc. London 269, 151–155. Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge. Nilsson, P., Fagerstro¨m, T., Tuomi, J., A˚stro¨m, M., 1994. Does seed dormancy benefit the mother plants by reducing sib competition? Evol. Ecol. 8, 422–430.
ARTICLE IN PRESS A. Valleriani / Theoretical Population Biology 70 (2006) 255–261 Philippi, T., Seger, J., 1989. Hedging one’s evolutionary bets, revised. TREE 4, 41–44. Ripa, J., Lundberg, P., Kaiala, V., 1998. A general theory of environmental noise in ecological food webs. Am. Nat. 151, 256–263. Rees, M., 1994. Delayed germination of seeds: a look at the effects of adult longevity, the timing of reproduction, and population age/stage structure. Am. Nat. 144, 43–64. Slatkin, M., 1974. Hedging one’s evolutionary bet. Nature 250, 704–705. Steel, J.H., Henderson, E.W., 1984. Modeling long-term fluctuations in fish stocks. Science 224, 985–987.
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Theoretical Population Biology 70 (2006) 255–261 www.elsevier.com/locate/tpb
Evolutionarily stable germination strategies with time-correlated yield Angelo Valleriani Max Planck Institute of Colloids and Interfaces, Theory Division D-14424 Potsdam, Germany Received 28 April 2005 Available online 12 August 2006
Abstract We investigate the effect of auto-correlated yield on the evolutionarily stable germination fraction of dormant seeds. By using both analytics and numerics, we first show that in a regime of small fluctuations a positive correlation reduces dormancy and a negative correlation enhances dormancy. By extending the numerical analysis we also show that in the regime of large fluctuations a more complex picture emerges where also negative correlations can reduce dormancy. r 2006 Elsevier Inc. All rights reserved. Keywords: Delayed germination; Environmental stochasticity; Auto-correlation; Dormancy; Seed bank; Evolutionarily stable strategy
1. Introduction Plant species living in deserts and strongly varying environments need to keep a permanent soil seed bank in order to avoid extinction (Cohen, 1966, 1967; Venable and Lawlor, 1980; Bulmer, 1984; Ellner, 1985a, b; Brown and Venable, 1986; Venable and Brown, 1988; Gutterman, 2002; Rees, 1994). This is achieved by means of delayed germination, which is a mechanism that allows seeds to remain dormant in the soil even if optimal conditions for germination are met. In this work, we describe delayed germination as an evolutionary strategy in stochastic environments (Cohen, 1966; Bulmer, 1984; Ellner, 1985a, b). According to this approach, it is the large level of the environmental variation entering through the between year changes of the yield Y, which is the average number of seeds per plant, that determines the degree of dormancy of the seeds. In order to simplify the formulation of models, it is usually assumed that all seeds produced after one given season share the same germination probability g so that, for large seed banks, g is the fraction of germinating seeds. The first study dedicated to the determination of the germination fraction as an evolutionary strategy was presented by Cohen (1966). There, the germination fraction Fax: +49 0 331 567 9612.
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was obtained as optimal strategy in a stochastic environment. The assumption of that work was that the dynamics of the seed bank is described by a multiplicative process without density dependence. Due to this assumption, it was possible to identify the optimal strategy as the value of the germination fraction that maximized the geometric average of the growth rate. Under the simple assumption that only two kinds of season occurs, namely good season with probability p and a season with no seed production with probability 1 p, Cohen (1966) showed that the optimal germination fraction is approximately equal to p. This study showed that under certain simplifying assumptions, there is a simple relationship between the optimal g and the statistical properties of the environment. This strategy was called bet-hedging (Cohen, 1966; Slatkin, 1974; Philippi and Seger, 1989). A further major step in the development of the theory was presented by Bulmer (1984) and Ellner (1985a, b), where density dependence effects have been included in the dynamics of the seed bank. The density dependence was introduced in terms of a decreasing seed production as function of the seedlings density. In these cases, the number of seeds in the seed bank fluctuates around a certain average. The nature of these fluctuations depended on the statistical properties of the environmental fluctuations. Since there was no long-term growth of the size of the seed bank, it was not possible to apply the method used by Cohen (1966) to compute the optimal germination strategy.
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In fact, what had to be done was to study this system by means of the invasibility analysis in order to find the evolutionarily stable strategy (Maynard Smith, 1982) for the system. It was then possible to show that large fluctuations of the environmental variables and of the seed bank size select for small evolutionarily stable germination fractions. Recently, Valleriani (2005) has also shown that the evolutionarily stable value of g depends on the moments of the distribution of the yield in a rather non-trivial way. In particular, it was shown that the second, third and fourth moment act differently in the determination of the evolutionarily stable germination fraction. Of interest here is another property of environmental stochasticity, namely the auto-correlation of its variables. It was stressed already some time ago (Halley, 1996) that environmental noise might be correlated in time and that the correlations should have an effect on the population dynamics, on the risk of extinction, and on community composition (Lawton, 1988; Steel and Henderson, 1984; Ripa et al., 1998; Halley and Kunin, 1999). Recently, Levine and Rees (2004) have shown that the autocorrelation properties of the year quality can influence the survival of rare annuals in a grass-dominated system. Their conclusion was that positive auto-correlation would not enhance coexistence whereas negative correlations would do. The basic mechanism was that positive yearto-year correlations in the stochastic variable would tend to produce long series of seasons with the same quality and favor the grasses. On the other hand, negative correlations would tend to produce short series and a strong year-toyear variability. Another study by Vasseur and Yodzis (2004) showed moreover that most, if not all environmental variables (e.g. precipitation, temperature) are auto-correlated, with properties of the correlation depending on the particular environment. Although Vasseur and Yodzis (2004) did not propose any particular general model on how to take into account their observation, they correctly stressed that the presence or absence of effects due to correlated noise must be taken into account in ecological modeling. In particular, Vasseur and Yodzis (2004) found that going from the inland towards the sea we have to expect an increase in the length of the auto-correlation. Thus, if we want to compare populations located at different geographical positions it is necessary and convenient to have a theory that is able to incorporate this important aspect of the environmental variables. The model system we want to consider here is given by annual plants in strongly stochastic and unpredictable environment like deserts. The most natural stochastic variable that enters those models is given by average yearly precipitation because this, at the end, influences the yield of the plants. Although we do not have a strong argument to say that average yearly precipitation must be autocorrelated, the work of Vasseur and Yodzis (2004) tells us that it is likely that it will be so.
We consider here the seed bank dynamics governed by the equation Sðt þ 1Þ ¼ SðtÞ½gf ðgSðtÞÞY ðtÞ þ ð1 gÞð1 dÞ,
(1)
where SðtÞ is the size of the soil seed bank at the beginning of season t. The stochastic variable Y ðtÞ is the average yield per adult plant in season t. The distribution of Y is given by the between year variation of the average yield per adult plant and in this and the next section we assume that it is a continuous variable. As we shall see later, in the numerical part of this work we will consider a discrete Y. The variable g in Eq. (1) is the fraction of SðtÞ that germinates and d is the fraction of non-germinating seeds, given by ð1 gÞSðtÞ, that become unviable during season t. The function f ðsÞ is responsible for density dependence and gives the fraction of germinating seeds s ¼ gSðtÞ that survive competition. We will assume (Ellner, 1985b; Nilsson et al., 1994; Mathias and Kisdi, 2002) that this function is given by the reciprocal yield model f ðsÞ ¼
K , K þs
(2)
where K is the carrying capacity of the system, namely the maximum density of adult plants supported by the environment. The previous studies on evolutionarily stable germination fraction assumed that the stochastic variable Y ðtÞ is not correlated. In this work instead we consider a correlated function as follows. Let us define yðtÞ ¼ Y ðtÞ Y , where Y is the average value of Y. We will then define M 2 ðnÞ ¼ hyðtÞyðt þ nÞi,
(3)
the two-point correlation function of yðtÞ, where M 2 ð0Þ is the variance of the distribution of Y, and where hfi denotes the long time average of the function f. In the following, we will sometimes also use the short form f to denote this time average. We would then recover the function for non-correlated yield when for n40 we have M 2 ðnÞ ¼ 0. In principle, M 2 ðnÞ can be positive or negative depending on whether the two values of y are positively or negatively correlated. The determination of the evolutionarily stable value of g occurs by exploiting the method of invasibility analysis (Bulmer, 1994). This method, already extensively discussed in the literature (Bulmer, 1984; Ellner, 1985a, b) provides the Evolutionarily Stable Strategy (ESS), which is given as solution of the equation (Bulmer, 1984; Ellner, 1985a) f ðgSðtÞÞY ðtÞ ð1 dÞ ¼ 0. (4) gf ðgSðtÞÞY ðtÞ þ ð1 gÞð1 dÞ To solve Eq. (4) it is necessary to numerically perform the time average over many generations of the dynamics given by Eq. (1) for many different values of g until the value that satisfies Eq. (4) is found (Bulmer, 1984; Ellner, 1985a). For this reason we will hereafter refer to the solution of Eq. (4) as the numerical solution and we will
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denote it with gESS . Nevertheless, the solution of Eq. (4) can be achieved with any degree of accuracy, the only limit being computational time and numerical precision. 2. Analytical approach In a recent work, we have presented an analytical solution that approximates well the numerical one obtained from Eq. (4) in absence of time correlations (Valleriani, 2005). Indeed, we showed that when fluctuations are small and when Y is symmetric around its average, the solution gESS of Eq. (4) can be approximated with the solution g of 1 ð1 dÞ½1 þ T 2 ¼ 0,
(5)
where hx2 ðtÞi hxðtÞxðt þ 1Þi (6) S2 is the second order contribution in the expansion of Eq. (4) in x defined as SðtÞ ¼ S þ xðtÞ and S is the average size of the seed bank. In absence of time correlations in Y we could show that the function T 2 becomes simply
imation is equivalent to state that in the regime x5K5gS the reciprocal yield model is well approximated by the saturated yield model (Ellner, 1985b). As further step in the determination of T 2 , we have to determine the dynamical equation for xðtÞ. This is given again by Eq. (1) after inserting SðtÞ ¼ S þ xðtÞ as xðt þ 1Þ ¼ gSðtÞf ðgSðtÞÞY ðtÞ ð1 vÞS þ vxðtÞ.
ð1 vÞ2 C2, ð1 þ vÞ
(7)
where v ¼ ð1 gÞð1 dÞ and C 2 ¼ M 2 ð0Þ=Y 2 . This allowed to use Eq. (5) to find an approximate solution of Eq. (4) and we could show that this solution is good when the distribution of Y is symmetric around the average and it is not too broad. We must notice here that a result similar to our Eqs. (5) and (7) was obtained also in Ellner (1985b). To obtain T 2 in presence of time correlations we must first look back at Eq. (1) together with Eq. (2) to compute the average size of the seed bank. By using the expansion in SðtÞ ¼ S þ xðtÞ, we first obtain the exact relation ð1 vÞS ¼ hgSðtÞf ðgSðtÞÞY ðtÞi,
(8)
where we now look for an approximate expression of the average in the rhs. To do this, we expand SðtÞ in the rhs around its average S and take into account the effect of the correlation between SðtÞ and Y ðtÞ. In the limit x5K5gS this leads to the first order approximation 2 K K ð1 vÞS ¼ KY 1 hxðtÞY ðtÞi þ , (9) þg gS gS where we see that the corrections to the first term are at most of the order K=ðgSÞ51 and we thus neglect them in this approximation. Therefore, in the regime of small fluctuations, we then discard the smaller term and we take the approximate value KY , (10) 1v which is the same result as it was obtained without timecorrelations (Valleriani, 2005). We notice here that Eq. (10) is equivalent to assume gSðtÞf ðgSðtÞÞ ¼ K. This approx-
S¼
(11)
By using now the approximation gSðtÞf ðgSðtÞÞ ¼ K, we get xðt þ 1Þ ¼ KyðtÞ þ vxðtÞ,
(12)
where we have also used Eq. (10) and the definition of yðtÞ ¼ Y ðtÞ Y . After this preamble, we are now ready to compute the numerator of T 2 . By recursively using Eq. (12), one obtains the identity
T2 ¼
T2 ¼
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hxðtÞxðt þ 1Þi ¼ vhx2 ðtÞi þ
1 K2 X vj M 2 ðjÞ, v j¼1
(13)
where M 2 ðnÞ was defined in Eq. (3). By means of the same technique we obtain also ! 1 X K2 2 j hx ðtÞi ¼ M 2 ð0Þ þ 2 v M 2 ðjÞ , (14) 1 v2 j¼1 P ~ ¼ 1 vj M 2 ðjÞ we have that so that by defining M 1 hx2 i hxðtÞxðt þ 1Þi ¼
~ 1v K 2 M 2 ð0Þ K 2 M . 1þv v 1þv
(15)
By dividing now by S we finally obtain T2 ¼
ð1 vÞ2 0 C , ð1 þ vÞ 2
(16)
where C 02 ¼ C 2
~ 1v M . Y2 v
(17)
We can compare the formal similarity between Eqs. (16) ~ will have an effect similar and (7) and see that a negative M ~ will have an effect to an increased C 2 and that a positive M similar to a decreased C 2 . To proceed further, we consider now the following specific form of the correlation function: M 2 ðnÞ ¼ M 2 ð0ÞLn ,
(18)
where jLjX1 is the parameter that determines the correlation length N ¼ 1= logðjLjÞ and L can be both positive and negative. By taking into account now Eq. (18) and the definition of ~ we can easily see that M ~ ¼ M 2 ð0Þ v , (19) M Lv so that now hx2 i hxðtÞxðt þ 1Þi ¼
K 2 M 2 ð0Þ L 1 , 1þv Lv
(20)
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where we see that the L-dependent factor is larger than unity for Lo 1 and is smaller than unity for L41. Putting it all together, we have that the equation producing the approximate value of the evolutionarily stable germination fraction can be derived from Eq. (5) and is given by ð1 vÞ2 0 1 ð1 dÞ 1 þ C 2 ¼ 0, (21) 1þv
addition, the moments of the distribution of Y do not depend on the specific values of chosen. With this construction, we have that L¼
1 , 2 1
where we see that for ! 0:5 we have a non-correlated series. By using this value of L we can determine the value of T 2 as function of q and :
where
3. Numerical analysis In this section, we describe an analysis where we compare the values of gESS and g . To this purpose, we need to make several simplifications. First of all, we simplify the stochastic variable Y ðtÞ, which was a continuous variable in the previous section, to take only two values: ( Y þ ¼ Y ð1 þ qÞ with probability pþ ; Y ðtÞ ¼ (23) Y ¼ Y ð1 qÞ with probability p ; where Y is the average yield and p ¼ 1 pþ . In the absence of time correlations, we would fix pþ ¼ 0:5 so that q2 Y 2 would give the variance of the distribution of Y. The variable 0pqp1 is therefore the coefficient of variation of the yield Y. When q ¼ 0 there are no fluctuations around the average whereas when q ¼ 1 the variable Y ðtÞ will take the value zero with probability p . We would like now to create an exponentially correlated time series where the probability that we observe Y ðtÞ ¼ Y þ at any arbitrary t is equal to 0:5 and where the values of the moments of the distribution of Y are the same as in the uncorrelated case. In order to create such a time series, we will make pþ dependent on the value of the yield at time t 1. It is easy to create such series by imposing ( if Y ðt 1Þ ¼ Y þ ; pþ ¼ (24) 1 if Y ðt 1Þ ¼ Y ; where 0oo1. With this prescription, 0oo0:5 will generate a negatively correlated series, 0:5oo1 will generate a positively correlated series, and ¼ 0:5 will generate a series without correlations. From the symmetry of the construction, it is possible to realize that the events Y þ and Y will occur with equal frequency. In
T 2 ¼ q2
2 2 ð1 vÞ2 , 1 vð2 1Þ 1 þ v
(26)
which we have checked against the value of T 2 determined from Eq. (6). We have also checked that the higher order terms (see also Valleriani, 2005) in expansion (5) are small compared to T 2 . After this check, Eq. (5) takes the final form 2 2 ð1 vÞ2 1 ð1 dÞ 1 þ q2 ¼ 0, (27) 1 vð2 1Þ 1 þ v from which we can compute g , to be compared with gESS . The results are summarized in Fig. 1. This figure refers to the regime of small q, where we expect that the solution of Eq. (27) is good. As one can see, we have in fact a rather good qualitative and quantitative agreement between gESS and g . Moreover, we see also that positive time correlations enhance germination and that negative time correlations enhance dormancy. For larger values of q, the approximated g computed from Eq. (27) is no more precise. In this regime, therefore, we must rely only on an analysis of gESS . This is shown in Fig. 2. There we clearly see that as q becomes large, the value of gESS as function of becomes flat and nonmonotonous. For q ¼ 1, moreover, it becomes a monotonous decreasing curve. In this particular case, therefore, the role played by the correlations is reversed in 1 0.9 0.8 Germination fraction
M 2 ð0Þ L 1 (22) C 02 ¼ Y2 L v is the squared of the modified coefficient of variation of Y. Eq. (21) is the main analytical result of this paper. In order to verify this result, we must compute the value of g from it and compare it with the value of gESS computed from Eq. (4). This is possible and easy if we consider a specific realization of the stochastic variable Y ðtÞ and its time auto-correlation as we shall see in the following section.
(25)
0.7 0.6 0.5
ESS for q=0.4 g* for q=0.4 ESS for q=0.5
0.4
g* for q=0.5 ESS for q=0.6
0.3
g* for q=0.6 zero correlation
0.2 0.1
negative correlation
positive correlation
0 0.1
0.2
0.3
0.4 0.5 0.6 Value of ε
0.7
0.8
0.9
Fig. 1. Comparison of gESS with g in the regime of small q. Other parameters are: d ¼ 0:1 and Y ¼ 500.
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ESS for q=0.8 ESS for q=0.9
0.8
ESS for q=1.0
Germination fraction
zero correlation
0.7 0.6 0.5 0.4 0.3 0.2 0.1
negative correlation
positive correlation
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Value of ε
Fig. 2. Evolutionarily stable strategy gESS for large q. Other parameters are: d ¼ 0:1 and Y ¼ 500. A behavior qualitatively similar to q ¼ 0:9 has been observed also for larger values of q.
comparison to the regime of small q. Here indeed, negative correlations select for higher germination fractions. The flattening of g as a function of for strong negative correlations is expected to happen whenever the variable v is very close to unity. This occurs when q is large (selecting for small g ) and d is small. Appendix A is devoted to a more detailed analysis of this rather technical aspect. 4. Conclusions In this work we have developed an equation to compute the approximate value of the evolutionarily stable germination fraction. The equation took into account a generic form of the auto-correlation of the continuous stochastic variable of the problem, namely of the yield. The result, given in Eq. (16), shows that positive correlations should favor germination and that negative correlations should enhance dormancy. The nature of this result is independent of the form of the correlation. Thus, this result holds both for colored and for exponential correlated noise. To verify our prediction, we have considered exponentially decaying positive and negative correlations. In the numerical part, we have also restricted our stochastic variable to take only two states. The prescription, given in Eqs. (23) and (24), ensures that the two states are visited with equal frequency in the average and nevertheless allows to tune the correlation by using just one parameter. To parameterize the changing broadness of the yield function we have used the parameter q. With this parameter we could therefore study the effects in very stable environments by taking q small. This is the regime were our analytical results were correct. Nevertheless, we could also study, at least numerically, the conditions of
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strongly varying environments by taking larger q and even putting q ¼ 1. These environments would resemble the desert conditions, where seasons with zero yield occur frequently. Despite these simplifications, we could see in Fig. 1 that our prediction is accurate in the range of parameters where our approximate solution is expected to hold. We can pretty well understand why negative correlations enhance dormancy in the small fluctuations regime. This is possible if we compare Eqs. (16) with (7). In fact, we see that negative correlations have the effect of increasing the effective size of the fluctuations, i.e. C 02 4C 2 , whereas positive correlations have the opposite effect. Intuitively, we can rephrase a similar situation studied by Kisdi and Mesze´na(1993) by saying that negative correlations tend to make a good year better and a bad year worse than without any correlation. In fact, the correlations in the yield induce a covariance between yield Y ðtÞ and the size of the seed bank SðtÞ. Since the covariance has the same sign of the correlation, after a good year, where seed production has been large, a bad year is more likely to follow and competition between seedlings will be large due to both the effect of the large number of seeds and to the low yield. After a bad year instead, a good year is likely to follow and competition between seedlings will be small due to both the effect of the smaller number of seeds and of the large yield. This implies that, compared to the case without time correlations, negative correlations tend to increase the fluctuations in year quality. A similar argument can be made also in the case of positive correlations. Furthermore, we have computed, by means of numerical averages, the evolutionarily stable germination fraction also outside the range of validity of our approximation. The results (see in Fig. 2) show that a flattening of the effect of the negative correlation occurs. We could demonstrate that this is expected to occur whenever both gESS and d are small. Since gESS is a decreasing function of q, we expected therefore that at large q and small d the derivative of gESS with respect to becomes zero. We could verify also this prediction in Fig. 3. A quite different behavior occurs whenever one of the two states of the yield takes the value zero. In Fig. 2 this is shown by the curve q ¼ 1. In this particular case, the negative correlation favors germination and positive correlations enhance dormancy. The intuitive explanation for this result is that with negative correlations the occurrence of zero-yield seasons is restricted to few or just one consecutive seasons, depending on the strength of the correlation. With positive correlations instead, long series of zero-yield seasons can occur with increased frequency as the strength of the correlation is increased. This long series of zero-yield seasons can severely reduce the size of the seed bank so that an enhanced dormancy is favored. The work presented here suggests that we should expect a dependence of the germination fraction on the autocorrelations of the main environmental variable of the model. A possible way to verify this prediction would be to
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1 0.9
ESS for q=0.8, d=0.5
Germination fraction
0.8
0.6 0.5 ESS for q=0.8, d=0.1
0.4
where v ¼ ð1 g Þð1 dÞ. To compute this derivative we would have to compute g from Eq. (27). However, even without knowing the exact value of g , we can see that dg lim ¼ 0. (A.3) v !1 d ¼0
0.3 ESS for q=0.8, d=0.01
0.1 0 0.1
0.2
0.3
(A.1)
holds for any v, where we recall that v ¼ ð1 gÞð1 dÞ. Due to the relation between g and v we then have dg 1 ð1 v Þ2 , (A.2) ¼ d ¼0 1 d 4
0.7
0.2
we obviously see that the relation dv qT 2 qT 2 ð1 vÞ2 ¼ ¼ d ¼0 q qv ¼0 4
0.4
0.5
0.6
0.7
0.8
0.9
Value of ε Fig. 3. Effect of mortality on gESS as dependent on . For large fluctuations and small mortality the effect of negative correlations becomes negligible.
consider a species distributed over a gradient of the correlation length of the yield Y. This suggestion has been inspired by the evidence provided in Vasseur and Yodzis (2004) where the correlation length of environmental variables was observed to change in a systematic way when going from the inland, dominated by series with short-range correlations, to the sea, dominated by series with long-range correlations. Yet, the work of Vasseur and Yodzis (2004) considers time series on the time scale of months, whereas the time scale of the population dynamics of the seed bank is one year. It is therefore unclear how much of the correlations seen in Vasseur and Yodzis (2004) are still visible at a larger time scale. Nevertheless, we believe that the empirical verification of the results presented here is a great challenge for field ecologists. Acknowledgments The author would like to thank Monika Schwager and Marcel Ausloos for suggesting several relevant references about correlated time series and their ecological relevance. Appendix A. Negative correlations and large fluctuations In this section we would like to understand the mechanism underlying the reduced effect of negative correlations for large fluctuations. To explain these results, we would again use T 2 and Eq. (27). Although Eq. (27) will never predict a decreasing g with , we can verify the conditions under which the derivative dg =d becomes zero at ¼ 0. Since we cannot compute the exact value of g from Eq. (27), we can use a method to see at least under which conditions this derivative changes from positive to zero. From Eq. (27)
This means that when both g and d are very small, so that v is close to unity, we should expect a small growth of g with . To test this hypothesis, we have computed gESS corresponding to q ¼ 0:8 for different values of d and plotted the result in Fig. 3. In conclusion, due to the fact that large q selects for small g we see that Eq. (A.3) correctly predicts that for large q and small d the effect of the negative correlations become weaker and weaker. References Brown, J.S., Venable, D.L., 1986. Evolutionary ecology of seed bank annuals in temporally varying environments. Am. Nat. 127, 31–47. Bulmer, M.G., 1984. Delayed germination of seeds: Cohen’s model revisited. Theor. Popul. Biol. 26, 367–377. Bulmer, M.G., 1994. Theoretical Evolutionary Ecology. Sinauer Associates Publishers, Sunderland, MA. Cohen, D., 1966. Optimizing reproduction in a randomly varying environment. J. Theor. Biol. 12, 119–129. 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 be made and the subsequent outcome. J. Theor. Biol. 16, 1–14. Ellner, S., 1985a. ESS germination strategies in randomly varying environments. I. Logistic-type models. Theor. Popul. Biol. 28, 50–79. Ellner, S., 1985b. ESS germination strategies in randomly varying environments. II. Reciprocal yield-law models. Theor. Popul. Biol. 28, 80–116. Gutterman, Y., 2002. Survival Strategies of Annual Desert Plants. Springer, Berlin Heidelberg. Halley, J.M., 1996. Ecology, evolution and 1=f -noise. TREE 11, 33–37. Halley, J.M., Kunin, W.E., 1999. Extinction risk and the 1=f family of noise models. J. Theor. Biol. 56, 215–230. Kisdi, E´., Mesze´na, G., 1993. Density dependent life history evolution in fluctuating environments. In: Yoshimura, J., Clark, C.W. (Eds.), Adaptation in Stochastic Environments. Lecture Notes in Biomathemaics, vol. 98. Springer, Berlin. Lawton, J.H., 1988. More time means more variation. Nature 334, 563. Levine, J.M., Rees, M., 2004. Effects of temporal variability on rare plant persistence in annual systems. Am. Nat. 164, 350–363. Mathias, A., Kisdi, E´., 2002. Adaptive diversification of germination strategies. Proc. R. Soc. London 269, 151–155. Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge. Nilsson, P., Fagerstro¨m, T., Tuomi, J., A˚stro¨m, M., 1994. Does seed dormancy benefit the mother plants by reducing sib competition? Evol. Ecol. 8, 422–430.
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