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Modeling coevolution in predator-prey systems Research conducted in partial fulfillment of the Honors Program in Biology By: Cristina Herren Advisor: Mark McPeek Submitted for consideration for The Neukom Prize for Outstanding Undergraduate Research in Computational Science May 14, 2012 1 Abstract Differential equations models used to describe the trajectories of interacting species often used constants for the vital rates of the species and the attack rate of a predator. However, the attack rate is the result of a physiological interaction between the predator and the prey, which depends on the phenotypic traits of the two species. In predator-prey systems, the traits related to prey capture and predator evasion are under strong selection, meaning that the phenotypic traits that mediate the predator-prey interaction are constantly evolving. Recent work on coevolution in predator-prey systems has demonstrated that phenotypic evolution occurs on time scales that are relevant to population dynamics, and thus should be included in models. These models assume that because all organisms are constrained by scarce resources, an increased allocation of resources to traits pertaining to either prey capture or predator evasion results in a decreased growth rate. In the models presented here, phenotypic traits change over time due to the local selection gradient on the trait, which is a function of the abundances and phenotypes of the two species. In this paper I analyze a differential equations model that allows for changes in the phenotypes and abundances of both species under two different assumptions about how attack rate changes as a function of the relative phenotypes of the predator and the prey. In the model where the predator attains its maximal attack rate by having a large trait value relative to that of the prey, the predator’s trait value must remain high in order for the predator to persist in the system. In systems where the predator maximizes its attack rate by having a trait that matches the trait of the prey, the system is more prone to oscillations because the prey can escape the predator by evolving its trait to be larger or smaller than that of the predator. Additionally, the frequency-dependent nature of the selection gradient generates oscillations in instances where a non-evolutionary model would reach a stable equilibrium. This is due to the optimal strategy of each species changing as a function of the phenotype of the other species. 2 INTRODUCTION: Anti‐predator defenses take on a variety of forms. Adaptations in prey species demonstrate the diversity of evolutionary strategies that lead to reduced predation. These strategies include synthesis of toxic compounds (Brodie and Brodie 1991, Hanifin et al. 1999), counterattacks on the predator (Pratt 1974, Moitoza and Phillips 1979), evasion by speed (Watanabe 1983, Watkins 1995, McPeek et al. 1996), and camouflage (Merilaita and Lind 2005, Ioannou and Krause 2009). Though the mechanisms are different, the consequence of each of these adaptations is to reduce mortality of the prey due to consumption by the predator. From a modeling standpoint, effective prey defenses decrease the attack rate of the predator. The population dynamics of a predator and a prey species are linked by the attack rate of the predator on the prey. In the Lotka‐Volterra framework of differential equation modeling, the attack rate is the proportion of the prey population that the predator can consume in a given time step (Barryman 1992, Gotelli 2001). However, the attack rate of the predator can change as a function of prey availability. The functional response of the predator describes the predator’s rate of prey consumption as the prey density varies (Solomon 1949, Murdoch 1973). Holling (1959) identified three functional responses that express the predator’s total prey consumption as a function of prey density. In Holling Type II and Type III functional responses, the total consumption of the predator saturates as prey density increases because of the time taken to “handle” each prey. This is one example of attack rates changing as a function of prey availability, because each predator consumes an increasingly smaller fraction of the total prey population as prey density increases. 3 However, the ability of a predator to capture its prey is also influenced by the phenotypic traits of the predator and the prey that contribute to the predator’s attack rate (Abrams 1995, Nuismer et al. 2005 and 2007). Thus, there are two different kinds of interactions that determine the strength of the predator‐prey relationship. The number of prey consumed by the predators depends on 1) the number of predators and their functional responses and 2) the effectiveness of the prey’s evasion strategy. However, the effectiveness of the prey’s defenses depends on the current phenotypic traits of the predator and the prey (Saloniemi 1993). Additionally, phenotypic and behavioral traits often change as a result of repeated interactions between two species (Preisser et al. 2005). Trait changes can occur rapidly in response to both environmental cues (i.e. phenotypic plasticity) (Reznick 1990, McCollum 1997, Van Buskirk 1999 and 2000, Van Buskirk and Schmidt 2000, Agrawal 2001, Krivan and Schmitz 2004), or over many generations in response to selective pressures (i.e. phenotypic evolution) (Abrams 1986, Abrams 1990, Reznick et al. 1990, Arnold 1994, Abrams 2000). Predator‐prey models with fixed parameters make the assumption that the species’ vital rates and interactions remain constant through time and ignore the effects of trait‐ dependent interactions (Bolker et al. 2003). This statement implies that neither population is under selection for any trait that affects the predator‐prey interaction (Werner and Peacor 2004). However, many empirical systems have demonstrated that evolutionary adaptations occur quickly enough to influence population dynamics (Hairston Jr. et al. 1999 and 2005, Yoshida et al. 2003 and 2007, Mougi 2012). These rapid evolutionary changes are especially common in predator‐prey and host‐parasite systems (Shertzer et al. 2002, Yoshida et al. 2003 and 2007, Mougi and Iwasa 2010, Morran et al. 2011). The predator’s 4 fitness, defined as its per‐capita growth rate, increases with increasing attack rate, which generates a selective pressure to maximize attack rate. Similarly, the prey’s per‐capita growth rate decreases with increasing vulnerability to the predator, which generates a selective pressure to minimize attack rate. Thus, selection on traits in the predator and the prey that contribute to the predator’s attack rate should be strong (Abrams 1995). This has been demonstrated in a number of empirical predator‐prey systems (Benkman 1999, Brodie and Brodie 1999, Abrams 2000, Benkman et al. 2001). Because organisms have finite resources available to them, there are trade‐offs in their ability to respond to biotic and abiotic constraints (Huston 1979, Smith and Huston 1989, Tilman 1990). For instance, prey will often change their foraging behavior in the presence of predators in order to balance the costs of predation with the costs of food scarcity (Krivan and Schmitz 2004). Thus, the effects of trait‐mediated interactions reduce population size, as the prey is forced to allocate a larger amount of its scarce resources to evasive behavior or defenses (Ramos‐Jiliberto 2003). In some cases, the effects of trait‐ mediated interactions reduce the prey population more than the effects of direct consumption by the predator (Krivan and Schmitz 2004, Preisser 2005). These trade‐offs hold true in the case that the trait is evolutionarily derived, as well as in the cases of behavioral and phenotypic plasticity (Arnold 1992, Brodie and Brodie 1999, Yoshida et al. 2003). Predators also experience phenotypic changes over time in response to the prey’s phenotypic evolution (Schaffer and Rosenzweig 1978, West et al. 1991). Due to pleiotropic effects, selection on the genes controlling the predator’s phenotype often reduces the population of the predator through simultaneous negative affects on other aspects that 5 contribute to the predator’s fitness (Abrams 1995, Abrams 2000, Werner and Peacor 2004). For instance, selection for a phenotypic trait that increases a predator’s attack rate on one prey species may decrease its ability to capture other prey, or may incur additional energetic costs to maintain that trait. These secondary effects of trait evolution demonstrate why the predator’s growth rate decreases with the evolution of novel traits. Because the phenotypes of the predator and the prey evolve over time and influence the abundances of both species, coevolutionary models allow for changes in the phenotype and population of the predator and the prey. In these models, the phenotypes of the species respond to selective pressures by changing in the direction of the local selection gradient. Lande (1982) and Abrams et al. (1993) demonstrated that the local selection gradient is a product of the additive genetic variance of the trait and the partial derivatives of a species’ abundance with respect to the trait. The additive genetic variance of a trait describes the ability of a phenotype to respond to selective pressures, and the partial derivative of abundance with respect to the trait describes the quantitative effect of a phenotypic trait on a species’ population. The direction of the change in the phenotype will always increase the species’ fitness, and the magnitude of the change of fitness is proportional to the strength of selection on the trait and the additive genetic variance of the phenotype. Assuming that the phenotypes are controlled by many genes of small effect, the change in the phenotype can be approximated by a continuous process that can be modeled with differential equations (Lande 1982). Because the phenotypes of the species affect the abundances of both the species, a stable equilibrium for the system is only achieved when both the abundances and the traits have reached equilibrium. Thus, models with coevolutionary dynamics exhibit oscillations 6 in some instances when the equivalent model without phenotype evolution would reach a stable equilibrium (Abrams 2000, Mougi and Iwasa 2011, Mougi 2012). This is due to the inherent frequency‐dependent selection, as the selection gradient of a species’ phenotype changes based on the mean values of the phenotypes of both species (Abrams et al. 1993, Saloniemi 1993, de Mazancourt and Dieckmann 2004). The fitness of the predator and the prey are determined by the phenotypic trait of the other species, meaning that the optimal phenotype of each species changes as the trait value of the other species evolves. Most models of coevolution have utilized one of two general approaches for generating the predator’s attack rate as a function of the predator and prey phenotypes. The first is that the attack rate grows larger as the predator increases its phenotype over that of the prey (Salionemi 1993, Nuismer et al. 2005, Mougi and Iwasa 2010). Instances where predator‐prey systems follow the unidirectional attack rate model include the following interactions: venom versus antivenom, speed versus speed, and armor versus strength (Nuismer et al. 2007, Mougi and Iwasa 2010). This model provides a unidirectional axis by which the prey can avoid the predator, because the prey can only reduce the pressure of predation by increasing its trait value. Models using a unidirectional axis of predator evasion utilize a logistic curve to represent the predator’s attack rate as a function of the relative difference between the predator and the prey’s phenotypes. Large relative differences in phenotype, which correspond to a larger quantitative trait value in the predator, produce the maximum possible attack rate. The second approach is that the predator must match the prey in some capacity in order to attain the maximum attack rate, and that attack rate declines if the predator’s phenotype is larger or smaller than that of the prey (Dieckmann and Law 1996, Gavrilets 7 1997, Abrams 2000, Nuismer et al. 2007, Mougi 2012). Thus, this model provides a bidirectional axis by which the prey can avoid predation because any mismatch between the predator and the prey’s traits lowers the attack rate. This model for the predator’s attack rate applies to interactions where the predator selectively attacks prey within a specific size range (such that the prey’s body size matches the predator’s jaw size) and to host‐parasite systems where a parasite must mimic the morphology of its host (Benkman 1999, Nuismer et al. 2007). Models assuming a bidirectional axis of predator evasion utilize a Gaussian curve to describe the predator’s attack rate as a function of the difference between the predator and the prey’s phenotypes. When the difference is zero, meaning that the predator perfectly matches the prey, the maximum attack rate is achieved. Previous studies of coevolutionary predator‐prey systems have found that stabilizing selection on the trait is necessary for oscillations to occur (Nuismer et al. 2005), but that strong stabilizing selection contributes to the stability of equilibria (Mougi 2012). Strong negative density‐dependence in the prey also leads to stable equilibria (Mougi and Iwasa 2011). In the case of persisting oscillations, similar and intermediate rates of evolution of the predator and the prey generate oscillations with the largest amplitude (Mougi and Iwasa 2010). In this analysis, I compare the results of predator‐prey coveolutionary systems where species interactions are mediated by traits that generate a unidirectional axis of predator evasion versus systems where the axis is bidirectional. Although it is possible to analyze the existence and stability of equilibria in coevolutionary systems analytically, the solutions are mathematically intensive and complex, so I have instead chosen to simulate the model over a broad range of plausible parameter space. 8 MODEL: Population Dynamics In the following, I simultaneously model the population dynamics and phenotypic evolution of a predator (P) interacting with a single prey (N). The population dynamics of the species are modeled in continuous time as functions of their per capita birth and death rates. These per capita demographic rates are also influenced by the mean value of one phenotypic trait of each species; these are modeled as continuous traits. The basic model for each species is then dN = N ⎡⎣ BN ( N ,Z N ) − DN ( N , P,Z N ,Z P ) ⎤⎦ dt dP = P ⎡⎣ BP ( N , P,Z N ,Z P ) − DP ( P,Z P ) ⎤⎦ dt (1) where the birth rate of the prey (BN) is a function of its abundance (N) and mean phenotype ( ), and the death rate of the predator (DP) is a function of its abundance (P) and mean phenotype ( ). Interactions between the species link the birth rate of the predator (BP) with the death rate of the prey (DN). The prey’s per capita birth rate is modeled as ( ( BN = c 1− γ Z N* − Z N ) ) − dN 2 (2) Given this function, the prey’s population abundance is limited in the absence of the predator by assuming that its per capita birth rate is a linearly decreasing function of its own abundance (i.e., negative density‐dependence). The parameter d defines the strength of density‐dependence. The first term of equation (2) defines the maximum birth rate when N ≈ 0. defines the phenotypic value giving the highest maximum birth rate, and c 9 is the maximum birth rate at as deviates farther from decline faster away from . This maximum birth rate declines as a quadratic function . Greater values of cause the maximum birth rate to . As with the prey’s birth rate, I model density‐dependence in the predator’s per capita death rate using the function ( ( D p = f 1+ θ Z P* − Z P ) ) + gP . 2 (3) The first term of equation (3) defines the minimum death rate at P ≈ 0. defines the phenotypic value giving the lowest minimum death rate, and f is the minimum death rate at . Analogous to the prey, the predator’s minimum death rate increases as a quadratic function as deviates farther from . The strength of the trade‐off between phenotypic evolution and predator birth rate is represented by θ, where larger values of θ cause the minimum death rate to increase more rapidly away from . The parameter g defines the strength of density‐dependence in the predator’s death rate. The predator and prey interact through the functional response that defines the rate at which predators kill prey (i.e. ) and the numerical response of the predator which defines how captured prey are converted into new predators (i.e. ). The attack rate function occurs in both the predator’s and the prey’s growth equation and provides the link that couples the abundances of the species. The death rate of the prey due to predation describes the amount of the prey population that is utilized as resources for the predator. Thus, the predator’s growth rate should be proportional to the amount of prey consumed. However, biomass is not perfectly assimilated through trophic levels, and energy is 10 required in order for the predator to reproduce. The conversion constant b is used in the predator’s birth rate function ( ) to represent the average reproductive capacity gained by a predator as a result of consuming one prey. Throughout this analysis I assume a linear functional response in which the attack coefficient is a function of the predator and prey phenotypes (i.e. DN = a ( Z N ,Z P ) P ), and the per capita numerical response of the predator is a constant fraction of prey killed (i.e. ). There are two main ways in which a prey’s phenotype can contribute to its evasion of a predator. The first is that the predator is most successful at consuming the prey when its trait is larger than the prey’s trait, such that the prey can only escape predation by increasing the value of its phenotype. The direction of selection on these traits due to predation is unidirectional and always selects for an increasing trait value. An example of this kind of attribute would be running speed. As the prey increases its running speed relative to that of the predator, the attack rate of the predator decreases. When the predator has a larger running speed than the prey, the attack rate is high, but when the prey is able to outrun the predator, the attack rate is low. When the predator and prey are closely matched, the attack rate is intermediate and changes most rapidly with incremental changes in the relative phenotypes of the species. Thus, the attack function in the unidirectional model is given by a logistic curve with the following equation: ⎛ a ⎞ a ( Z N ,Z P ) = ⎜ ⎝ 1+ e− α Δ ⎟⎠ The difference between the predator’s and prey’s phenotypes is Δ ( (4) ). The predator approaches its maximum attack rate (a) when Δ is large. The steepness of the attack 11 function near Δ = 0 is controlled by the scaling factor α, where larger values of α increase the steepness of the function (Figure 1). Figure 1. The effective attack rate of the predator based on the difference between the predator and prey phenotypes, with a = 1 and α = 3. The second way the phenotypes of the two species could affect attack rate is if the predator must match the prey in some aspect in order to capture the prey. The direction of selection on these traits due to predation can be to either increase or decrease the phenotype of the prey, and depends upon the current value of the predator’s trait. An example of such a characteristic would be a predator’s jaw size matching the prey’s body size. In this case, the prey could avoid predation by being either smaller or larger than the body size preferred by the predator. Under this assumption, the attack rate is maximized at Δ = 0 and declines with increasing or decreasing values of Δ. The attack curve in the bidirectional model is given by a Gaussian function with the equation a ( Z N ,Z P ) = ae 12 ⎛ Δ⎞ −⎜ ⎟ ⎝ β⎠ 2 (5) where β is a scaling parameter for the steepness of the function (Figure 2). Larger values of β decrease the steepness of the function. Figure 2. Effective attack rate as a function of the difference between the predator and prey phenotypes, where a = 1 and β = 1 Phenotypic Evolution To allow the phenotypes to evolve, two additional equations are required to describe the change in the quantitative traits of the predator and prey. The magnitude and direction of the change in these quantitative traits are a result of the local selection gradient. Lande (1982) and Abrams et al. (1993) demonstrated that the local selection gradient for either species is the additive genetic variance multiplied by the gradient of the growth rate of the species with respect to its quantitative traits. The additive genetic variance is a constant describing the rate of change of the trait and is a product of the variance in the phenotypic trait across the population and the heritability of the trait. The gradient of the population growth rate is a vector of partial derivatives where each entry is a function for the rate of change of the population’s growth rate with respect to the 13 quantitative phenotypic trait (Lande 1982). This model considers one phenotypic trait, so the corresponding selection gradient is a vector of length 1. In a differential equations model, the gradient of the population growth rate with respect to the quantitative trait is easy to evaluate because the growth rates of the species are explicitly stated in the model equations. Taking the partial derivative of the population growth rate of the prey yields two terms, one of which describes the change in growth rate due to predation and the other of which describes the change in growth rate due to increasing mortality as a result of phenotype evolution. The selection gradient of the predator is similar, with terms describing the change in the predator’s growth rate due to attack rate and increased mortality as a result of phenotype evolution. The sign of the sum of the costs and benefits to phenotype evolution determines whether the incremental change in phenotype in the next generation will be toward or away from the phenotype where the maximum birth rate occurs genetic variance of the trait ( . Multiplying this sum by the additive or ) gives the realized change in phenotype from one generation to the next. Unidirectional Prey ⎡ ⎤ dZ N aPe− α Δ * ⎥ ⎢ = VZ − 2cγ (Z N − Z N ) 2 N ⎢ ⎥ −α Δ dt ⎢⎣ 1+ e ⎥⎦ ( ) (6) Predator ⎡ ⎤ dZ P baN α e− α Δ * ⎥ ⎢ = VZ − 2 f Θ(Z P − Z P ) 2 P ⎢ ⎥ −α Δ dt 1+ e ⎢⎣ ⎥⎦ ( ) 14 Bidirectional Prey Predator 2 ⎛ Δ⎞ ⎡ ⎤ −⎜ ⎟ β ⎝ ⎠ ⎢ ⎥ dZ N −2aPΔe * = VZ ⎢ − 2c γ (Z − Z ) N N ⎥ N dt β2 ⎢ ⎥ ⎢⎣ ⎥⎦ ⎡ ⎢ −2baN Δe dZ P = VZ ⎢ P dt β2 ⎢ ⎢⎣ ⎛ Δ⎞ −⎜ ⎟ ⎝β⎠ 2 ⎤ ⎥ − 2 f Θ(Z P − Z P* ) ⎥ ⎥ ⎥⎦ (7) Coupling equations (1) with the respective trait change equations (6) or (7) comprise the differential equations necessary for the two models to allow the population dynamics to change as a function of the phenotypes of the two species while the phenotypes themselves evolve. RESULTS: I evaluated each model under the following conditions: the predator’s quantitative trait value where its lowest death rate occurred was higher than the prey’s quantitative trait value for maximizing birth rate ( ), the predator’s minimum death rate occurred at a trait value lower than that of the prey’s maximum birth rate ( the two growth rate maximizing trait values were equal ( ), and ). For the cases where the trait values of the species were different, I used differences of 0.15, 0.5, and 2.0 quantitative trait units for the separation of and ( ). Within each of these categories of , I varied each parameter incrementally to assess the effect of the parameter on the long‐term behavior of the system. I chose to analyze the system at a combination of parameters that 15 generated damped oscillations so that any change in the model due to parameter changes would be immediately apparent. Effects of Parameters on System Behavior: Dynamic Behavior of Equilibria In both models, parameters that increased the growth rate of either species were destabilizing to the system when the values were increased. When other parameters in the model were fixed, the system rapidly approached equilibrium when the prey’s maximum birth rate, c, was small. As the value of c increased, the system began to overshoot the eventual equilibrium, and both the abundances and the trait values of both species entered a regime of damped oscillations. As c increased further, the fluctuations before the eventual equilibrium persisted for a longer amount of time. When the prey’s maximum birth rate was large, the system exhibited persisting oscillations (i.e. a stable limit cycle) in the abundances and the trait values of the species. The maximum attack rate of the predator, a, and the predator’s conversion rate b, determine the birth rate of the predator based on the abundance of the prey. When increasing the value of either of these two parameters, the birth rate of the predator also grows larger. Both a and b induce the same progression of system dynamics as the prey’s birth rate, c, when the values are increased. The abundances of the predator and the prey initially approach a stable equilibrium rapidly, then enter a region of parameter space where the system exhibits damped oscillations in the approach to equilibrium, and finally the system oscillates perpetually. Any parameter that decreases the growth rate of one species without affecting the growth rate of the other species is stabilizing to the system. When the negative density‐ 16 dependence of the prey is weak, meaning d is small, the system oscillates in both the traits and abundances of the two species. When the negative density‐dependence is moderate (i.e., as d is increased), the oscillations decrease over time, and the system eventually reaches a stable equilibrium. If d is large (i.e., negative density‐dependence is strong), the trait values and abundances of the two species rapidly approach a stable equilibrium. The case of the negative density‐dependence of the predator, represented by f, is the same. When negative density‐dependence in the predator is weak (i.e., f is small) the abundances and traits of the two species oscillate in a stable limit cycle. As f increases and the negative density‐dependence in the predator becomes stronger, the abundances and the trait values of the two predator and the prey show damped oscillations around a stable equilibrium. When f is large, the trait values and abundances of both species rapidly approach equilibrium. The predator’s density‐independent mortality rate, g, parallels the behavior of d and f such that increasing values of g stabilize the system. The parameters that affect the rate of evolution can be stabilizing or destabilizing to the system depending on the rate of evolution of the other species. The effects of the evolutionary parameters also differ slightly in the two models. In both models, increasing the additive genetic variances of the populations, and , shortens the period of oscillations when the system is unstable and reduces the time taken in reaching equilibrium when the system is stable. Increasing the additive genetic variances of the species in tandem effectively speeds up the dynamics of the system such that the same dynamics occurs on a shorter time span. However, if the additive genetic variance is large for one species and small for the other, the system will be stable. This is equivalent to giving one species a much faster rate of evolution than the other species, if all other 17 parameters are equal. In these scenarios, either the prey’s rate of phenotypic evolution is rapid enough to effectively evade the predator, or the predator is able to evolve its phenotype to effectively capture and regulate the prey. Making the additive genetic variances of the two species comparable in magnitude makes the system increasingly unstable, as neither species is able to evolve their phenotype quickly enough to escape the constraints imposed by the other species. The parameters that describe the costs of phenotypic evolution in terms of the prey’s birth rate, rate, , and the costs of phenotypic evolution in terms of the predator’s death , have similar dynamics. In the unidirectional model, if the cost to one population is much greater than to the other population, either the prey will be able to evolve effective defenses or the predator will be able to regulate the prey population (Figure 3). However, the bidirectional model does not have a stable equilibrium when the cost of evolution is much higher in the prey than in the predator. Even when the cost of evolution in the prey is high, the system oscillates so long as the predator is able to track the prey’s trait value and impose strong costs of predation. If the cost to the predator is much larger than the cost to the prey, the system is stable. In both models, when the rates of phenotypic evolution in the predator and the prey are roughly comparable, the system is unstable because the predator is able to effectively track the prey’s phenotype and the prey is able to change its phenotype once the predator has evolved. 18 Figure 3: Bifurcation diagrams showing regions of stable equilibrium and regions of persisting oscillations in the abundances of the predator and the prey. Data comes from the unidirectional model with parameters c = 3, d = 0.05, a = 0.2, b = 0.1, f = 0.1, g = 1, ! = 2.2, " = 0.15, Z N* = 1, Z P* = 0, vZ N = .1, vZ P = .1 The parameters that scale the steepness of the attack function also have a strong influence on the stability of the predator‐prey system. In the unidirectional model, 19 controls the slope of the attack function when the quantitative trait values of the predator and the prey are equal, meaning Z N = Z P . An increasing value of means that the attack function changes more quickly, such that the predator gains a greater advantage over the prey for the same incremental change in the predator’s quantitative trait value. When small the system is stable and rapidly approaches equilibrium, but as is increases the system becomes unstable and oscillates. In the bidirectional model, side of Z N = Z P . When controls the steepness of the attack function on either is high, the function is less steep and there is little change in the attack rate for each incremental movement in trait space. At low values of unstable, and as increases the traits and abundances first enter a region of damped oscillations and eventually rapidly approach equilibrium when the system is The difference of the predator’s minus the prey’s is large. (Z*P ! Z N* = " * ) also strongly affects the stability of the models. In both the unidirectional and bidirectional attack models, the system was stable when stable as was much less than zero and became less approached zero. This corresponds to scenarios where the prey has its optimal birth rate at a trait value that is much higher than that of the predator. In the unidirectional system, this would correspond to a prey that is stronger or faster than the predator it interacts with. In a bidirectional model, this corresponds to a prey that is much larger in body size than the predator’s preferred prey size. In the bidirectional model, values of much larger than zero also contribute to system stability. If a predator attacks its prey based on how close the animal is to its preferred body size, being smaller than that size is equally as beneficial for a prey as being larger. However, in the unidirectional model, 20 increasing the value of near = 0 is initially destabilizing and causes oscillations to grow in amplitude and decrease in period. This area of parameter space is very dynamic in terms of the increased attack rate gained by the predator for every small increase in its relative phenotype. In this region of , the predator is able to run slightly faster than the prey, which confers a large advantage for each increase in the predator’s phenotype over that of the prey. Past small increases in , larger values of contribute to stability in the unidirectional system as well as the predator is able to maintain a high attack rate and phenotypic evolution in the prey has little benefit. Location of Equilibria In both models, increasing the maximum attack rate of the predator, a, decreased the equilibrial abundance of the prey when the system reached a stable equilibrium. With increasing values of maximum attack rate, the predator’s equilibrium abundance initially increased until intermediate values of a, but then decreased due to a lower abundance of the prey. This same pattern in abundances versus attack rate is seen in the analogous predator‐prey population model that does not allow for evolution. Identifying patterns in the equilibrial trait values of the two species is more complex, since the phenotypes of minimal cost ( Z N* and Z P* ) are parameters that are subject to change. Thus, I describe the patterns in the locations of the trait values as converging toward or diverging away from Z N* or Z P* . Increasing the predator’s maximum attack rate resulted in the equilibrial trait values of the predator and the prey diverging from their phenotypes of minimum cost. Increasing the conversion efficiency of the predator, b, decreased the equilibrium abundance of the prey and increased the equilibrial abundance of the predator. Similar to increasing the maximum attack rate of the predator, larger values of b caused the species’ 21 trait values to diverge from Z N* and Z P* , respectively. Increasing the maximum birth rate of the prey, c, caused increases in the equilibrial abundances of both species, though the slope of the increase was larger in the predator than in the prey. Larger values of c caused the species’ trait values to converge toward Z N* and Z P* at equilibrium. Increasing the density‐ independent death rate of the predator, f, caused the equilibrial abundance of the predator to decrease and the equilibrial abundance of the prey to increase. However, the species’ traits both converged toward Z N* and Z P* . Increasing the density‐dependent death rate of the predator, g, showed the same pattern as increases in f, though the initial increase in prey abundance and decrease in predator abundance were more pronounced. Increasing the cost of evolution to the prey, γ , decreased the equilibrial abundance of the prey and increased the equilibrial abundance of the predator in both models. The trait values of both species converged toward their phenotypes of lowest cost. However, the unidirectional and bidirectional models differed in how the predator’s cost of evolution, Θ , affected the equilibria of the systems. In both models, as Θ increased, the prey’s equilibrial abundance increased while the predator’s equilibrial abundance decreased. In the bidirectional model, the predator’s trait value at equilibrium converged toward Z P* whereas the prey’s trait value diverged away from Z N* . In the unidirectional model, the prey’s trait value converged toward Z N* under most circumstances, but under some conditions (such as Z N* ≈ Z P* ), the prey’s trait slightly diverged from Z N* . The predator’s trait value at equilibrium always tracked that of the prey, such that the predator’s Z P* had little effect on the eventual trait value of the predator. 22 The shape parameters used in the attack rate functions in the two models also affected the location of the abundance and trait value equilibria. When the attack function in the unidirectional model grew steeper, meaning that α increased, the predator’s equilibrial abundance decreased. The prey’s abundance initially grew over increasing values of α , but then decreased at higher values of α . As the steepness of the attack function increased, the equilibrial trait values of the predator and the prey both diverged from their phenotypes of lowest cost. As the steepness of the attack function in the bidirectional model increased, meaning that β grew larger, the predator’s equilibrial abundance increased while the prey’s equilibrial abundance decreased. However, the trait values of the predator and the prey converged toward Z N* and Z P* as β increased. Unidirectional Model: In the unidirectional model at equilibrium, the predator’s trait was higher than the prey’s trait in all the parameter space explored. Under the condition that the prey’s decrease in birth rate as a result of phenotype evolution, , is roughly equal to the predator’s increase in death rate, , the cost to the prey for phenotypic evolution is much larger than the cost to the predator. The given by total cost functions of phenotypic evolution are for the prey and . These terms are analogous, except that the cost for the prey is in terms of its birth rate, c, whereas the cost to the predator is in terms of its density‐dependent mortality rate, f. However, the prey’s birth rate per unit time is a much larger value than the predator’s death rate per unit time (up to one to two orders of magnitude), and so the total cost of phenotypic evolution, in terms of growth rate, is larger for the prey than for the predator if is roughly equal to 23 . The prey’s birth rate is necessarily larger than the predator’s density‐dependent death rate in systems with both species present because the predator’s conversion of prey into new predators is not perfectly efficient. For the predator to have any region of positive per capita growth, the availability of prey must be much greater than the predator’s death rate. Disregarding the population effects of evolution, abN must be greater than fP for at least small values of P, which is only achieved under the condition that c ! f . In a system without evolution, the equilibrium predator abundance is abc − fd , which is only positive when abc > fd . In the dg + a 2b coevolutionary model, a is the maximum possible attack rate and f is the minimum possible death rate, meaning that the condition of abc > fd becomes more difficult to satisfy as the predator’s phenotype evolves away from Z P* . Thus, c must be much greater than f in this model in order to ensure that the predator has a positive equilibrium value. The phenotypes of the predator and the prey reach equilibrium when the costs to evolution equal the benefits of evolution. Thus, in the unidirectional model, the trait value at which the prey’s benefits of phenotypic evolution equal the costs of trait evolution occurs at a lower quantitative value than for the predator. Making the cost to the predator numerically equal to the cost to the prey involves either decreasing c or , or increasing f or . Changing any of these parameters enough to equalize the costs of evolution of the two species decreases the equilibrium abundance of the predator and eventually drives the predator out of the system. Bidirectional Model: In the bidirectional model, when the growth rate maximizing quantitative trait values of the predator and the prey are the same ( 24 ) the system always reached an equilibrium with . When this occurs, the values of , , and are all equal to zero. These terms are all involved in the equations for the change in the phenotypes of the predator and the prey, and consequently the change in the phenotypes of the species are also equal to zero ( = = 0). Without phenotypic change, the system follows the dynamics of a predator‐prey model without evolution. In the case that the predator’s phenotypic optimum and the prey’s phenotypic optimum did not occur at the same quantitative trait value (either there was a value of or ) that produced the largest divergence in the prey’s trait from its birth‐maximizing phenotype when the system reached equilibrium (Figure 4). Figure 4: Deviation of and from and increases. c = 2, d = .1, a = .25, b = .2, f = .1, g = 0, 25 as the difference between and * = 5, = .05, = .15, Z N = 0 This occurs because the total divergence from is equal to the integral of the trait change equation from the starting point until the system reaches equilibrium. Maximizing this integral means finding the region of quantitative trait space where the difference between the selective pressures and the costs of evolution is largest; when this quantity is large, the phenotype of the prey will change rapidly. Using a Gaussian function for attack rate and a quadratic function for costs in terms of birth rate, the following figure shows that this difference is largest between zero and the maximum or minimum of the derivative of the attack function. Although the fitness surface and the benefits to phenotypic evolution change depending on the predator’s current phenotypic value, results of the model showed that a starting value of where the benefit of evolution was much greater than the cost to evolution led to the greatest amount of prey evolution when the system reached equilibrium. Notably, this point occurred at intermediate levels of predation where the prey’s abundance was not strongly affected by the presence of the predator (Figure 5). Figure 5: Selective pressure acting to move away from preventing quantitative trait divergence from 26 . and costs of evolution Oscillatory Dynamics in the Two Models: The relationship between the trait and abundance oscillations in these two models are fundamentally different. In the unidirectional model, the prey trait and population cycles are nearly in phase, meaning that peaks in the prey’s abundance and trait values occur at nearly the same time (Figure 6). However, the predator trait and population cycles have a phase shift of approximately half a period, such that abundance maxima and trait value minima occur at roughly the same time. In any given oscillation, the prey’s trait value is the first variable to reverse direction. The sharp increase in the prey’s trait marks the beginning of each oscillation. Soon after, the prey’s population also increases as the prey is able to effectively avoid capture. Then, the predator is under strong selective pressure to increase its trait value in order to capture the prey, and the predator’s quantitative trait value increases quickly. However, this rapid evolution in the predator has the cost of increased mortality, and the predator’s population declines as its trait evolves. When the predator’s trait value is large enough such that the prey suffers heavy predation, the cost of maintaining a high phenotype in the prey is not offset by the benefits of predator evasion, and the prey’s trait value decreases. The prey’s population decreases with increased predation, and the predator’s population increases due to a heightened attack rate. Additionally, the predator’s trait value decreases because the predator can attain a similarly high attack rate with a lower trait value because the attack rate is a function of the relative phenotypes of the two species. 27 Figure 6: Abundance and trait value oscillations in the unidirectional model. Blue indicates the abundance and trait value of the predator, whereas green indicates the abundance and trait value of the prey. In contrast, the abundances of the predator and prey cycle twice as quickly as the trait values of the species the bidirectional model (Figure 7). The abundances of the predator and the prey increase before each minimum or maximum in the prey’s trait value and then decline until immediately before the next minimum or maximum. In a bidirectional system where the predator’s ability to capture the prey is based on how closely the predator matches the prey’s quantitative trait value, the prey’s trait is the first variable to show a dramatic increase. The prey’s phenotype changes in response to the moderate level of predation present due to the high abundance of predators. At this time, the benefits of the extreme trait value of the prey do not balance the costs of this extreme phenotype. When the prey’s trait increases, the attack rate of the predator initially increases as the prey’s phenotype gets closer to the preferred phenotype of the predator. The predator’s abundance increases because its attack rate grows larger without the cost of the predator’s trait evolving. However, the prey also benefit from their evolution back toward their birth rate maximizing trait value because they are released from the high cost (in terms of birth rate) of maintaining an extreme phenotype. The prey’s abundance then 28 also increases, and reaches its maximum when the prey has an intermediate trait value. At this point in the cycle, the costs to maintaining this phenotype are relatively small and the predator has not yet evolved to track the prey’s trait value, so there is little predation. However, as soon as the prey’s trait value surpasses the predator’s current preferred trait value the predator’s phenotype begins to evolve in the opposite direction in order to track the prey’s current trajectory. When the cost to the prey of having a large phenotype outweighs the benefits of predator avoidance, the prey’s trait value will reverse and the same dynamics will repeat when the prey’s trait decreases. The population dynamics have the same period of oscillation as the trait values, but the abundances of the predator and the prey have two maxima and two minima within each period. Each time the direction of the prey’s trait evolution reverses, the prey’s trait and the predator’s trait are closely matched for a short period of time. This increases the population of the predator due to an increased attack rate and increases the population of the prey due to smaller costs of their phenotype in terms of their birth rate. Thus, each change in direction of the phenotype corresponds to an abundance peak in the two species, which means that abundance peaks occur twice as often as trait peaks. However, the two abundance maxima within each period are not equal. The two peaks are of different magnitude and change size depending on the value of magnitude of relative to and the . The mismatch in the size of the abundance peaks is determined by the amount of asymmetry in the costs of evolution in various parts of the trait value cycle. The trait oscillations of both the predator and the prey are centered around the prey’s , meaning that the prey always incur symmetric costs to evolution whether their trait is increasing or decreasing because it deviates the same amount from its 29 in either case. When = 0 the predator’s is also in the center of the oscillations, and the predator incurs symmetric costs as well. In this case, the peaks in species’ abundances are identical because both species have symmetric costs of phenotype evolution around . However, if is not equal to zero, the predator incurs asymmetric costs because increasing its trait value to track the prey brings it closer to its direction and further from its in one in the opposite direction. Thus, the effect of trait evolution on the abundance of the predator depends on whether the prey trait is increasing or decreasing. This asymmetric cost leads to the difference in abundance peaks in the predator, which contributes to difference in the abundance peaks in the prey. Changing relative to changes the duration of the abundance peaks and can generate erratic abundance cycles when the cost of evolution to one species is much greater than the cost to the other species. Figure 7: Abundance and trait value oscillations in the bidirectional model. Blue indicates the abundance and trait value of the predator, whereas green indicates the abundance and trait value of the prey. Fitness Surfaces During Oscillations: Although the oscillations in the two models are different, the fitness surfaces of the prey exhibited the same pattern when the trait values oscillated. The prey’s fitness surface 30 is bimodal when the system is in a stable limit cycle (Figure 8). Each local maximum in the fitness surface corresponds to a phenotype where the prey would have increased reproductive success when compared with the success of another prey with a slightly higher or lower phenotypic trait value. These two maxima correspond to the phenotypes where the prey maximizes fitness by either 1) maintaining a high birth rate or 2) effectively avoiding predation. Figure 8: Fitness surface of the prey in an oscillating system Selection on a phenotypic trait will always act to increase per capita reproduction, meaning that the change in the trait value from one generation to the next will act to increase the average fitness of the population. Because the movement of the phenotype is constrained to be continuous along the fitness surface and to always increase fitness, the prey in this model cannot traverse local fitness minima. However, because the growth rate of the prey depends on its own phenotype and the phenotype of the predator (as well as their abundances), the optimal strategy of the prey changes as the phenotype of the predator changes. Thus, the fitness of the prey is frequency dependent based on the 31 predator’s phenotype, and the shape of the fitness surface changes depending on the current value of . In systems where oscillations occur, the shape of the fitness surface changes over time such that the second fitness peak grows larger until it extends out to the point where the prey’s phenotype currently resides. The slope of the local fitness surface, and therefore the direction of the local selection gradient, change sign when this occurs. The prey then begin to reverse the direction of their evolution because their fitness increases when evolving their phenotype to head toward the other fitness peak. When the second fitness peak extends outward and encapsulates the prey’s current trait value, the location of the fitness trough crosses to the other side of the prey’s trait value on the phenotype axis. The fitness surface is still bimodal, but the prey now follow the path toward the second fitness maximum, as they are no longer prevented by the fitness trough from evolving toward the larger fitness peak (Figure 9). 32 Figure 9: The fitness surfaces of the prey in an oscillating system when the trait is increasing (panel 1) and when the trait is decreasing (panel 2). The current value is shown by the star. In the case of damped oscillations, the prey have the same bimodal fitness surface, but the oscillations grow smaller over time because the prey switch strategies earlier in quantitative trait space as time goes on. This is a result of the fitness surface changing more rapidly due to the predator’s increased ability to track the prey as compared to models where the traits and abundances perpetually oscillate. When the oscillations cease, the prey’s traits reaches equilibrium at a stable fitness minimum directly in the center of the fitness trough separating the two peaks in the fitness surface (Figure 10). Though movement in either direction would increase the fitness of the prey, the point is stable because the fitness surface changes rapidly enough around this equilibrium that the previous phenotype (the stable fitness minimum) has greater fitness than the newly derived phenotype once the trait has changed (Abrams et al. 1993). In this system, if the prey’s trait were to evolve in either direction from the stable fitness minimum, the prey would have increased fitness for a brief period of time before the predator’s trait evolved in response. However, when the predator’s trait value changes, the fitness surface of the prey 33 also changes. In the case of a stable fitness minimum, the previous phenotype of the prey has increased fitness over the current phenotype of the prey after this fitness surface change, and the prey evolves toward its previous phenotype. Though the point is a minimum in the prey’s current fitness surface, it is a maximum in the dynamic system because any deviation from that point induces rapid changes in the fitness surface that lead the prey to evolve back toward its previous trait value. The predator’s trait, however, equilibrates at a fitness maximum in a system where there are damped oscillations. Figure 10: Equilibrium trait value of the prey in a system exhibiting damped oscillations. c = 3, d = .1, a = .25, b = .2, f = .1, g = 0, α = 1.49, = .05, = .10 DISCUSSION: Dynamic Behaviors of Coevolutionary Models Coevolutionary dynamics can generate oscillatory behavior in predator‐prey systems where the same system, lacking evolution, would reach equilibrium. This occurs 34 because phenotypic evolution in the two species allows the predator to regulate the prey to low levels through a combination of density mediated and trait mediated interactions. Previous investigations of predator‐prey dynamics have demonstrated that oscillations in these systems are most likely to occur when a predator is able to decrease the prey’s population size to well below its equilibrial abundance in the absence of the predator (Maynard Smith and Slatkin 1973; Ginzburg 1986, Turchin 2001, Holt p. 139). The results of this coevolutionary model largely support this conclusion. The population‐related parameters that augmented the predator’s growth rate in relation to that of the prey were the attack rate, a, and the conversion rate of prey into predators, b. Increasing the values of either of these parameters destabilized the system and led to oscillations. The parameters that decreased the predator’s growth rate were the predator’s density‐independent death rate, f, and the predator’s density‐dependent death rate, g. When either of these terms increased, the system became more stable. These models also show that increasing the prey’s birth rate, c, or decreasing the strength of the prey’s negative density‐dependence, d, induces oscillations in the system. This is in accordance with the theory of the paradox of enrichment, which predicts that higher prey productivity can lead to instability in predator‐prey systems (Rosenzweig 1971, Roy et al. 2007). In the case of the paradox of enrichment, the increased birth rate of the prey causes the system to shift from having stable equilibria to exhibiting persisting oscillations in the abundance of the predator and the prey. Empirical studies of predator‐ prey systems have demonstrated a crossing of the Hopf bifurcation in algae‐rotifer systems corresponding to an increase in nutrient input into the system (Fussman et al. 2000). Reaching this bifurcation point corresponds to the place at which the system’s dominant 35 eigenvalue transitions from having a negative real part to having only imaginary parts, meaning that the long‐term trajectories of the system switch from reaching stable equilibria to oscillating indefinitely. These models suggest that the same dynamics are present in predator‐prey models that allow for evolution. The evolutionary parameters, however, were more complex in their effects on system stability. When the prey have low costs to evolution, meaning γ is small in comparison to Θ , the prey is able to avoid predation pressure due to a low attack rate of the predator, and the system equilibrates with a large prey population and a small predator population. These results largely support previous findings of predator‐prey models, as the system becomes unstable as the predator’s rate of evolution increases relative to that of the prey, meaning that the predator increases its ability to regulate the population of the prey via predation. When the costs of evolution are similar in both species, the prey evolve rapidly when under strong pressure from the predator. However, in order to induce cycles in trait values, the predator must be able to evolve rapidly enough to consistently impose high predation costs on the prey. Thus, the prey is able to temporarily evade the predator, but the predator’s cost to evolution is low enough that the predator can successfully track the prey’s trait value. This provides a plausible explanation for why intermediate and similar rates of evolution produce oscillations of the largest magnitude in predator‐prey systems (Mougi and Iwasa 2010). In the unidirectional system, when the predator is able to efficiently exploit the prey due to a low cost of evolution of its phenotype (i.e. Θ is small), there is a stable equilibrium for the traits and abundances of both species. This result is unexpected, as the predator in this case is able to strongly limit the prey’s abundance. In this scenario, the prey is 36 regulated at such a low level that its per capita birth rate is nearly at its maximum, c. With a high birth rate, a small population of prey can sustain a large population of predators. This is often the case for the stable equilibrium that occurs in the unidirectional model when Θ is smaller than γ . Though this region of stability was only found in the unidirectional model, it is possible that this pattern emerges in the bidirectional model under some conditions. The bidirectional model under these same conditions ( Θ is much smaller than γ ) demonstrated persisting oscillations, but the number of parameters in the model prevents a complete investigation of all possible parameter space. Though the two models behave similarly in response to many parameters, the locations of the trait equilibria are strongly dependent on the specific attack rate used in the model. The unidirectional system is much more likely to produce patterns similar to the “arms‐race” description of coevolution. This framework of evolutionary models describes coevolution as a race between the predator and the prey, with each perpetually evolving to increase its phenotype (Rosenzweig 1973, Van Valen 1973, Slobodkin 1974, Dawkins & Krebs 1979). The unidirectional model provides a single option for the predator to increase its attack rate and for the prey to decrease pressure from predation; in both species this strategy is to increase the trait value. This unidirectional pressure resulting from the shape of the attack rate curve selects for higher trait values in both species, which is in accordance with the arms race analogy. However, in the bidirectional model, the prey have two options for avoiding predation: they can increase or decrease their trait value, so long as it evolves away from the mean trait value of the predator. Similarly, the predator can attain an identical attack rate if its phenotypic value is the same amount above or below the prey’s phenotypic value. 37 The bidirectional attack rate curve therefore contributes to a coevolutionary model where arms races are less likely to occur. Consequently, the bidirectional model is more prone to oscillatory behavior than is the unidirectional model because both the predator and the prey have two evolutionary strategies that result in an identical attack rate. Trait oscillations are the result of a change in the evolutionary strategy of the prey. The prey can maximize their growth rate in one of two ways: they can effectively avoid predation by phenotypic evolution or they can optimize their birth rate by incurring low costs of evolution. The prey will follow the trajectory of their current strategy until the alternate strategy allows for higher instantaneous fitness. In terms of the fitness surface of the prey, the prey’s trait value will move toward the closer fitness peak until the slope of the local selection gradient changes sign. For example, the prey will evolve an increasing trait value until the point where their birth rate is so negatively affected by phenotypic evolution that the tradeoff between evasion and birth rate selects for a higher birth rate. In the bidirectional model, the prey can evade predation by having a larger or smaller phenotype than the predator. Thus, switching between strategies will occur frequently because the prey have low costs of predation when following either strategy. The selective pressure from a decreased birth rate is easily offset by the dual benefits of optimizing birth rate and avoiding predation when switching to the alternate strategy. This is not true in the unidirectional model because switching to a strategy that optimizes birth rate in the prey incurs high costs of predation, and so switching occurs less frequently. However, it is impossible to directly compare the tendency toward oscillation between the two models because the attack rate functions utilize different shape parameters. 38 Comparison to Previous Coveolutionary Models These models largely support the findings of previous coevolutionary models. Stabilizing selection is a necessary condition for oscillations in both models, but strong stabilizing selection contributes to stable system dynamics. This is because oscillations require that two evolutionary strategies exist, and that the prey’s better strategy changes depending on the trait value of the predator. The presence of stabilizing selection creates a tradeoff between birth rate and predator evasion in the prey, which generates the two evolutionary strategies. However, when the strength of this selection is strong, the prey species is confined to follow a strategy of low phenotypic evolution, which prevents oscillations from occurring. Additionally, when the parameters governing the evolutionary abilities of the two species ( γ , Θ , and the additive genetic variances of the species) are comparable in the predator and the prey, oscillations are most likely to occur and the amplitudes of the oscillations are large. The results of this coevolutionary model are different from the outcomes of similar coevolutionary models under some circumstances. Firstly, previous analyses have found that the maximum growth rate of the prey species has no effect on the stability of a bidirectional system (Mougi 2012). However, this model demonstrated multiple instances where increasing the value of c altered the behavior of the system from reaching a stable equilibrium to exhibiting persisting oscillations. In fact, every parameter could potentially induce oscillations in the system. Additionally, a stability analysis performed in a previous study of a bidirectional system suggested that predator evolution is stabilizing because models where only the predator evolves are always stable (Mougi 2012). In the model presented here, when the 39 predator and the prey are both capable of evolution, decreasing the predator’s cost of evolution can induce oscillatory dynamics. In this bidirectional model, decreasing values of Θ causes oscillations in the trait values and abundances of both species. Reducing the predator’s cost to phenotype evolution allows the predator to more closely track the prey’s evolving phenotype, which results in a higher attack rate. The increased pressure from predation, even at high trait values of the prey, causes the prey to switch evolutionary strategies more frequently, resulting in oscillations in the trait values and abundances of both species. One notable difference in the model presented here and the model used in Mougi, 2012, where the analysis was performed, is the death rate function of the predator. This model utilizes a quadratic function to describe the increase in the predator’s death rate due to phenotypic evolution, whereas the previous model uses a function from the Gaussian family. The difference in the shape of the curves may be responsible for some of the discrepancies in the outcomes of the two bidirectional models. Applications to Empirical Systems These models predict that predator‐prey systems where the predator attains its maximal attack rate by matching the phenotype of the prey should be more prone to oscillations in abundance and phenotypic traits than systems where the predator must increase its trait value over that of the prey. Traits of the prey that generate the corresponding bidirectional attack rate function include body size, coloration, and chemical signals. Previous studies of predator‐prey systems provide examples of oscillatory behavior in systems mediated by these kinds of traits. Phenotypic cycling has been demonstrated in plant‐herbivore interactions where the susceptibility of a plant to predation by an insect depends upon the insect’s ability to metabolize the suite of 40 compounds produced by the plant (Berenbaum and Zangrel 1998). This study, and others, have demonstrated that rapid evolution occurs on time scales relevant to ecological interactions, and can strongly impact the abundances of predator and prey species (Hairston et al. 2005, Strauss et al. 2008, terHorst et al. 2010). However, because many evolutionary changes occur on a timescale too long for any individual to directly observe, the fossil record may serve as a way to test the predictions of these models. One specific hypothesis is that species known to suffer from heavy predation may experience more frequent changes in body size than more resistant prey. The likelihood of oscillation in a unidirectional system is strongly dependent on the steepness of the attack rate curve as a function of the relative phenotypes of the predator and the prey. When the curve is steep, oscillations are more likely to occur. An empirical example of a unidirectional system where the attack rate responds strongly to changes in the relative phenotypes of the two species is the algae‐rotifer system comprised of Chlorella vulgaris and Brachionus calyciflorus. Some phenotypes of the algae produce a compound that is highly toxic to the rotifer, which causes a decrease in pressure from predation as the rotifers are unable to consume the toxic resource (Yoshida et al. 2003). However, when the predator population is low and there is little pressure from predation, the non‐toxic phenotype is more competitively successful because its growth rate is faster than that of the toxic phenotype (Yoshida et al. 2003). Unidirectional systems where the attack curve is less steep should equilibriate with the predator having a more extreme phenotype than the prey. In addition to toxicity, traits of the prey that might generate this attack rate curve include speed and strength. In systems where the attack rate is mediated by these traits, the phenotype of the prey over 41 time is indicative of the amount of predation inflicted upon the prey species. Under circumstances where there is low predation, the prey’s trait value is relatively close to its trait value of minimum cost, but increases as the pressure from the predator increases. The prey’s trait value serves as an indicator of predation pressure in the unidirectional system but not in the bidirectional system because the prey can only avoid predation in this system only by increasing their phenotype. The unidirectional system equilibriates with Δ > 0 , which is a region of the graph where the attack rate is saturating as a function of Δ . Thus, unlike the bidirectional system, the degree of phenotypic evolution at equilibrium reflects the pressure of evolution due to the attack rate of the predator. This pattern of net directional selection due to predation in unidirectional systems has been demonstrated using the burst speed of tadpoles as the prey’s phenotypic trait of interest. Tadpoles with high burst speeds evade predation much more than tadpoles with an average burst speed, indicating strong selection for an increasing phenotype (Watkins 1995). At high levels of predation, when this directional selection is strong, the phenotype of the prey should increase correspondingly. These models also indicate that predator‐prey systems are stable when the prey species has a much faster rate of evolution than the predator species. This can occur if the prey’s generation time is much shorter than that of the predator or if the prey have a lower cost than the predator of evolving its phenotype. In the case of mosquito larvae preying upon protozoa, the generation time of the predator is more than two orders of magnitude longer than that of the prey (terHorst et al. 2010). In accordance with the predictions of these models, the system shows convergence upon a stable equilibrium in the predation experiments carried out in terHorst et al. 2010, and the negative impact of the predator on 42 the prey population diminishes strongly over the course of the experiments. These models suggest this stability is due to rapid evolution in the prey leading to a decreased attack rate of the predator and the lack of corresponding rapid evolution in the predator. Both the unidirectional and bidirectional models predict that systems exhibiting damped oscillations will equilibriate with the prey at a stable fitness minimum, providing the assumptions of the model are met. However, some of these assumptions, such as random mating in the prey and a lack of stochasticity, are unlikely to be upheld in empirical systems. When these conditions are relaxed, these models provide a starting point for conceptualizing the evolution of polymorphisms in prey species. The fitness surface of the prey at equilibrium after a period of damped oscillations is analogous to the fitness surface of a population under disruptive selection. In both cases, the individuals with the mean phenotype of the population have the lowest fitness (Rueffler et al. 2006). Previous studies have demonstrated that conditions of disruptive selection favor assortative mating between the prey, as fitness increases as the phenotype of the individuals deviates further from the mean phenotype (Benkman 1999, Kopp and Hermisson 2006). Thus, future coevolutionary models may find it of interest to allow for a degree of non‐random mating in the prey species as to observe the trajectories of the system under the conditions where these models show damped oscillations. Additionally, both empirical (Rundle et al 2003, Nosil and Crespi 2006) and theoretical studies (Meyer and Kassen 2007, Reed and Janzen 1998) have indicated that a predator can impose disruptive selection upon a prey species, and that this selection can lead to polymorphisms in the prey species. However, it is unclear how the frequency‐dependent nature of the fitness surface of the prey would affect 43 the evolution of prey populations with diverging phenotypes, even after accounting for stochasticity and non‐random mating. Limitations of the Models Although models can be useful for offering general insights into complex biological systems, there are limitations on the generalizability of such models, including those presented here. These models pertain to a predator‐prey system where the predator is a specialist and the prey suffers predation from only one predator. Specific assumptions of this model include treating the predator and prey traits as continuous variables, a lack of inducible defenses and phenotypic plasticity, constraining the additive genetic variances to be constant throughout the model, and using the mean trait value of the predator and prey species to represent the phenotypes of each population. Additionally, this model assumes that the phenotypes of the predator and the prey are uncorrelated with the fitnesses of the species, except for imposing a higher mortality rate in the predator and a lower birth rate in the prey. This assumption is invalid in cases where the phenotype otherwise affects birth rate in the prey, such as if larger body size increases fecundity. Lastly, these models imply spatial homogeneity and a lack of refuges for the prey, as neither of these aspects is explicitly addressed in the equations. 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