The role of the Allee effect in species packing

THEORETICAL

POPULATION

BIOLOGY

27, 27-50

(1985)

The Role of the Allee Effect in Species

Packing

F. A. HOPF Optical

Sciences

Center,

University

of Arizona,

Tucson,

Arizona,

Ithaca,

New

85721

AND

F. W. HOPF Department

of Biology,

Cornell

Received

University,

York

June 1983

The species-packing model of May and MacArthur is moditied to include a commonly-expected influence of sexual reproduction, namely a systematic diminishing of the rate of increase in a population when it becomes rare (called the “Allee effect”). This effect causes discreteness, i.e., a tiniteness to the density of species found along a resource axis. The species separate in a manner that relates to their intrinsic capacities to utilize the resources. Also discussed is the issue of species diversity gradients, and how the question of species discreteness might apply to it. The model with the Allee effect is in reasonable accord with island diversity patterns, but is minimally applicable to longitudinal gradients. Environmental stochasticity is modelled with noise terms governed by widely varying timescales. However, the resulting stochastic extinction is found neither to generate discrete distributions by itself, nor to have substantive effects on the discrete distributions generated by the Allee effect. C 1985 Academic Press, Inc.

I. INTRODUCTION Darwin’s (1979, p. 172) discussionof the “dilemma of transitional forms” begins with the question: “Why ... do we not everywhere see innumerable transitional forms? Why is not all nature in confusion instead of the species being, as we seethem, well defined?” What was missing was a reason why speciesshould become discrete and finite, rather than a continuum of types. MacArthur (1972, pp. 3444) addressed a similar problem with a model based on a Malthusian law of growth that is a linear function of the limiting resources.The resourcesare depleted by the population that consumesthem and this depletion is described by a logistic term. This model allows stable distributions of specieswhose resource utilization can be arbitrarily similar to each other (May, 1973, p. 148). This result describesa stable continuum of types, similar to the community structure envisioned by Darwin. May 27 0040.5809/85

$3.00

Copyright Q 1985 by Academic Press, Inc. All rights of reproduction m any form reserved.

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HOPF

(1973, pp. 109-156) proposed that stochastic effects would act as a mechanism causing some populations to grow and others to go to extinction. He modified MacArthur’s model accordingly by including noise terms in the equations (we denote this modified model by “Mm’). Simultaneously, a series of laser systems were being studied (Parks et al., 1968; Parks, 1969; Litvak, 1970), which obey the MM model with fluctuations, and give, empirically, a continuum of types. Strict constraints can be placed on these systems to force discreteness, but these constraints have no analog in biology. Since then many experiments (on many systems) have been performed, and all show continuous distributions, even in the presence of fluctuations. Thus, if May’s stochastic extinction argument is logically sound, the laser systems indicated above would behave differently than they do. Also, it has been recognized among biologically oriented theorists that stochasticity, in the form modeled by May (1973), is not a straightforward cause of discreteness (see Feldman and Roughgarden, 1975. pp. 204-206; Turelli, 198 1, and references therein). We are primarily trying to show that there is a biological rationale for why species should be discrete. We base our discussion on the MM model in the specific form given in MacArthur (1972, p. 37, Eq. (3)), which can be parameterized in terms of birth and death rates rather than r and K. For reasons explained in Section II, we cannot use Lokta-Volterra parameterization. By using the MM model, we can skip much of the analysis, including the most difficult algebraic parts, since they are found in May (1973, Chap. 6). We modify the MM model by adding an Allee effect (see below for definition), which adds the effect of sexual reproduction. The essential assumptions of our model are as follows: (a) In a sexually reproducing population there is a nonzero cost for being rare that is not compensated for by any benefit from rarity. (b) This cost is reflected in a decrease in birth rate, and hence in fitness, that is continuous and monotone as the population decreases. Biologically interesting cases have the property that the fitness decreases most rapidly when the populations are rare, and least rapidly when they are common. We call this decrease in fitness an “Allee” (1939) effect, which is consistent with the terminology in May (1973, p. Sl), Watt (1968, pp. 290-295), and Asmussen (1979) (we also call the Allee effect a “cost of rarity” for ease of discussion). “Fitness” describes the expected rate of increase of a population, in the sense of Fisher (1958, p. 37). The qualification in assumption (a) acknowledges that there are advantages to rarity, for example, in escaping predation by apostatically searching predators, but it is unclear how important such benefits are relative to the

ALLEE

EFFECT

IN SPECIES

PACKING

29

cost of being rare (see Levin and Segel, 1982, for a discussion of the benefits of rarity). The first sentence of assumption (b) gives the sufficient conditions for a mechanism to cause discreteness. However, only a few functions consistent with the sufficient conditions are biologically reasonable. The distinction between a cost of rarity and stochastic extinction in the sense of May is subtle but important. The former is a deterministic effect in which extinction occurs when titnesses become negative. The latter asserts that a population can go extinct due to random processes, despite a nonnegative fitness. Our argument is not that stochastic extinction does not occur in biology, but simply that, if modelled by noise terms, it does not have the consequences that May (1973) asserts. A simple case of the cost of rarity is the difficulty in finding a mate, and this is used for purposes of discussion (as done, e.g., in Asmussen, 1979). We acknowledge that in real populations this is only part of the cost of rarity, and it need not be the most important one. A social animal has little difficulty finding a mate, but rarity can present many other difficulties (see Allee, 1939). In any case, only the qualitative aspects of the Allee effect are of major importance, except in a few limiting cases. The cost of rarity is discussed in detail in Bernstein et al. (1984) and in Asmussen (1979). The terms “competitive exclusion” (Gause, 1934) and “limiting similarity” (MacArthur and Levins, 1967) have been used to describe the mechanisms which will lead to discreteness. However, as Hardin (1960) and Abrams (1983) point out, the terms have been used in a number of ways. We use both terms to describe the final results of a calculation, not as fundamental principles. Competitive exclusion refers to a result in which extinction has occurred; limiting similarity indicates a result in which the surviving species differ in a substantive way with respect to resource use. Both are used in the context of an “everything else being equal” (ceteris paribus) hypothesis: the species differ only in location on the resource axis (see Hopf and Hopf, in preparation, for further, extensive discussion). “Species packing” is used in this paper to describe both the pattern and the process. The discussion on the subject of species packing in MacArthur (1972) and May (1973) is presented in conjunction with a discussion of species diversity patterns along longitudinal gradients (arctic to the tropics), and on small vs large islands. We discuss how different parameterizations of the Allee function, or different “costs of rarity,” might apply to this issue. In all cases, stochasticity, modelled by adding noise terms, is found to have minimal effects on species diversity. Note, however, that we are not proposing that a model based on single trophic levels does or does not lead to real patterns of species packing, which is a matter of controversy (see Bowers and Brown, 1981; Simberloff and Boecklen, 198 1; Hopf and Brown, 1984; and Schoener, 1982, for a recent overview). In fact, the model with the Allee effect gives results that are difficult to reconcile with observed longitudinal gradients.

30

HOPF

AND

HOPF

Lastly, as argued in Bernstein et al. (1984), the cost of rarity is associated with sexual reproduction. Therefore, we predict that organisms with different reproductive styles (e.g., parthenogenesis vs outcrossing sexuals) should have different population ecologies. Hopf and Hopf (in preparation) discuss this question in detail, and find that clonal organisms do not appear to show a limiting similarity.

II. THE MODEL AND THE STABILITY ANALYSIS Our model is based a variation of MacArthur’s model (1972, p. 37, Eq. (3)), which we parameterize in terms of birth (b) and death (d) rates, rather than r (r = b - d) and K. In order to incorporate the Allee effect, which affects only the birth term, we must use this variation. The difficulties of Lokta-Volterra parameterizations are indicated below. The notation that we use follows as much as possible the usage in May (1973), MacArthur (1972), and Levins (1968). We denote by p(s) the amount of resource at point s, where s, i.e., the type of resource, is identified by its location on a resource axis that is, in principle, continuous. For example, p(s) (for a tree) could describe the unoccupied land area at the altitude s on a mountain. The value of p(s) depends on the abundance of the species that use them; p,,(s) describes the available resources when all species are absent. Throughout this work, p,,(s) is taken to be a continuous function of s, but the model allows patchy resources. We assume that there is no variation in type within a species (this assumption is relaxed in the fifth section). The genotype is denoted by y, and the numerical value of y signifies the resource(s) to which the species is optimally adapted. E.g., y = 1OOOm means that the species y is optimally adapted to the resources available at an altitude of 1OOOm. Each species is intrinsically capable of utilizing resources other than those for which it is optimally adapted. The utilization function describing this intrinsic capacity is denoted f(x), and is taken to be the same for all species. It enters the equations in the form f(s - y), indicating the capacity of the species optimally adapted to resource y to utilize resource s. To describe this,f(x) has a maximum at x = 0 (i.e., at s =y, where the species is optimally adapted), and decreases smoothly and monotonically with increasing 1x1. For simplicity we also assume that the function f is normalized, i.e., that the integral off(x) over x is unity. The density (number per unit area) of the species y is denoted N,. The equation describing the change in NY reads z

= (ba(N,,) R, - d) N,,

ALLEE

EFFECT

IN

SPECIES

PACKING

31

where t denotes time and R, =.i’”

-cc

ds/-+)f(s

-Y>.

(2)

Here, R, (which changes in time) denotes the total density of resources available to species y at any instant in time. The term p(s)f(y - s) in Eq. (2) is the product of the resources available at s times the intrinsic capacity f(y - s) of the species y to utilize the resources. The Allee effect is introduced through the function denoted a(N,). This function is taken to be the same for all species. It is taken to be a monotonically increasing function of N, and takes the value of unity in the limit that N, becomes very large. The density regulation of the populations occurs through a diminishing of the resources by the populations. For simplicity we have chosen to regulate populations by a drop in birth rate. The results are unchanged if the mortality is allowed to increase as well, i.e., with a mortality term like d/R,, but the resulting algebra is more complicated. We take p(s) to be a logistic expression of the form P(S) =Po(s> -ye

P

-Y> Ny,

(3)

Equation (3) incorporates a basic assumption that is used in the MM model, namely that a species diminishes the resources in direct proportion to the amount that they are utilized. Summations written as in Eq. (3) are to be understood to run over all species present in the community. Any two species compete for the resources to the degree to which they share the same set of resources. Following May (1973, pp. 139-150), we now develop the community stability properties of the model described by Eqs. (l)-(3). The function p&) is taken to be a constant independent of s (in particular we choose units such that p&) = 1). Consider the case in which the species are distributed along the resource axis with a uniform spacing Y, each species occurs at a point y = mY, where m is any integer (see Fig. 1 for an illustration). May

FIG. 1. A schematic of the species distribution. The species are uniformly spaced along the resource continuum with a separation Y. The bell curves describe the intrinsic capacities of the species to utilize the resources.

65312711

3

32

HOPF

AND

(1973, p. 170-from Whittaker, 1969) type of distribution along an altitudinal there is a steady state solution, in which We denote by K the steady state value given implicitly by

HOPF

gives an empirical example of this gradient. With these assumptions, all species have the same density. of any one population, where K is

ba(K)( 1 - KP) - d = 0. Here P is a Y-dependent coefficient p=

(4)

defined by 2

m=-*,m

fW>.

(5)

It is straightforward to test the stability of this solution by using Taylor’s expansion about the steady state. We write N,=K$n,, where ny @ K is an infinitesimal reads

(6)

perturbation.

Taylor’s

expansion formally

CYR

%=bDn,(L-PK)+bo(K)~n,.$~ dt

Y’

Y’

,

(7)

PI=0

where n = 0 means one sets all of the small perturbations to zero. The constant D is the slope of the Allee function at carrying capacity and is defined as D = Wx) -2F

x=K

*

(8)

The dependence of D on Y occurs though the carrying capacity K, which is determined by Eqs. (4) and (5). The second term in Eq. (7) is the complicated one. It is in the form of a community stability matrix (the partial derivative) whose structure is given in May (1973, p. 148) times the population displacements ny,, times an overall positive constant h(K) which alters the magnitudes but not the signs of the resulting stability eigenvalues. This term is treated in detail in May (1973, pp. 148-149). We merely summarize his results. The eigenvalues of the second term are always negative. This says that if the Allee function is missing (the first term in Eq. (7) is zero), then the community is stable no matter how dense the species packing is. The eigenvalues do, however, become very small as soon as Y is small compared to the width off(x). The first term in Eq. (7) describes the effect of the Allee function. We call the Allee effect “strong” or “weak” according to whether D is large or small. This term is just ny times some constants. Therefore, when written as a

ALLEE

EFFECT

IN

SPECIES

PACKING

33

matrix, it takes the form of a constant times the unit matrix. Its effect is to shift all the eigenvalues of the previous matrix by the amount bD( 1 -PK). The maximum stability eigenvalue is the sum of a positive term bD(1 - PK) coming from the Allee effect (the first term) and a negative term coming from the MM model (the second term). As Y decreases, the resources are partitioned among more and more species, and the carrying capacity for each species (K) gets smaller. We have assumed (see assumption (b) in Sect. I) that the Allee function a(x) is such that its slope gets larger as x = K decreases, so the positive term increases as Y decreases. As noted above, the absolute value of the negative term decreases to zero as Y decreases. The character of the two terms in Eq. (7) guarantees that as Y decreases, there is some critical value Y = Ymi, past which the maximum stability eigenvalue is positive, and the community is unstable. Hence the Allee effect insures against an arbitrarily dense species packing. Note that we would not have achieved a finite species packing had we parameterized MacArthur’s (1972, p. 37) equations in terms of r = b - d and K rather than b and d. Multiplying r by a(N,) involves a modification of both the birth and death terms in Eq. (I), and yields a situation in which there is a benefit to rarity (lowered death rate described by da(N,)) which exactly compensates the cost of rarity (lowered birth rate described by ba(N,,)). In this case the contributions to the eigenvalues from the birth and death terms exactly cancel, and the Allee effect has no influence on stability of the community. The main results of our analysis lead to the same conclusions deduced by May from the (hypothesized) destabilization term due to environmental stochasticity. One can replace May’s (1973, p. 151) term a2/k with the first term in our Eq. (7) and use May’s Fig. 6.3 to see that Ymin depends weakly on D except in the case when D is very large. For large D, Y,,,i, depends strongly and monotonically on D. The variable Yin our theory, when written in real units, is exactly the same quantity as May’s (1973, p. 140) term “d”, and “w” is the width of f(x) in real units (all references to May’s terminology are in quotes). In the dimensionless units of the fourth section, our Y is measured in units of “w,” and is hence the same as May’s ratio “d/w.” From the analysis given above, we find Y,,,i, - 1 (in dimensionless units), except in the limit of large D, in which case Ymin is much larger than one. This is the result May attempted to prove using the stochastic-extinction argument. (Note that we are only implying that this answer is the mathematically correct one if the Allee function is included. The MM model neglects the influence of other trophic levels, and so leaves open the question of whether Y < 1 is possible.) For Y > Y,,,,,, a community may be stable to invasion of a species (y’) that is not initially a part of the community. This invasion is most favorable if the optimum utilization of the invading species is positioned exactly at the

HOPF AND HOPF

34

midpoint between two existing species, i.e., for y’ =y + Y/2, since at this point resource depletion by neighboring speciesin minimal. The community is unstable if y’ has a positive fitness, which occurs when ba(O)( 1 - P’K) - d > 0,

(9)

where the constant P’ is p’=

1

(10)

f(Im + f l 0

In the limit that Y is very large compared to the width off(x), each term in Eq. (10) goes to zero and P’ -+ 0. The community then can be unstable if ba(0) - d > 0. However, ba(0) -d < 0 may be realistic for outcrossing sexually reproducing species, because even empty communities would be stable to invasion of very low density populations (e.g., one individual). A successfulinvasion requires sufficiently large populations. We define Y,,,,,, if it exists, to be the largest value of Y such that the community is stable to invasion according to the criterion of Eq. (9). Even in the limit of a weak Allee effect, Ymin and Y,,,,, usually differ by a factor of two or more. Thus the analysis predicts a range of stable communities rather than a unique one. For cases in which the community is established from random starting conditions, we show later that it is highly likely that the community will have Y w Y,,,i,. For all but very large D, this reduces to May’s (1973, pp. 165-l 70) assertion that a community of competing species should be characterized by “d/w - 1.” It is not, however, reasonable to assume that all real communities were established randomly. An ameliorating change in climate might take a community from a case of large D to small D, in which case we expect “d/w > 1” or “d/w & 1” to characterize the stable community. Conversely, we have results (not shown here) that indicate that when a community goes from modest D to larger D, as would be expected in a deteriorating environment, then mass extinction is possible such that “d/w 3 1.”

III. THE PARAMETERIZATION

OF THE ALLEE

FUNCTION

In this section we discuss the specific Allee function that we use in the numerical studies. In addition we discuss the biological rationale for the choice, and the biological interpretations of the parameters. We parameterize the Allee function as

(11)

ALLEE EFFECTIN

SPECIES PACKING

35

The constant y < 1 is the value of the Allee function in the limit of zero density, i.e., a(O) = y, and a(co) = 1. The biological rationale for this form is as follows (using the example of finding a mate). A female may have a severe problem finding a male when very rate, with a large decreasein fitness due to the unlikelihood of breeding. Hence for N < N, the fitness becomes smaller. As the density increases,many males can be encountered. However, encounters beyond the one that leads to mating are superfluous. Thus the function is taken to level off for N > N, (i.e., the Allee effect saturates). The parameter y describesthe ease with which an organism can colonize an empty habitat. For organisms capable of self-fertilization or asexual reproduction, a single individual can colonize. In such casesthe value of y is such that Eq. (9) reads by-d > 0 in the limit that all populations are zero, because invasion when rare is allowed. Organisms that reproduce through sexual union of two individuals cannot establish a population with a single individual. In this case y = 0, and by-d < 0, i.e., invasion when rare is forbidden. If y = 1, then a(N) = 1, which is used to remove the Allee effect from some of our calculations. This can also be applied to populations which never reproduce sexually. The coefficient N, denotes the value of the population at which the Allee function is exactly half way between its value at N = 0 and its value of unity when N is very large. In biological terms, N, is a measureof the density at which the Allee effect has a substantial impact on growth rate (e.g., when finding a mate becomesa significant problem. The coefficient D in Eq. (8) which is the slope of the Allee function, is obtained from Eq. (11) as D = Not1 - Y> (K + No)’ ’

(12)

Note that K is the carrying capacity determined self-consistently from Eq. (4). Since 1 - y > 0 and all other parameters are positive definite then D > 0. The form in Eq. (11) is a poor one for investigating very large D, becauseEq. (12) has a maximum at K = N, , at which point D = (1 - y)/N, .

IV. NUMERICAL

ANALYSIS

In this section we present the results of numerical calculations of Eqs. (l)-(3), and generalizations thereof (i.e., Eqs. (A5) and (A6) in Appendix A). The purposes of the numerical calculations are as follows: (1) to find out whether the results of the stability analysis of Section II can be applied to resource distributions with finite bounds (they can if used with caution); (2) to seewhether stochastic fluctuations, modelled as noise terms,

36

HOPF

AND

HOPF

are important in determining species packing (they are not); (3) to find out how community initiation influences species packing (random immigration reliably produces communities whose density is near the maximum stable point); (4) to determine the effect of the parameterization of the Allee function (it plays a minor role except in very severe cases); (5) to see if genetic variability, allowing character displacement, affects the results (there is very little change in the answers). Finally, the stability analysis of Section II deals only with the special case of equal separations along the resource spectrum. Moreover it ignores the possibility of more complex population dynamics such as stable limit cycles. The numerical study confirms that we were correct in ignoring such complications. In the numerical work, the functions have specific forms. The resource utilization function is f(x) = --& exp(-x2).

(13)

The width of the resource function is the unit of measure of the variables s and y. By choosing this unit, the species separation along the resource continuum (Y), is measured relative to the intrinsic capacity to use the resources. The birth and death rate are b = 2 and d = 1, so r ’ = 1 (r = b - d) becomes the unit of time. We use the following values of y: y = 0 (obligate outcrossing); y = 0.55, i.e., by -d = 0.1 > 0 (partial selling or facultative parthenogenesis); y = 1 (obligate asexual reproduction, or no Allee effect). The resource distribution function p,,(s) is unity except at the boundaries, at which point it falls off linearly to zero; it is illustrated in Fig. 2a. The width of the flat region of p,Js) is 9 units, and the full width of the dropoff to zero is 3 units. Note that the fitnesses are less than or equal to zero at the half point of the resources, so that species can only survive over a resource range of 12 units. The resource utilization function is illustrated in Fig. 2b. All populations and resources are measured in the same dimensionless units. Species utilization of the resources is absorbed into the units of the

FIG. 2. (a) The resource continuum p,,(s) used in most of the numerical calculations. When random resources are used, this curve gives the time average of the resources. (b) The resource utilization curve f(s - y) used in the calculations.

ALLEE

EFFECT

IN SPECIES

PACKING

37

populations. The populations are made dimensionless by measuring them relative to the nominal carrying capacity K,, defined as the carrying capacity of a single asexual species living alone at the center of the resource continuum. K, is given by K, = 1 - d/b, and takes the value K, = 0.5 with the values of b and d used here. The parameter IV,, in the Allee function is N,,/K, in the dimensionless units, and is varied over several orders of magnitude. The species distribution and resource continuum are discretized in steps of 0.2, as is the time step. The differential equations are integrated using a firstorder predictor method. The choice of numerical technique is based on experience in integration of the analogous laser problems (Litvak, 1970; Parks et al., 1968; Parks, 1969; Menegozzi and Lamb, 1978), and the expected accuracy is approximately 5%. Some choice of initial distribution is made (see below), and Eqs. (l)-(3) are then iterated until a final distribution is reached. We regard a species as extinct if its population falls below 2 x 10d3 (called the “extinction point”). Without the Allee effect, an extinction point is essential since rare populations recover and stochastic extinction cannot take place. Also, use of an extinction point greatly speeds up the numerical caiculation by ignoring essentially extinct species, which otherwise decrease indefinitely in calculations with an Allee effect. When initiating a population distribution, we turn off the extinction feature until all populations are given an opportunity to get started. An illustration of the results of a numerical calculation is shown in Fig. 3. The parameters used are given in the figure caption. The Allee effect is initially turned off by setting y = 1. The resulting distribution is the continuum shown in Fig. 3a. Then the Allee function is turned on by setting y= 0.55, using the distribution in Fig. 3a as the initial condition. The distribution changes to a discrete set of spikes, as shown in Fig. 3b. This represents a finite number of species (6 in this case) separated in resource

FIG. 3. An example of a numerical calculation. The resource continuum and utilization function are shown in Fig. 2, 6 = 2, d = 1; the initial population is uniform with NY = 0.2 x 10m3. The step size in time is 0.2 (a step of 0.1 was also used to confirm numerical accuracy). The step in resource space is 0.2. (a) No Allee function is used (i.e., y= 1). (b) Starting with the final distribution shown in (a) the Allee function was given N, = 0.004, y = 0.55, resulting in a final distribution shown in (b).

38

HOPF

AND

HOPF

space by some amount determined by the width of the resource utilization. We made several calculations of this type, using different values of N,. For smaller N, the final distribution remains the same (i.e., spikes), but the time needed to reach it increases substantially. Thus the major consequence of weakening the Allee effect is to slow down the rate of competitive exclusion. For N, larger than shown the rate of competitive exclusion speeds up, and for large enough N, the distribution becomes more sparse. This is discussed in detail later. Figure 3 shows that the Allee effect is sufficient to cause species discreteness. We now show that it is necessary, at least within the range of alternative effects that we investigated. These effects are achieved by varying the parameters enumerated at the beginning of this section, as detailed below: A. Random vs Deterministic

Resources

(i) Random resources. The resource function p,(s) varies randomly in time at each grid point. The probability distribution ranges uniformly from zero to twice the value indicated in Fig. 2a. Thus the time average value of pO(s) is given in Fig. 2a. Various cases were considered in which different time intervals elapsed before pO(s) was allowed to change. This interval was fixed in any one calculation, but varied between calculations over a range of one to several hundred time steps. This tested resource variation on timescales from shorter to very much larger than r-‘. (The limits on the timescales are those that correspond to the Ito and Stratonovich constructions of diffusion equations (Roughgarden, 1979, pp. 373-386). We have shown that careful inclusion of these limits (Feldman and Roughgarden, 1975) has no effect on the results.) (ii) Deterministic resources. The resources p,,(s) are fixed in time, usually with the value given in Fig..2a. B. Initial

Populations

(i) Discrete. Every Yth species along the distribution is started near carrying capacity with all others starting slightly above the “extinction point” at N, = 2 x 10e3. (ii) Smooth. N,, = 2.02 x 10-3.

All species were

started

at the same value; usually

(iii) Random initial populations. Each species starts with a random initial value. Unless otherwise specified the distribution of these values is uniform over an interval 0 to 2 X 10P3. The “extinction point” is set to zero for a time of several hundred I-’ before being reset to the conventional value.

ALLEE

EFFECT

IN

SPECIES

39

PACKING

(iv) Random immigration. All species start with an initial population of zero. At each time interval, if the species has not already immigrated, the random number generator determines whether it will do so. If allowed, the immigrant starts with a fixed value Nyi. For y = 0.55, we use 2.02 x 10P3. For y = 0, the value of NYi is 10% larger than the value needed to give a positive fitness in the absence of competition. The probability of immigration is kept very low since otherwise this degenerates into case (ii) or (iii). The extinction point is treated as in case (iii). C. Adaptation

If adaptation (character displacement) is not specified, we use the model described in Section III, using Eqs. (l)-(3). If specified, we use the model discussed in Appendix A. The populations are assumed to be genetically heterogeneous. A single diploid locus with two alleles A and a is assumed to control adaptation to the resource. An Aa individual of species y has an TABLE

Summary

Allee

I

of the Results of the Calculations Done to Assess the Role of the Allee Effect on Species Packing

Adaptation

Initial

populations

Final populations

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

No No No No No No No Yes Yes Yes Yes

Deterministic Random Deterministic Random Deterministic Random Deterministic Deterministic Random Deterministic Random

Smooth Smooth Random Random Discrete Discrete Random immigration Smooth Smooth Random Random

Discrete Discrete Discrete Discrete Discrete Discrete Discrete Discrete Discrete Discrete Discrete

No No No No No No No No No

No No No No No No No Yes Yes Yes

Deterministic Random Deterministic Random Random Deterministic Random Deterministic Random Random

Smooth Smooth Random Random Discrete Random immigration Random immigration Smooth Smooth Random

Continuous, smooth Continuous, smooth Continuous, noisy Continuous, noisy Continuous” Continuous, noisy Continuous, noisy Continuous, smooth Continuous, smooth Continuous, noisy

* Calculation not taken to final condition. Nofe. “No Allee” signifies y = 1.

40

HOPF

AND

HOPF

optimal resource at y. An AA individual has an optimum one grid point higher, and an aa individual has an optimum one grid point lower. Mating is random, and the argument of the Allee function is the total population of species y. The initial populations are taken to be purely heterozygous. The results of the calculations are summarized in Table I. The parameters of these calculations are the same as in Fig. 3 unless otherwise specified. The interpretation of the result works as follows. “Discrete” signifies a distribution as in Fig. 3b and 4d. There are always at least two, and in most cases four, species that have been driven to extinction in between the ones that survive. “Continuous” signifies that, except at the edges of the distribution, there are no more than one or two, and usually zero, species that are either extinct, or in danger of stochastic extinction. “Smooth” signifies a distribution as in Fig. 3a and 4a. “Noisy” signifies a distribution that has a very jagged appearence, but which is basically continuous, as in Fig. 4b. This last distribution is as close to a discrete distribution as we ever obtained without the Allee function. The linear scale of the plot disguises the fact that there are rare species in between the spikes that are not in danger of stochastic extinction in any finite time. Thus the distribution is continuous. Figure 4c shows a case in which a population was discrete initially, but became continuous in the absence of an Allee effect. The random resources used here speeded up the destabilization of the discrete distribution. Instead of stochastic extinction, the intervening populations grew at the expense of the initial species. a

c

b

d

FIG. 4. Typical final species distributions found in the study summarized in Table I. (a) A continuous and smooth distribution arising from an initially smooth distribution with random resources and no Allee effect. (b) A continuous and “noisy” distribution from a random immigration scenario with deterministic resources and no Allee. (c) A continuous distribution from an initial discrete distribution, with random resources, and no Allee effect. This calculation was stopped well before the final configuration was reached. (d) A discrete distribution with random resources, random initial populations, and an Allee effect with the same parameters as in Fig. 3b.

ALLEE

EFFECT

IN SPECIES

41

PACKING

We see then that random resources, which incorporate the notion of environmental stochasticity, play no role in the existence of a discrete speciespacking distribution. Neither does the incorporation of different starting scenarios, nor does adaptation. Therefore, within the limit of the effects considered, the Allee effect is both necessary and sufficient for a discrete species packing. We now turn the problem around. We take the Allee effect as necessary for discrete packing, and ask how various scenarios might determine the final species packing density. The Allee effect predicts a wide range of stable packing densities between those unstable to the Allee effect and those unstable to invasion. It is therefore not surprising that the actual distributions could be influenced by the specific scenario. Random resources are found to play no role in determining the final answer. Instead, initiation is most important, with adaptation playing a minor but perceptible role. The calculations are summarized in Fig. 5, which deals with the case y = 0.55, and in Fig. 6, which deals with the case y = 0. In each figure, the vertical axis denoted Y is the mean separation of the species in the final

UNSTABLE

TO

INVASION -

4-

‘.

STABLE

Y

-

DR 9

‘\

‘\

.\. ./

DA

=A’,

0

I 1e4

t

t

’ No

I&

t

13

FIG. 5. Summary of the results of the species packing densities found with y = 0.55 in the Allee function. The horizontal axis is the value of N,,. The vertical axis is the average separation in resource space between the species, i.e., the inverse of the species-packing density. The solid curves are the values of Y,i, and Y,,,. The enclosed portion of the graph is stable according to the analysis of Section III. The upper portion is unstable to the invasion of a rare species. The lower portion is destabilized by the Allee effect. The boldface letters are located at the value of Y found in the numerical calculation. For clarity, the various cases have been spread out horizontally. The arrows on the horizontal axis give the proper horizontal location on the figures. The conditions under which the calculations were made are summarized in Table I. The broken portions of the curve signify values of N,, such that the analysis of Section III cannot be used to predict the final packing density.

42

8J I

HOPF

AND

HOPF

5 4 Y

3

STABLE

2

&

I-

UNSTABLE

TO ALLEE

FIG. 6. Summary of the results of the species packing densities found with y = 0 in the Allee function. As in Fig. 5, arrows on the axes serve to locate the correct value of N, when the points have been spaced apart for clarity. See the caption of Fig. 5 for the notation. In this case, all species distributions, even empty ones, are stable to invasion of rare species.

distribution of a calculation. The species packing density is then the inverse of Y. The solid curves are the extremes of the stability region found from the analysis in Section II. In the portion of the curve in Fig. 5 broken up by dots, the analysis cannot be used to predict the species distribution. The stability eigenvalues from Section II become vanishingly small, both in the stable and in the unstable regime, and the results are strongly influenced by the resource boundary. We prefer not to assign any significance to results in this regime. The results of the numerical runs are indicated by a bold face letter on the figure. The conditions which define the boldface label are given in the table that appears on the figure. A series of calculations were made which do not appear in the figure. These started with a species distribution with a uniform spacing (Y = Y, see Sect. II) to see if it had the stability properties implied by the analysis of Section II. Random resources did have a small influence in this case, in that stable values of Y that lie within a few percent of the boundary can be destabilized by random resources. This is, however, such a fine detail that it is unlikely to be signifigant in real biota. Aside from this case, and the portion of Fig. 5 discussed above, the analysis in Section II appeared to be quite adequate to predict the stability of the community (there is a small change due to adaptation discussed below). Compare the cases labelled I and E in Fig. 5. These are random-initiation calculations that measure the degree of artifact due to the arbitrary choice of the length of the resource continuum. In E we extend the length of the continuum slightly, and an additional species colonizes the edge. Hence the

ALLEE

EFFECT

IN SPECIES

PACKING

43

vertical separation between the I and E can be taken as a rough measure of the expectation of such artifact. Next compare the following pairs of results as they appear in the figures: D vs R, D, vs R,, and I, vs C, in Fig. 5; and I vs C in both figures. These are all comparisons of cases with random and deterministic resources. The results are essentially the same for each pair. Hence we conclude that environmental stochasticity, modelled as noise terms, plays no significant role. Now consider random initiation (labelled I, I,, E, C, and C,) and random immigration (labelled M). These should be contrasted with cases D and R which have deterministic (“smooth”) initiation. In all cases random initiation and immigration lead to a more dense distribution than deterministic initiation, usually with a packing density close to the minimum density allowed by the Allee efect. The calculations subscripted A used the adaptation model of Appendix A. In this case compare the same label with and without the subscript (e.g., C vs C,). In all but one case there is an increase in species density when adaptation is included, and this increase is large and systematic enough to conclude that it is meaningful. In two cases on Fig. 5, the density is greater than that allowed by the analysis of Section II. This increase is small but significant, and is not simply an artifact of our crude way of measuring the packing density. There are two cases in Fig. 6 in which the final distributions are very sparse compared to the minimum packing density, namely the point D at Nil - 3 x lO-2 and I at N, - 7 x 10P2. These give the largest value of N, at which it is possible to establish a nonempty community using the deterministic and random initiation scenarios, respectively. Above these thresholds, mass extinction of the entire community takes place. We have done a limited study in which we establish stable communities below this threshold, and then increase N, to a value above the threshold. The result is normally just a typical sparse population found for large N,,, but in a few cases the entire community goes extinct. Lastly, we would like to note how the Allee effect influences the temporal aspects of competitive exclusion. Often, when the species first grow to carrying capacity, species diversity exceeds that permitted by the Allee effect. At that point, the influence of the Allee effect on the relative fitness is weakest and competitive exclusion is slowest. As the excess species are driven to lower density, the influence of the Allee effect increases and the rate of competitive exclusion speeds up. This gives a characteristically sudden aspect to the final extinction.

44

HOPF ANDHOPF

V. APPLICATION OBSERVED

OF THE ALLEE EFFECT TO DIVERSITY GRADIENT

In this section, we discuss whether the species-packing analysis can be applied to observed patterns in speciesdiversity (MacArthur, 1972), such as those along longitudinal gradients (arctic to the tropics), or gradients in island size. Specifically we are concerned whether the numerical packing results from different strengthed Allee effects (see Sect. III) can be used to describe these gradients. There is no necessary connection between species packing (how, or whether, species sort themselves), and species diversity (the number of species). An environment with a low packing density on each resource axis may have a greater diversity than another environment due to more axes. We concentrate only on the way speciespack along a single resource axis to see whether one can account for all or part of the diversity gradient. However, the parameter variation used to study strong Allee effects has mathematical peculiarities, and therefore all conclusions stated here are tentative. The biological rationale for an environmental influence on the Allee effect goes as follows. The carrying capacity used to evaluate D is the one at the time of mating. For many species that live in cyclical but predictable environments, mating occurs at the low ebb of the population cycle, i.e., at the lowest value of K. For example, many birds go through a population decline in winter or migration, and mate in the following spring. More extreme environmental cycles result in a more extreme Allee effect. In some of our numerical calculations, there is a marked increase in the separation between speciesas N, becomeslarge (our K is the mean carrying capacity, so we must vary N, to model cyclic influences). This is equivalent to a drop in speciesdiversity, and it decreasesby as much as a factor of five (see Fig. 6, random immigration cases labelled M, which are the most reasonablebiologically). This may play some role in explaining the observed gradient in speciesdiversity from tropical to arctic environments. However, there are many speciesfor which a larger D may not be relevant. In addition MacArthur (1972, p. 209) has observed population cycles in the tropics. The Allee effect might also be applied to diversity patterns on islands (MacArthur, 1972). The decrease in K with land area is clear. In addition, fitness can be influenced by the absolute numbers of a population, not just by density alone. For example, fluctuations in sex ratios would cause a predictable decrease in average fitness for small populations. Bradley and May (1978) give a single-speciescalculation resulting in an Allee effect due to these fluctuations. Clearly hermaphrodites would not be effected by this, but other facts such as inbreeding might mitigate against rarity. In summary, the Allee effect only predicts substantive changes in species diversity in the limit of very low carrying capacities. Thus, it may be

ALLEE EFFECT IN SPECIES PACKING

45

reasonably applicable to island patterns, which show a similar sort of effect. Brown and Gibson (1983, Fig. 15.4, p. 444) show a plot of log species number vs log area that drops off for small areas and levels off for large areas. Our results are, at best, marginally applicable to observed longitudinal diversity gradients. Brown (1981, Fig. 1, p. 884) shows a linear relationship of diversity with resource, which is not consistent with the results in Section III. As resources increase, the Allee effect predicts an increase in the number of individuals per species, not a strong increase in the number of species. This points out the weakness of models with a single trophic level. Preliminary calculation in which higher trophic levels have been included show that the continuous solution remains unstable in the presence of the Allee effect, which is our main point. However, there can be considerable enrichment of stable solutions, and one can find cases in which May’s “d/w N 1” rule breaks down completely. Some of these calculation show a linear increase in diversity with resources, as shown by Brown (see above). There is an empirical body of work by Rohde (1979, and references therein) regarding the role of the Allee effect in determining the community structure of parasites. These communities could be modelled by a single trophic level. Rohde considers the spatial rather than the resource aspect of the community, and thus its relationship to our calculation is uncertain.

VI. RELATION

TO OTHER CALCULATIONS OF SPECIES PACKING

AND CONCEPTS

Calculations of limiting similarity based on population models go back to the work of MacArthur and Levins (1967). The discussion of limiting similarity from MacArthur and Levins (for an updated version see Roughgarden, 1979, pp. 540-549) ostensibly conflicts with our derivation insofar as they obtain limiting similarity without the Allee effect and we do not. This conflict is not new to our model. The same conflict exists with respect to Roughgarden’s (1979, pp. 529-537) clone selection model for resource partitioning within a species. The MacArthur-Levins (1967) calculation is a special case of the MM model in which two species are placed in the flat region of the resource curve illustrated in Fig. 2. There are limitations preventing a third species from invading on the resources between the two, but nothing limits invasion on resources outside the two. Thus the MacArthur-Levins community is unstable by the criterion used in this paper, because their community permits invasion. The community is stable by their criterion because they limit invasion to a subset of species along the resource grade. More recent work has considered the role of genetic changes leading to character displacement (Roughgarden, 1976, 1979, ‘pp. 492-494, 554-556,

46

HOPF ANDHOPF

and references therein). Our adaptation model in Appendix A allows for the occurence of a small amount of character displacement. The model is rudimentary compared to other cases that have been studied, which may explain why it has such a minor influence on the results. However, we are not trying to use our adaptation calculation to deny the role of character displacement in determining packing configurations. Instead, we show that displacement cannot destabilize the continuous solution, either by itself, or in conjunction with stochastic processes. See Abrams (1983) for further discussion of the role of limiting similarity in Lokta-Volterra models.

VII.

SUMMARY

In summary, we have shown that, when the Allee effect is incorporated into the species-packing model of May and MacArthur, species sort out into discrete distributions along a resource grade in a manner that depends systematically on their intrinsic capacities to utilize resources. Without the Allee effect, one does not obtain such results. Random fluctuations, modelled as noise terms, play a very minor role. They do not generate discrete distributions when the Allee effect is removed, nor do they substantially influence the final packing density when the Allee effect is included. The community analysis with an Allee effect predicts a range of stable community densities. The observed final density is largely determined by the historical pattern of colonization. Random colonization systematically gives a spacing along the resource grade that is close to the minimum stable spacing predicted by the analysis. The terminology “strength” or “weakness” of the Allee effect can be used, e.g., to describe the cost of finding a mate at carrying capacity. For a species in which sexual reproduction is very infrequent, or in very benign environments, this cost could be very small. However, the analysis shows that the Allee effect still results in a discrete species pattern, although the rate of competitive exclusion of rare species is greatly slowed down. We cannot explore this limit numerically due to limits on computer time. The analysis in Section II indicates that the final patterns of diversity should not be strongly influenced by the numerical details of a weak Allee effect. Strong Allee effects speed up the rate of competitive exclusion enormously and result in a substantial decrease in species diversity. The strength of the Allee effect is inversely proportional to the carrying capacity, and thus environments with low carrying capacity should have low diversity. It is possible that seasonal environmental changes, if they result in a low carrying capacity at mating time, may be of some significance in causing the decrease in diversity observed in cyclical environments such as the arctic (MacArthur, 1972, Chaps. 7-9). However, we are skeptical that our analysis is capable of

ALLEE

EFFECT

IN

SPECIES

41

PACKING

describing more than a part of observed diversity gradients. It is more likely that diversity gradients require explicit recognition of the influence of higher trophic levels. On islands, the carrying capacity decreases with area. This means that factors such as sex-ratio fluctuations will decrease fitness as area decreases, acting like an Allee effect. In that case the strong-Allee limit would seem to be relevant to decreased species diversity on small islands.

APPENDIX

A

In this appendix we sketch the model used to allow rudimentary genetic diversity within a species so that it can adapt to its niche. We refer to such a change as character displacement. We assume that there are two alleles, labeled a and A. An aA individual of species y has an optimal resource utilization at the resource s =y. An AA individual has an optimum at y + dy, where dv is the grid size of the resource spectrum, and an au individual has optimal utilization at y - dy. Thus each species can change its maximum resource utilization by one grid point up or down the resource spectrum. Let us denote by py the fraction of gene a in the population (1 -py is the fraction of A), and by NY,,, the number of individuals of species y of type au (similarly define Ny,aA and NY,,,). Then

N, = Nw + Nwx.4+ NY,,, .

(Al)

Let us now assume the following: (1)

mating is completely random within

a species;

(2)

at all instants in time the number of matings is very small;

(3) the population then adiabatically follows the composition implied by the instantaneous value of py. Models arising from such assumptions are standard and are found, for example, in Crow and Kimula (1970, pp. 190-192). The final assumption is only valid if py varies slowly, which is verified by the numerical results. Under these assumptions we have N y,aa =%P:,

WI

N y,aA = 2Ny py(l -P,),

(‘43)

N y.an = NJ1 -pyj2.

(A41

and

48

HOPF AND HOPF

The equation for N,, then reads %=

(WN,)b:R,-A,

+ 2~,(1

-P,JR, + (1 -~y)*~~+~~l- 4Ny>

645)

where R, is defined in Eq. (9). The fraction py is Py =

N y,Ila + 0.5Ny,a.4 NY

’

from which it is straightforward to find that

dPY _ =

-”

%.,,ldt + .5dN,,,,/dt NY.,, + .5N,,,,

dN,ldt NY

.

G47)

The three terms in the bracket [ .a. ] in Eq. (A5) give the contributions via the birth terms to the populations of the au, aA, and AA types, respectively. From this one can read off the rate of change of the N,, (i.e., dN,,,,/dt) and can substitute it into Eq. (A7) to obtain WN&P,R,-A,+

(1 -P,)R,J

-d++-

dN fdt

.

WV

Y

Equations (A5) and (A8) are solved together modification of Eq. (lo), which reads

with

an appropriate

~6)=P&)- z: [f(s-Y’>N,,,,A +f(s-Y’ +4)Ny,,,, +.W-Y’ -4VLd The sum in the square bracket is the total population resource utilization is optimal at the value y’.

649)

of individuals

whose

ACKNOWLEDGMENTS F. A. Hopf would like to specially thank H. Bernstein, H. Byerly, R. Michod, and K. Vemulapalli for assistance throughout the entire study. This paper is one of a series of papers discussing the similarities and differences in the role of Darwinian evoluttion in biology, physics, and chemistry. The specific parameterization of the equations in Section II of this paper come directly from an earlier paper in this series (Bernstein et al., 1983). Other papers are included in the reference list (Bernstein et al., in preparation). This research was supported in part by the National Science Foundation under Contract PHY-8104982.

ALLEE

EFFECT

IN SPECIES

PACKING

49

REFERENCES ABRAMS, P. 1983. The theory of ALLEE, W. C. 1939. “The Social

limiting similarity, Ann. Rev. Ecol. Syst. 14, 359-76. Life of Animals,” Heinemann, London. ASMUSSEN, M. J. 1979. Density dependent selection II. The Allee effect, Amer. Nut. 114, 796-809. BERNSTEIN,H., BYERLY, H. C., HOPF, F. A., MICHOD, R. E., AND VEMULAPALLI, C. K. 1983. The Darwinian dynamic, Biol. Quart. 58, 185-207. BERNSTEIN, H., BYERLY, H. C., HOPF, F. A., AND MICHOD, R. E. 1984. Sex and the emergence of species, J. Theor. Biol., in press. BOWERS, M. A., AND BROWN. J. H. 1981. Body size and coexistance in desert rodents: Chance or community structure? Ecology 63. 391-400. BRADLEY, D. J., AND MAY, R. M. 1978. Consequences of helminth aggregation for the dynamics of schistosomiasis, Trans. Roy. Sot. Trop. Med. Hyg. 72, 262-273. BROWN,J. H. 1981. Two decades of homage to Santa Rosalia: Towards a general theory of diversity, Amer. Zool. 21, 877-888. BROWN, .I. H., AND GIBSON, A. C. 1983. “Biogeography,” Mosby, Toronto. CROW. J. C., AND KIMURA, M. 1970. “An Introduction to Population Genetics Theory,” Harper & Row, New York. DARWIN, C. 1979. “The Origin of Species,” 6th ed., Avenel. New York. FELDMAN, W. F., AND ROUGHGARDEN,J. 1975. A population’s stationary distribution and chance of extinction in a stochastic environment with remarks on the theory of species packing, J. Theor. Biol. 7, 197-207. FISHER, R. A. 1958. “The Genetical Theory of Natural Selection,” Dover, New York. CAUSE, G. F. 1934. “The Struggle for Existance,” Williams & Wilkins, Baltimore. GRANT, V. 1981. “Plant Speciation,” 2nd. ed., Columbia Univ. Press, New York. HOPF, F. A., AND BROWN,J. H. 1984. The bullseye method for testing randomness, Ecology, in press. HOPF, F. A., AND HOPF, F. W. Limiting similarity and sexual reproduction. LEVIN, S. A., AND SEGEL, L. A. 1982. Models of the influence of predation on aspect diversity in prey populations, J. Math. Biol. 14, 253-284. LEVINS. R. 1968. “Evolution in Changing Environments,” Monographs in Population Biology, No. 2, Princeton Univ. Press, Princeton, N.J. LITVAK, M. M. 1970. Linewidths of a Gaussian broadband signal in a saturated two-level system, Phys. Rev. A2, 2107. MACARTHUR. R. H. 1972. “Geographical Ecology; Patterns in the Distributions of Species,” Harper & Row, New York. MACARTHUR, R. H., AND LEVINS, R. 1967. Limiting similarity, convergence, and divergence of coexisting species, Amer. Nat. 101, 377-385. MAY, R. M. 1973. “Stability and Complexity in Model Ecosystems,” Monographs in Population Biology No. 6, Princeton Univ. Press, Princeton, N.J. MENEGOZZI, L. N., AND LAMB, W. E., JR. 1978. Laser amplification of incoherent radiation, Phys. Rev. A 11.

PARKS J. H., RAO, D. R., AND JAVAN, A. 1968. A high-resolution study of the Cn, + Bn,(O, 0) stimulated transitions in N,, Appl. Phys. Lett. 13, 142. PARKS, J. H. 1969. “A High-Resolution Study of Stimulated Ultraviolet Transitions in Molecular Nitrogen C3n, + B3n,(0, 0) JS70A.” Doctoral thesis, Massachusetts Institute of Technology, Cambridge, Mass. ROHDE, K. 1979. A critical evaluation of intrinsic and extrinsic factors responsible for niche restriction in parasites, Amer. Nat. 114, 648-671.

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ROUGHGARDEN,J. 1976. Resource partitioning among competing species-A Coevolutionary approach, J. Theor. Biol. 9, 388-424. ROUGHGARDEN, J. 1979. “Theory of Population Genetics and Evolutionary Ecology: An Introduction.” Macmillan Co., New York. SCHOENER, T. M. 1982. The controversy over interspecific competition, Amer. Sci. 70, 586-595. SIMBERLOFF, D., AND BOECKLEN, W. 1981. Santa Rosalia reconsidered: Size ratios and competition, Evolution 35, 1206-1228. TURELLI, M. 1981. Niche overlap and invasion of competitors in random environments I: Models without demographic stochasticity, J. Theor. Biol. 20, l-56. WATT, K. E. F. 1968. “Ecology and Resource Management; a Quantitative Approach,” McGraw-Hill, New York.