The paradox of enrichment in an adaptive world

Paradoxically, enrichment can destabilize a predator-prey food web. While adaptive dynamics can greatly influence the stability of interaction systems, few theoretical studies have examined the effect of the adaptive dynamics of interaction-related traits on the possibility of resolution of the paradox of enrichment. We consider the evolution of attack and defence traits of a predator and two prey species in a one predator-two prey system in which the predator practises optimal diet use. The results showed that optimal foraging alone cannot eliminate a pattern of destabilization with enrichment, but trait evolution of the predator or prey can change the pattern to one of stabilization, implying a possible resolution of the paradox of enrichment. Furthermore, trait evolution in all species can broaden the parameter range of stabilization. Importantly, rapid evolution can stabilize this system, but weaken its stability in the face of enrichment.     

Fitness minimization and dynamic instability as a consequence of predator–prey coevolution

We analyse dynamic models of the coevolution of continuous traits that determine the capture rate of a prey species by a predator. The goal of the analysis is to determine conditions when the coevolutionary dynamics will be unstable and will generate population cycles. We use a simplified model of the evolutionary dynamics of quantitative traits in which the rate of change of the mean trait value is proportional to the rate of increase of individual fitness with trait value. Traits that increase ability in the predatory interaction are assumed to have negative effects on another component of fitness. We concentrate on the role of equilibrial fitness minima in producing cycles. In this case, the mean trait of a rapidly evolving species minimizes its fitness and it is chased around this equilibrium by adaptive evolution in the other species. Such cases appear to be most likely if the capture rate of prey by predators is maximal when predator and prey phenotypes match each other. They are possible, but less likely when traits in each species determine a one-dimensional axis of ability related to the interaction. Population dynamics often increase the range of parameter values for which cycles occur, relative to purely evolutionary models, although strong prey self-regulation may stabilize an evolutionarily unstable subsystem.     

Fitness minimization and dynamic instability as a consequence of predator-prey coevolution

Summary We analyse dynamic models of the coevolution of continuous traits that determine the capture rate of a prey species by a predator. The goal of the analysis is to determine conditions when the coevolutionary dynamics will be unstable and will generate population cycles. We use a simplified model of the evolutionary dynamics of quantitative traits in which the rate of change of the mean trait value is proportional to the rate of increase of individual fitness with trait value. Traits that increase ability in the predatory interaction are assumed to have negative effects on another component of fitness. We concentrate on the role of equilibrial fitness minima in producing cycles. In this case, the mean trait of a rapidly evolving species minimizes its fitness and it is chased around this equilibrium by adaptive evolution in the other species. Such cases appear to be most likely if the capture rate of prey by predators is maximal when predator and prey phenotypes match each other. They are possible, but less likely when traits in each species determine a one-dimensional axis of ability related to the interaction. Population dynamics often increase the range of parameter values for which cycles occur, relative to purely evolutionary models, although strong prey self-regulation may stabilize an evolutionarily unstable subsystem.     

Evolution towards oscillation or stability in a predator–prey system

We studied a prey–predator system in which both species evolve. We discuss here the conditions that result in coevolution towards a stable equilibrium or towards oscillations. First, we show that a stable equilibrium or population oscillations with small amplitude is likely to occur if the prey''s (host''s) defence is effective when compared with the predator''s (parasite''s) attacking ability at equilibrium, whereas large-amplitude oscillations are likely if the predator''s (parasite''s) attacking ability exceeds the prey''s (host''s) defensive ability. Second, a stable equilibrium is more likely if the prey''s defensive trait evolves faster than the predator''s attack trait, whereas population oscillations are likely if the predator''s trait evolves faster than that of the prey. Third, when the adaptation rates of both species are similar, the amplitude of the fluctuations in their abundances is small when the adaptation rate is either very slow or very fast, but at an intermediate rate of adaptation the fluctuations have a large amplitude. We also show the case in which the prey''s abundance and trait fluctuate greatly, while those of the predator remain almost unchanged. Our results predict that populations and traits in host–parasite systems are more likely than those in prey–predator systems to show large-amplitude oscillations.     

Unique coevolutionary dynamics in a predator-prey system

In this paper, we study the predator-prey coevolutionary dynamics when a prey's defense and a predator's offense change in an adaptive manner, either by genetic evolution or phenotypic plasticity, or by behavioral choice. Results are: (1) The coevolutionary dynamics are more likely to be stable if the predator adapts faster than the prey. (2) The prey population size can be nearly constant but the predator population can show very large amplitude fluctuations. (3) Both populations may oscillate in antiphase. All of these are not observed when the handling time is short and the prey's density dependence is weak. (4) The population dynamics and the trait dynamics show resonance: the amplitude of the population fluctuation is the largest when the speed of adaptation is intermediate. These results may explain experimental studies with microorganisms.     

ADAPTIVE RESPONSES OF PREDATORS TO PREY AND PREY TO PREDATORS: THE FAILURE OF THE ARMS-RACE ANALOGY

This paper analyzes a number of relatively general models of predator-prey adaptation and coadaptation. The motivation behind this work is, in part, to evaluate the “race analogy” that has been applied in analyzing predator-prey coevolution. The models are based on the assumption that increased investment in predation-related adaptations must be paid for by decreased adaptation to some other factor. Increased investment in predation-related adaptations by the prey lowers the predator's functional response, and increased investment by the predator increases the functional response. The models are used to determine how each species should respond to an increase in the predation-related investment of the other species. Several broad classes of population-dynamics models and several alternatives for the cost of predation-related adaptation are investigated. The results do not support the general applicability of the race analogy. In the type of model analyzed in greatest detail here, predator and prey adaptations combine multiplicatively in determining the predator's capture-rate constant. In such models, prey usually increase investment in predator avoidance or escape when predators increase their investment in capture. However, predators often do not change or decrease their investment in response to an increase in the prey's investment. The direction of the predator's response depends on the particular parameter that pays the cost of increased predation investment, the shape of the cost-benefit functions, and the assumptions about the population dynamics of the predator-prey system. Similar models are used to determine whether increased investment by one species should increase the rate of incorporation of mutations that improve the predation-related adaptations of the other species. The arms-race analogy also fails for this case. The results cast doubt on the usefulness of Dawkins and Krebs (1979) “life-dinner” principle.     

Slight phenotypic variation in predators and prey causes complex predator-prey oscillations

Predator-prey oscillations are expected to show a 1/4-phase lag between predator and prey. However, observed dynamics of natural or experimental predator-prey systems are often more complex. A striking but hardly studied example are sudden interruptions of classic 1/4-lag cycles with periods of antiphase oscillations, or periods without any regular predator-prey oscillations. These interruptions occur for a limited time before the system reverts to regular 1/4-lag oscillations, thus yielding intermittent cycles. Reasons for this behaviour are often difficult to reveal in experimental systems. Here we test the hypothesis that such complex dynamical behaviour may result from minor trait variation and trait adaptation in both the prey and predator, causing recurrent small changes in attack rates that may be hard to capture by empirical measurements. Using a model structure where the degree of trait variation in the predator can be explicitly controlled, we show that a very limited amount of adaptation resulting in 10–15% temporal variation in attack rates is already sufficient to generate these intermittent dynamics. Such minor variation may be present in experimental predator-prey systems, and may explain disruptions in regular 1/4-lag oscillations.     

Evolutionary responses of foraging-related traits in unstable predator–prey systems

The evolutionary responses of predators to prey and of prey to predators are analysed using models for the dynamics of a quantitative trait that determines the capture rate of prey by an average searching predator. Unlike previous investigations, the analysis centres on models and/or parameter values for which the two-species equilibrium is locally unstable. The instability in some models is driven by the predators non-linear functional response to prey; in other models, the cycles are a direct consequence of evolutionary response to selection acting on the trait. When the values of predator and prey traits combine multiplicatively to determine the capture rate, the predators trait shows only a transient response to changes in the preys trait in stable systems. However, when the population densities exhibit sustained oscillations, predators often evolve an increased long-term mean capture rate in response to an increased prey escape ability. Under the multiplicative model, prey in stable systems always evolve increased escape ability in response to an increased predator capture a     

Predator-prey coevolution driven by size selective predation can cause anti-synchronized and cryptic population dynamics

Population dynamics and evolutionary dynamics can occur on similar time scales, and a coupling of these two processes can lead to novel population dynamics. Recent theoretical studies of coevolving predator-prey systems have concentrated more on the stability of such systems than on the characteristics of cycles when they are unstable. Here I explore the characteristics of the cycles that arise due to coevolution in a system in which prey can increase their ability to escape from predators by becoming either significantly larger or significantly smaller in trait value (i.e., a bidirectional trait axis). This is a reasonable model of body size evolution in some systems. The results show that antiphase population cycles and cryptic cycles (large population fluctuation in one species but almost no change in another species) can occur in the coevolutionary system but not systems where only a single species evolves. Previously, those dynamical patterns have only been theoretically shown to occur in single species evolutionary models and the coevolutionary model which do not involve a bi-directional axis of adaptation. These unusual dynamics may be observed in predator-prey interactions when the density dependence in the prey species is strong.     

Comparing the effects of rapid evolution and phenotypic plasticity on predator-prey dynamics

Ecologists have increasingly focused on how rapid adaptive trait changes can affect population dynamics. Rapid adaptation can result from either rapid evolution or phenotypic plasticity, but their effects on population dynamics are seldom compared directly. Here we examine theoretically the effects of rapid evolution and phenotypic plasticity of antipredatory defense on predator-prey dynamics. Our analyses reveal that phenotypic plasticity tends to stabilize population dynamics more strongly than rapid evolution. It is therefore important to know the mechanism by which phenotypic variation is generated for predicting the dynamics of rapidly adapting populations. We next examine an advantage of a phenotypically plastic prey genotype over the polymorphism of specialist prey genotypes. Numerical analyses reveal that the plastic genotype, if there is a small cost for maintaining it, cannot coexist with the pairs of specialist counterparts unless the system has a limit cycle. Furthermore, for the plastic genotype to replace specialist genotypes, a forced environmental fluctuation is critical in a broad parameter range. When these results are combined, the plastic genotype enjoys an advantage with population oscillations, but plasticity tends to lose its advantage by stabilizing the oscillations. This dilemma leads to an interesting intermittent limit cycle with the changing frequency of phenotypic plasticity.     

Consequences of behavioral dynamics for the population dynamics of predator-prey systems with switching

This article explores how different mechanisms governing the rate of change of the predators preference alter the dynamics of predator-prey systems in which the predator exhibits positive frequency-dependent predation. The models assume that individuals of the predator species adaptively adjust a trait that determines their relative capture rates of each of two prey species. The resulting switching behavior does not instantaneously attain the optimum for current prey densities, but instead lags behind it. Several mechanisms producing such lags are discussed and modeled. In all cases examined, our question is whether a realistic behavioral lag can significantly change the dynamics of the system relative to an analogous case in which the predators switching is effectively instantaneous. We also explore whether increasing the rate parameters of dynamic models of behavior results in convergence to the population dynamics of analogous models with instantaneous switching, and whether different behavioral models produce similar population dynamics. The analysis concentrates on systems that undergo endogenously generated predator-prey cycles in the absence of switching behavior. The average densities and the nature of indirect interactions are often sensitive to the rate of behavioral change, and are often qualitatively different for different classes of behavioral models. Dynamics and average densities can be very sensitive to small changes in parameters of either the prey growth or predator switching functions. These differences suggest that an understanding of switching in natural systems will require research into the behavioral mechanisms that govern lags in the response of predator preference to changes in prey density.     

Adaptive changes in prey vulnerability shape the response of predator populations to mortality

Simple models are used to explore how adaptive changes in prey vulnerability alter the population response of their predator to increased mortality. If the mortality is an imposed harvest, the change in prey vulnerability also influences the relationship between harvest effort and yield of the predator. The models assume that different prey phenotypes share a single resource, but have different vulnerabilities to the predator. Decreased vulnerability is assumed to decrease resource consumption rate. Adaptive change may occur by phenotypic changes in the traits of a single species or by shifts in the abundances of a pair of coexisting species or morphs. The response of the predator population is influenced by the shape of the predator's functional response, the shape of resource density dependence, and the shape of the tradeoff between vulnerability and food intake in the prey. Given a linear predator functional response, adaptive prey defense tends to produce a decelerating decline in predator population size with increased mortality. Prey defense may also greatly increase the range of mortality rates that allow predator persistence. If the predator has a type-2 response with a significant handling time, adaptive prey defense may have a greater variety of effects on the predator's response to mortality, sometimes producing alternative attractors, population cycles, or increased mean predator density. Situations in which there is disruptive selection on prey defense often imply a bimodal change in yield as a function of harvesting effort, with a minimum at intermediate effort. These results argue against using single-species models of density dependent growth to manage predatory species, and illustrate the importance of incorporating anti-predator behavior into models in applied population ecology.     

Simulation model of a predator-prey system comprised ofPhytoseiulus persimilis andTetranychus urticae. II. Model sensitivity to variations in the life history parameters of both species and to variations in the functional response and components of the numerical response

A detailed sensitivity analysis of a model of a predator-prey system comprised of Tetranychus urticae and Phytoseiulus persimilis was performed. The aim was to assess the relative importance of the life history parameters of both species, the functional response, and the components of the numerical response. In addition, the impact of the initial predator-prey ratio and the timing of predator introduction were tested. Results indicated that the most important factors in the system were relative rates of predator and prey development, the time of onset of predator oviposition, and the mode of the predator's oviposition curve. The total oviposition of the predator, the effect of prey consumption on predator oviposition, and predator searching were important under some conditions. Factors of moderate importance were the adult female predator's functional response, total prey oviposition, the mode of the prey's oviposition curve, abiotic mortality of the pre-adult predator, and the effect of prey consumption on predator development and on the immature predator's mortality. Factors of least importance were the variances of the predator's and prey's oviposition curves, the abiotic mortality of the adult predator, the abiotic mortality of the pre-adult and adult prey, the functional response of the nymphal and adult male predators, and the effect of prey consumption on adult predator mortality. The sex ratios had little effect, except when the proportion of female predators was very low. The initial predator-prey ratio and time of predator introduction had significant impacts on system behavior, though the patterns of impact were different.     

Adaptive evolution of foraging-related traits in a predator-prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.     

Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits

Summary We present models of adaptive change in continuous traits for the following situations: (1) adaptation of a single trait within a single population in which the fitness of a given individual depends on the population's mean trait value as well as its own trait value; (2) adaptation of two (or more) traits within a single population; (3) adaptation in two or more interacting species. We analyse a dynamic model of these adaptive scenarios in which the rate of change of the mean trait value is an increasing function of the fitness gradient (i.e. the rate of increase of individual fitness with the individual's trait value). Such models have been employed in evolutionary game theory and are often appropriate both for the evolution of quantitative genetic traits and for the behavioural adjustment of phenotypically plastic traits. The dynamics of the adaptation of several different ecologically important traits can result in characters that minimize individual fitness and can preclude evolution towards characters that maximize individual fitness. We discuss biological circumstances that are likely to produce such adaptive failures for situations involving foraging, predator avoidance, competition and coevolution. The results argue for greater attention to dynamical stability in models of the evolution of continuous traits.     

Revisiting the Role of Individual Variability in Population Persistence and Stability

Populations often exhibit a pronounced degree of individual variability and this can be important when constructing ecological models. In this paper, we revisit the role of inter-individual variability in population persistence and stability under predation pressure. As a case study, we consider interactions between a structured population of zooplankton grazers and their predators. Unlike previous structured population models, which only consider variability of individuals according to the age or body size, we focus on physiological and behavioural structuring. We first experimentally demonstrate a high degree of variation of individual consumption rates in three dominant species of herbivorous copepods (Calanus finmarchicus, Calanus glacialis, Calanus euxinus) and show that this disparity implies a pronounced variation in the consumption capacities of individuals. Then we construct a parsimonious predator-prey model which takes into account the intra-population variability of prey individuals according to behavioural traits: effectively, each organism has a ‘personality’ of its own. Our modelling results show that structuring of prey according to their growth rate and vulnerability to predation can dampen predator-prey cycles and enhance persistence of a species, even if the resource stock for prey is unlimited. The main mechanism of efficient top-down regulation is shown to work by letting the prey population become dominated by less vulnerable individuals when predator densities are high, while the trait distribution recovers when the predator densities are low.     

Cannibalism in an age-structured predator-prey system

Recently, Kohlmeier and Ebenhöh showed that cannibalism can stabilize population cycles in a Lotka-Volterra type predator-prey model. Population cycles in their model are due to the interaction between logistic population growth of the prey and a hyperbolic functional response. In this paper, we consider a predator-prey system where cyclic population fluctuations are due to the age structure in the predator species. It is shown that cannibalism is also a stabilizing mechanism when population oscillations are due to this age structure. We conclude that in predator-prey systems, cannibalism by predators can stabilize both externally generated (consumer-resource) as well as internally generated (agestructure) fluctuations.     

Rapid prey evolution and the dynamics of two-predator food webs

Traits affecting ecological interactions can evolve on the same time scale as population and community dynamics, creating the potential for feedbacks between evolutionary and ecological dynamics. Theory and experiments have shown in particular that rapid evolution of traits conferring defense against predation can radically change the qualitative dynamics of a predator–prey food chain. Here, we ask whether such dramatic effects are likely to be seen in more complex food webs having two predators rather than one, or whether the greater complexity of the ecological interactions will mask any potential impacts of rapid evolution. If one prey genotype can be well-defended against both predators, the dynamics are like those of a predator–prey food chain. But if defense traits are predator-specific and incompatible, so that each genotype is vulnerable to attack by at least one predator, then rapid evolution produces distinctive behaviors at the population level: population typically oscillate in ways very different from either the food chain or a two-predator food web without rapid prey evolution. When many prey genotypes coexist, chaotic dynamics become likely. The effects of rapid evolution can still be detected by analyzing relationships between prey abundance and predator population growth rates using methods from functional data analysis.     

The evolution of rates of successful and unsuccessful predation

Summary The question, how will evolutionary change in a predator or in its prey change the ratio of the rate of successful captures to the rate of unsuccessful capture attempts is addressed. I argue that this ratio is not a good index of the predator's adaptation to prey capture, because decreased costs of capture attempts or increased assimilation efficiency (both favored by natural selection in the predator) will usually reduce the ratio. In addition, the evolution of increased ability to capture prey may lead to a reduction in the success/failure ratio. Analysis of several simple models suggests that this result is robust. The presence of unsuccessful predation does have an important influence on the evolution of predator traits that increase its rate of encounter with the prey; the presence of unsuccessful predation may cause the predator to increase its adaptations for prey detection in response to an increase in the prey's ability to avoid detection.