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.     

Predation, competition, and nutrient recycling: a stoichiometric approach with multiple nutrients

A model for two competing prey species and one predator is formulated in which three essential nutrients can limit growth of all populations. Prey take up dissolved nutrients and predators ingest prey, assimilating a portion of ingested nutrients and recycling or respiring the balance. For all species, the nutrient contents of individuals vary and growth is coupled to increasing content of the limiting nutrient. This model was parameterized to describe a flagellate preying on two bacterial species, with carbon (C), nitrogen (N), and phosphorus (P) as nutrients. Parameters were chosen so that the two prey species would stably coexist without predators under some nutrient supply conditions. Using numerical simulations, the long-term outcomes of competition and predation were explored for a gradient of N:P supply ratios, varying C supply, and varying preference of the predator for the two prey. Coexistence and competitive exclusion both occurred under some conditions of nutrient supply and predator preference. As in simpler models of competition and predation these outcomes were largely governed by apparent competition mediated by the predator, and resource competition for nutrients whose effective supply was partly governed by nutrient recycling also mediated by the predator. For relatively small regions of parameter space, more complex outcomes with multiple attractors or three-species limit cycles occurred. The multiple constraints posed by multiple nutrients held the amplitudes of these cycles in check, limiting the influence of complex dynamics on competitive outcomes for the parameter ranges explored.     

Selective feeding by protozoa: model and experimental behaviors and their consequences for population stability

Selective feeding by zooplankton can have profound consequencesfor the stability of grazer and prey populations, as demonstratedby the behavior of plankton dynamics models. We present an analyticalapproach—calculation of prey ratio trajectories—thatreveals unambiguously whether selective feeding behavior isstabilizing (i.e. provides a refuge for preferred prey speciesat low prey concentrations) or destabilizing (i.e. results inelimination of prey populations). Prey ratio trajectories werecalculated for three modeled selective feeding behaviors. Constantselection was consistently destabilizing, while selection behaviorsthat changed inresponse to either prey ratio or prey abundancecould be stabilizing. Batch culture experiments with four protozoangrazer species (three ciliates, one heterotrophic dinoftagellate)demonstrated that protozoa fed selectively in every case, weaklypreferring the larger of the two algal species offered. Stabilizingselection was observed only in the experiment with Favella sp.,however, meaning that only this species altered its selectionbehavior in response to changing experimental conditions. Becauseprotozoa are the major grazers of phytoplankton in many planktonicsystems, our findings indicate that the use of selective feedingbehaviors to stabilize plankton dynamics models needs carefulevaluation. The modeling and graphical techniques presentedhereare a tool for linking further exploration of selective feedingbehaviors with the development of planktondynamics models.     

On the structure and stability of mutualistic systems: Analysis of predator-prey and competition models as modified by the action of a slow-growing mutualist

Two basic models of mutualism are presented in which interactions among three species lead to mutualism between two of them. The models represent 2-species predator-prey or competition systems in which a third species acts as a mutualist with either the predator, the prey, or one of the competitors. The models include the assumptions that there is a cost of associating with the mutualist and that the mutualist population grows much more slowly than the other two populations. Special cases of these two models correspond to six qualitatively different types of mutualistic benefit, all of which are known to occur in nature: deterring predation, increasing prey availability, feeding on (or competing with) a predator, increasing competitive interactions, decreasing competitive interactions, and feeding on (or competing with) a competitor. These models and their special cases are subjected to a local stability analysis. The results show that mutualism based upon deterring predation, competing with a predator, or decreasing competitive interactions enhances local stability, while mutualism based upon increasing prey availability or increasing competitive interactions reduces local stability. These results clearly reject the idea that mutualism is an inherently unstable process, and reinforces the idea that each different kind of mutualism will have to be considered separately. Compared to 2-species models of mutualism, the 3-species models provide a more realistic representation of the structure of many mutualistic systems, the mechanisms by which one species benefits another, and the regulation of the interaction.     

Multiple predator effects result in risk reduction for prey across multiple prey densities

Investigating how prey density influences a prey’s combined predation risk from multiple predator species is critical for understanding the widespread importance of multiple predator effects. We conducted experiments that crossed six treatments consisting of zero, one, or two predator species (hellgrammites, greenside darters, and creek chubs) with three treatments in which we varied the density of mayfly prey. None of the multiple predator effects in our system were independent, and instead, the presence of multiple predator species resulted in risk reduction for the prey across both multiple predator combinations and all three levels of prey density. Risk reduction is likely to have population-level consequences for the prey, resulting in larger prey populations than would be predicted if the effects of multiple predator species were independent. For one of the two multiple predator combinations, the magnitude of risk reduction marginally increased with prey density. As a result, models predicting the combined risk from multiple predator species in this system will sometimes need to account for prey density as a factor influencing per-capita prey death rates.     

Dynamics and responses to mortality rates of competing predators undergoing predator-prey cycles

Two or more competing predators can coexist using a single homogeneous prey species if the system containing all three undergoes internally generated fluctuations in density. However, the dynamics of species that coexist via this mechanism have not been extensively explored. Here, we examine both the nature of the dynamics and the responses of the mean densities of each predator to mortality imposed upon it or its competitor. The analysis of dynamics uncovers several previously undescribed behaviors for this model, including chaotic fluctuations, and long-term transients that differ significantly from the ultimate patterns of fluctuations. The limiting dynamics of the system can be loosely classified as synchronous cycles, asynchronous cycles, and chaotic dynamics. Synchronous cycles are simple limit cycles with highly positively correlated densities of the two predator species. Asynchronous cycles are limit cycles, frequently of complex form, including a significant period during which prey density is nearly constant while one predator gradually, monotonically replaces the other. Chaotic dynamics are aperiodic and generally have intermediate correlations between predator densities. Continuous changes in density-independent mortality rates often lead to abrupt transitions in mean population sizes, and increases in the mortality rate of one predator may decrease the population size of the competing predator. Similarly, increases in the immigration rate of one predator may decrease its own density and increase the density of the other predator. Proportional changes in one predator's birth and death rate functions can have significant effects on the dynamics and mean densities of both predator species. All of these responses to environmental change differ from those observed when competitors coexist stably as the result of resource (prey) partitioning. The patterns described here occur in many other competition models in which there are cycles and differences in the linearity of the responses of consumers to their resources.     

Can environmental variation generate positive indirect effects in a model of shared predation?

Classic models of apparent competition predict negative indirect effects between prey with a shared enemy. If predator per capita growth rates are nonlinear, then endogenously generated periodic cycles are predicted to generate less negative or even positive indirect effects between prey. Here I determine how exogenous mechanisms such as environmental variation could modify indirect effects. I find that exogenous variation can have a broader range of effects on indirect interactions than endogenously generated cycles. Indirect effects are altered by environmental variation even in simple models for which the per capita growth rate of the predator species is a linear function of population densities. Temporal variation that affects the predator attack rate or the conversion efficiency can lead to large increases or decreases in the indirect effects between prey, dependent on how prey populations co-vary with the environmental variation. Positive indirect effects can occur when the period of environmental variation is close to the natural period of the biological system and shifts in subharmonic resonance occur with the addition of the second prey. Models that include nonlinear numerical responses generally lead to indirect effects that are sensitive to environmental variation in more parameters and across a wider range of frequencies.     

Alternative causes for range limits: a metapopulation perspective

All species have limited distributions at broad geographical scales. At local scales, the distribution of many species is influenced by the interplay of the three factors of habitat availability, local extinctions and colonization dynamics. We use the standard Levins metapopulation model to illustrate how gradients in these three factors can generate species' range limits. We suggest that the three routes to range limits have radically different evolutionary implications. Because the Levins model makes simplifying assumptions about the spatial coupling of local populations, we present numerical studies of spatially explicit metapopulation models that complement the analytical model. The three routes to range limits give rise to distinct spatiotemporal patterns. Range limits in one species can also arise because of environmental gradients impinging upon other species. We briefly discuss a predator–prey example, which illustrates indirect routes to range limits in a metacommunity context.     

Stable periodic solutions for two species,density dependent coevolution

This paper studies a four dimensional system of time-autonomous ordinary differential equations which models the interaction of two diploid, diallelic populations with overlapping generations. The variables are two population densities and an allele frequency in each of the populations. For single species models, the existence of periodic solutions requires that the genotype fitness functions be both frequency and density dependent. But, for two species exhibiting a predator-prey interaction, two examples are presented where there exists asymptotically stable cycles with fitness functions only density dependent. In the first example, the Hopf bifurcation theorem is used on a two parameter, polynomial vector field. The second example has a Michaelis-Menten or Holling term for the interaction between predator and prey; and, for this example, the existence and uniqueness of limit cycles for a wide range of parameter values has been established in the literature.     

Non-consumptive effects of larval Salamandra on crustacean prey: can eggs detect predators?

Predators affect prey populations not only by prey consumption but also in nonconsumptive ways including modifying prey behavior. I tested the effects of fire salamander larvae (Salamandra infraimmaculata) on populations of co-occurring crustacean species in artificial outdoor pools. I also tested whether these effects were due entirely to prey consumption by Salamandra larvae or alternatively to some nonconsumptive effect. The soil (containing crustacean eggs) added to the artificial pools was collected from a dried-out temporary pool that is inhabited by Salamandra during the early part of the hydroperiod. I randomly assigned the pools to one of three treatments: control, free Salamandra, or caged Salamandra. Free salamander larvae could roam the entire pool and prey upon crustaceans. Caged salamander larvae were placed within a cage with having 250-μm mesh windows. They could not prey upon the crustaceans but could, for example, influence them by chemical cues. Densities of the three dominant crustacean species (Arctodiaptomus similis, Ceriodaphnia quadrangula and Cyzicus sp.) were drastically reduced in both salamander treatments compared to the control. Crustacean densities, however, were not significantly different between the two salamander treatments. One plausible explanation is that crustacean eggs can detect the presence of this predator via chemical cues and delay hatching. Received: 4 March 1996 / Accepted, 23 October 1996     

Host-dependent differences in prey acquisition between populations of a kleptoparasitic spider Argyrodes kumadai (Araneae: Theridiidae)

Abstract.  1. A kleptoparasitic spider, Argyrodes kumadai , is known to use phylogenetically unrelated host species in different regions – Cyrtophora moluccensis (Araneidae) in south-west Japan and Agelena silvatica (Agelenidae) in north-east Japan. The work reported here examined whether differences in host characters affect prey acquisition of A. kumadai .
2. Field surveys showed that prey-biomass capture rate of Argyrodes was significantly higher in populations parasitising Cyrtophora than in populations parasitising Agelena . Although Argyrodes appeared to catch fewer prey within Cyrtophora webs, they were able to feed upon substantially larger prey.
3. Differences in prey-biomass capture rate were found to reflect differences in host traits rather than regional differences in potential prey availability. Individuals in populations parasitising Cyrtophora were observed to acquire prey via a number of foraging tactics that included stealing wrapped food bundles, feeding upon prey remains and, in the case of large prey items, feeding together with the host. In contrast, individuals in populations parasitising Agelena were only ever observed to feed upon small prey items ignored by its host.
4. This variability in prey acquisition between kleptoparasite populations reflected different opportunities for feeding within their respective host webs – opportunities that were primarily determined by the foraging behaviour of the host. One key trait associated with host foraging behaviour was host-web structure, namely the presence/absence of a retreat.     

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.     

Environmental factors influencing spotted hyena and lion population biomass across Africa

The spotted hyena (Crocuta crocuta Erxleben) and the lion (Panthera leo Linnaeus) are two of the most abundant and charismatic large mammalian carnivores in Africa and yet both are experiencing declining populations and significant pressures from environmental change. However, with few exceptions, most studies have focused on influences upon spotted hyena and lion populations within individual sites, rather than synthesizing data from multiple locations. This has impeded the identification of over‐arching trends behind the changing biomass of these large predators. Using partial least squares regression models, influences upon population biomass were therefore investigated, focusing upon prey biomass, temperature, precipitation, and vegetation cover. Additionally, as both species are in competition with one other for food, the influence of competition and evidence of environmental partitioning were assessed. Our results indicate that spotted hyena biomass is more strongly influenced by environmental conditions than lion, with larger hyena populations in areas with warmer winters, cooler summers, less drought, and more semi‐open vegetation cover. Competition was found to have a negligible influence upon spotted hyena and lion populations, and environmental partitioning is suggested, with spotted hyena population biomass greater in areas with more semi‐open vegetation cover. Moreover, spotted hyena is most heavily influenced by the availability of medium‐sized prey biomass, whereas lion is influenced more by large size prey biomass. Given the influences identified upon spotted hyena populations in particular, the results of this study could be used to highlight populations potentially at greatest risk of decline, such as in areas with warming summers and increasingly arid conditions.     

Metapopulation dynamics for spatially extended predator–prey interactions

Traditional metapopulation theory classifies a metapopulation as a spatially homogeneous population that persists on neighboring habitat patches. The fate of each population on a habitat patch is a function of a balance between births and deaths via establishment of new populations through migration to neighboring patches. In this study, we expand upon traditional metapopulation models by incorporating spatial heterogeneity into a previously studied two-patch nonlinear ordinary differential equation metapopulation model, in which the growth of a general prey species is logistic and growth of a general predator species displays a Holling type II functional response. The model described in this work assumes that migration by generalist predator and prey populations between habitat patches occurs via a migratory corridor. Thus, persistence of species is a function of local population dynamics and migration between spatially heterogeneous habitat patches. Numerical results generated by our model demonstrate that population densities exhibit periodic plane-wave phenomena, which appear to be functions of differences in migration rates between generalist predator and prey populations. We compare results generated from our model to results generated by similar, but less ecologically realistic work, and to observed population dynamics in natural metapopulations.     

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.     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.     

Dynamic versus instantaneous models of diet choice

We investigate the dynamics of a series of two-prey-one-predator models in which the predator exhibits adaptive diet choice based on the different energy contents and/or handling times of the two prey species. The predator is efficient at exploiting its prey and has a saturating functional response; these two features combine to produce sustained population cycles over a wide range of parameter values. Two types of models of behavioral change are compared. In one class of models ("instantaneous choice"), the probability of acceptance of the poorer prey by the predator instantaneously approximates the optimal choice, given current prey densities. In the second class of models ("dynamic choice"), the probability of acceptance of the poorer prey is a dynamic variable, which begins to change in an adaptive direction when prey densities change but which requires a finite amount of time to approach the new optimal behavior. The two types of models frequently predict qualitatively different population dynamics of the three-species system, with chaotic dynamics and complex cycles being a common outcome only in the dynamic choice models. In dynamic choice models, factors that reduce the rate of behavioral change when the probability of accepting the poorer prey approaches extreme values often produce complex population dynamics. Instantaneous and dynamic models often predict different average population densities and different indirect interactions between prey species. Alternative dynamic models of behavior are analyzed and suggest, first, that instantaneous choice models may be good approximations in some circumstances and, second, that different types of dynamic choice models often lead to significantly different population dynamics. The results suggest possible behavioral mechanisms leading to complex population dynamics and highlight the need for more empirical study of the dynamics of behavioral change.     

Selection-delayed population dynamics in baleen whales and beyond

While it is known that population cycles are driven by delayed density-dependent feedbacks, the search for a common feedback mechanism in natural populations with cyclic dynamics has remained unresolved for almost a century. To identify the existence and cause of delayed feedbacks I apply six age- and sex-structured population dynamics models to seven species of baleen whales (suborder Mysticeti) that were heavily depleted by past commercial whaling. The six models include a predator–prey model with killer whale (Orcinus orca) as the predator, and five singe-species models based on (1) exponential growth, (2) density-regulated growth, (3) density-regulated growth with depensation, (4) delayed density-regulated growth and (5) selection-delayed dynamics. The latter model has a density-regulated growth rate that is accelerated and decelerated by the intra-specific natural selection that arises from the density-dependent competitive interactions between the individuals in the population. Essential parameters are estimated by a Bayesian statistical framework, and it is shown that baleen whales have a delayed recovery relative to density-regulated growth. The time-lag is not explained by depensation, or by interactions with prey or predators. It is instead resolved by a selection-delayed acceleration of the intrinsic growth rate. The results are discussed in relation to the literature on cyclic dynamics, and it is noted (1) that selection-delayed dynamics is both theoretically and empirically sufficient for cyclic population dynamics, (2) that it is widespread in natural populations owing to the widespread occurrence of otherwise unexplained phenotypic cycles in populations with cyclic dynamics, and (3) that there is a lack of empirical evidence showing that predator–prey interactions is a sufficient cause for the cyclic dynamics of natural populations. The conclusion stresses the importance of intra-specific delays in cyclic dynamics, and suggests that it is the acceleration of the growth rate, and not the growth rate itself, that is determined by the density-dependent environment.     

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.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.