Introduction

It has long been known that natural selection can reduce the equilibrium population size of a species if it involves traits that increase an individual’s performance at the expense of other individuals in the population (Haldane 1932). For example, sexual selection/competition and the adaptive evolution of ability in contest competition are known to be capable of decreasing population size, even to the point of extinction (Abrams and Matsuda 1994; Matsuda and Abrams 1994; Rankin et al. 2011; Ferriere and Legendre 2013); Le Galliard et al. (2005) and Denison et al. (2003) provide empirical examples of such decreases. Evolution of the within-host growth rate of parasites is also known to reduce population size in some systems (e.g., Decaester et al. 2007). However, the ecological parameters involved in simple models of population growth of free-living organisms do not fall into these categories. In standard continuous-time models in community ecology, these universal parameters are resource conversion efficiencies, resource attack rates, and mortality (loss) rate (see eq. [1a]). Increased conversion efficiency and decreased mortality are usually assumed to increase population size. This underlies the literature on evolutionary rescue (Gomulkiewicz and Holt 1995), whereby adaptive evolution prevents extinction. Adaptive evolutionary increase in a consumer’s attack rate may decrease its population size if its resource is self-reproducing and is overexploited, that is, below the abundance that maximizes the resource population growth rate. However, widespread occurrence of overexploitation is often discounted (e.g., Slobodkin 1974; Dawkins and Krebs 1979). Both field and laboratory studies examining overexploitation are nearly nonexistent (Abrams 2003). At the other end of the spectrum, some theoretical works that argue for evolutionary extinction (“suicide”) have made the unrealistic assumption that attack rates increase indefinitely without individual-level costs (e.g., Webb 2003; Parvinen and Diekmann 2013).

Perhaps because of the lack of empirical studies, many recent discussions of the interaction between ecological and evolutionary processes (eco-evolutionary dynamics; e.g., Schoener 2011) have not considered when or how frequently adaptive processes decrease, or maladaptive processes increase, the population size of the evolving species. These two possibilities should occur under the same circumstances, and both will be referred to here as “adaptive decline.” Discussions of maladaptive change seem particularly likely to assume that population size necessarily decreases as a result (i.e., that adaptive decline—maladaptive increase—does not occur). This assumption is built into the mechanism of mutational meltdown (Lynch et al. 1995) and underlies most of the literature on inbreeding depression. A recent book on eco-evolutionary dynamics, while acknowledging the possibility of adaptive decline, claims that “many populations and species have recently undergone range contractions and/or numerical declines … and both of these effects must ultimately stem from maladaptation” (Hendry 2017, p. 174; italics mine).

The present article uses simple consumer-resource models to examine the population-level effects of evolutionary change on the consumer. All biological species are consumers of resources, and consumer-resource models are the simplest ecological description reflecting universal processes influencing their population growth. Early works examining the impact of evolution on population size used simple single-variable models of density-dependent population growth (Charlesworth 1971; Roughgarden 1971). The results of these analyses were shown to be inconsistent with growth based on consumption of dynamic resources by Matessi and Gatto (1984) and are inconsistent with the wide range of potential responses of consumer population size to increased environmentally caused mortality revealed in several simple consumer-resource models (Abrams 2009a, 2009b, 2009c, 2009d). Nevertheless, the logistic or similar single-species models of density-dependent growth are still popular in eco-evo studies, constituting the main part of Hendry’s (2017, chap. 7) extensive review of eco-evo effects on population dynamics.

Many of the consumer-resource models considered here have been analyzed in a purely ecological context to show that unfavorable ecological change in one or more consumer parameters can increase consumer population size (including Abrams 2002, 2009d, 2014; Matsuda and Abrams 2004; Abrams and Matsuda 2005; Liz and Pilarczyk 2012; Sieber and Hilker 2012; Cortez and Abrams 2016). Most of these studies have focused on the increase in population size with increased mortality, termed a hydra effect (Abrams and Matsuda 2005). They clearly imply that decreases in population size due to decreased mortality are possible. Similar counterintuitive changes occur following any parameter shift that directly affects only the population growth of the focal species, at least in stable systems (Cortez and Abrams 2016), producing a generalized hydra effect. For example, increased (reduced) resource conversion efficiency decreases (increases) population size in consumer species that exhibit counterintuitive responses to altered mortality. The studies of the hydra effect referenced above employed models in which consumer fitness is equivalent to the per capita growth rate of an individual. In other words, the ecological parameter (e.g., mortality rate) that characterizes a consumer individual is independent of the mortality-determining trait values of other individuals of that species. This equivalence implies that adaptive evolution and favorable environmental change that produce similar parameter changes will produce the same direction of response in the equilibrium population density. Nevertheless, the ecological literature on hydra effects has not explicitly drawn conclusions about the analogous evolutionary process and has very seldom been cited in the evolutionary literature. Agrawal (2010), Agrawal and Whitlock (2012), Fronhofer and Altermatt (2015), and Osmond et al. (2017) appear to be the only four evolutionary articles that cited any of the above-referenced articles on the environmentally driven analogue.

Empirical evidence regarding the effect of evolution of basic ecological traits on population size is surprisingly sparse and inconclusive. Nicholson (1960) demonstrated that selection increased population size in laboratory populations of blowflies. However, Nicholson (1954) had earlier shown, in related work on the same system, that removal of individuals could increase population size, implying that evolution of lower mortality should decrease it. In one of the few subsequent studies, Bull et al. (2006) showed that phage populations sometimes decreased in response to natural selection. Nevertheless, treatments of adaptive evolution decreasing population size or maladaptive evolution increasing it are lacking in many current evolution textbooks (e.g., Charlesworth and Charlesworth 2010; Herron and Freeman 2014) and in a recent encyclopedia of evolution (Losos 2014). Admittedly, population increases due to maladaptation must have an upper limit; maladaptive changes of a sufficiently large magnitude are bound to make a population disappear. The extreme nature of many of the environmental changes that have been studied may be responsible for the fact that examples of populations decreasing due to maladaptation exist (e.g., Bell and Gonzalez 2009; others reviewed in Hendry 2017), while increases have apparently not been reported.

Knowing how evolution changes population size is clearly important for understanding and predicting trends in population size, so it contributes to both resource management and conservation biology. Moreover, the relationship between adaptive evolution and population size affects most long-term interspecific interactions in community ecology and should be a central concept in the study of eco-evolutionary dynamics (Fussmann et al. 2007; Kinnison and Hairston 2007; Schoener 2011; Ellner 2013; Hendry 2017). This article argues that adaptive decline is likely to occur too often to be ignored when assessing the cause of a change in population size. In evolutionary biology, understanding the effects of trait changes on population size is often required to understand the coevolution of interacting species (Abrams 1986; Cortez 2018). Given the realization that evolutionary change is often rapid, the sign structure of interspecific interactions can also be affected by adaptive decline (Abrams 1987).

The following section presents two variants of a simple, widely used model that appears to argue against adaptive decline. The remainder of the article explores weaknesses in this argument. I begin by discussing the meaning of the question, How does evolution change population size? The interpretation of this question has a major impact on predictions of the probability of adaptive decline. The next section investigates consumer-resource models that include realistic connections between the basic ecological parameters that were assumed to be independent in the first section. These models exhibit adaptive decline for a wider range of traits and/or conditions. This is followed by an examination of models that allow adaptive change in prey (resource) defensive traits; prey evolution can have a particularly strong effect in bringing about adaptive decline in its predator. All of these cases of adaptive decline are driven by a broader than usual definition of overexploitation. The penultimate section briefly reviews previous results showing adaptive decline in some models with more species or different types of feeding links. The final section (“Discussion”) summarizes the results and open questions and suggests how to resolve those questions.

An Argument against Adaptive Decline

The model used in this section involves the simplest possible description of consumer dynamics, characterized by linear functional and numerical responses. Resource dynamics are described by the two most commonly used models; logistic and chemostat dynamics. These systems are covered in many textbooks (e.g., Case 2000), and Matessi and Gatto (1984) presented an analysis similar to the following. Consumer population (N) growth as a function of resource abundance (R) is given by

Here and in later sections, I assume purely individual-level selection. The analysis also assumes that the additive genetic variance in the evolving trait is small enough that evolution can be approximated by fitness gradient dynamics (Iwasa 1991; Abrams et al. 1993); thus, the rate of change in a trait, x, affecting b, c, and/or d is proportional to $(\partial /\partial x)(\mathit{bcR}-d)$.

The two models for resource (R) dynamics in the presence of the consumer (N) are

The equilibrium resource density ($R^{*}$) is d/(bc) in both cases. Adaptive evolution always favors increases in b and c and decreases in d, as the derivative of per capita growth rate with respect to parameter change is positive in all cases. This means that evolution always reduces $R^{*}$. It is easily verified (Matessi and Gato 1984) that expressions (2a) and (2b) always increase with larger b or smaller d (unless $k=0$; see below). The abiotic model equilibrium ($N_1^{*}$) increases with larger c (unless $E=0$; see below). However, $N_2^{*}$ increases with c only if $c<2\mathit{kd}/\left(\mathit{br}\right)$; it decreases if $c>2\mathit{kd}/\left(\mathit{br}\right)$. This reversal point corresponds to the resource density that maximizes the resource population’s total growth rate in the absence of the consumer ($R^{*}=r/\left(2k\right)$). Values of c that produce a smaller $R^{*}$ represent overexploitation. Abiotic resources cannot be overexploited. These qualitative results extend to forms that are more general than equations (1b) and (1c); that is, resource loss rate may be any increasing function of R in the abiotic model, and density-dependent reduction in growth may be any increasing function of R in the biotic model (appendix, “Models with Direct Consumer Density Dependence and Biotic Resource Growth”). If most resources are abiotic or if most biotic resources are not overexploited, a genetic change in the consumer that alters any parameter(s) in a fitness-increasing direction will usually increase consumer population size. It should be noted that prey organisms can have effectively “abiotic” dynamics when only those individuals that do not contribute to reproduction (e.g., senescent individuals) are vulnerable to the predator (MacArthur 1972).

Interpreting the Question, How Does Evolution Change Population Size?

The above question is not well posed. It is necessary to specify a starting point for measuring population size and trait values and to specify what causes change. The previous section has assumed a scenario in which the system is at an ecological equilibrium and is perturbed by an adaptive genetic change that has strictly positive effects on one or more ecological parameters. However, many traits are characterized by trade-offs, with positive effects on some parameter(s) and negative effects on others. Opposite fitness effects on different ecological parameters have been demonstrated for many ecologically important traits, such as body size and temperature tolerance (Yodzis and Innis 1992; Gilbert et al. 2014; Amarasekare 2015). These trade-offs help maintain genetic variation via frequency-dependent fitness (e.g., Charlesworth and Hughes 2000). Genetic or environmental changes that displace such traits from the initial equilibrium—or a change in the equilibrium itself—require a different approach.

Environmental change can occur within an occupied area or when individuals move into an unoccupied habitat characterized by different environmental characteristics. Both cases are followed by attainment of a new ecological equilibrium and subsequent evolution, which will generally change population size from its initial ecological equilibrium (in real systems the timescales of ecology and evolution likely overlap, so measurement of the character throughout the process may be needed to separate these effects). Evolution may also restore a previous trait equilibrium following temporary evolutionary forces that have displaced the trait (temporary gene flow or drift due to a temporary population bottleneck). This section will treat cases where the original trait produces opposing fitness effects on different parameters.

Here a different framework is needed. Given equation (1a) and the assumption of character change at a rate proportional to the gradient of individual fitness with respect to the individual’s character (Iwasa et al. 1991), an evolutionary equilibrium value of an evolving trait x occurs where the derivative of individual per capita growth rate, $b\left(x\right)c\left(x\right)R-d\left(x\right)$ with respect to x evaluated at equilibrium ($R^{*}=d/\left(\mathit{bc}\right)$), is zero; that is,

For both equations (1a) and (1b) and equations (1a) and (1c), $N^{*}$ decreases with x at the value specified by equation (4), which is denoted $x^{*}$; see equations (A1a) and (A1b) in the appendix. This result implies that for small enough changes reducing x below $x^{*}$ (i.e., smaller c) increases $N^{*}$, while a larger x decreases $N^{*}$; the equilibrium trait and c are larger than the value that maximizes population size. The slope of the relationship between $N^{*}$ and x at evolutionary equilibrium approaches zero only as $N^{*}$ itself approaches zero (appendix, “Various Derivations Relating to Equations (1)–(3)”). In general, there is no reason to assume that a perturbation to the trait or consumer fitness function is more likely to increase or decrease the equilibrium x relative to that of the original system. In many cases knowing the details of both the system and the perturbation allows the subsequent direction of change in x to be deduced. However, averaged across many systems with many types of perturbations, it is likely that the approach to the new equilibrium involves a decrease in $N^{*}$ in approximately half of all cases.

Figure 1 provides numerical examples based on the abiotic resource system consisting of equations (1a) and (1b). Here the equilibrium is always stable. The trait x influences both the attack rate and the mortality rate ($c=c_{0}x/\left(1+\alpha x\right)$) and $d=d_0+d_{1}x$). The figure shows population size as a function of x and both the evolutionary equilibrium and population-maximizing values of x. The two panels represent environments characterized by different baseline mortality rates, d₀, which produce different evolutionary equilibrium x values ($x^{*}$). In figure 1A ($d_0=0.1$), the population-maximizing x is 0.0925, while $x^{*}=0.447$. Most displacements of x below $x^{*}$ produce a subsequent net increase in $N^{*}$, and all involve $N^{*}$ decreasing over the later phase of the evolutionary trajectory. Figure 1B involves a higher d₀ (=0.4), which leads to a smaller difference between the population sizes at the maximizing and equilibrium values of x. Displacements of x to values less than the population-maximizing value involve an initial evolutionary increase in $N^{*}$, followed by a decrease. A net population increase may occur for the largest negative displacements. For example, in figure 1A with $x^{*}=0.447$, a net increase requires an initial shift to $x<0.0338$. Barring evolutionary rescue, extinction occurs at $x=0.0228$. Thus, a displaced $x<0.0338$ involves a significantly increased risk of extinction. Even ignoring this possibility, the figure 1A example implies that 97.4% of the range of possible downward displacements of x result in a net adaptive decline; the corresponding range for figure 1B is 88.9%. The corresponding fractions of possible downward displacements producing a strictly monotonic decrease in $N^{*}$ are 83.6% for figure 1A and 65.9% for figure 1B.

As shown in the appendix (“Figure 1 Derivations”), the two panels of figure 1 reflect a general result for this model; that is, greater resource reduction by the consumer leads to a larger ratio of the maximum consumer population size to the population size at $x^{*}$. This is because reduction in $R^{*}$ produces indirect frequency dependence in fitness that is greater for large reductions. Figure 1 also illustrates how population size changes as it evolves from its $x^{*}$ in a previous environment to a different $x^{*}$ in a new environment following colonization. For example, if individuals from the $d_0=0.1$ (fig. 1A) environment with $x^{*}=0.447$ were introduced into an uninhabited $d_0=0.4$ environment (fig. 1B), the population would initially attain an ecological equilibrium of $N=4.782$ but would subsequently decrease to the eco-evolutionary equilibrium of $N^{*}=3.783$ as x increased to 0.894. The reverse introduction (from $d_0=0.4$ to $d_0=0.1$) would cause a population increase from 7.022 to 12.219 as x decreased from 0.894 to 0.447.

The type of scenario considered in this section likely characterizes most cases where adaptive evolution affects $N^{*}$ on a timescale that is easily measured for multicellular organisms. Unconditionally beneficial genotypes with large positive effects are rare, but environments change frequently. Temporary but extreme environmental change can produce large evolutionary changes in ecologically important characters, which subsequently evolve back to their original equilibrium. More permanent environmental change often causes evolution to a new equilibrium. Thus, the “response to altered environment” scenario seems likely to underlie most studies of character shifts due to natural selection in the field (e.g., Endler 1986; Kingsolver et al. 2001). This argues that even in the absence of overexploitation as it is traditionally defined, adaptive decline may still be common.

Nongenetic Parameter Linkages in More Realistic Models with Biotic Resources

More realistic representations of the consumer-resource interaction provide additional mechanisms by which adaptive changes in b or d decrease population size. Some of these mechanisms arise when equations (1a) and (1c) are altered by changing the linear functional response to one that exhibits saturation, such as Holling’s type 2 or type 3 responses. Jeschke et al. (2004) showed that such saturating responses occur in the large majority of consumers studied. In biotic resource models these responses make it possible for increases in d or decreases in b to increase consumer population size; these results were implicit in Rosenzweig and MacArthur (1963) and were discussed explicitly in Abrams (2002). Linkage occurs because changes in b or d alter $R^{*}$, which decreases the effective attack rate (i.e., the capture rate per unit time per unit resource density by an average consumer). This effective rate is $c/\left(1+\mathit{hcR}\right)$ in the case of a type 2 response with handling time h. Changes in other parameters (b, d) may also alter the effective attack rate under equation (1a) if adaptive phenotypic plasticity and/or behavior affect c. An important case is the behavioral trade-off between lowering mortality and increasing the attack rate on the resource, which is frequently observed (e.g., Lima and Dill 1990; Werner and Anholt 1993; Werner and Peacor 2003). This allows adaptive evolutionary changes in traits affecting only b or d to increase the effective attack rate and, thus, decrease $N^{*}$ over part of the range where the resource is overexploited. Similar parameter linkage may occur with dynamic shifts in energy allocation between different physiological functions (Nisbet et al. 2000).

This effect of trait coupling can be explored in its simplest setting by assuming that both c and d in equations (1a) and (1c) are affected by a phenotypically plastic (e.g., behavioral) allocation trait. I define the trait, y, such that attack rate c is the trait value multiplied by a scaling factor c₀. Mortality, d, is given by a baseline rate d₀ plus a plastic trait-dependent component given by a scaling factor d₁ multiplied by a positive increasing function, f(y). The function is assumed to have a positive second derivative ($f^{″}>0$) so that an equilibrium characterized by stabilizing selection exists. Using the fitness gradient approach to describe the behavioral dynamics of y (Abrams et al. 1993) means that equations (1a) and (1b) are supplemented by

Here the parameters b, c₀, d₀, and d₁ may evolve. The effects of adaptive evolutionary change in each consumer parameter are given in the appendix (eqq. [A3a]–[A3d]), where it is shown that $N^{*}$ increases with adaptive change in either b or d₀. The effect of c₀ on $N^{*}$ depends on overexploitation; equilibrium resource densities less than r/2k (overexploitation) bring about a negative effect of larger c₀ on $N^{*}$. The impact of adaptive change (a decrease) in d₁ is more complicated, but it must be negative when k is sufficiently small (i.e., density dependence is weak). Thus, the evolution of d₁ produces adaptive decline when the consumer reduces resource abundance sufficiently. Many traits affect both d₀ and d₁; this is likely for traits determining activity times and for traits influencing body size. A large enough relative effect on d₁ produces adaptive decline. Abrams (1992, 2014) analyzed a related model with behavioral plasticity that couples c and d in a system with a nonlinear consumer numerical response. This model can exhibit adaptive decline as a result of evolution of a larger number of consumer parameters than in the simpler model considered above (Abrams 2014).

The models treated in this section show that although overexploitation is still needed for adaptive decline under the beneficial genetic change scenario, it can occur for traits that affect mortality or conversion efficiency, provided plasticity produces an indirect effect on the attack rate.

Adaptive Decline in a Predator Driven by Adaptation of Its Prey

Previous work has often viewed prey (resource) adaptation as a factor preventing predators (consumers) from overexploiting prey in spite of evolution of the predator’s attack rate (Slobodkin 1974; Dawkins and Krebs 1979). However, this ignores the negative effects of antipredator traits on prey population productivity. Two models are used to show that prey evolution often generates adaptive decline in the predator. The first is a two-prey analogue of equations (1a) and (1c); it consists of a predator, two prey, and a lower-level resource shared by both prey. This “diamond food web” has been analyzed many times in a purely ecological context (e.g., Leibold 1996; Noonburg and Abrams 2005). The model is given in the appendix (eqq. [A4]). The second model in this section assumes that the predator has a type 2 functional response and that the prey trait increases defense at the expense of reduced maximum growth rate (Abrams and Matsuda 1997; Yoshida et al. 2003; Jones and Ellner 2007).

The diamond food web model is equivalent to evolution in a single asexual prey species with two types that differ in predator susceptibility and another fitness component. Equilibrium predator density ($P^{*}$) is independent of its death rate or its conversion efficiencies, provided these do not cause extinction of one prey species. $P^{*}$ always decreases with proportional increases in the predator’s attack rates of both prey or with increases in its attack rate on the more vulnerable prey (eq. A4e). The direction of these responses of $P^{*}$ to parameters are identical to the direction of change of $N^{*}$ in equations (1a) and (1c) with density-independent ($k=0$) growth of its single prey. However, in the diamond food web, a sufficiently large change in any predator parameter eventually causes the extinction of one prey species; adaptive predator changes in mortality or conversion efficiency favor the better-defended prey. Once this is the only prey species present, further adaptive changes in the predator will then bring about adaptive decline only if the predator’s effective attack rate increases and the remaining prey is overexploited. The predator responses in this model are equivalent to those in a three-species web in which the two prey species compete with the product of their competition coefficients equal to 1 (Abrams and Matsuda 2005).

Abrams and Matsuda (1997) studied predator-prey models with a type 2 predator functional response and an evolving prey. However, neither this nor later works on this topic (Yoshida et al. 2003; Jones and Ellner 2007) discussed the impact of prey evolution on the equilibrium predator population’s dependence on it own parameters. Abrams and Matsuda (1997) employed a fitness gradient model to describe the dynamics of the prey’s growth-vulnerability trait, z. If there is a linear trade-off between the prey’s per capita growth rate at low density and its vulnerability to (i.e., attack rate by) the predator, as in Abrams and Matsuda (1997), adaptive decline is driven by four of the five predator parameters, provided that (1) the prey’s equilibrium trait value does not reach its maximum or minimum and (2) sustained cycles do not alter the direction of response in mean population size of the predator. Slightly changing the notation in Abrams and Matsuda (1997), denoting the trait value by z and generalizing the form of prey density dependence, the model becomes

Figure 2 illustrates the entire relationship between predator mortality rate and predator population size for two sets of values of the remaining parameters. At low mortality rates (d), the prey trait z is fixed at zero, so no adaptive response to the predator occurs. In figure 2A, there is a zone of (slight) adaptive decline for $d<0.1111$, where the equilibrium (with $z=0$) is unstable; this is attributable to the type 2 predator response. This does not occur in figure 2B, which has a smaller handling time. In both panels, $N^{*}$ increases with increasing d (or decreases with decreasing d) over the entire range over which z responds to d. The dashed line in both panels shows the potential effect of an upper limit to z; such an upper limit always exists when adaptation involves a shift in the abundances of two competing prey phenotypes. Adaptive decline above this upper limit is always due to overexploitation combined with parameter linkage due to the type 2 response rather than to prey adaptation. The frequency dependence in prey evolution (which is caused by the type 2 response) magnifies the productivity-reducing effect of its evolution and thus magnifies the adaptive decline in the predator.

Similar effects have been shown in a number of related models. Abrams and Matsuda (2005) and Abrams (2009a) explored the impact of environmentally caused increases in predator mortality for several versions of the diamond food web, which also represents a three-species food chain with two asexual clones in the middle species. Abrams and Matsuda (2005) allowed resource partitioning between the competing prey, and Abrams (2009a) illustrated the effects of population cycles on the response of mean population density to increased mortality in some unstable systems. The increased population size driven by greater mortality revealed in both studies implies adaptive evolutionary decline caused by predator evolution of lower d or higher b. Pinti and Visser (2019) studied a more complicated model of adaptive vertical migration of predator and prey and note the possibility of adaptive decline in predator populations due to evolution of its attack rate.

The various models and results reviewed above support the argument that adaptive change in the prey often makes adaptive decline in the predator more likely (contra Slobodkin [1974] and Dawkins and Krebs [1979]).

Adaptive Decline in Other Models

Many other plausible models for consumer-resource interactions exist, and such interactions are usually embedded in larger food webs. Further work is required to fully explore the eco-evolutionary change in these systems, but several cases may be understood on the basis of the comparable conditions required for a hydra effect, the ecological analogue of adaptive decline.

Direct negative effects of consumer density on consumer per capita growth rate are absent from the above models but are common in nature (Abrams and Ginzburg 2000; Heath et al. 2013). They can be added to equation (1a) by making either (or both) the consumer death rate (d) or its attack rate (c) a function of N. The consequences for adaptive decline are treated briefly in the appendix (“Models with Direct Consumer Density Dependence and Biotic Resource Growth”). Direct predator density dependence does not change the fact that increased c decreases $N^{*}$ when $R^{*}$ is below the point of maximum productivity. Consumer density-dependent death rates were included in several models of hydra effects by Cortez and Abrams (2016); in the cases examined, these terms did not eliminate the hydra effects that occurred in their absence, again implying the same for adaptive decline. Note also that direct consumer density effects do not qualitatively alter the “displaced trait” scenario considered above; the change in population size is still determined by whether the displaced value of x lies above or below its equilibrium.

Moving beyond simple consumer-resource models, some models of simple food webs with three or more trophic levels, with competing consumer species, or with stage-structured populations have been shown to exhibit hydra effects in relatively simple models (Abrams and Vos 2003; Abrams and Quince 2005; Abrams and Cortez 2015b; Cortez and Abrams 2016). For example, Abrams and Vos (2003) showed that hydra effects could occur in the middle and top species in a three-species food chain where the middle (prey) species has adaptive change in an allocation trait affecting both its predator vulnerability and resource capture rate. Evolution of species-specific parameters other than mortality in the model would also exhibit adaptive decline, based on Cortez and Abrams (2016). These results for three-level systems mean that many prey species are also expected to experience adaptive decline, as any prey species in a two-level model in reality would have dynamics that depend on those of its own resources, for which a three-level model is often more appropriate.

Webs having four or more species and other configurations have received limited attention. Abrams (2012) and Abrams and Cortez (2015a) consider two-consumer/two-resource models and show that adaptive evolution of the relative attack rates on different resources during character displacement will frequently reduce the population size of the evolving species. Yodzis (1988) studied indirect ecological effects in simple food web models based on 16 different empirically determined food webs. He found hydra effects (which he termed negative self-effects) for some parameter estimates in 27% of all of the webs. All of these are cases in which adaptive evolutionary changes in b or d would produce adaptive decline based on the ecological results of Cortez and Abrams (2016). The population-level consequences of ecological changes in attack rates have received less attention than changes in mortality, but a variety of cases of population decrease in response to favorable change in attack rates have been demonstrated in systems with two competing consumer species (Abrams 2003, 2004).

Of course, some food webs make adaptive decline less likely for some traits. For example, a specialist, strictly food-limited predator on the consumer will prevent any change in the equilibrium consumer population size following any evolutionary change that causes no extinctions and only affects parameters of the consumer-resource part of the food chain. However, adding a fourth-level predator, additional consumers, and/or connections of consumer attack rate to the consumer’s vulnerability to the predator (Abrams and Vos 2003) can all allow adaptive decline to occur in the consumer species, in spite of its specialist predator.

There are still many unknowns about factors influencing the occurrence of adaptive decline. In most species, there are likely to be multiple traits affecting different combinations of parameters and, in many cases, altering each other’s selection gradients. Such multiple trait models and the possible complications arising from the dynamic instability of their concurrent evolutionary change cannot be treated here, and very little is known about these issues. In size-, stage-, or age-structured populations different measures of abundance (numbers vs. biomass) may change in opposite directions due to evolution (de Roos and Persson 2013). Such situations lead to some ambiguity in the definitions of both hydra effects and adaptive decline. However, these circumstances also lead to similar ambiguity in the definition of all interspecific interactions. It is not known how the measure of population size affects the frequency of adaptive decline.

In spite of these unknowns, there is as yet no clear evidence that alternative models or those with more species will weaken the general case for adaptive decline.

Discussion

Associate Editor: Benjamin M. Bolker

Editor: Daniel I. Bolnick