This article explores the combined evolutionary and ecological responses of resource uptake abilities in a generalist consumer to exploitative competition for one resource using a simple 2-resource model. It compares the sizes of ecologically and evolutionarily caused changes in population densities in cases where the original consumer has a strong or a weak trade-off in its abilities to consume the two resources. The analysis also compares the responses of the original species to competition when the competitor's population size is or is not limited by the shared resource. Although divergence in resource use traits in the resident generalist consumer is expected under all scenarios when resources are substitutable, the changes in population densities of the resources and resident consumer frequently differ between scenarios. The population of the original consumer often decreases as a result of its own adaptive divergence, and this decrease is often much greater than the initial ecological decrease. If the evolving consumer has a strong trade-off, the overlapped resource increases in equilibrium population density in response to being consumed by a generalist competitor. Some of these predictions differ qualitatively in alternative scenarios involving sustained variation in population densities or nutritionally essential resources.
Most consumers use more than one resource. As a result, the simplest of such systems—one that consists of a single-consumer species that uses two resources—is the subject of considerable ecological and evolutionary theory. This food web (or community module, sensu Holt ) has frequently been used to explore apparent competition and its evolutionary consequences (Holt 1977; Abrams 2000). When augmented by a second consumer species, this system was the basis of early models of character displacement and niche shift of consumer species (Lawlor and Maynard Smith 1976; Abrams 1986). Extensions of these models were later used to examine evolutionary diversification in a single-consumer lineage in a varying environment (Egas et al. 2004, 2005; Abrams 2006a,b,c). Nevertheless, much remains to be learned about the interaction of population dynamical and evolutionary processes in this module. Such interactions are likely to occur following the introduction of a second consumer into the 1-consumer-2-resource system. The present article deals with the population-level consequences of evolutionary change within the resident consumer in this scenario.
Previous research has devoted more attention to the trait evolution in competitive systems than to the population-level consequences of that evolution. However, those consequences are important, both in their own right and for understanding the indirect evolutionary responses of other species that interact with the module. Three related questions considered here are as follows: (1) How does evolution of the resident consumer modify the initial “ecological” changes in population density caused by a competitor?; (2) How is the answer to question (1) affected by the trade-off experienced by the resident consumer?; (3) How is the answer to question (1) affected by population regulation in the introduced competitor? The starting point of the scenario is a 1-consumer-2-resource food web in which a trait that determines the consumer's relative use of the two resources is at an evolutionary equilibrium. Much previous work has addressed evolution in the single-consumer system (Lawlor and Maynard Smith 1976; Lundberg and Stenseth 1985; Abrams 1986; Schreiber and Tobiasson 2003; Rueffler et al. 2004, 2006a; de Mazancourt and Dieckmann 2004), so the analysis begins with a review of what is known about this case. The treatments of the second and third questions contrast two alternative assumptions. In the case of question (2), the dynamics under weak versus strong trade-offs is considered. Question (3) is explored by comparing an invading competitor that is a resource-limited specialist with an invader whose population density is independent of its consumption of the resource it shares with the resident. In each of these dichotomies, the second alternative has received relatively little attention. Strong trade-offs have been discussed by theoreticians in the context of single-consumer systems in some previous works (e.g., Schreiber and Tobiasson 2003; Rueffler et al. 2004, 2006a; de Mazancourt and Dieckmann 2004; Ma and Levin 2006; Abrams 2006b,c), but the contrasting population-level consequences of the two trade-off categories have not received much attention. Competitors whose populations are largely independent of their consumption of shared resources have also been ignored.
Much analysis of evolution in single-consumer systems with a strong trade-off has assumed that it will produce evolutionary branching (e.g., Geritz et al. 1998; Doebeli and Dieckmann 2000). Here, I assume that branching in the original consumer is either prevented by the constraints of its genetic system (such constraints arise in a variety of different genetic scenarios; Dieckmann and Doebeli 1999; Bürger 2005), or simply has not occurred at the time of invasion. Rueffler et al. (2006b) review evolutionary outcomes other than branching in systems characterized by branching point equilibria, and Schreiber et al. (2011) recently analyzed the population-level consequences of evolution in a 1-consumer-2-resource system with a strong trade-off. Branching did not occur in their model because the trait was determined by many independent loci of small effect, which is a common situation for ecologically important traits in multicellular organisms.
Most ecological experiments are of short duration, and there is a growing realization that evolution often changes population sizes significantly over intermediate-length time periods that are longer than the duration of most experiments, but short enough to be important for most predictions in applied ecology (Schoener 2011). Measuring the changes in population densities attributable to adaptive changes in traits has been a major component of many studies of predator–prey systems and food chains. (See theory in Abrams (1995) and empirical reviews by Werner and Peacor (2003) and Preisser et al. (2009).) However, there has been much less work on the population-level consequences of evolutionary change in consumer-resource models of interspecific competition. The results presented here suggest that evolutionary change of a given species following introduction of a competing consumer species may often produce much larger decreases in the population size of that consumer than the initial reduction due to resource depletion. Results also show that the responses of the resource populations that are the object of competition often change direction as the consequence of the resident consumer's evolution.
DYNAMICS OF A SYSTEM WITH A SINGLE GENERALIST CONSUMER SPECIES AND TWO RESOURCES
This section describes the basic 1-consumer-2-resource model with adaptive change in the consumer's trait specifying relative consumption of the two resources (the “choice” trait). The trait is specified by a value x, with 0 ≤x≤ 1. This is a measure of specialization on resource 1; x= 1 corresponds to the maximum per capita attack rate of the consumer on resource type 1 and minimum on resource type 2, while x= 0 corresponds to the maximum per capita attack rate on resource type 2 and minimum attack rate on resource type 1. The two attack rates, C1(x) and C2(x) are respectively an increasing and a decreasing function of x. For simplicity, I also assume that the second derivatives of C1 and C2 have the same sign as each other and that sign applies to all x. The two resources have population densities R1 and R2, while the consumer density is N1. A subscript 1 on the consumer density is used because in subsequent sections N1 and N2 will be used to denote the resident (evolving) and introduced (competing) consumer. The resources are assumed to be nutritionally substitutable for the resident consumer, and functional and numerical responses are linear. The per capita death rate of the consumer species i is di, the per capita growth rate of resource i is fi (which is a decreasing function of Ri), and the value (conversion efficiency) of resource i to consumer 1 is bi. The choice trait, x, changes in a direction given by the sign of, and at a rate proportional to the magnitude of, the derivative of the consumer's individual per capita growth rate with respect to the trait (Abrams et al. 1993). The quantity v scales the rate of evolutionary (adaptive) change, and is the additive genetic variance in models of quantitative trait evolution (Iwasa et al. 1991). The dynamic model is
In numerical simulations, additional terms are added to equation (1d) to prevent evolution of x beyond the boundary values of 0 and 1. As in Abrams and Matsuda (2004), these terms are extremely small for intermediate values of x, but become significant and positive as x approaches 0, and significant and negative as x approaches 1. (See Abrams and Matsuda (2004) for more details.)
The analysis of this and subsequent models combines some qualitative analytical results (which apply to systems with stable equilibria) with numerical analysis of particular models. The latter illustrate the magnitudes of the predicted responses to competition and are required to understand cases that lack stable equilibria. The impacts of the competitor are measured in two ways; the first examines the consequences of a small change in density or a key demographic parameter of the competitor, while the second compares the equilibrium or long-term mean densities with the competitor present to those with the competitor absent. These two perturbations can produce responses characterized by different signs (Abrams 1987b).
Small changes near equilibrium are analyzed first. Setting the left-hand sides of equations (1a–d) to zero defines the equilibrium conditions for all four variables. The distinction between weak and strong trade-offs is determined by the sign of the second derivative of the consumer per capita growth rate with respect to x (the first derivative of the quantity in parentheses on the right hand side of equation [1d]). A positive second derivative of fitness with respect to the trait (b1C1″R1+b2C2″R2 > 0) at an equilibrium having an intermediate x indicates disruptive selection (a strong trade-off) in the vicinity of the equilibrium value. If, for example, C1(x) =xm, and C2(x) = (1 –x)m, larger values of m increase the second derivative at the generalist phenotype of x= (1/2); here C1″=m(m– 1)(1/2)m− 2; C1″= 0, 2, and 3, respectively, for m= 1, 2, and 3. Larger values of m also imply that the generalist is more disadvantaged relative to the specialist; if m= 1, 2, and 3, the generalist has a capture rate that is, respectively, 1/2, 1/4, and 1/8 that of a specialist.
Some aspects of systems like equations (1) have been analyzed previously. Weak trade-offs were examined by Lawlor and Maynard Smith (1976) and by Abrams (1986); neither included trait dynamics, but they assumed that the trait would reach equilibrium at a point where x satisfied the condition for dx/dt= 0 from equation (1d). They showed that there is a single evolutionarily stable generalist strategy, unless one of the resources is much rarer, less nutritious, or more difficult to catch than the other. With very unequal resources, there is a single stable specialist equilibrium using the better resource. In contrast, a strong trade-off makes it possible for a generalist and either one or both specialists to be alternative evolutionary equilibria in some systems (Rueffler et al. 2004, 2006a; de Mazancourt and Dieckmann 2004). In general, higher (positive) values of the second derivative of consumer fitness favor specialist outcomes, as does greater asymmetry in the evolutionary maximum resource intake rates. When the generalist equilibrium is locally stable under a strong trade-off, it represents an evolutionary branching point (de Mazancourt and Dieckmann 2004; Rueffler et al. 2004, 2006a; Ma and Levin 2006). The outcome of evolution can be altered in cases with population cycles (Egas et al. 2004; Abrams 2006b; Schreiber et al. 2011). Persistent oscillations may occur when the adaptive rate parameter v is sufficiently large, again given a strong trade-off (Abrams 2006b; Schreiber et al. 2011). Cycles may also occur in related models because of a type-2 consumer functional response or because of environmental forcing of one or more parameters. These possibilities are considered briefly in a later section.
The present analysis of the impact of competition from another consumer is restricted to cases in which persistence of a generalist is a potential outcome of equations (1). A generalist outcome is favored by (1) having a trade-off in which the generalist consumption rates are not too far below those of the specialist (this of course applies to all cases with weak trade-offs); (2) relatively equal potential contributions of the two resources to consumer fitness; (3) high efficiency of resource use (where “high efficiency” means that the consumer causes a large proportional reduction in resource abundance). The third condition ensures that resource depletion generates sufficiently strong frequency dependence to make the generalist equilibrium locally stable. If these conditions are not met, the generalist state is an evolutionary repeller, and x approaches 0 or 1. Rapid evolution can also make the generalist state unstable (Abrams 2006b; Schreiber et al. 2011), but this is associated with cycles in x above and below the generalist state; it is only considered in the “alternative assumptions” section.
Two competitive scenarios are considered: (1) the introduction of a resource-limited specialist competitor, and (2) the introduction of a generalist competitor whose density is independent of its consumption of the one resource it shares with the resident (generalist) consumer. In case (2), consuming a large number of other, similarly nutritious and similarly abundant resources could make the competitor's density approximately independent of that of the shared resource. Alternatively, the density of the competitor may be determined by a distinct essential resource not shared with consumer 1 (e.g., nesting sites). For each scenario, both strong and weak trade-offs in the resident consumer are considered. In all cases, the effect of evolution of the resident consumer on equilibrium (or average) population densities is compared to the purely ecological effects of the competitor.
SCENARIO 1: A RESOURCE-LIMITED SPECIALIST COMPETITOR FOR ONE RESOURCE
A full model of competition with a resource-limited specialist competitor for resource 2 augments equations (1) by the following:
In addition, an extra consumption term, C22N2, must be subtracted from the quantity inside the outer parentheses of equation (1c), which gives the per capita growth rate of resource 2. (The assumption that resource 2 is the shared resource is arbitrary and does not affect the results.) A necessary condition for such a specialist to invade the generalist system is that its resource requirement for zero population growth must be less than the equilibrium density of resource 2 in the presence of consumer 1. The addition of consumer 2 simplifies the analysis of consumer 1's evolution because the equilibrium R2 must be d2/(b22C22). If the equilibrium point remains stable, the effect of increasing consumer 2 may be determined from equations (1a, b, d), by analyzing how a reduction in R2 affects x, R1, and N1.
Scenario 1A: Resident generalist has a weak trade-off
The stability analysis in Appendix 1 shows that, if it exists, an equilibrium with the resident at an intermediate x is stable, given a weak trade-off. The effect of reduced R2 (caused by the introduced competitor) may be found by implicit differentiation of the equilibrium conditions for equations (1a) and (1d) with respect to −R2. This gives a pair of linear equations for and . Here, I initially assume that the trade-off is weak, so that b1C1″R1+b2C2″R2 < 0. The resulting formulas are
Together with equation (1b), these two expressions determine how the abundance of the first consumer changes with a small decrease in the abundance of the second resource:
Because R2=d2/(b22C22) at equilibrium, the right-hand sides of equations (2) are proportional to the effect of a reduced mortality rate of consumer species 2.
The directions of change for R1 and N1 given by equations (2b, c) are identical to a comparable system where x cannot evolve. The numerator of equation (2a) must be negative because C1 increases with x while C2 decreases with x (C1′ > 0, C2′ < 0), and the denominator is negative because of the weak trade-off. This implies that a reduced R2 (due to competition) increases the resident consumer's per capita attack rate on resource 1, and consequently decreases its attack rate on R2. This response represents divergence from the specialist competitor. Equation (2b) implies that R1 must increase in response to the decrease in R2, which means that there must be a decrease in consumer density that is proportionately greater than the increase in attack rate on resource 1 caused by the divergence. This response implies apparent competition (Holt 1977) if the resources are living organisms (i.e., prey). Equation (2c) confirms that, given a weak trade-off, a competitive reduction in R2 decreases the abundance of the generalist (N1). This follows because both of the terms in the square brackets in equation (2c) must be negative. These two terms measure the relative changes in consumer 1's population due to the direct ecological effect of the resource reduction (the first term) and the evolutionary change in its consumption rates (second term), provided that it remains a generalist. The fact that the second term is negative means that, regardless of the type of resource growth (fi), evolution of consumer species 1 reduces its own density below what it would have been in the absence of that evolutionary divergence. Because the signs of the derivatives given by equations (2) are determinate (given a weak trade-off), they predict the direction of change following the gain or loss of the introduced competitor, as well as the changes caused by a small change in its mortality.
Scenario 1B: Resident generalist has a strong trade-off
The outcome of evolution in this case is that consumer 1 always diverges and specializes on resource 1 (x= 1) following successful invasion by the specialist competitor for resource 2. This follows from previous work on evolutionary branching in this type of system (e.g., Rueffler et al. 2004, 2006a). The specialist competitor prevents the density of resource 2 from changing as a function of consumer 1's trait, thus removing the frequency dependent stabilization of the generalist equilibrium. The corresponding changes in the abundances of consumer 1 and resource 1 are somewhat more difficult to establish because the shift from an interior to a boundary equilibrium (x= 1) does not allow the techniques applied in deriving equations (2). However, some conclusions can be drawn when the initial system is symmetric; when b1=b2=b; C1(x) and C2(1 –x) are identical; and f1(R1) and f2(R2) are identical. This ensures that x= 1/2 is the generalist equilibrium, and interpretable stability conditions for that equilibrium may be derived (see Appendix 1). C(1/2) will be denoted Cg, and C1(1) =C22≡Cs The equilibrium resource 1 in the absence of the competitor is d1/(2bCg) and, after specialization due to the competitor, R1=d1/(bCs). A strong trade-off implies 2Cg < Cs, so R1 must decline due to the evolutionary specialization. This implies that, given a strong trade-off, the resources interact via apparent mutualism rather than apparent competition on an evolutionary time scale, as noted before by Schreiber et al. (2011).
An expression for the change in N1 due to the presence of N2 can also be determined analytically if the system is symmetric. Given the above assumptions, the specialist consumer 2 is only just able to invade the system with consumer 1 when its resource requirement d2/(bCs) is just below the corresponding equilibrium density of resource 2 when consumer 1 is present at equilibrium; (d1/(2bCg)). In this case, consumer 2 has almost no impact on N1 in the absence of evolution, and has an ecological equilibrium density only slightly above zero. However, the resulting evolutionary specialization of the first consumer can reduce its density by a large amount. The equilibrium conditions above imply that consumer 1's population changes from
at its original evolutionary equilibrium to
at evolutionary equilibrium with the introduced competitor. A strong trade-off ensures that (1/C) decreases by at least a factor of 1/2 after divergence (eq. 3b). Unless the trade-off is extremely strong, the per capita growth rate f will be similar before and after divergence, because the argument of f (the equilibrium R1) will only decrease slightly. A very strong trade-off and a resource growth function that declines quickly with R could make equation (3b) larger than equation (3a), but these conditions also are likely to lead to instability of the generalist equilibrium, which means that the original generalist condition of the resident would not be observed. Thus, it seems unlikely that evolution would reverse consumer 2's negative ecological effect on consumer 1. Extensive numerical explorations using commonly used resource growth functions (chemostat, theta-logistic [Gilpin and Ayala 1973]) have not revealed any examples of consumer 1 increasing in size as a result of the combined ecological and evolutionary responses to the specialist competitor.
Scenario 1C: The evolutionary modification of the ecological effect for both types of trade-off
The purely ecological response to competition can be determined by solving for the equilibrium density of consumer 1 in the presence and absence of the specialist competitor, given that consumer 1's trait x has been fixed at its original equilibrium value. This may then be compared to the two-consumer evolutionary equilibrium to determine whether ecological or evolutionary responses are larger. The total change in consumer 1's population at evolutionary equilibrium is given by the difference between equations (3a) and (3b). A general formula for the purely ecological response may be obtained by solving for the equilibrium N1 in the ecological system with the trait x fixed at its evolutionary equilibrium value. If we again simplify the system by assuming that it is symmetrical, that b1=b2=b22=b, and that C22=C1(1) =Cs, the ecological equilibrium N1 is the right-hand side of the following inequality, which gives the conditions for the evolutionary equilibrium of the resident consumer to be larger than the ecological:
If the specialist consumer 2 is just barely able to invade the generalist equilibrium of species 1, the equilibrium density of R2 produced by the generalist (d1/(2bCg)) is very slightly larger than the equilibrium requirement of the specialist invader (d2/(bCs)). This makes the right-hand side of inequality (4) nearly equal to the right-hand side of inequality (3a). This in turn implies that the evolutionary response reduces N1 below the purely ecological equilibrium. Because the ecological response of consumer 1 is very small, almost the entire competitive effect is evolutionary. Figure 1 shows how the ecological and evolutionary components of the resident consumer's population response to the competitor depend on the latter's efficiency for both strong (Fig. 1B) and weak (Fig. 1A) trade-offs. The example in Figure 1 assumes that consumer 1 ingests biotic (self-reproducing) resources, and, because of the relatively high resource carrying capacity, density dependence is relatively weak. Both of these favor relatively large ratios of evolutionary to ecological decline in the resident consumer.
The evolutionary change in population density of consumer 1 depends on consumer 1's trade-off and the shape of the resource growth function f. The largest ecological effect occurs when consumer 2 reduces R2(=d2/(bCs)) to near-zero density. This makes condition (4) equivalent to a comparison of the consumer densities in a single resource system with a high consumption rate (Cs; left-hand side of condition ) versus a low consumption rate (Cg; right-hand side). Condition (4) is then always satisfied (evolution counteracts the ecological decline in abundance) for a resource with abiotic (e.g., chemostat) growth, again given the assumption of a very efficient specialist competitor. Condition (4) is satisfied for biotic (self-reproducing) resources that are not overexploited (i.e., have equilibrium densities greater than those that maximize Rf) for all attack rates C≤Cs. Biotic resources that are much lower than their carrying capacity when at equilibrium with the resident consumer will cause consumer 1 to decrease as the result of its adaptive divergence from the specialist competitor. If resource 2 is much closer to carrying capacity before the competitor's introduction (an inefficient consumer 1), then evolution will counteract (although not completely reverse) the ecological decline in N1 caused by N2. Recall that a high efficiency (low equilibrium R) is a requirement for a locally stable generalist equilibrium under a strong trade-off, and this means that the evolutionary response of consumer 1 to consumer 2 is most likely to enhance the ecological decline in N1.
SCENARIO 2: A GENERALIST COMPETITOR
A highly generalized competitor that consumes resource 2 and many other resources (but does not consume resource 1) may have a different effect on adaptive change in the first consumer's choice trait (x) than that of a resource-2 specialist. Here, I concentrate on the simplest case where the population of consumer 2 is independent of the amount of resource 2 that it consumes. This could be because resource 2 is a small fraction of its diet, or does not contribute significantly to consumer 2's nutrition, or because consumer 2's population is limited by a resource other than food. The competitor has a population density that is determined by the unspecified resources, and the equilibrium conditions for the original two resources and consumer 1 require that equation (1c) be replaced by
The density of consumer 2 has been scaled so that the attack rate of consumer 2 on resource 2 (C22) is unity; thus, C22 does not appear in equation (5). Because of its lack of a numerical response, the second competitor has the same impact on the growth of resource 2 as any cause of resource mortality/loss whose per capita rate is independent of R2. This means that the loss term could be considered as part of the per capita growth function, f2, and it also implies that strong trade-offs can be consistent with a locally stable generalist equilibrium. Determining the evolutionary response in the first consumer's trait x to a change in the density of the second (nonevolving) consumer requires that the system with equations (1a, b, d) and equation (5) be set to equilibrium, then differentiated with respect to N2, and finally solved to give the responses of the two other populations and of the trait x. This results in the following set of partial derivatives with respect to N2:
where . Q is equal to the determinant of the Jacobian matrix of the dynamic system divided by vN1R1R2. A necessary requirement for a stable equilibrium for a four-dimensional dynamic system is that the determinant must be positive, so Q > 0. Note that the first of the two terms that make up the expression Q is negative when the equilibrium represents a fitness minimum (the trade-off is strong). The second term must be positive. To satisfy the stability requirement in this strong trade-off case, its curvature cannot be too great. In addition, the population size of consumer 1 cannot be too small relative to resource densities; this would prevent the strong effects of that consumer on its resources that are required to stabilize a fitness-minimizing equilibrium. Thus, high enough death rates of consumer 1 prevent a stable fitness-minimizing generalist equilibrium under a strong trade-off in the absence of consumer 2; there will be evolution to one or the other specialist type. Given a positive Q, x must increase with an increase in N2; that is, divergence always occurs. This does not depend on the shape of the trade-off. In addition, the stability results presented in Appendix 1 show that stability of the intermediate equilibrium in a symmetrical system ensures that the first consumer must decrease with an increase in the density of the second; equation (6d) must be negative. However, the responses of the resources have different signs, depending on the sign of the second derivative of the trade-off.
If the trade-off is weak, equations (6) imply that R2 decreases, R1 increases, and N1 decreases in response to increased N2; these responses all have the same sign as they would in a system with no evolution. However, if the trade-off is strong, the responses of both resources are the opposite of what occurs under either a weak trade-off or a situation without evolution; R2 increases and R1 decreases. These responses are also opposite in direction to what occurs when the competitor is a resource-limited specialist. The increase in R2 represents a “hydra effect” (sensu Abrams and Matsuda 2005); the population of the resource ultimately increases in response to greater per capita mortality. The decrease in R1 is what would be expected based on apparent competition, given the increase in R2. However, traditional apparent competition theory would also predict that greater mortality of resource 2 decreases its abundance, and therefore increases R1. Figure 2A shows the responses of the trait and the two resource densities in a system with logistic resource growth and a weak trade-off (m= 0.75); resource 2 decreases and resource 1 increases. Figure 2B represents a comparable scenario that differs from Figure 2A primarily in having a larger trade-off exponent (m= 1.75); it shows opposite responses of the two resources to increases in the competitor's density (i.e., resource removal rate). However, in panel B, R2 increases rather than decreasing over most of the range of densities. The resource growth equation in Figure 2B is theta-logistic (Gilpin and Ayala 1973) with an exponent of θ= 2. A large value of θ (weaker density dependence) favors a large hydra effect. Figure 2B shows that consumer 1 becomes specialized on resource 1 once the removal rate of resource 2 exceeds approximately 0.8. This system has an alternative, locally stable equilibrium at which consumer 1 is specialized on resource 1; this exists for all removal rates shown. A third alternative (specialization on resource 2) occurs and is locally stable for the full range of removal rates allowing persistence of consumer 1 (again not shown). Smaller values of m (1 < m < 1.75) and/or smaller θ produce alternative specialist attractors over narrower ranges of removal rates, and smaller hydra effects.
It is important to consider what happens when the competitor causes x to increase to 1; that is, consumer 1 becomes completely specialized on resource 1. This may occur under strong or weak trade-offs given a large enough intake by consumer 2, and it means that equations (6) cannot be used to determine responses to N2. However, the resulting equilibrium densities are easily determined; resource 2's density is the solution of f(R2) =N2; resource 1 is d1/(b1Cs), and N1= (1/Cs)f(d1/(b1Cs)). R2 may be either larger or smaller than its original equilibrium with the generalist consumer 1, depending on the magnitude of N2. If N2 is only slightly above the threshold where specialization of consumer 1 occurs, R2 will still be larger than its generalist equilibrium. A sufficiently large N2 will decrease R2 below the generalist equilibrium density. Resource 1 can be larger than its original (precompetition) density if the trade-off is weak. The net effect of the introduced competitor on the equilibrium N1 was negative in all numerical results, although it is still an open question whether this is completely general.
The ecological and evolutionary responses may again be separated, and it is again true that the evolutionary response can be much larger than the ecological one. A major difference between specialist and generalist competitor scenarios is that specialization of consumer 1 does not evolve under strong trade-offs in response to a generalist competitor whose population density is sufficiently low. The ecological and evolutionary responses are shown in Figure 3 for cases with weak and strong trade-offs. The change in N1 caused by the evolutionary change in x is quite small for small to moderate densities of consumer 2, but it is larger than the ecological change for a wide range of higher densities of N2, which cause near-extinction resource 2 on an ecological time scale. This near-extinction is what causes the ecological equilibrium N1 to become independent of N2 at larger N2. The assumption here is that the evolutionary change in consumer 1 is rapid enough to prevent resource 2 from disappearing completely. Thus, the solid line, reflecting evolutionary change, still exhibits a competitive effect of consumer 2 on consumer 1 via reduction in resource 2. (Even when evolution is very slow, the same outcome can be produced by a very low rate of input of resource 2 from outside the system being modeled.) The additional decline in N1 caused by evolution is typically smaller in cases where consumer 1 is less efficient (e.g., has a higher death rate) than in Figure 3.
Alternative Models and Scenarios
A GENERALIST COMPETITOR WHOSE DENSITY IS AFFECTED BY SHARED RESOURCES
A more widely applicable model of the second scenario considers a competitor whose population density at equilibrium is affected by consumer 1's use of the shared resource(s), but which can maintain itself on other resources. Although there is insufficient space to give a full analysis here, aspects of both of the above two scenarios can be observed. A simple case assumes the second consumer uses both resource 2 and a third resource; it is limited by its summed consumption of resources 2 and 3, and has the same trade-off between exclusive and overlapped resources that is experienced by consumer 1. This leads to a six-dimensional dynamical system, or seven-dimensional if the competitor also evolves. If both trade-offs are weak and both species evolve, divergence always occurs. Determining whether evolution increases or decreases population size is more complicated. In general, when resources are biotic, consumers are efficient, and the trade-offs are weak, evolution decreases consumer population sizes below their ecological equilibria following sympatry. Evolutionary enhancement is more common under these circumstances when consumers are less efficient and/or when both consumers evolve. However, the enhancement is generally the result of evolution of the other consumer; each consumer's divergence increases the density of its competitor, offsetting the generally negative effects of that competitor's divergence on its own density. When both consumers evolve and one or both have strong trade-offs, a wide variety of outcomes occur. The most interesting case is when both species have strong trade-offs. A range of simulations using a power-law trade-off with an exponent >1 suggests that specialization and generalization can both occur. An equilibrium with intermediate trait values for both species was never observed. The evolutionary outcome is likely to be two specialists when the trade-off exponent is considerably larger than unity and/or the consumer's efficiencies are low (so that they have little impact on the resource populations). In these cases, if the original resident species is a generalist, a second consumer introduced at low densities will always evolve to become a specialist on resource 3 if it has a reasonably high attack rate on resource 3 and if resource 3 has a high enough carrying capacity. This outcome eliminates interspecific competition.
SYSTEMS WITH AN UNSTABLE EQUILIBRIUM
At least three mechanisms can generate instability in the food web model considered here. The first is rapid evolutionary (or other adaptive) change combined with a strong trade-off. Abrams (1999, 2006b) pointed out the potential for cycles to be driven by rapid evolution combined with a fitness-minimizing trade-off, and Schreiber et al. (2011) have analyzed the unstable dynamics in the scenario in greater detail. Cycles in this case are driven by an alternation in consumer preference. The introduction of a resource-limited specialist competitor for resource 2 has an effect similar to what is described above for stable systems, with the original generalist becoming a specialist on resource 1. One difference in this case is that the cycles driven by the rapidly adapting generalist make it more difficult for the resource-2 specialist to invade the system. (Its death rate or some other aspect of its efficiency must be lower for a successful invasion.) This is because the specialist is adversely affected by the periodic shortages of its sole resource. If it does invade, the subsequent divergence (specialization) of consumer 1 eliminates the cycling.
The other two mechanisms that can generate sustained fluctuations are the intrinsic consumer-resource dynamics (usually driven by a saturating functional response) or environmental forcing of dynamics. The presence of a type-2 functional response or a decelerating numerical response in a cycling system can transform the fitness versus trait relationship from one that represents a fitness minimum in a stable system to one that maximizes fitness (Abrams 2006b). Numerical results show that the introduction of a specialist competitor in such a system need not result in complete specialization of consumer 1, as is true for fitness-maximizing equilibria under stable conditions. Divergence of consumer 1 becomes more pronounced, the larger the reduction in the overlapped resource caused by the competitor. When there is cyclic or chaotic coexistence of a generalist consumer 1 with a specialist consumer 2, the responses to competition given by equations (2) need not apply. However, the outcomes observed in numerical investigations all exhibited divergence; consumer 1 increased its average attack rate on resource 1 as a consequence of the invasion of a specialist consumer 2. Resource 2 is much less likely to increase in abundance with higher mortality than in stable systems, and the abundance of resource 2 generally decreases more than that of resource 1 with increases in N2. These results are related to the fact that cycles often shift the temporal mean fitness surface so that the equilibrium trait maximizes rather than minimizing temporal mean per capita growth rate.
A complete account of the impacts of nonequilibrium dynamics on ecological and evolutionary variables in this system requires a much more extensive analysis. However, cycling does not alter the finding that evolution frequently reduces the population density of the evolving consumer, and this decrease can exceed that caused directly by the reduction in resources.
NUTRITIONALLY ESSENTIAL RESOURCES
Appendix 2 presents a general equilibrium analysis for a consumer of two different essential resources. The purely evolutionary response to a competitor for one resource in this scenario was treated in Abrams (1987a); this work predicted convergence in attack rates (an increase in the attack rate of resource 2 and a decrease in the attack rate of resource 1).
The qualitative shift (convergence) is not affected by the nature of the trade-off between the two attack rates. Appendix 2 shows that the population densities of consumer 1 and the two resources change as follows. Under competition from a specialist, resource 2 declines and resource 1 increases, but consumer 1 may increase or decrease depending on whether it initially overexploits resource 1. An increase in the density of a generalist competitor following increased abundance of its “competitor” occurs when the exclusive resource is overexploited. The single-consumer system often exhibits alternative equilibria, and changes in competition may produce jumps between alternative equilibria (Abrams and Shen 1989). The changes in densities that accompany such jumps are not described by the formulas presented in the Appendix 2, and more analysis of this case is required for a more complete description of potential responses to competition. Nevertheless, it is not difficult to find numerical examples where the introduction of a specialist consumer initially decreases the resident's population size, but the resident's evolutionary response reverses this, causing the population of the generalist consumer to increase above its original density.
The simple models analyzed here suggest two aspects of competitive systems that do not seem to be widely appreciated. The first is that the evolutionary divergence associated with competition for substitutable resources very often decreases the population density of the evolving species. In what is likely to be the most common situation in natural systems—a resident competitor with a weak trade-off and a shift that does not produce complete specialization—evolutionary divergence always causes the resident's population to decrease, regardless of the form of resource population growth or the precise form of the trade-off. At least in the simple models considered here, the population decrease caused by evolution is often much greater than that caused by the population dynamics before the adaptive shift in character values occurs. This suggests that the ultimate change in population size caused by the addition or loss of a competitor may be much greater than one would expect from short-term observations or experiments. It also suggests that delayed declines in native species and delayed increases in invaders may sometimes be due to adaptive change in the native in response to competition from the invader, rather than adaptation of the invader.
A second major message from this work is that the combined evolutionary and ecological responses to a competitor for one type of resource are likely to differ significantly between systems where the competitor is a resource-limited specialist and one where the competitor's population is not limited by the overlapped resource. The latter case includes systems where the competitor is sufficiently generalized that its consumption of the overlapped resource has a minimal impact on its own population size. When the competitor's dynamics are resource-independent, the overlapped resource increases with increase in the population of that competitor, provided that the resident (evolving) consumer has a strong trade-off. This increase in the overlapped resource is driven by the adaptive divergence of the original consumer, which shifts to use less of that overlapped resource.
In the context of a system with one resource, it is already known that evolution of resource consumption rates does not in general maximize consumer population size (Abrams 1983; Matessi and Gatto 1984; Mylius and Diekmann 1995). This is a consequence of the frequency dependence inherent in the evolution of traits that increase resource exploitation. When high-exploiters are rare, they have a large advantage that erodes as they increase in frequency and the resource declines in abundance. The result can be an increase or decrease in consumer population depending on resource dynamics. Nonmaximization of population size has also been shown for models of interspecific competition (Abrams 1986, 1990). However, nonmaximization does not imply that evolution always reduces population size from an ecological equilibrium with a recently introduced competitor. The single-resource theory mentioned above does not apply in a simple manner to systems with two or more resources and a trade-off between attack rates; this is because the diverging resident consumer is concomitantly increasing and decreasing its consumption of different resources. Thus, in some cases, the evolutionary response of the resident generalist increases its population size relative to its ecological equilibrium with the invader, even though decreases are more common. Unlike single-resource situations, a decrease in population size associated with specialization in the two-resource scenario considered here does not require self-reproducing resources; under weak trade-offs with a resource-limited specialist competitor, an evolutionary decrease in population size occurs regardless of the resource per capita growth function, f. Evolutionary change in traits that cause an absolute increase in the exploitation of all resources is more analogous to the previous single-resource theory. Evolution of such traits can even lead to self-extinction as a consequence of purely intraspecific competition (Matsuda and Abrams 1994).
It is also known that evolution can greatly alter the outcomes of ecological interactions, even changing their signs (Abrams 1987b; de Mazancourt et al. 2005). The analysis presented here suggests that this is likely to often be true of competition and apparent competition. It adds support to the view that evolutionary responses should be considered in assessing the potential consequences of species additions and deletions (Abrams 1996).
The results presented here reemphasize the importance of the trade-off experienced by generalist consumer phenotypes; particularly, whether it is weak or strong. The number of documented examples of strong trade-offs is still relatively small. In those few cases where strong trade-offs have been shown experimentally (e.g., Bolnick 2004), they have been measured over short time intervals. It is generally not known whether a temporally averaged fitness function measured over a longer interval would change what appears to be strong trade-offs to weak ones. Strong trade-offs most often produce larger magnitude divergence in response to competition from a resource-limited specialist than do weak trade-offs. In addition, when the trade-off is strong, the magnitude of the divergence is independent of the population size of the competitor at its initial ecological equilibrium, which is not the case for weak trade-offs. When the specialist competitor is only just able to invade the resident system, the ultimate reduction in population density in the resident may be almost entirely due to its own evolutionary response to the introduced specialist. Linear and weak-but-nearly-linear trade-offs can result in similar responses, but weak trade-offs with pronounced nonlinearity are characterized by evolutionary responses that are smaller in maximum magnitude.
In the generalist competitor scenario described here, the resident's exclusive resource increases in density in response to decreased mortality of its “apparent competitor” (resource 2). This positive effect of a beneficial perturbation to resource 2's growth on the density of resource 1 is inconsistent with the traditional idea of apparent competition. However, the two resources still change in opposite directions, which is an alternative defining characteristic of apparent competition. The apparent contradiction arises because of the hydra effect involving the overlapped resource (R2).
The adaptive change considered above is evolutionary. However, virtually the same model may be used to represent behavioral responses that determine a trade-off between attack rates on two resources (Abrams 2010; see also Abrams and Matsuda 1993). Aside from being more rapid, adaptive behavioral shifts differ from evolution mainly in that strong trade-offs are likely to produce between-individual polymorphism rather than a single generalist phenotype (Abrams and Matsuda 2004; Abrams 2010). Nevertheless, introducing a specialist competitor for one resource in such a system is still expected to produce complete specialization on the exclusive resource in the resident consumer. Although there is no separation of ecological and adaptive time scales in this case, it is still possible to decompose the population decrease in the resident consumer into an ecological component due to the competitor's abundance and an adaptive component due to change in the trait of the resident species. If the behavior is sufficiently well understood or if the behavior can be experimentally disabled, it is not difficult to determine empirically what part of the ultimate population response can be attributed to that behavior. If the system has a stable equilibrium point, the proportion of the resident's decline due to its own adaptive change is identical to that predicted by the above analysis of evolutionary change. The implication is again that a significant fraction (sometimes the vast majority) of the decline subsequent to introduction of the competitor is due to adaptive change of the resident consumer. The scenario described in this paragraph also describes what would occur if the resident had undergone branching before the invasion; the resident's descendent lineage that shared a resource with the invader would go extinct, and the equilibrium would again consist of two specialists.
Associate Editor: R. Burger
I thank the Natural Sciences and Engineering Research Council of Canada for financial support for this project. I also thank the Mathematical Biosciences Institute of Ohio State University for their April 2011 workshop on “Coevolution and the Ecological Structure of Plant-Insect Communities,” which spurred me to formulate these ideas. Caroline Tucker provided very useful comments on an earlier draft of the manuscript. I particularly thank Eva Kisdi for her very detailed comments on the penultimate draft.
The local stability of equilibria was determined applying the Routh-Hurwitz criteria to the characteristic polynomial of the Jacobian Matrix for the dynamical system in question. The matrix for equations (1) is
where primes denote derivatives with respect to the single argument of each of the functions; fi′ < 0, C1′ > 0, and C2′ < 0. One requirement for local stability of a particular equilibrium is that the trace of this matrix be negative; R1f1′+R2f2′+v(b1R1C1″+ b2R2C2″) < 0. Instability can occur with weak resource density dependence, a sufficiently strong trade-off, and/or a sufficiently large rate constant for evolutionary change (v). The subsystem without change in the trait does not cycle. The characteristic polynomial of A1 has the form , with the following coefficients:
Stability requires that all four coefficients be positive, and that both a1a2 > a3, and a1a2a3 > a32+a12a4. This leads to unwieldy formulas for the full set of stability conditions unless the system is simplified in some way. If the trade-off is linear or weak (Ci″≤ 0), or if the evolutionary rate constant (v) is zero, it is not difficult to show that the Routh-Hurwitz criteria are always satisfied. The criteria also simplify considerably if evolution is slow (v close to zero) and the system is symmetrical, in which case x= 1/2 is the intermediate equilibrium trait value and b1=b2; C1=C2; C1′=−C2′; R1=R2; f1=f2; and f1′=f2′ at that equilibrium. This intermediate equilibrium is a repeller when the condition a4 > 0 is violated. This condition may be simplified to the following expression in the symmetrical system:
Here, C1 and its derivatives are evaluated at x=½. For a given death rate of consumer 1, equation (A2) specifies a maximum trade-off curvature (e.g., maximum exponent of xm) consistent with stability; for a particular trade-off, it specifies a maximum death rate, d1, consistent with stability.
The system with a fixed-density competitor is equivalent to equations (1) with a change in the form of f2; thus the matrix (A1) still applies. The requirement that consumer 1 have a smaller population size at equilibrium when N2 increases from a density of zero (eq. 6d) follows from equation (A2) when the system is symmetrical.
The system with a resource-limited specialist competitor becomes five-dimensional and asymmetric, so analytical stability conditions become effectively uninterpretable. However, the evolutionary equation in this system can be analyzed under a separation of time scales for ecological variables and x. This confirms that, given a strong trade-off, an intermediate x is never an attractor when the competitor is present, and that the specialist equilibrium is stable.
RESULTS FOR ESSENTIAL RESOURCES
The model with essential resources is characterized by a numerical response of consumer 1, which is an increasing function of its intake rate of the more limiting resource. The generalist is colimited by both resources when its intake rate of the first is β times its intake of the second. If β= 1 in a symmetrical system, x=½ is an equilibrium, but two additional equilibria are possible if the resource has biotic growth (Abrams and Shen 1989); one where C1 is high and R1 is low, and the other where C1 is low and R1 is high. At each of these other potential equilibria, the relative values of C2 and R2 are opposite those for resource 1. Previous work has shown that, given a trade-off between attack rates, the expected response to a consumer that only eats one of the two resources is convergence toward that consumer (Abrams 1987a; Fox and Vasseur 2008). For the symmetrical generalist described in this paragraph, convergence means that, in shifting to its evolutionary equilibrium with the R2 specialist, species 1's trait, x decreases. The type of equilibrium analysis performed in the main article provides additional results, giving the change in population densities of the original consumer and the two resources.
Given a specialist, resource-limited competitor, the dynamic equations are similar to equation (1) except that the per capita birth rate is proportional (with constant b, which is assumed equal to unity in the following) to the minimum of C1R1 or βC2R2. The rate of change of the trait, x, is assumed to be proportional to the difference between these quantities: dx/dt=v(βC2R2−C1R1). Abrams and Shen (1989) analyze the stability of the equilibrium of this system given logistic resource growth and the limiting case of very rapid trait dynamics. The equilibrium values are specified by
The second consumer has its population density and per capita death rate scaled so that C22 and b22 are set to unity. The impact of the death rate of the specialist consumer can be determined by differentiating the above equilibrium conditions with respect to d2. The final equation, (A3e) implies that R2=d2 can be substituted in equations (A3a–d), and the partial derivatives of R1, x, N1, and N2 with respect to d2 can be determined to examine the impact of a quantitative change in the competition from a specialist that is already present. The impact of greater competition can be determined by examining the derivatives with respect to −d2:
Equation (A4a) requires that the original consumer (species 1) converge toward the specialist competitor; that is, decreased mortality of the specialist (d2) decreases x. The other determinate relationship is that the first resource increases in response to the decreased mortality of consumer 2 (eq. A4c). The original consumer will decrease with decreasing d2 when its exclusive resource (resource 1) is not overexploited (f1+R1f1′ < 0) and will increase if R1 is overexploited. The specialist competitor may increase or decrease in response to a decline in its own mortality. Thus, it is possible for raising the death rate of the competitor to both raise the competitor's density (a hydra effect) and decrease the density of the generalist consumer 1, while at the same time consumer 1's trait diverges from that of the specialist (x becomes larger). Numerical results have confirmed that these outcomes occur for some parameters. A complete analysis of this system is considerably more complicated than equations (A4) indicate because alternative equilibria are often present, and changes in the competitor's mortality rate (d2) can cause shifts from one equilibrium to another. It is also possible to have priority effects between the consumers, and for many parameters, the two consumer species will not coexist for any mortality rate of the specialist. These details require a separate treatment in a future article.
Results change relatively little if competitor for resource 2 has a population size that is independent of that consumption. In this case, the density N2 becomes a parameter, but the dynamic equations for x, R1, R2, and N1 are identical to the specialist competitor scenario. The responses of the rest of the system to a change in the specialist density are given by
and Q > 0 at a stable equilibrium. These results mean that the original consumer converges toward the specialist when the specialist density increases. In addition, the exclusive resource (R1) increases and the shared resource (R2) decreases; consumer 1 increases if the exclusive resource is overexploited and decreases if the exclusive resource is not overexploited. These are similar to the corresponding responses to the specialist competitor given by equations (A4). Once again, the possibility of jumps between alternative equilibria requires further analysis. The direction of change in consumer 1's density with an increase in consumer 2 can abruptly reverse when there are three equilibria, and the system shifts between equilibria as a result of the perturbation in N2.