The effects of interacting species on predator-prey coevolution

THEORETICAL

POPULATION

BIOLOGY

39, 241-262 ( 199 1)

The Effects of Interacting Species on Predator-Prey Coevolution PETER

A.

ABRAM

Department of Ecology, Evolution and Behavior, University of Minnesota, 318 Church SI., SE, Minneapolis, Minnesota 55455

Received December 21, 1989

Simple optimization models are used to explore the effects that an additional specieshas on predator-prey coevolution. The additional speciesis either an alternative prey or a higher-level predator that consumes the first predator species. Results indicate that a variety of effects on predator-prey coevolution are possible. In general, the additional species does not decouple evolutionary changes in the predator and the prey, and often leads to stronger coupling. The manner in which predator and prey adaptations combine to determine the predator’s capture rate has a major effect on how an additional species influences the coevolutionary responsesof the original pair of species. 6 1991 Academic Press, Inc.

I.

INTRODUCTION

It is widely believed that evolutionary change in a given specieswill often bring about evolutionary changes in other species with which it interacts (Futuyma and Slatkin, 1983; Thompson, 1982, 1989; Nitecki, 1983; Stenseth and Maynard Smith, 1984; DeAngelis et al., 1985). When such induced change occurs reciprocally between two species, it is known as coevolution. I refer to evolutionary changes in other species that are produced by evolutionary change in a given species as coevolutionary responses, regardless of whether there is reciprocal coevolution. Clearly, reciprocal coevolution can only be understood if the nature of coevolutionary responses is understood for each species. Recent analysis of the coevolution of a predator and prey (Abrams, 1986a,b) and of competitors (Abrams, 1986c, 1987a,b, 1990a,b) have shown that the direction of coevolutionary responsesis often counterintuitive. To date, most models of coevolution have considered interaction between a single pair of species. There has been a smaller number of models that have considered the interaction of large numbers of species (Levin and Segel, 1982; Stenseth and Maynard Smith, 1984). However, there has been an almost total neglect of intermediate cases in which a 241 0040-5809/91 $3.00 Copyright 0 1991 by Academic Press, Inc All rights of reproductmn in any form reserved.

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relatively small number of species interact. These cases are important for two reasons: (i) Studies of food webs (Pimm, 1984; Cohen, 1989) have shown that most species typically have strong interactions with small to moderate numbers of others; and (ii) the relevance to the real world of theory based on interacting pairs of species can only be assessedif we understand the effects of a small number of additional species. The present paper begins the task of developing coevolutionary theory for the case of a small number of interacting species greater than two. It examines how the coevolution of a single predator and a single prey is altered by the addition of a third species;either a predator of the predator, or a second prey species.These caseswere chosen for two reasons: (1) both are common in natural communities; and (2) there has already been considerable conjecture about the effects that these additional species might have on the coevolutionary interactions of a predator-prey pair (Vermeij, 1982, 1983, 1987; Futuyma, 1983). The most commonly expressed view seemsto be that additional interacting speciesdecouple the coevolution of a given pair; i.e., they reduce or eliminate the coevolutionary response in species A to an evolutionary change in species B. This view has been used to explain the lack of clear evidence for pairwise coevolution in communities with many species(Futuyma, 1983). The following quotations from Vermeij (1982) present the logic behind the decoupling view for the two types of additional speciesI consider here: if the predator feeds on more than one prey species, the short term response of the predator might be to ignore the increasingly well-defended prey in favor of easier victims, so that no increase in predator efficiency with respect to the defended prey takes place. Selection for predatory improvement is likely to be far stronger from the predator’s own enemies or social interactions than from the prey. (Vermeij, 1982, p. 712)

II. A REVIEW OF PREDATOR-PREY COEVOLUTION The analysis of three-species systems below is based on the two-species predator-prey models analyzed in Abrams (1986a). The predator is assumed to have a linear functional response; i.e., the number of prey eaten per unit time by an average predator is given by a constant, C, multiplied by prey density, I’. Two alternative representations of the consumption rate constant C are used: (i) Multiplicative; i.e., C= C,C,, where C, is a measure of the predator’s capture abilities and C, is a measure of the prey’s escape abilities, and (ii) additive; i.e., C=C,+C,+C,. If C,>O in the additive model, the predator will capture some prey even in the absenceof adaptations for doing so. Note that in each case, greater predator ability

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243

to capture prey implies a larger C,, while increased prey escape abilities imply a smaller C,. Both C, and C, are nonnegative in the multiplicative model; C, could take negative values in the additive model, provided that it did not produce a negative C. The multiplicative model may be a better approximation of reality when the adaptations determine encounter rates, while the additive model may be better when the adaptations relate to post-encounter success(Abrams, 1986a). Predator and prey population dynamics are given by the expressions dP/dt = P[ g,( CV) - DJ

(14

dV/dt = V[f,( V) - D,] - CPV.

(lb)

gp is a nondecreasing function which describes the rate of reproduction as a function of food intake. f, is a decreasing function that describes the density-dependent components of prey per capita population growth. D, and D, are density-independent components of the per capita death rates in predator and prey. Natural selection is assumed to maximize individual fitness, which is generally equivalent to maximizing the per capita growth rate. I assume that there is a cost to improving predation-related adaptations, so that fitness is maximized by an optimal balance of costs and benefits. For the predator, achieving a greater C, is assumed to imply a greater D,; for the prey, a lower C, entails a greater D,. Other possible tradeoffs are analyzed in Abrams (1986a). Abrams (1986a) showed that if C in Eq. (1) is represented by a multiplicative combination of predator and prey traits, a change in prey escape abilities (C,) does not change the optimal C, for the predator, because increased prey population density exactly compensates for the decreased catchability of individual prey. However, an increase in the predator’s capture abilities always increases the optimal prey escapeability. The additive model is mentioned briefly in Abrams (1986a), and is analyzed in the Appendix of this paper. This model of the consumption rate constant predicts that an increase in prey escape abilities can have the effect of increasing or decreasing optimal predator capture abilities. If the birth rate function g, is a linear function of food intake, increasing prey escapeability increases the evolutionarily optimal predator capture ability under the additive model. This model predicts that increases in the predator’s capture ability may increase or decreasethe optimal prey escapeability, depending upon whether increases in predator capture ability increase or decreasethe equilibrium predator population. This and other counterintuitive conclusions result from the cost of increasing predation or escape abilities. Because of these costs, decreasing such abilities will always have some compensating benefits. An increase in predator ability may alter the

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PETERA.ABRAMS

cost-benefit balance so that a decreasein escapeability may be favored. As the Appendix shows, the predator’s population size rather than predator capture abilities determine the nature of the cost-benefit balance for prey under the additive model. An important assumption of Eq. (1) is that the predator population is limited solely by its consumption of prey; predator population density has no direct effect on predator fitness. Such density effects can alter some of the results described above (Abrams, 1986a). In summary, it is clear that an interacting predator and prey population which are both evolving adaptively, need not always exhibit evolutionary responsesto evolutionary changes in the other species.Thus, the questions addressed below are (1) If there is no coevolutionary response in the twospecies system, will there be one in the three-species system? and (2) If there is a coevolutionary response in the two-species system, does the additional speciesdecreasethe magnitude of this response? An additional species can potentially affect predator-prey coevolution via several interrelated mechanisms. In the context of the simple models examined here, the two primary mechanisms are (i) a change in population dynamics, which affects equilibrium population levels of predator and prey and the responses of population densities to changes in particular characters; and (ii) the possible introduction of additional direct selection pressure on the predator’s prey-capture ability. A third specieswill directly affect selection on the original predator-prey relationship if there is a correlation between ability to capture the original prey species and ability to interact with the third species. In the following analyses, both mechanisms are examined for both additive and multiplicative models of the capture rate constant. III. EFFECTOF A HIGHER-LEVEL PREDATOR ON PREDATOR-PREYCOEVOLUTION

The presenceof a higher-level predator means that the system being considered becomesa three-speciesfood chain with a prey at the base, a lowerlevel predator in the middle, and a higher-level predator at the top. I assume that only the bottom two species exhibit coevolutionary change. The subheadings below consider different models of the capture rate constant of the lower-level (evolving) predator: multiplicative and additive. I assume that the per capita population growth rate of the higher-level predator is determined by its food (the lower-level predator), but is independent of its own density. I also assume that the lower-level predator’s adaptations for capturing prey and avoiding/escaping the higher-level predator are independent. The effectsof altering these two assumptions are discussed at the end of each section.

PREDATOR-PREY

A. Multiplicative

245

COEVOLUTION

Model for the Consumption Rate Constant

The presence of a higher-level predator changes the original system of Eq. (1) in two ways. There is an additional equation describing the dynamics of the higher-level predator’s population size, S,

dsldt = SCg,(C,f’) - o,l,

Pa)

where C, and D, again represent a capture rate constant and a densityindependent component of the death rate. g, is a nondecreasing function describing net per capita reproduction as a function of food intake. Equation (2a) assumesa linear functional response by the higher predator. The results presented below do not depend critically on this assumption, provided that the nonlinear functional response does not cause sustained population cycles. The addition of the higher-level predator changes Eq. (la) to dP/dt=P[(g,(C,C,V)-D,]-C,PS.

(2b)

The prey equation is still given by Eq. (lb). I assume that both two- and three-speciessystems do, in fact, have a stable equilibrium. General conditions for stability are difficult to interpret, and may depend on the rates at which C, and C, evolve. Numerical results have shown that local stability is at least a common result when simple functional forms are assumed for the components of the model. 1. Coevolutionary change in the predator. First consider the coevolutionary response of the predator to improved prey escape ability (decreased C,). The Appendix (following Abrams (1986a)) shows that the coevolutionary response in the predator’s prey-capture rate is given by ac

,ac po

= v

(v+

C,

WdCJg;+

[-Cyg;-PD,/aC;]

C,C,

J’g;)

’

(3)

where primes denote derivatives with respect to the argument of the function (e.g., gb =dg,/d(C,C, I’)), and the subscript o indicates that the derivatives is evaluated at the optimum. When there is no higher-order predator, Eqs. (2) imply that the first factor in the numerator of expression (3) is zero, so X,,/iX, = 0 (see Abrams, 1986a). When the higher predator is present, the equilibrium P is a constant, fixed by the parameters of (2a), and the first factor in the numerator of Eq. (3) becomes V+ C,C, P/f L. This quantity is positive if V is above the value that maxi-

246

PETER A. ABRAM

mizes the prey’s population growth function, F’l;, and is negative if V is below this value. The sign of expression (3) may be assessed by noting that the denominator must always be positive (by the second derivative condition for C,, to maximize fitness: Abrams, 1986a), and the second factor in the numerator is positive unless the birth function gP has a large negative second derivative. Thus, if g, is relatively linear, K’,,/X, will be positive when V is large, and negative when V is small. If g6 is large and negative, the conditions for a negative K,,/K, will be reversed. A negative X,,/K, represents a predator response that counteracts the original prey evolution. Regardless of the sign and magnitude of g;, the presence of the higher predator causes a coevolutionary response of the lower predator to changes in its prey; no response would occur in its absence. 2. Coevolutionary change in the prey. The optimal evolutionary response of the prey to evolution of improved predator capture abilities in Eqs. (2) is always an increase in escape abilities. This occurs because an increase in C, always increases the predation rate experienced by a prey individual (C,P), and this always decreases the optimal C,. This is justified by the results in Abrams (1986a) for the case of fixed predator population size. The direction of the coevolutionary response is thus the same as in the one-predator, one-prey system. It is also of interest to compare the magnitudes of the optimal prey response to increased C, in systems having or lacking the higher-order predator. These magnitudes are increasing functions of a/X,(C,P). In the system without the higher predator, p may increase or decrease with an increase in C,. P decreases when the prey are overexploited (i.e., at a population size below that which maximizes P’f,). Thus, whenever the prey are overexploited in the two-species system, the change in predation pressure (C,P) with increasing capture abilities will be greater in the presence of a higher predator than in its absence. As a result, the coevolutionary response in the prey will be increased because of the higher predator. (In general, overexploitation of prey increases the probability that a predator-prey system will be unstable. However, systems with linear functional responses are generally stable, even when the prey are overexploited.) 3. Alternative assumptions. It is also important to consider whether the presence of evolutionary correlations between C, and C, might make it less likely that prey evolution would induce a compensatory evolutionary change in the predator. In this case, an increase in C, not only increases D,, but also alters C,. The equation defining the predator’s optimal strategy is then

c,vg;-ao,/ac,-Sac,/ac,=o.

(4)

PREDATOR-PREY

247

COEVOLUTION

Implicit differentiation of the above equation reveals the effect of a decreased C, on the optimum C,:

acpo_ (&$I+ g;c,c, w+ c"(awc") - @wq(~~/~c") a*D,lac~+S(a2c,lac~)-~,v2~b: ac,

(5) .

This expression differs from expression (3) in the addition of one term to each of the numerator and denominator. Neither of these two terms has a determinate sign. Consider first the -(X,/X,)(a$X,) term in the numerator. Vermeij (1982, 1983) argues that X,/X, will often be negative, based on the fact that traits that aid prey capture (e.g., crabs claws) can also be used in defense. @/laC, has the same sign as 8+ C,(ap/laC,), which is also a factor in the first term of the numerator. Thus, if Vermeij is correct, the evolutionary correlation will increase the magnitude of the numerator of (5) when the growth function g, is relatively linear, and decrease it when g, is strongly concave (gb + giC,C, V < 0). The additional term in the denominator of (5) has the sign of a’C,/aCt. This may be either positive or negative, but negative values seem more likely; increasing adaptations to catch prey may increase escape ability from higher predators, but should not increase escape ability at an accelerating rate. A negative a’C,/aC~ will increase the magnitude of (5). Thus, it seemsmore likely that evolutionary correlations between C, and C, will increase rather than decrease the magnitude of the predator’s response to a prey that is better able to escape. An alternative scenario (suggested by a reviewer) is that adaptations to increase prey capture (C,) make the lower-level predator more susceptible to the higher-level predator (C, becomes larger). This is a common assumption of behavioral models of foraging under risk (Abrams, 1984; Ives and Dobson, 1987). This assumption reverses the predictions made above. The magnitude of (5) will be greater than that of (3) when g, is strongly concave, but not when it is linear. Another possible modification of the above analysis is to consider a higher-level predator that has a fixed population density. If the predator has a linear functional response with fixed population density, then it simply represents another density-independent mortality factor. The addition of S will then be equivalent to adding a constant to the function D,. This will increase the equilibrium prey population size and decrease predator population size, but will not affect the equations determining the nature of coevolutionary response. The predator’s optimum C, will not change in response to changes in C,. If the higher predator has a nonlinear functional response, the results become more complex, since the lower predator’s population density will enter into the formula for the optimum C,. In this case, equilibrium prey availability will change with a change in

248

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A. ABRAM

C, ; the resulting change in the optimum C, will depend on the nature of the change in availability, the shape of the g, curve, and the second derivative of the higher predator’s functional response. A detailed analysis is not presented here. B. Additive Model of’ the Consun~ption Rate Constunt The Appendix outlines the nature of coevolutionary change in a system with one predator and one prey. The question here is whether the qualitative features of this analysis are changed by the presence of a higherlevel predator. The general model is still described by equations (2), except that the consumption rate constant is C, + C, + C,. 1. Induced coevolutionary change in the prey. Recall from Section A that a higher-level predator whose population is regulated by food consumption will maintain the lower-level predator at a constant density. Fom the Appendix, the response of the prey to increased C, is given by

ac,,jac, = -(a~/dc,)/(a20./act). Equation (6) is zero when the equilibrium P is a constant that depends only on the growth parameters of the higher-level predator. Thus, the addition of the higher predator effectively decouples the prey from responding to evolutionary changes in the predator’s capture ability. 2. Coevolutionary change in the predator. Expression (A4) is a general formula for the optimal predator response to altered prey escape abilities. Although C, may change in either direction in response to a decreased C,, C, will increase in the two-species system if g, is linear. The addition of the higher predator makes the sign of 13p/K, indeterminate, even when g, is linear. The higher predator may thus cause X,,/X, to be positive. In the case of a linear g,, the higher predator makes it less likely that the lower predator will increase capture ability in response to increased prey escape ability (decreased C,). However, given a nonlinear g,, the higher predator may cause either an increase or a decrease in K’,,/X,. 3. Other alternatives. If S is held constant by others factors, it effectively acts as another source of density-independent mortality on the lower predator. As a result both predator and prey coevolutionary responses are identical in direction to those in a system with S = 0. The final situation to consider is that in which there are evolutionary correlations between C, and C,; Will this decrease the probability that the predator responds positively to evolutionary changes in the prey? (It has no effect on the prey’s coevolutionary response.) Equation (A3), which

PREDATOR-PREY

249

COEVOLUTION

specifies the predator’s optimal strategy, is changed by the addition of a term -$X,/K,). The general expression for X,,/X, then becomes

[g’,dt/ac,+ VgL( v+ c(av/dC,))- (a~/ac,)(ac,/ac,)] ’ [a*D,/acz,-v*g;:+ ~(a’c,/ac;)]

(7)

where C = C, + C, + C,. aL?/laC,is given by (l/C,) g;( p+ C(a??/laC,)). If (X,/X,) is negative, as Vermeij (1982) suggests, and g; < 0 (as seems likely), then the third term in the numerator of expression (7) is likely to offset the second term. However, this may either increase or decrease the magnitude of the numerator as a whole. a2C’,/aC~ is likely to be negative (see discussion of Eq. (5)) which will increase the magnitude of expression (7). However, as noted in the discussion of expression (5), there are other possibilities. Thus, it is difficult to generaize about the effects of evolutionary correlations on the overall magnitude of dC,,/X,. However, there is no reason to believe that such correlations generally decrease its magnitude. Table I summarizes the effects of a higher-level predator on the coevolutionary responsesof each species. TABLE I Effects of a Higher-Level Predator on the Coevolutionary in a Predator-Prey System” Type of consumption rate constant

Responses

Predator response

Prey response

Multiplicative; no correlations with other traits

+ or - (0)

+ (+) Magnitude may be increased

Additive; no correlations with other traits

+ or - (+ or -) less likely to be + than in two-species system if g, linear

O(+or-)

Effect of evolutionary

correlations

on induced responses

Multiplicative; C, and C, correlated

Usually increases magnitude of response

No effect

Additive; C, and C, correlated

Can increase or decrease magnitude of response

No effect

Note. Signs in parentheses give the responses expected in a system with no higher-level predator. a + denotes that the coevolutionary response of the species counteracts the change in the evolutionary parameter of the other species. 0 denotes no response. - indicates that the coevolutionary response increases the change in the consumption rate constant caused by changes in the other species.

2.50

PETER

A.

ABRAMS

IV. ALTERNATIVE PREY This section uses models similar to those analyzed above the determine whether the decoupling argument is logically valid when the third species is an alternative prey species rather than a higher-level predator. The analysis is again divided on the basis of how predator and prey adaptations combine to determine a capture rate constant. In addition, analyses are separated based on the presence or absence of evolutionary correlations between the predator’s ability to capture different types of prey. A. Multiplicative

Combination of Components of the Consumption Rate Constant; Ability to Capture the Second Prey Independent of Ability to Capture the First

In this case, the main effect of a second prey is on the change in equilibrium population density that occurs in the first prey when it increases its escape ability. A general population dynamics model for analyzing this problem has the form

dV,ldt= V,.f,(V,)-D,,V,-C,C,V,P

(84

dVJdt=

(8b)

V,f,(V,)-D,,V,-C,V,P

dP/dt = P( g,[ C, C, V, ] + C, V, - Dp).

(8~)

The population dynamics of the prey’s resources are not explicit in this model, but they implicitly determine the form of the prey population growth functions, fi. C, may be a function of predator- and prey-controlled factors. However, because prey species 2 is assumed not to evolve, C, need not be decomposed. 1. Predator response to altered prey escapeability. The optimum value of C, is determined by the same equation as in the single-prey case, but the expression for the equilibrium abundance of this prey differs. Assuming that the system has a stable equilibrium, the equilibrium prey abundances are determined by the two equations,

gp(C, Cv V, + Cz VA = D, (f, -D,,)/C,C,=(f,-D,,)lC,.

Pa) Pb)

If prey 1 evolves a decreased C,, what is the effect on the optimum C,? As in the single-prey system, this is determined by the sign of 13/X,(c, P,) when g, is linear; if the total availability of prey 1 increases as its escape

PREDATOR-PREY

251

COEVOLUTION

ability decreases,then the predator should respond by evolving a larger C,. From Eqs. (9) it follows that awac”

= CGCp(f2 - Da)

-cc;

w2llCwl+

c;w21.

(10)

+ c’,w21.

(11)

Thus, I/, + C,(138,/8C,) is given by

c2cc,c”(f*

- D”2) + c2 w;1/cw;

If g, is linear, X,,/aC, has the same sign as expression (1 l), and if g, is nonlinear, the sign is that of (11) multiplied by (g;, + C,C, P, gi). The denominator of (11) is negative, but the numerator depends upon the relative magnitudes of C,C,fi and C, P,f; ; if the former is smaller in magnitude, then X&K’, is positive (given a linear g,). This means that the predator should decreasethe level of (costly) adaptations for capturing prey 1 in response to an increase in prey l’s escapeability (decreased C,). However, if C,C,f2 is larger in magnitude than C, P, f ;, then the predator responds to an increase in prey l’s escapeability by increasing its capture-related adaptations. These rather abstract results can be clarified by an example in which both prey specieshave logistic population growth. If g, is the identity function and the two-prey populations have logistic growth with parameters ri and Ki, the expressions for equilibrium prey abundances become

P, =

D,r2CpCvK,-rzCpCvCZK1K2+rlC~K1K2 rzCiCzK, +rlC:K2

(124

P* =

D,r,C,KZ-rlCpC,C2K1Kz+r2C~CtK,K2 r2C:CtK1 +r,CiK,

(12b)

The change in the availability of prey 1 with an increase in its escapeability is given by a/aC,(C, P,):

\ &CA= ”

+KzG(~~C~-~~~C,C,)I [r,C’ECtK,

+r,CiK,]’

/ ’

(13)

If the two prey have identical r’s and K’s, this simplifies to &(C,O= ”

C;[-KC;C~+KC;+2D,C,C,-2CJ,C,K] pz’,ct + c:]’

(14)

Figure 1 illustrates this derivative for a variety of relative of C, and C,. If

252

PETER

A. ABRAMS

0.3 0.2

.-t!

0.1

z 2 & 0

0 -0.1

-0.2 -0.3 -0.4 0.05

0.55

1.05

1.55

2.05 c2

2.55

3.05

3.55

4.05

FIG. 1. The derivative, dC,,/K’,, in a two-prey system as a function of the capture rate constant on the second prey, Cz. Cz is assumed to be independent of C,, and the derivative is given by expression (14). The four lines give the derivative for different values of C, (i.e., C,C,); in the lowest line (diamonds) C, = 3D/K; the triangles correspond to C, = 2D/K, “+” symbols indicate C, = 3D/K; squares indicate C, = SD/4K. The maximum C, that will allow coexistence of both prey is 4.0 if C, = 2D/K; 4.5 if C, = 3D/K or 3D/2K, and 6.25 if C, = 5D/4K. A negative derivative means that the predator increases its adaptations for catching the first prey in response to that prey increasing its escape abilities.

C2 approaches 0 (i.e., the second prey is an insignificant part of the predator’s diet) Eq. (14) approaches 0. If C, is nonzero, but is small, then (14) is negative if the first prey is overexploited (has an equilibrium density less than K/2) and positive if it is not. However, a small CZ implies that the magnitude of the coevolutionary response of the predator to prey 1 is small. Expression (14) is always negative for a range of C2 values that includes the case in which both prey are caught equally well (C, = C,). A sufficiently large C, will always make expression (14) positive, but this may not occur until C, is greater than the maximum value that will allow coexistence of the two prey. If C, > CTK/(C, K- D,), prey species 1 will be excluded by “apparent competition” (Holt, 1977). As Fig. 1 shows, expression (14) is positive and large in magnitude only when C, is relatively small and C, is relatively large. For most possible parameter values, expression (14) is negative, implying that greater escape ability of prey 1 actually increases its availability to the predator. This means that, unlike the single-

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253

prey system, the predator evolves increased abilities to capture prey 1 in response to this prey’s increased escape abilities. 2. Response of prey 1 to increased C,. The next question to consider is the effect of the second prey on the evolutionary response of the first prey to a predator that is better able to capture it. Given a tradeoff between D,, and C,, the general equation for the optimal C, is identical to that in the case of a single-prey system; X,,/X, = -a/X,( C, p)/( a2D,, /X t ). The sign of this quantity is determined by its numerator (the denominator must be positive). The equilibrium P is determined by setting Eq. (8a) equal to zero; this may then be used to derive the expression

aiac,(c,B) = (lic,)cf’,(aP,/ac,)i,

(15)

where af,/X, is negative and is given by Eq. (10). Like the analogous derivative for the single-prey system, expression (15) must be positive. Prey 1 decreasesC, in response to increased C,, regardless of the presence of the second prey. The question is then whether the evolutionary response of C, to C, is increased by the presence of the second prey. Quantitative results may be derived by assuming specific functional forms for the prey growth equations; I again assume identical logistic growth of both prey. Assume also without loss of generality that the D,, are zero. The resulting expression for predation pressure, C,p, is C,B=rCp[KCpC,-Dp+C2K]/[K(C~CfI+C~)]

(16)

and the derivative of (16) with respect to C, is r[D,C~C~-KC~C~C,$2KC~C,C,-D,C~+C~K] K[C;C; + C;12

(17)

If CZ is close to zero, the increase in predation pressure, C,P with a change in C, (given by (17)) is very small. Thus, for small Cz there is a smaller coevolutionary response in prey 1 due to the second prey. However, the magnitude of expression (17) increases for intermediate values of C, ; if C,C, = C,, then the magnitude of the (17) is always greater than the corresponding formula for a single-prey system. It should be quite common for coevolutionary responses in one prey to evolutionary changes in its predator to become larger becauseof the presenceof a second prey species. In any case, it is clear that the addition of a second specieswill not, in and of itself, decreasethe response of the first to a more skillful predator.

254

PETER

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B. Multiplicative Consunzption Rate Constunt; Evolutionury Correlations het~~~renAbilities to Cupture the Two PreJ, There may often be evolutionary correlations between the abilities of a predator to capture different prey. Traditionally, these correlations have been assumed to be negative in models of adaptive resource use (e.g., Lawlor and Maynard Smith, 1976; Holt, 1983; Abrams, 1986~). The effect of a second prey on the predator’s evolutionary response to a single-prey species may again be examined without adopting a particular model of prey population dynamics. The predator’s fitness will again be assumed to be given by g,(C,C, V, + C, V,) - D,. Cz is actually a product of predator and prey-controlled factors, but can be assumed to ‘be solely a function of predator traits, because the second prey is assumed to have evolutionarily fixed escape abilities. C2 will be assumed to be a decreasing function of C,; this implies that at least some of the traits that aid capture of one prey type decrease the capture rate of the other. In general, a larger C, may imply a larger D, as well as a smaller C,, but possible effects on D, will be ignored here to simplify the analysis. The optimal value of C, is then given by ac,/ac,

= -c,

P, /P2.

(18)

The intuitive notion is that a smaller value of C, will cause the predator to increase its adaptations for consuming the second prey. This notion is correct when population densities of the two prey remain constant, because a decreased C, will then decrease the magnitude of the RHS of Eq. (18) decreasing the optimal C, (Lawlor and Maynard Smith, 1976). However, in general, both prey will undergo changes in equilibrium population densities when the first prey becomes better at escape. The “intuitive” results will only occur if the RHS of Eq. (18) decreases in magnitude as the result of decreased C,; i.e., a/X,(C, lir/p,) > 0. This inequality implies that the ratio of the equilibrium availability of prey 1 to that of prey 2 decreases when prey 1 becomes better at escape. If the two prey have growth functions ,f, and fi, then the formula for this derivative may be found from Eqs. (8) and (18):

(5 ~~“~,~~~~:P*+c,c”c,P,llc*P,~‘~+c,c,~~l P:[c;.f’, + c;c;yJ ac, (V2)

.

(19)

This may again be illustrated for the case of logistic prey growth functions; given equal r’s and K’s for the two prey, formula (19) becomes [C,D,][K(C;-C;Ct-2C,C,C,)+2C,C,D,] [D,C*C,C,C*K+ KC;C:)]’

’

(20)

PREDATOR-PREY

255

COEVOLUTION

Expression (20) is illustrated in Fig. 2. If C, is very small, (20) has the sign of D, - C,C,K/2. Therefore, if the first prey is overexploited, (D, < C,C, K/2), the relative availability of prey 1 to the predator actually increases when it becomes more difficult to capture. However, as Fig. 2 shows, if C, is small, any coevolutionary responses will be small. If C, = C,C, (the two prey are exploited equally) then expression (20) is generally negative. (It is negative when either prey alone can support a predator population; i.e., if C, C,K> D,. This seemslikely if the predator-prey system is to persist.) A sufficiently large C2 will make expression (20) positive, but this value may be larger than the maximum that will allow coexistence of the two prey (see discussion under expression (14)). Expression (20) is positive if C2 > C, + (l/K) J2C, K(C, K- Dp). (214 This value of C, is less than the maximum consistent with coexistence (C,=CfK/(C,K-D,)) if C,<(2D/K)-(D;/K).

@lb)

2.5 a 2 a 1.5

a

0

a

1

-2 -2.5 0 q

cl=

3

0.25

0.5 +

0.75 cl

= 2

1

1.25 c2 0

1.5

1.75

Cl =1.5

2

2.25

2.5 a

2.75

Cl = 1.25

FIG. 2. The derivative, dC,,/dC,, in a two-prey system as a function of the capture rate constant on the second prey, C2. Cz is assumed to be a decreasing function of C,, and the derivative is given by expression (20). The four curves correspond to the different values of C, illustrated in FIG. 1, and the maximum Cz values are identical to those given under FIG. 1. As before, a negative derivative means that the predator increases its adaptations for catching the first prey in response to that prey increasing its escape abilities.

65313913.2

256

PETER

A. ABRAMS

Both of these inequalities must be satisfied if the predator is to evolve so as to switch away from a prey which has increased its escape ability. Inequality (21b) implies that the equilibrium density of prey 1 in the absence of prey 2 must be significantly greater than K/2, and inequality (21a) means that Cz must be significantly greater than C, These conditions are satisfied for the right-hand portions of two of the curves illustrated in Fig. 2. However, the general conclusion of this analysis is that, for most permissible parameter values, the predator should increase its ability to capture prey 1 in response to that prey improving its escape abilities. It should be remembered that there is no coevolutionary response of the predator to a change in C, in the single-prey system. The analysis therefore implies that the presence of a second prey usually (but not always) produces a coevolutionary response in the predator which counteracts improvements in prey l’s escape ability, and which would not have occurred in the absence of prey 2. C. Additive Model; No Evolutionary

Correlations

1. Coevolutionary response of predator to prey 1. Given an additive model with Cz independent of Cl, the predator’s optimal C, is determined by the same equation as in the single-prey case (Eq. (A3) in the Appendix). Equation (A4) then defines X,,/K,. When the birthrate function, g,, is linear,
aWac, = cw-,-

c, w;iim-;

+ cm,

(22)

where C, = C, + C, + C,. Because the prey per capita growth functions, J are decreasing, expression (22) must be negative. This implies that, just as in the single-prey situation, the predator should increase capture adaptations when the prey increase escape adaptations. This conclusion is reversed in both one- and two-prey systems if gg is negative and large in magnitude. Does the presence of the second prey increase the magnitude of the predator’s response (which is proportional to af,/X,)? Again assume the two-prey species have equivalent logistic growth functions and a predator

PREDATOR-PREY

257

COEVOLUTION

has a linear g,. The formula for dP,/dC, may be derived from Eq. (11) after changing the expression for the capture rate constant. This yields aP,/ac,=

[D,(c~-c~)+K(c;c,-2c,c~-c;)]/(c~+c;)2,

(23)

where C, is again given by the additive combination of components. If expression (23) is evaluated at C2 = 0, one obtains the relevant formula for a single-prey system. Some algebra will show that the (negative) magnitude of the derivative is smaller in the two-species system. Thus, a second prey reduces the magnitude of the predator’s coevolutionary response to changes in the first prey. 2. Coevolutionary response of the prey. Again assumethat the prey’s per capita growth rate function has a parameter D, that increases as C, decreases.The formula for aC,,/aC, given by (A2) in the Appendix is-still valid when a second prey is added to the system. This means that the prey’s response has the opposite sign of alj/X,, which in turn can be derived from the equilibrium condition, p = f2/C2,

=wc,)C-c,f2-

w2f;l

+Gf;l

CGf>

.

(24)

The sign of (24) is indeterminate. In the simple example of prey that have identical logistic growth functions, ap/jaC, is given by [K(-C;-2C,C,+C;)+2D,C,]/[K(C:+C;)]*.

(25)

The numerator of expression (25) is equal to the numerator of expression (20) divided by CzD,. Therefore, the previous discussion and Fig. 2 show that dp/X, is usually negative, unless the alternative prey (2) is much more easily caught than the first prey. It follows that X,,/X, is usually positive. Thus, the most common response of the prey is to reduce the level of costly adaptations for escapewhen the predator becomesbetter at capture. In the corresponding one-prey system, this response occurred if and only if the prey was overexploited (Appendix). D. Additive Model; Negative Correlations between the Two Capture Rate Constants

In this analysis, as in Section B, I assume that the cost of an increased C, for the predator is solely a reduced C,. The conditions for an optimum C, are

ac,jac,= -P,p2

and

a*c,lac;
(2% b)

258

PETER A. ABRAMS

Thus, the effect of C, on the optimum C, is given by

(7CpJL1C,, = [ P,((;‘P&?xy - v&G, /CT, ,]/[ P;(a’cJac;)].

(27)

This formula may be expanded as

ac PO -

ac,-

-1

i 8;(aT,jac;) ) xIc:P:f’;+(cfP2+C1C2Pl).fZ-CIC2PIP2f;l c*Grl

(28)

+ cm

Although the sign of this expression is indeterminate, the fact that two of the three terms in the numerator are positive (and the denominator is always negative) makes it likely that C,, will increase when C, decreases. TABLE II Effects of an Alternative Prey Species on the Coevolutionary in a Predator-Prey System” Type of consumption rate constant

Responses

Predator response

Prey response

Multiplicative C, vs. D, tradeoff

+ or - (0) + more likely in logistic growth example

Larger or smaller than in two-species system

Multiplicative C, vs. C, tradeoff

+ or - (two-species) not applicable) + more likely in logistic growth example

Same as above

Additive C, vs. D, tradeoff

+or-(+or-) Magnitude smaller when two prey present

+or-(+or-) - more likely because of second prey in logistic growth example

Additive C, vs. Cz tradeoff

+ or - (two-species not applicable) + more likely

Same as above

+ (+I

Note. Signs in parentheses give the responses expected in a system with no alternative prey. ’ + denotes that the coevolutionary response of the species (predator or prey 1) counteracts the change in the evolutionary parameter of the other species. 0 denotes no induced response. - indicates that the response increases the change in the consumption rate constant caused by changes in the other species.

PREDATOR-PREY

259

COEVOLUTION

The above general result may again be illustrated using the case of two prey with identical logistic growth. In this case, a/X,[ - p1/p2] is D&D&-C;K-C:K)+K’C,(C;+C;-2C,C,) -[D,Cz-C,C,K+C;K]2

’

(29)

Unless C2 is sufficiently larger than C,, this expression is positive, implying that the optimum C, increases when C, decreases.This is qualitatively similar to the case with a single-prey type (Appendix). The magnitudes of the coevolutionary responses in one- and two-prey systems are difficult to compare, since we have assumed different costs to an increased C, in the two cases.If D, is only slightly affected by C,, then C, may have its maximum possible value in the single-species system for almost all possible values of C,. This would then mean that the magnitude of coevolutionary change in the two-species system would be larger than in the single-prey system. The analysis of the coevolutionary response in the prey following predator evolution is identical to that in the preceding section (1V.C). Table II contains a summary of the coevolutionary responsesin all of the models with an alternative prey.

V.

DISCUSSION

This paper has analyzed a wide variety of situations. Tables I and II summarize the nature of coevolutionary responsesin all of these cases.It is clear that the “decoupling” argument is not generally valid; additional species of the types studied here often increase the magnitude of the coevolutionary response in predator or prey (or both) to evolutionary changes in the other species.Higher-order predators and alternative prey have been proposed to be agents that prevent the predator from responding to the prey (Vermeij, 1982). However, the only case (of those analyzed here) for which the decoupling argument is valid involves the higher predator eliminating the prey’s response to its predator when adaptations combine additively. In the majority of the cases analyzed, additional species either cause a coevolutionary response in the predator where one would not otherwise occur, or increase the magnitude of the predator’s response. The decoupling argument may sometimes explain why a predator with alternative prey does not respond to counteract evolutionary improvements in a single-prey species’ escape abilities. One prerequisite for the “decoupling” argument is that there be a coevolutionary response in the absence of the alternative prey, and this does not occur under the multi-

PETERA.ABRAMS

260

plicative model when predators limit prey population size. There may be a counteracting coevolutionary response in the additive predator-one prey model. This response may be reduced or eliminated in the presence of an alternative prey species, but this requires a number of conditions on the shapes of functions and magnitudes of parameters in the model. The overall results make it hard to understand why there are so few documented cases of evolutionary responses to prey among predatory species (e.g., see Endler, 1986, Table 5.1). The models analyzed here represent only a small subset of the possibilities, even under the simple assumptions of homogeneous populations and linear functional responses that have been adopted here. The mechanism of population regulation in the predator can have a major effect on the direction of coevolutionary responses (Abrams, 1986a). However, a higher-level predator produces effects that are similar to those produced by direct density dependence in the predator. It is therefore unlikely that additional density dependence would significantly change the arguments advanced here. Different types of costs of increased predation-related adaptations result in different coevolutionary responses in predator-prey models (Abrams, 1986a). Various types of between-individual variation can affect the dynamics of predator-prey systems (Taylor, 1984), and may have effects on coevolutionary responses. Nonlinear functional responses may lead to predator-prey limit cycles, and the effects of such cycles on coevolutionary processes are largely unknown. Optimization approaches, like that used here, do not take into account the dynamics of gene frequency change. Clearly, investigation of more complex models is desirable. However, there is no obvious reason why the “decoupling” argument would be saved by adding these complications to simple models. The models considered here do not represent true three-species coevolution. In all cases, the third species was assumed not to evolve. It is likely that many more possibilities could arise if there was the potential for coevolutionary responses in all three species. APPENDIX: PREDATOR-PREYCOEVOLUTIONWITH MULTIPLICATIVE AND ADDITIVE CONSUMPTIONRATE CONSTANTS A. Multiplicative Expression (3) in the text gives the effect of a change in C, on the optimal C,. It is derived as follows. The condition for the optimal C, is that the derivative of predator fitness (g,(CV) - DP) with respect to C, be zero. This yields the requirement that C, Vgb-dD,/K, =O. Taking the derivative of this equation with respect to C, and solving for X,/X, yields Eq. (3) in the text.

261

PREDATOR-PREYCOEVOLUTION

B. Additive The optimal prey strategy in this case is given by

a/ac,{g,(c,R)-~,-(~,+~,+~,)p~=

-ao,lac,-P=O.

(Al)

(In addition, the second derivative of prey fitness with respect to C, must be negative.) The expression is identical if the prey per capita population growth function is represented by f,( V) rather than g,(C, R). The effect of a greater capture ability in the predator (larger C,) is found by implicit differentiation of (A 1):

ac,,lac,= -aP/ac,l(a2o,lac;).

(AZ)

The second derivative condition assures that the denominator of expression (A2) is positive. Thus, expression (A2) is positive if and only if dp/X, is negative. In turn, ap/ldC, is negative if and only if the prey are overexploited (a/aV( Vf,) is positive). Thus, when prey are overexploited, dC,.,/X, is positive, and the prey will become worse at escapein response to the predator becoming better at capture. The optimal predator strategy is determined by

a/ac,{gp[-co+cp+cv)

VI-

~~)=g;-

aD,jac,=o.

(A3)

The effect of altered prey escape ability on the optimal C, is found by implicit differentiation of (A3), giving ac,,,ac,

= ~g;(aVacv) + &;(P+

WV + c, + cm%c))~~

[a%,/ac;-

(A4)

Pg;l

The sign of expression (A4) may be positive, negative, or zero. If g, is linear function of food intake, the sign of (A4) must be that of av/K,, which is always negative (the equilibrium prey density from Eq. (1) in the text simply D,/(C, + C, + C,)). In this special case, the predator will evolve increased C, in response to the prey evolving decreased C,. ACKNOWLEDGMENTS This paper benetitted from the comments of the two referees and from discussions with many individuals at Simon Fraser University and the University of British Columbia. REFERENCES ABRAMS, P. A. 1984. Foraging time optimization 8&96.

and interactions

in food webs. Am. Nat. 124,

262

PETER

A. ABRAMS

AHRAMS. P. A. 1986a. Adaptive responses of predators to prey and prey to predators: The failure of the arms race analogy. Elolutio~? 40, 1229-1240. ABRAMS, P. A. 1986b. Is predator-prey coevolution an arms race? TREE Rep. 1, 108-I 10. ABRAMS. P. A. 1986~. Character displacement and niche shift analyzed using consumerresource models of competition. Tllc~or. Pop. Biol. 29, 107-160. ABRAMS, P. A. 1987a. Alternative models of character displacement. I. Displacement when there is competition for nutritionally essential resources. Evolution 41, 651-661. ABRAMS. P. A. 1987b. Alternative models of character displacement. II. Displacement when there is competition for a single resource. Am. Not. 130, 271-282. ABRAMS. P. A. 1990a. Mixed responses to resource densities and their implications for character displacement. Euol. Ecol. 4, 93-102. ABKAMS, P. A. 1990b. Adaptive responses of generalist herbivores to competition; convergence or divergence. Evol. Ecol. 4, 103-l 14. ABRAMS, P. A. 1990~. Ecological vs. evolutionary responses to competition, Oikvs 57, 147-l 5 1. COHEN, J. 1989. Food webs and community structure, in “Perspectives in Ecological Theory” (J. Roughgarden, R. M. May, and S. A. Levin, Eds.), pp. 181-202, Princeton Univ. Press, Princeton, NJ. DEANGELIS. D. L., KITCHELL, J. A., ANU POST, W. M. 1985. The influence of naticid predation on evolutionary strategies of bivalve prey: Conclusions from a model, Am. Nut. 126, 8 17-842. ENDLER, J. A. 1986. “Natural Selection in the Wild,” Princeton Univ. Press, Princeton, NJ. FUTUYMA, D. J. 1983. Evolutionary interactions among herbivorous insects and plants, in “Coevolution” (D. J. Futuyma and M. Slatkin, Eds.), pp. 207-231, Sinauer, Sunderland, MA. FUTUYMA, D. J., AND SLATKIN, M., Eds. 1983. “Coevolution,” Sinauer, Sunderland, MA. HOLT, R. D. 1977. Predation, apparent competition, and the structure of prey communities, Theor. Pop. Biol. 12, 197-229. HOLT, R. D. 1983. Optimal foraging and the form of the predator isocline, Am. Nut. 122, 521-541. IVES, A. R., AND DOBSON, A. P. 1987. Antipredator behavior and the population dynamics of simple predator-prey systems, Am. Nat. 130, 431447. LAWLOR, L. R., ANU MAYNARD SMITH, J. 1976. The coevolution and stability of competing species, Am. Nat. 10, 79-99. LEVIN, S. A., AND SEGAL, L. A. 1982. Models of the influence of predation on aspect diversity of prey populations, J. Math. Biol. 14, 253-284. NITECKI, M.. Ed. 1983. “Coevolution,” Univ. of Chicago Press, Chicago, IL. PIMM, S. L. 1984. The complexity and stability of ecosystems, Nature 307, 321-326. STENSE:TH,N. CHR., AND MAYNARD SMITH, J. 1984. Coevolution in ecosystems: Red Queen evolution or stasis? Evolution 38, 870-880. TAYLOK, R. J. 1984. “Predation,” Chapman-Hall, New York. THOMPSON, J. N. 1982. “Interaction and Coevolution,” Wiley, New York. THOMPSON, J. N. 1989. Concepts of coevolution, TREE 4, 179-183. VERMEIJ, G. J. 1982. Unsuccessful predation and evolution, Am. Nat. 120, 701-720. VERMEIJ, G. J. 1983. Intimidate associations and coevolution in the sea, in “Coevolution” (D. J. Futuyma and M. Slatkin, Eds.), pp. 311-327, Sinauer, Sunderland, MA. VERMEIJ, G. J. 1987. “Escalation and Evolution,” Harvard Univ. Press, Cambridge, MA.