Inducible defenses of prey and inducible offenses of predators are drastic phenotypic changes activated by the interaction between a prey and predator. Inducible defenses occur in many taxa and occur more frequently than inducible offenses. Recent empirical studies have reported reciprocal phenotypic changes in both predator and prey. Here, we model the coevolution of inducible plasticity in both prey and predator, and examine how the evolutionary dynamics of inducible plasticity affect the population dynamics of a predator–prey system. Under a broad range of parameter values, the proportion of predators with an offensive phenotype is smaller than the proportion of prey with a defensive phenotype, and the offense level is relatively lower than the defense level at evolutionary end points. Our model also predicts that inducible plasticity evolves in both species when predation success depends sensitively on the difference in the inducible trait value between the two species. Reciprocal phenotypic plasticity may be widespread in nature but may have been overlooked by field studies because offensive phenotypes are rare and inconspicuous.
Recently, several studies have documented the co-occurrence of inducible offenses and defenses (i.e., reciprocal phenotypic plasticity) in tightly coupled predator–prey systems, including in ciliates (Kopp and Tollrian 2003b), marine crabs and snails (Smith and Palmer 1994; Trussell 1996; Edgell and Rochette 2009), and salamander larvae and frog tadpoles (Kishida et al. 2009a). For example, in a system comprising predatory salamander larvae and prey tadpoles, the salamanders broaden their gape when tadpoles are present so that they can efficiently consume them, whereas in the presence of these predators, the tadpoles enlarge their body volume to make their consumption more difficult (Kishida et al. 2006). Kishida et al. (2007) studied the genetic and geographic variation of the prey tadpoles’ defense level induced by predatory salamander larvae and demonstrated that the inducible defense morph has a strong genetic basis. Moreover, they observed substantial genetic differentiation in tadpoles between a region with predatory salamanders and another without them (Kishida et al. 2007). These observations suggest that the levels of defense and offense expressed by inducible phenotypes can evolve. Because each instance of inducible plasticity itself generates temporal and spatial variation in the predator–prey interaction, reciprocal phenotypic plasticity may be more common in nature than has been thought. Whether reciprocity in antagonistic phenotypic plasticity is widespread in nature is an interesting question (Agrawal 2001; Jeschke 2006; Agrawal et al. 2007; Mougi and Kishida 2009; Kishida et al. 2010), but our understanding of the prevalence of reciprocal phenotypic plasticity in predator–prey systems and of the evolution of antagonistic phenotypic plasticity is very limited, partly because of the lack of a theoretical framework for explaining their coevolution.
Here, we study the coevolutionary dynamics of an inducible defense and an inducible offense in a predator–prey system. The population dynamics of a predator–prey system is characterized by a timescale intermediate between the fast timescale of phenotypic plasticity and the slow timescale of evolution. We thus present a new coevolution model that describes the system dynamics at three different timescales: phenotypic plasticity, population dynamics, and evolution. Phenotypic changes due to phenotypic plasticity occur more quickly than population size changes, whereas the magnitudes of the phenotypic changes (the defense level induced in prey and the offense level induced in the predator) evolve more slowly. Although a few previous studies have studied phenotypic plasticity from a coevolutionary viewpoint (van Baalen and Sabelis 1999; Gardner and Agrawal 2002; Lima 2002), there is no study that explicitly considers three time scales. Our model results show that inducible plasticity evolves in both prey and predator when predation success depends sensitively on the difference in their trait values, as in the example of gape-limited predation (Kishida et al. 2006). Moreover, after inducible plasticity has coevolved in the two species, the proportion of offensive predators in the predator population is predicted to be smaller than that of defensive prey in the prey population, and the offense level tends to be lower than the defense level. These coevolutionary consequences suggest that the occurrence of reciprocal phenotypic plasticity in nature may often be overlooked because the offensive phenotype is rare and inconspicuous.
DYNAMICS OF POPULATION SIZE
We consider the following population dynamics of a predator and a prey:
where X and Y are prey and predator population sizes, respectively. (i∈X, Y) is the fitness in each species averaged over normal and induced phenotypes, represented by
where wXn and wXd are the fitness of the normal prey and the defensive prey, respectively, and wYn and wYo are the fitness of the normal predator and the offensive predator, respectively. fi and 1 −fi(i∈X, Y) are the respective proportions of the induced and normal phenotypes in the population of each species. fi is changed by phenotypic plasticity, as explained later. The fitness of the normal prey, defensive prey, normal predator, and offensive predator are given as follows:
where rn and rd are the per capita growth rate of the normal and defensive prey, respectively; K is the carrying capacity of the prey; aij is the capture rate of the predator with phenotype i(i∈n, o) with respect to the prey with phenotype j (j∈n, d); gn and go represent conversion efficiency, a parameter that relates the predator's birth rate to prey consumption, of the normal and offensive predator; and d is the death rate of the predator.
The induced phenotypes can have a defense or offense trait level higher than that of the normal phenotypes. We denote the defense and offense trait levels by u and v, respectively, and these may be increased by phenotypic plasticity. A normal predator is less likely to capture a prey that expresses the inducible defense than a normal prey, that is, ann > and(u). In contrast, a predator that expresses the inducible offense is more likely to capture the normal prey, that is, aon(v) > ann. The capture rate of the offensive predator attacking the defensive prey is determined by the difference between u and v, that is, aod(u, v).
In our scenario, and(u) decreases with increasing u, and aon(v) increases with increasing v. aod(u, v) depends on the difference between u and v. We assume that the capture rate is a sigmoidal function, and(u) =a/(1 +eθu), aon=a/(1 +e−θv), and aod(u, v) =a/(1 +eθ(u-v)), where a is the basal prey capture rate of the predator and θ is the shape parameter of the function. If the prey's trait level u is much greater than the predator's trait level v, the prey can effectively escape predation, because aod is very small. In contrast, if the predator's trait v is much greater than the prey's trait u, then the capture rate aod is close to a. As θ increases, the function becomes closer to a step function of (u−v) (Fig. 1B). When predation success depends sensitively on the difference in the inducible trait values, as in gape-limited predation, the value of θ should be large. We assume that ann is equal to aod(0, 0).
The cost of developing the trait in each species is modeled by assuming that rd and go are decreasing functions of u and v, respectively (trade-off functions). The rates of decrease (−r′d and −g′o) indicate the strength of the cost constraint on the prey and the predator, respectively (the prime indicates the first derivative of function with respect to u or v). In this article, we adopt the simple functions rd=rn(1 −uρX) and go=gn(1 −vρY). For ρ > 1, =1, or <1, the shape of the trade-off is convex, linear, or concave, respectively. A large value of ρ means a weak trade-off.
DYNAMICS DETERMINING THE PROPORTIONS OF INDUCED PHENOTYPES
We consider the dynamics of phenotypic changes between the normal and induced phenotypes via phenotypic plasticity in each species (i.e., fi (i∈X, Y)). We assume that the population mean proportions of the induced phenotypes, fX and fY, follow the adaptive dynamics given by
Following this model, the proportions of the induced phenotypes move toward the values that achieve the highest fitness. Equations (4) were chosen to be similar to the evolutionary dynamics of natural selection and mutation (Abrams et al. 1993; Hofbauer and Sigmund 1998). The first term on the right-hand side describes the adaptive phenotypic plasticity. Its speed is equal to the fitness gradient multiplied by the factor λifi(1 −fi), where λi (i∈X, Y) represents the speed of adaptation in each species. We assume that the proportions of the induced phenotypes change faster than population size. The fitness is a function of the proportion of the induced phenotype of the focal individual . Fitness may also depend on the population mean proportion fi (Iwasa et al. 1991). The factor fi(1 −fi) keeps the mean proportion to within 0 < fi < 1. The term −ɛfi+ɛ(1 −fi) (i∈X, Y), in which ɛ is a small positive constant, prevents the mean proportion from remaining at one of the boundary values (0 or 1). In this model, we assume that fi reaches the level maximizing the mean fitness for the sake of tractability, to reduce the number of free parameters. We will also numerically explore alternative model, density-dependent induction (see the section Results).
EVOLUTIONARY DYNAMICS OF DEFENSE AND OFFENSE TRAIT LEVELS
The population mean values of the defense and offense trait levels of the induced phenotypes, u and v, follow the evolutionary dynamics given by
which are based on quantitative genetic dynamics formulations (Iwasa et al. 1991; Day and Proulx 2004; Yamauchi and Yamamura 2005). In equations (5), (i∈X, Y) represents the speed of evolution, which is equal to the additive genetic variance divided by the generation time of each species. The evolutionary dynamics are slower than the population dynamics, that is, << 1 (i∈X, Y). The averaged prey and predator fitness, and , respectively, are as given by equation (2). Equations (5) indicate that the rate of adaptive change in defense and offense trait levels is proportional to the selection gradient. If the selection gradient is positive (negative), selection pushes the population toward a higher (lower) defense or offense trait level. At evolutionary equilibrium, the right side of equations (5) becomes zero.
We use these six differential equations–equations (1), (4), and (5)—to study the coupled coevolutionary, population, and plasticity dynamics of predator and prey. We performed the analysis of the dynamics based on the separation of time scales. However, we found nothing useful to improve the understanding of the model's behavior. Hence we excluded these from the article. In addition, we can obtain the equilibrium condition mathematically (Appendix S1). However, the full condition is too complex to understand. We directly calculate the six differential equations. In a special case in which the both species have no plasticity (i.e., fX=fY= 1), we can interpret the equilibrium condition easily (see the next paragraph).
The coevolutionary system in which the phenotypic plasticity is neglected (i.e., fX=fY= 1) has been analyzed previously (Mougi and Iwasa 2010). In that system, the values of constitutive defense and offense traits at equilibrium (u* and v*) are determined by the strength of cost constraint (−r′d and −g′o). When the cost of defense is equal to or smaller than that of offense, the equilibrium value of defense trait is equal to or higher than that of offense trait (u*≥v*), and vice versa. The condition, u* > v* and/or tends to cause a stable equilibrium. In addition, the coevolutionary dynamics depend on the value of the shape parameter θ of the capture rate function (Fig. 1C). When θ is small, u and v evolve to small values because there is no much reward in increasing the level of offense/defense. In contrast, when θ is large, u and v evolve to large values because there is much reward in increasing the level of offense/defense, meaning coevolutionary arms races (Fig. 1C). Note that u*=v* is satisfied in this simulation because the cost parameters are assumed to be same.
We focus on the question: does coevolution lead to interaction systems in which no, either, or both species have induced phenotypes? We run numerical simulations of differential equations—equations (1), (4), and (5)—during a sufficiently long time to obtain the coevolutionary outcomes. The initial conditions do not influence the result. The coevolutionary dynamics depend on the value of the shape parameter θ of the capture rate function (Fig. 1A). As θ increases from a very small value to a very large one, we can recognize four phases. For very small values of θ, the proportion of the induced phenotype fi is one in both species, implying as the evolutionary consequence that each species has a constitutive defense or offense (phase I in Fig. 1A). This phase is consistent with the case in which plasticity is neglected (Fig. 1C). For larger values of θ, the proportion of the defensive phenotype fX takes on a value between zero and one but the proportion of the offensive phenotype fY remains one, implying that only the prey species exhibits plasticity (inducible defense) (phase II in Fig. 1A). When θ exceeds some threshold value, the proportion of the induced offensive phenotype fY drops abruptly to a value intermediate between zero and one, and the proportion of the defensive phenotype remains between zero and one. Hence, both prey and predator show induced phenotypes (inducible defense and offense) (phase III in Fig. 1A). In this phase, fX > fY. A sharp transition of fY occurs in a very narrow region between phase II (in which fX < fY= 1) and phase III (in which fX > fY). In other parameter conditions, we also see a sharp transition (Figs. S2–S5). When θ is very large, fi oscillates (phase III′ in Fig. 1A). In no-parameter region of this model does the predator alone display phenotypic plasticity. The coevolutionary dynamics in each phase are shown in Figure S1.
The same pattern holds for wide value ranges of the ratios (speed of evolution) and of ρX/ρY (strength of the cost trade-off) (Figs. 2, S2, and S3). However when the ratios and ρX/ρY are small, the proportion of the prey population displaying the inducible defense is high over a wide parameter range, whereas the predator displays no plasticity even when θ is large (i.e., phases III and III′ do not occur; Fig. 2).
The result also depends on the shape of trade-off function. In the above example, we assumed that the trade-off curve was convex (ρ > 1). When the trade-off curve is concave (ρ < 1), we find that over a much wider parameter range the proportions of the induced phenotypes fi are between zero and one in both species, with that of the defensive phenotype higher than that of the offensive phenotype (fX > fY), over a wide range of values of θ (Figs. 3, S4, and S5).
The relative magnitudes of the trait values u and v follow a general rule: u tends to be larger than v over a wide range of values of θ, and the difference (u−v) is particularly large when only the prey exhibits phenotypic plasticity (Fig. 1A, phase II; see also Figs. S2–S5). This trend u > v which is apparent in phases I–III seems to be associated with a stable population dynamics. In the system in which the plasticity is neglected (i.e., fX=fY= 1), u=v at equilibrium when the cost constraint is same between species. Even in such a condition, however, u can be larger than v in the system with plasticity.
The phases I–III can be explained in the light of the equilibrium condition of fi (Appendix S1). The phase I is likely to occur when the levels of defense and offense are low. This implies that both species should have defense/offense constitutively when the cost of defense and offense is small (because of small trait values). The phase II is likely to occur when the offensive phenotype is beneficial (offense is effective but not costly) and the defense is effective to the normal predator. The phase III is likely to occur when the defense is effective compared with the offense. These interpretations seem to correspond to the simulation result.
The difference in time scale between plasticity and evolution does not much influence the main result. When the time scale between two processes are very similar, the size of phase III (III′) region is very narrow and that of phase II (II′) region is very broad (Fig. 4A). In contrast, when the time scale between two processes are very different, the size of the phase III region is very narrow but the phase III′ region is very broad (Fig. 4C). Note that the result in the cases of λi > 10 is same with that in λi= 10. In other words, our main result is likely to occur when the time scales are clearly different.
We also examined to what extent this result depended on the values of the parameters a, d, and K. We found that an increase in a tended to decrease the size of the phase III region but to increase that of the phase III′ region, whereas an increase in d tended to decrease the size of the phases II′ and III′ regions. An increase in K did not change the result.
In this system, the induction probability is determined so as to maximize the fitness. To examine a generality of the result, we also considered alternative model, density-dependent induction, in which the proportion of prey's defensive phenotype increases with an increase in the predator's total population size and the proportion of predator's offensive phenotype increases with an increase in the population size of prey's defensive phenotype. We used the specific functions, ) and ), where γX(γY) is the population size of predator (defensive prey) at which induction reaches half its maximum probability, and βX(βY) is a shape parameter of the induction and decay functions. This type of induction function has been used in earlier inducible defense/offense models (Vos et al. 2004; Mougi and Kishida 2009). The result shows that both species are likely to show phenotypic plasticity when θ is large (phase III and III′), and the proportion of defensive phenotype is higher than that of offensive phenotype (Figs. S6–S9). However, the result also shows some differences between them. In the density-dependent induction model, (1) there is no phase I, (2) the predator has not constitutive offense (not fY= 1 but fY= 0) in phase II, and (3) the offense level is higher than defense level in phase III (III′). We also confirmed that the main result is not influenced by the parameters, γi and βi. Full understanding of the behavior of the density-dependent induction model would require further investigation in the future.
We examined how the levels of induced defense and offense phenotypes coevolve by predator–prey interaction and how the coevolution influences the population dynamics. The result showed that inducible plasticity can evolve in both predator and prey species when predation success depends sensitively on the difference in the inducible trait values between the two species. In addition, they showed that once inducible plasticity has coevolved in both species, the proportion of defensive prey in the prey population at coevolutionary equilibrium tends to be larger than the proportion of offensive predators in the predator population, and, furthermore, the defense level of those prey is greater than the offense level of the offensive predators.
These results are explained by the following mechanisms. A low value of θ is not likely to drive coevolutionary arms races (the levels of defense and offense evolve toward low values) because there is no much reward in increasing the level of offense/defense (phase I). In this case, increasing the proportion of defense/offense phenotype is not very costly, hence the defense/offense are constitutively expressed. In contrast, higher values of θ are likely to drive arms races (the levels of defense and offense evolve toward high values) because an increase in the defense or offense levels can much increase the fitness (phase III). However, high levels of defense and offense are very costly, thus they need to change to low cost phenotypes. In this situation, the appearance of offensive phenotypes induces the defensive phenotypes with high defense level. This results in a decrease of the proportion of offensive phenotypes (sharp transition in phase III) because offensive phenotype is not very effective due to a low proportion of easy-to-eat normal prey and a low success of predation on a defensive prey. Consequently, the selection pressure on the predator's offense trait decreases. This intuitive explanation for the behavior of the model holds only when the level of induction is the level maximizing the mean fitness. In the other case where induction is density dependent, we cannot really apply the same reasoning. Understanding what is going on under other kinds of dynamics of the induction level would require further investigation.
Inducible defenses are widespread, occurring in many taxa (reviewed by Tollrian and Harvell 1999). In contrast, inducible offenses have been reported much less frequently (Smith and Palmer 1994; Kopp and Tollrian 2003a; Kishida et al. 2006). The lack of evidence for inducible offenses can be explained by the results of our analyses. First, in our model, we never observe an evolutionary equilibrium in which the predator species has an inducible offense and the prey species has no inducible defense. This result can be explained intuitively by noting that a predator without an inducible offense can prey on a normal prey even when the prey species has an inducible defense, whereas it is beneficial to the prey to have an inducible defense even when the predator has no inducible offense. Therefore, inducible defenses can be expected to occur more frequently than inducible offenses in natural systems.
Second, our analysis suggested that inducible offenses might have been overlooked even if they exist. There are two possible reasons: (1) when both species have induced phenotypes, the fraction of defensive preys is greater than the fraction of offensive predators. In addition, (2) a stable equilibrium of predator–prey dynamics is likely to be realized when the offense of the predator is lower than the defense of the prey (u* > v*), which implies inconspicuousness of the inducible offenses as functional traits.
The results of this analysis predict that the predator and prey are likely to exhibit highly effective offensive and defensive phenotypes when the shape parameter θ of the capture rate function curve is large. This finding implies that marked antagonistic inducible offenses and defenses should be observed when predation success depends sensitively on the difference in the inducible trait values, as in gape-limited predation (Kishida et al. 2006). The reason for this result can also be explained intuitively: inducible plasticity is favored if the predator can attack its prey more effectively (or the prey can protect itself from the predator) by a small change to its morphology. This prediction is apparently supported by some clear examples of reciprocal phenotypic plasticity reported recently. For example, in the predator–prey interaction between larval salamanders and frog tadpoles, the prey size that the predator can capture is strongly controlled by the predator's gape size. In this system, in response to the presence of the other species, predatory salamanders broaden their gape 1.4-fold so that they can more efficiently consume their tadpole prey, whereas tadpoles enlarge their body volume twofold to protect themselves from predation by the salamanders (Kishida et al. 2006). Moreover, predation on any tadpoles by salamander larvae without the offensive phenotype is rarely successful (O. Kishida, unpubl. ms.). In a ciliate predator–prey system, another well-known example of reciprocal phenotypic plasticity (Kopp and Tollrian 2003b), predators without the offensive phenotype also have very low predation success (Kopp and Tollrian 2003a).
Another prediction of the analysis is that the coevolution and persistence of inducible defense and offense are likely when the prey has an effective inducible defense. In the ciliate system, the inducible defense of the prey ciliates (i.e., enlargement of body size) effectively protects them from being swallowed by the predator ciliates even when those predators intensify their inducible offense (i.e., enlargement of their capturing organ) (Kopp and Tollrian 2003b). Also in the amphibian system, experimental evidence has indicated that the inducible defense of the prey tadpoles is highly effective against the offensive predators in a natural pond habitat (Kishida et al. 2009b). Our model analysis predicts in these predator–prey coevolution is likely to lead to stable population dynamics (Mougi and Kishida 2009; Mougi and Iwasa 2010).
These examples show the evolutionary and ecological consequences of focused antagonistic phenotypic plasticity in morphological traits. Although morphological traits present particularly clear examples of phenotypic plasticity, this theoretical framework is applicable to behavioral or physiological traits as well. For example, predators may produce offensive chemical substances better able to break down barricades to such substances induced in the prey in the presence of the predators. Similarly, prey may increase protective behaviors, for example by reducing their activity in the presence of predators displaying inducible offenses (Lima 2002). Further research on plasticity in various types of traits will likely reveal arms-race-like reciprocal phenotypic plasticity in additional predator–prey systems. The predictions of our model can also be tested by examining ecological aspects of predator–prey interactions and reciprocal phenotypic plasticity in greater detail, for example, by comparing the performance of inducible defenses between related prey species interacting with predators with high- or low-performance inducible offenses.
The model presented in this article highlights the importance of the interaction between phenotypic plasticity and evolution as a driver of predator–prey population dynamics and lays a foundation for further analyses of the coevolutionary dynamics of phenotypic plasticity in a predator–prey system.
Associate Editor: S. Gandon
We are very grateful to Dr. P. A. Abrams for his valuable comments on a previous version of our manuscript. This study was supported by Grants-in-Aid for a Research Fellow of the Japan Society for the Promotion of Science Research Fellowship for Young Scientists (no. 20*01655) to AM and (no. 22770011 and no. 2187001) to OK, and by a Grant-in-Aid (B) to YI.