The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs

This article studies the effects of adaptive changes in predator and/or prey activities on the Lotka-Volterra predator-prey population dynamics. The model assumes the classical foraging-predation risk trade-offs: increased activity increases population growth rate, but it also increases mortality rate. The model considers three scenarios: prey only are adaptive, predators only are adaptive, and both species are adaptive. Under all these scenarios, the neutral stability of the classical Lotka-Volterra model is partially lost because the amplitude of maximum oscillation in species numbers is bounded, and the bound is independent of the initial population numbers. Moreover, if both prey and predators behave adaptively, the neutral stability can be completely lost, and a globally stable equilibrium would appear. This is because prey and/or predator switching leads to a piecewise constant prey (predator) isocline with a vertical (horizontal) part that limits the amplitude of oscillations in prey and predator numbers, exactly as suggested by Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey systems. Prey and predator activities in a long-term run are calculated explicitly. This article shows that predictions based on short-term behavioral experiments may not correspond to long-term predictions when population dynamics are considered.     

Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence

We analyze the transient dynamics of simple models of keystone predation, in which a predator preferentially consumes the dominant of two (or more) competing prey species. We show that coexistence is unlikely in many systems characterized both by successful invasion of either prey species into the food web that lacks it and by a stable equilibrium with high densities of all species. Invasion of the predator-resistant consumer species often causes the resident, more vulnerable prey to crash to such low densities that extinction would occur for many realistic population sizes. Subsequent transient cycles may entail very low densities of the predator or of the initially successful invader, which may also preclude coexistence of finite populations. Factors causing particularly low minimum densities during the transient cycles include biotic limiting resources for the prey, limited resource partitioning between the prey, a highly efficient predator with relatively slow dynamics, and a vulnerable prey whose population dynamics are rapid relative to the less vulnerable prey. Under these conditions, coexistence of competing prey via keystone predation often requires that the prey's competitive or antipredator characteristics fall within very narrow ranges. Similar transient crashes are likely to occur in other food webs and food web models.     

Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators

We synthesize previous theory on ideal free habitat selection to develop a model of predator movement mechanisms, when both predators and prey are mobile. We consider a continuous environment with an arbitrary distribution of resources, randomly diffusing prey that consume the resources, and predators that consume the prey. Our model introduces a very general class of movement rules in which the overall direction of a predator's movement is determined by a variable combination of (i) random diffusion, (ii) movement in the direction of higher prey density, and/or (iii) movement in the direction of higher density of the prey's resource. With this model, we apply an adaptive dynamics approach to two main questions. First, can it be adaptive for predators to base their movement solely on the density of the prey's resource (which the predators do not consume)? Second, should predator movements be exclusively biased toward higher densities of prey/resources, or is there an optimal balance between random and biased movements? We find that, for some resource distributions, predators that track the gradient of the prey's resource have an advantage compared to predators that track the gradient of prey directly. Additionally, we show that matching (consumers distributed in proportion to resources), overmatching (consumers strongly aggregated in areas of high resource density), and undermatching (consumers distributed more uniformly than resources) distributions can all be explained by the same general habitat selection mechanism. Our results provide important groundwork for future investigations of predator-prey dynamics.     

Antagonistic coevolution over productivity gradients

This study addresses the question of how spatial heterogeneity in prey productivity and migration act to determine geographic patterns in antagonistic coevolution with a predator. We develop and analyze a quantitative coevolutionary model for a predator-prey interaction. If the model is modified appropriately, the results could broadly apply to multispecies communities and to herbivore-plant, parasite-host, and parasitoid-host associations. Model populations are distributed over a gradient in prey birth rate (as a measure of productivity). Each population, in each patch, is made up of a suite of strains. Each strain of the predator has a certain ability to successfully attack each strain of the prey. We consider scenarios of isolated patches, global migration, and stepping-stone (i.e., local) migration over a linear string of patches. The most pervasive patterns are the following: investments in predator offense and prey defense are both maximal in the patches of highest prey productivity; when there are no constraints on maximal investment, mean predation evolves to highest levels in the most productive patches; similarly, the predator has a greater impact (measured as the percentage reduction in prey density) on the prey population in high productivity patches as compared with low productivity ones-in spite (even after evolution) of prey abundance being highest in the most productive patches; and migration has the net effect of shunting relatively offensive and defensive strains from productive patches to nonproductive ones, potentially resulting in the elimination of otherwise rare, low-investment clones. A modification of the model to gene-for-gene type interactions predicts that generalist strains (in terms of the range of strains the predator can exploit or the prey can fend off) dominate in productive areas of the prey, whereas specialists prevail in marginal habitats. Assuming a wide range of productivities over the prey's geographical distribution, the greatest strain diversity should be found in habitats of intermediate productivity. We discuss the implications of our study for adaptation and conservation. Empirical studies are in broad accord with our findings.     

Habitat choice in predator-prey systems: spatial instability due to interacting adaptive movements

The role of habitat choice behavior in the dynamics of predator-prey systems is explored using simple mathematical models. The models assume a three-species food chain in which each population is distributed across two or more habitats. The predator and prey adjust their locations dynamically to maximize individual per capita growth, while the prey's resource has a low rate of random movement. The two consumer species have Type II functional responses. For many parameter sets, the populations cycle, with predator and prey "chasing" each other back and forth between habitats. The cycles are driven by the aggregation of prey, which is advantageous because the predator's saturating functional response induces a short-term positive density dependence in prey fitness. The advantage of aggregation in a patch is only temporary because resources are depleted and predators move to or reproduce faster in the habitat with the largest number of prey, perpetuating the cycle. Such spatial cycling can stabilize population densities and qualitatively change the responses of population densities to environmental perturbations. These models show that the coupled processes of moving to habitats with higher fitness in predator and prey may often fail to produce ideal free distributions across habitats.     

Reversed predator–prey cycles are driven by the amplitude of prey oscillations

Ecoevolutionary feedbacks in predator–prey systems have been shown to qualitatively alter predator–prey dynamics. As a striking example, defense–offense coevolution can reverse predator–prey cycles, so predator peaks precede prey peaks rather than vice versa. However, this has only rarely been shown in either model studies or empirical systems. Here, we investigate whether this rarity is a fundamental feature of reversed cycles by exploring under which conditions they should be found. For this, we first identify potential conditions and parameter ranges most likely to result in reversed cycles by developing a new measure, the effective prey biomass, which combines prey biomass with prey and predator traits, and represents the prey biomass as perceived by the predator. We show that predator dynamics always follow the dynamics of the effective prey biomass with a classic ¼‐phase lag. From this key insight, it follows that in reversed cycles (i.e., ¾‐lag), the dynamics of the actual and the effective prey biomass must be in antiphase with each other, that is, the effective prey biomass must be highest when actual prey biomass is lowest, and vice versa. Based on this, we predict that reversed cycles should be found mainly when oscillations in actual prey biomass are small and thus have limited impact on the dynamics of the effective prey biomass, which are mainly driven by trait changes. We then confirm this prediction using numerical simulations of a coevolutionary predator–prey system, varying the amplitude of the oscillations in prey biomass: Reversed cycles are consistently associated with regions of parameter space leading to small‐amplitude prey oscillations, offering a specific and highly testable prediction for conditions under which reversed cycles should occur in natural systems.     

The impact of mortality on predator population size and stability in systems with stage-structured prey

The relationships between a predator population's mortality rate and its population size and stability are investigated for several simple predator-prey models with stage-structured prey populations. Several alternative models are considered; these differ in their assumptions about the nature of density dependence in the prey's population growth; the nature of stage-transitions; and the stage-selectivity of the predator. Instability occurs at high, rather than low predator mortality rates in most models with highly stage-selective predation; this is the opposite of the effect of mortality on stability in models with homogeneous prey populations. Stage-selective predation also increases the range of parameters that lead to a stable equilibrium. The results suggest that it may be common for a stable predator population to increase in abundance as its own mortality rate increases in stable systems, provided that the predator has a saturating functional response. Sufficiently strong density dependence in the prey generally reverses this outcome, and results in a decrease in predator population size with increasing predator mortality rate. Stability is decreased when the juvenile stage has a fixed duration, but population increases with increasing mortality are still observed in large areas of stable parameter space. This raises two coupled questions which are as yet unanswered; (1) do such increases in population size with higher mortality actually occur in nature; and (2) if not, what prevents them from occurring? Stage-structured prey and stage-related predation can also reverse the 'paradox of enrichment', leading to stability rather than instability when prey growth is increased.     

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.     

Spatial dynamics of communities with intraguild predation: the role of dispersal strategies

I investigate the influence of dispersal strategies on intraguild prey and predators (competing species that prey on each other). I find an asymmetry between the intraguild prey and predator in their responses to each other's dispersal. The intraguild predator's dispersal strategy and dispersal behavior have strong effects on the intraguild prey's abundance pattern, but the intraguild prey's dispersal strategy and behavior have little or no effect on the intraguild predator's abundance pattern. This asymmetry arises from the different constraints faced by the two species: the intraguild prey has to acquire resources while avoiding predation, but the intraguild predator only has to acquire resources. It leads to puzzling distribution patterns: when the intraguild prey and predator both move away from areas of high density, they become aggregated to high-density habitats, but when they both move toward areas of high resource productivity, they become segregated to resource-poor and resource-rich habitats. Aggregation is more likely when dispersal is random or less optimal, and segregation is more likely as dispersal becomes more optimal. The crucial implication is that trophic constraints dictate the fitness benefits of using dispersal strategies to sample environmental heterogeneity. A strategy that affords greater benefits to an intraguild predator can lead to a more optimal outcome for both the intraguild predator and prey than a strategy that affords greater benefits to an intraguild prey.     

Predation risk perception and food scarcity induce alterations of life-cycle traits of the mosquito Culex pipiens

Abstract.  1. Predators may affect prey populations by direct consumption, and by inducing defensive reactions of prey to the predation risk. Food scarcity frequently has effects on the inducible defences of prey, but no consistent pattern of food–predation risk interaction is known.
2. In this study the combined effect of food shortage and predation-risk perception in larvae of the mosquito Culex pipiens was investigated. Water exposed to the aquatic predator bug Notonecta glauca was used as a source of predation intimidation. Mosquito larvae were reared in three different media containing either no predator cues or the cues of N. glauca that had been fed on either C. pipiens larvae or on Daphnia magna . Food was provided in favourable or limited amount for these set-ups.
3. The results showed that chemical cues from the predators fed with prey's conspecifics caused a decreased survival, delayed pre-imaginal development, and reduction in body size of emerged mosquitoes, whereas chemical cues from predators fed with D. magna caused only delayed development. Food scarcity significantly exacerbates the negative effect of the predator cues on pre-imaginal development of C. pipiens . Effects of the cues on larval development and body size of imagoes are significantly stronger for females than for males.
4. The present study suggests that when food is limited, predators can affect population dynamics of prey not only by direct predation, but also by inducing lethal and sublethal effects due to perception of risk imposed by chemical cues. To understand the effects of predators on mosquito population dynamics, environmental parameters such as food deficiency should be considered.     

Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation

Interplay between predator and prey is a complex process in ecosystems due to its nature. The population dynamics can be affected by many extrinsic and intrinsic factors. In this paper, we make an attempt to uncover the effects from environmental disturbances when populations are subject to habitat complexity and aggregation effect. We firstly propose a stochastic predator-prey model with habitat complexity and aggregation efficiency for prey. We then mathematically analyze the model, to demonstrate the existence, uniqueness and the stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the existence of ergodic stationary distribution of the solution. We also establish sufficient conditions under which either only predator population dies out or the entire predator-prey model becomes extinct. Our theoretical and numerical results indicate that: (1) the environmental noises are disadvantage for the survival of biological populations; (2) when the density of prey is greater than one, prey aggregation can heighten the capability of predator species to capture prey and reduce the effect of environmental fluctuations, while when the density of prey is less than one, the results are opposite; (3) habitat complexity is propitious to the survival of prey population and may seriously threaten the persistence of the predator population.     

Gregarious behaviour of evasive prey

Gregarious behavior of potential prey was explained by Hamilton (1971) on the basis of risk-sharing: The probability of being picked up by a predator is small when one makes part of a large aggregate of prey. This argument holds only if the predator chooses its victims at random. It is not the case for herds of evasive prey in the open, where prey's gregarious behavior, favorable for the fast group members, makes it easier for the predator to home in on the slowest ones. We show conditions under which gregarious behavior of the relatively fast prey individuals leaves slowest prey with no other choice but to join the group. Failing to do so would signal their vulnerability, making them a preferred target for the predator. Analysis of an n+1 player game of a predator and n unequal prey individuals clarifies conditions for fully gregarious, partially gregarious, or solitary behavior of the prey.     

A functional response model of a predator population foraging in a patchy habitat

1. Functional response models (e.g. Holling's disc equation) that do not take the spatial distributions of prey and predators into account are likely to produce biased estimates of predation rates. 2. To investigate the consequences of ignoring prey distribution and predator aggregation, a general analytical model of a predator population occupying a patchy environment with a single species of prey is developed. 3. The model includes the density and the spatial distribution of the prey population, the aggregative response of the predators and their mutual interference. 4. The model provides explicit solutions to a number of scenarios that can be independently combined: the prey has an even, random or clumped distribution, and the predators show a convex, sigmoid, linear or no aggregative response. 5. The model is parameterized with data from an acarine predator-prey system consisting of Phytoseiulus persimis and Tetranychus urticae inhabiting greenhouse cucumbers. 6. The model fits empirical data quite well and much better than if prey and predators were assumed to be evenly distributed among patches, or if the predators were distributed independently of the prey. 7. The analyses show that if the predators do not show an aggregative response it will always be an advantage to the prey to adopt a patchy distribution. On the other hand, if the predators are capable of responding to the distribution of prey, then it will be an advantage to the prey to be evenly distributed when its density is low and switch to a more patchy distribution when its density increases. The effect of mutual interference is negligible unless predator density is very high. 8. The model shows that prey patchiness and predator aggregation in combination can change the functional response at the population level from type II to type III, indicating that these factors may contribute to stabilization of predator-prey dynamics.     

Coevolutionary alternation in antagonistic interactions

Coevolution between parasites and hosts or predators and prey often involves multiple species with similar kinds of defenses and counter-defenses. Classic examples include the interactions between phytophagous insects and their host plants, thick-shelled invertebrates and their shell-crushing predators, and ungulates and their predators. There are three major hypotheses for the nonequilibrium coevolutionary dynamics of these multispecific trophic interactions: escalation in traits, cycles in traits leading to fluctuating polymorphisms, and coevolutionary alternation. The conditions under which cycles and escalation are likely to occur have been well developed theoretically. In contrast, the conditions favoring coevolutionary alternation-evolutionary fluctuations in predator or prey preference driven by evolutionary shifts in relative levels of prey defense and vice versa-have yet to be identified. Using a set of quantitative coevolutionary models, we demonstrate that coevolutionary alternation can occur across a wide range of biologically plausible conditions. The result is often repeated, and potentially rapid, evolutionary shifts in patterns of specialization within networks of interacting species.     

Field evidence of trait-mediated indirect interactions in a rocky intertidal food web

Studies on the implications of food web interactions to community structure have often focused on density-mediated interactions between predators and their prey. This approach emphasizes the importance of predator regulation of prey density via consumption (i.e. lethal effects), which, in turn, leads to cascading effects on the prey's resources. A more recent and contrasting view emphasizes the importance of non-lethal predator effects on prey traits (e.g. behaviour, morphology), or trait-mediated interactions. On rocky intertidal shores in New England, green crab ( Carcinus maenas ) predation is thought to be important to patterns of algal abundance and diversity by regulating the density of herbivorous snails ( Littorina littorea ). We found, however, that risk cues from green crabs can dramatically suppress snail grazing, with large effects on fucoid algal communities. Our results suggest that predator-induced changes in prey behaviour may be an important and under-appreciated component of food web interactions and community dynamics on rocky intertidal shores.     

How population loss through habitat boundaries determines the dynamics of a predator–prey system

The increased persistence of predator–prey systems when interactions are distributed through the space has been acknowledged by both empirical and theoretical studies. One salient feature of predator–prey interactions in heterogeneous space, for example, is the existence of cycles with reduced amplitude when compared with a homogeneous landscape. Although the role of spatial interactions in shaping the dynamics of predator–prey systems has been extensively studied, still very few works have focused on the effects of habitat loss and fragmentation on these systems. In this work, we study the population dynamics of a predator–prey system in a single finite habitat with flux at the boundaries. Species movement and growth are described through a reaction–diffusion model with Rosenzweig–MacArthur type local interactions. Conforming with the existing literature, we find that the reduction of habitat size, or increasing of species movement rates equivalently, has the potential to decrease the amplitude of oscillations and even bring the system to a steady coexistence equilibrium above a threshold. We observe, however, situations in which this trend is reversed. This occurs when species movement rates and response at patch boundaries interact to induce non-trivial patterns of species distributions. These distributions are characterized by anti-correlation between predator and prey, creating then spatial refugia for prey. Our results highlight the role of population loss through habitat boundaries in determining the dynamics of predator–prey interactions.     

Pulsed-resource dynamics increase the asymmetry of antagonistic coevolution between a predatory protist and a prey bacterium

Temporal resource fluctuations could affect the strength of antagonistic coevolution through population dynamics and costs of adaptation. We studied this by coevolving the prey bacterium Serratia marcescens with the predatory protozoa Tetrahymena thermophila in constant and pulsed-resource environments for approximately 1300 prey generations. Consistent with arms race theory, the prey evolved to be more defended, whereas the predator evolved to be more efficient in consuming the bacteria. Coevolutionary adaptations were costly in terms of reduced prey growth in resource-limited conditions and less efficient predator growth on nonliving resource medium. However, no differences in mean coevolutionary changes or adaptive costs were observed between environments, even though resource pulses increased fluctuations and mean densities of coevolving predator populations. Interestingly, a surface-associated prey defence mechanism (bacterial biofilm), to which predators were probably unable to counter-adapt, evolved to be stronger in pulsed-resource environment. These results suggest that temporal resource fluctuations can increase the asymmetry of antagonistic coevolution by imposing stronger selection on one of the interacting species.     

Effects of vole fluctuations on the population dynamics of the barn owl <Emphasis Type="Italic">Tyto alba</Emphasis>

Many predator species feed on prey that fluctuates in abundance from year to year. Birds of prey can face large fluctuations in food abundance i.e. small mammals, especially voles. These annual changes in prey abundance strongly affect the reproductive success and mortality of the individual predators and thus can be expected to influence their population dynamics and persistence. The barn owl, for example, shows large fluctuations in breeding success that correlate with the dynamics in voles, their main prey species. Analysis of the impact of fluctuations in vole abundance (their amplitude, peaks and lows, cycle length and regularity) with a simple predator prey model parameterized with literature data indicates population persistence is especially affected by years with low vole abundance. In these years the population can decline to low owl numbers such that the ensuing peak vole years cannot be exploited. This result is independent of the length and regularity of vole fluctuations. The relevance of this result for conservation of the barn owl and other birds of prey that show a numerical response to fluctuating prey species is discussed.     

Dynamics of predator and modular prey: effects of module consumption on stability of prey–predator system

In traditional models of predator–prey population dynamics, it is usually assumed that consumed prey are immediately removed from the population. However, in plant–herbivore interactions, damaged plants are generally alive after attacks by herbivores. This can result in successive or simultaneous attacks by multiple predators on a single prey item (here, the term prey is expanded to include plants). We constructed a mathematical model with two time scales, taking into account predation processes within a generation, considering post‐predation survival and the modularity of prey. We assumed that a prey item can be divided into modules and that it can be fed on by multiple predators or parasitized by multiple parasites at the same time. The model includes two essential factors: the modularity of prey for predators (n) and the detaching/attaching ratio of predators to prey (ε). Based on the formulae, we revealed a general property of realistic dynamics in plant–herbivore and host–parasite interactions. The analysis showed that the model could be approximated by models with the type I, type II or Beddington–DeAngelis functional responses by taking appropriate limits to the situations. When modularity is low or the detaching/attaching ratio is high, population dynamics tend to be stabilized. These stabilizing effects may be related to interference competition among predator individuals or increases in free prey modules and free predator individuals. In the limit of high modularity, the ratio of the attached prey modules to the total prey modules becomes negligible and the dynamics tend to be destabilized. However, if quantity and quality of prey modules are negatively correlated, the equilibrium is likely to be stabilized at high modularity as long as it remains feasible. These results suggest that considering post‐predation survival and modularity of prey is crucial to understand predator–prey interactions.     

PREY ADAPTATION AS A CAUSE OF PREDATOR-PREY CYCLES

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.