Impacts of predation on dynamics of age-structured prey: Allee effects and multi-stability

With a series of mathematical models, we explore impacts of predation on a prey population structured into two age classes, juveniles and adults, assuming generalist, age-specific predators. Predation on any age class is either absent, or represented by types II or III functional responses, in various combinations. We look for Allee effects or more generally for multiple stable steady states in the prey population. One of our key findings is the occurrence of a predator pit (low-density ??refuge?? state of prey induced by predation; the chance of escaping predation thus increases both below and above an intermediate prey density) when only one age class is consumed and predators use a type II functional response ??this scenario is known to occur for an unstructured prey consumed via a type III functional response and can never occur for an unstructured prey consumed via a type II one. In the case where both age classes are consumed by type II generalist predators, an Allee effect occurs frequently, but some parameters give also rise to a predator pit and even three stable equilibria (one extinction equilibrium and two positive ones??Allee effect and predator pit combined). Multiple positive stable equilibria are common if one age class is consumed via a type II functional response and the other via a type III functional response??here, in addition to the behaviours mentioned above one may even observe three stable positive equilibria????double?? predator pit. Some of these results are discussed from the perspective of population management.     

Egg-eating age-structured predators in interaction with age-structured prey

In this paper, we consider a system of integrodifferential equations which models a predator-prey system with both species (predator and prey) age-structured and predators living only on the eggs of prey. The present model is a generalization of the model given in [20]. The existence, stability, and instability of nonnegative equilibria is studied assuming a general fecundity rate function for the prey. With a special choice of fecundity rate function for the predator it is shown here that a large maturation period m of the predator leads to stability. This seems to be contrary to the usual rule of thumb that increasing delays in growth rate responses cause instabilities.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

The influence of vigilance on intraguild predation

Theoretical work on intraguild predation suggests that if a top predator and an intermediate predator share prey, the system will be stable only if the intermediate predator is better at exploiting the prey, and the top predator gains significantly from consuming the intermediate predator. In mammalian carnivore systems, however, there are examples of top predator species that attack intermediate predator species, but rarely or never consume the intermediate predator. We suggest that top predators attacking intermediate predators without consuming them may not only reduce competition with the intermediate predators, but may also increase the vigilance of the intermediate predators or alter the vigilance of their shared prey, and that this behavioral response may help to maintain the stability of the system. We examine two models of intraguild predation, one that incorporates prey vigilance, and a second that incorporates intermediate predator vigilance. We find that stable coexistence can occur when the top predator has a very low consumption rate on the intermediate predator, as long as the attack rate on the intermediate predator is relatively large. However, the system is stable when the top predator never consumes the intermediate predator only if the two predators share more than one prey species. If the predators do share two prey species, and those prey are vigilant, increasing top predator attack rates on the intermediate predator reduces competition with the intermediate predator and reduces vigilance by the prey, thereby leading to higher top predator densities. These results suggest that predator and prey behavior may play an important dynamical role in systems with intraguild predation.     

Diet choice and predator—prey dynamics

Summary We compare the dynamics of predator-prey systems with specialist predators or adaptive generalist predators that base diet choice on energy-maximizing criteria. Adaptive predator behaviour leads to functional responses that are influenced by the relative abundance of alternate prey. This results in the per capita predation risk being positively density-dependent near points of diet expansion. For a small set of parameter values, systems with adaptive predators can be locally stable whereas systems with specialist predators would be unstable. This occurs mainly when alternate prey have low enough profitability that predators cannot sustain themselves indefinitely when feeding on alternate prey. Local stability of systems with adaptive predator behaviour is inversely related to the goodness of fit to optimal diet choice criteria. Hence, typical patterns of partial prey preference are more stabilizing than perfect optimal diet selection. Locally stable systems with adaptive predators are often globally unstable, converging on limit cycles for many initial population densities. The small range of parameter combinations and initial population densities leading to stable equilibria suggest that adaptive diet selection is unlikely to be a ubiquitous stabilizing factor in trophic interactions.     

Modeling direct positive feedback between predators and prey

Predators can have positive impacts on their prey through such mechanisms as nutrient mineralization and prey transport. These positive feedbacks have the potential to change predictions based on food web theory, such as the assertion that enrichment is destabilizing. We present a model of a simple food web, consisting of a resource, a consumer, and its predator. We assume that the predator has a direct positive effect on the consumer, by increasing the rate at which the consumer acquires resources. We consider two cases: the feedback strength is a saturating function of predator density, or it is proportional to the encounter rate between predators and prey. In both cases, the positive feedback is stabilizing, delaying or preventing the onset of oscillations due to enrichment. Positive feedback can introduce an Allee effect for the predator population, yielding multiple stable equilibria. Strong positive feedback can yield counterintuitive results such as a transient increase in consumer density following the introduction of predators, and a decrease in the resource pool following enrichment.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Generalist and specialist predators that mediate permanence in ecological communities

 General dynamic models of systems with two prey and one or two predators are considered. After rescaling the equations so that both prey have the same intrinsic rate of growth, it is shown that there exists a generalist predator that can mediate permanence if and only if there is a population density of a prey at which its per-capita growth rate is positive yet less than its competitor’s invasion rate. In particular, this result implies that if the outcome of competition between the prey is independent of initial conditions, then there exists a generalist predator that mediates permanence. On the other hand, if the outcome of competition is contingent upon initial conditions (i.e., the prey are bistable), then there may not exist a suitable generalist predator. For example, bistable prey modeled by the Ayala–Gilpin (θ-Logistic) equations can be stabilized if and only if θ<1 for one of the prey. It is also shown that two specialist predators always can mediate permanence between bistable prey by creating a repelling heteroclinic cycle consisting of fixed points and limit cycles. Received 10 August 1996; received in revised form 21 March 1997     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Impacts of Foraging Facilitation Among Predators on Predator-prey Dynamics

Whereas impacts of predator interference on predator-prey dynamics have received considerable attention, the “inverse” process—foraging facilitation among predators—have not been explored yet. Here we show, via mathematical models, that impacts of foraging facilitation on predator-prey dynamics depend on the way this process is modeled. In particular, foraging facilitation destabilizes predator-prey dynamics when it affects the encounter rate between predators and prey. By contrast, it might have a stabilizing effect if the predator handling time of prey is affected. Foraging facilitation is an Allee effect mechanism among predators and we show that for many parameters, it gives rise to a demographic Allee effect or a critical predator density in need to be crossed for predators to persist. We explore also the effects of predator interference, to make the picture “symmetric” and complete. Predator interference is shown to stabilize predator-prey dynamics once its strength is not too high, and thus corroborates results of others. On the other hand, there is a wide range of model parameters for which predator interference gives rise to three co-occurring co-existence equilibria. Such a multi-equilibrial regime is rather robust as we observe it for all the functional response types we explore. This is a previously unreported phenomenon which we show cannot occur for the Beddington–DeAngelis functional response. An interesting topic for future research thus might be to seek for general conditions on predator functional responses that would produce multiple co-existence equilibria in a predator-prey model.     

Stabilizing Effect of Cannibalism in a Two Stages Population Model

In this paper we build a prey–predator model with discrete weight structure for the predator. This model will conserve the number of individuals and the biomass and both growth and reproduction of the predator will depend on the food ingested. Moreover the model allows cannibalism which means that the predator can eat the prey but also other predators. We will focus on a simple version with two weight classes or stage (larvae and adults) and present some general mathematical results. In the last part, we will assume that the dynamics of the prey is fast compared to the predator’s one to go further in the results and eventually conclude that under some conditions, cannibalism can stabilize the system: more precisely, an unstable equilibrium without cannibalism will become almost globally stable with some cannibalism. Some numerical simulations are done to illustrate this result.     

Predator‐induced morphological defences as by‐products of prey behaviour: a review and prospectus

Predator‐induced morphological defences (PIMDs) are ubiquitous. Many PIMDs may be mediated by prey behaviour rather than directly cued by predators. A survey of 92 studies indicated 40 that quantified prey behaviour, all of which document positive associations between defence production and activity reduction. Thus, PIMDs are associated with changes in prey activity, which could have caused the morphological change. We propose two possible mechanisms: 1) decreased activity reduces feeding rate, resulting in lower growth and morphological change; and 2) activity reduction conserves energy, which is reallocated for growth, subsequently changing morphology. Resource availability also causes similar morphological change to predator presence, suggesting confounding effects of resources and predators with current methodology. Future studies should estimate food ingestion, assimilation efficiency, and growth rate in the presence and absence of predators, crossing predator presence with resource levels. Not all PIMDs will be behaviourally‐mediated, but consideration of causal linkages between prey behaviour and PIMDs is warranted.     

Immigration can destabilize tri-trophic interactions: implications for conservation of top predators

Top predators often have large home ranges and thus are especially vulnerable to habitat loss and fragmentation. Increasing connectance among habitat patches is therefore a common conservation strategy, based in part on models showing that increased migration between subpopulations can reduce vulnerability arising from population isolation. Although three-dimensional models are appropriate for exploring consequences to top predators, the effects of immigration on tri-trophic interactions have rarely been considered. To explore the effects of immigration on the equilibrium abundances of top predators, we studied the effects of immigration in the three-dimensional Rosenzweig-MacArthur model. To investigate the stability of the top predator equilibrium, we used MATCONT to perform a bifurcation analysis. For some combinations of model parameters with low rates of top predator immigration, population trajectories spiral towards a stable focus. Holding other parameters constant, as immigration rate is increased, a supercritical Hopf bifurcation results in a stable limit cycle and thus top predator populations that cycle between high and low abundances. Furthermore, bistability arises as immigration of the intermediate predator is increased. In this case, top predators may exist at relatively low abundances while prey become extinct, or for other initial conditions, the relatively higher top predator abundance controls intermediate predators allowing for non-zero prey population abundance and increased diversity. Thus, our results reveal one of two outcomes when immigration is added to the model. First, over some range of top predator immigration rates, population abundance cycles between high and low values, making extinction from the trough of such cycles more likely than otherwise. Second, for relatively higher intermediate predator migration rates, top predators may exist at low values in a truncated system with impoverished diversity, again with extinction more likely.     

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.     

一个具有寄生虫病感染和恢复的捕食模型(英文)

本文假设感染的食饵有恢复率和对捕食者有收获,研究了一个对部分食饵和全部捕食者具有寄生虫病感染的捕食模型.用定性理论证明了边界和正平衡点的稳定性.结论表明恢复率和收获率对正平衡点的稳定性有影响.     

Inferring Predator Behavior from Attack Rates on Prey-Replicas That Differ in Conspicuousness

Behavioral ecologists and evolutionary biologists have long studied how predators respond to prey items novel in color and pattern. Because a predatory response is influenced by both the predator’s ability to detect the prey and a post-detection behavioral response, variation among prey types in conspicuousness may confound inference about post-prey-detection predator behavior. That is, a relatively high attack rate on a given prey type may result primarily from enhanced conspicuousness and not predators’ direct preference for that prey. Few studies, however, account for such variation in conspicuousness. In a field experiment, we measured predation rates on clay replicas of two aposematic forms of the poison dart frog Dendrobates pumilio, one novel and one familiar, and two cryptic controls. To ask whether predators prefer or avoid a novel aposematic prey form independently of conspicuousness differences among replicas, we first modeled the visual system of a typical avian predator. Then, we used this model to estimate replica contrast against a leaf litter background to test whether variation in contrast alone could explain variation in predator attack rate. We found that absolute predation rates did not differ among color forms. Predation rates relative to conspicuousness did, however, deviate significantly from expectation, suggesting that predators do make post-detection decisions to avoid or attack a given prey type. The direction of this deviation from expectation, though, depended on assumptions we made about how avian predators discriminate objects from the visual background. Our results show that it is important to account for prey conspicuousness when investigating predator behavior and also that existing models of predator visual systems need to be refined.     

Evolutionary branching and long-term coexistence of cycling predators: critical function analysis

It is well known that two predators with different functional responses can coexist on one prey when the system exhibits nonequilibrium dynamics. In this paper, we investigate under which conditions such coexistence is evolutionarily stable, and whether the two predators may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in handling time, and hence in the shape of their Holling type II functional response. Longer handling times are costly in terms of lost foraging time, but allow the predator to extract more nutrients from the prey and therefore to produce more offspring per consumed prey. In the analysis, we apply a new method to accommodate arbitrary trade-off functions between handling time and offspring production. Contrary to previous results obtained assuming a particular trade-off [Kisdi, E. and Liu, S., 2006. J. Evol. Biol. 19, 49-58], we find that evolutionary branching of handling time is possible, although it does not appear to be very likely and can be excluded for a class of trade-offs. Evolutionarily stable coexistence of two predators occurs under less restrictive conditions, which are always satisfied when the trade-off function has two strongly concave parts connected by a convex piece.     

Multi-behavioral strategies in a predator–prey game: an evolutionary algorithm analysis

Behavioral games between predators and prey often involve two sub-games: 'pre-encounter' games affecting the rate of encounter between predators and prey (e.g. predator–prey space games, Sih 2005 ), and 'post-encounter' games that influence the outcome of encounters (e.g. waiting games at prey refugia, Hugie 2003 , and games of vigilance, Brown et al. 1999 ). Most models, however, focus on only one or the other of these two sub-games.
I investigated a multi-behavioral game between predators and prey that integrated both pre-encounter and post-encounter behaviors. These behaviors included landscape-scale movements by predators and prey, a type of prey vigilance that increases immediately after an encounter and then decays over time ('ratcheting vigilance'), and predator management of prey vigilance. I analyzed the game using a computer-based evolutionary algorithm. This algorithm embedded an individual-based model of ecological interactions within a dynamic adaptive process of mutation and selection. I investigated how evolutionarily stable strategies (ESS) varied with the predators' learning ability, killing efficiency, density and rate of movement. I found that when predators learn prey location, random prey movement can be an ESS. Increased predator killing efficiency reduced prey movement, but only if the rate of predator movement was low. Predators countered ratcheting vigilance by delaying their follow-up attacks; however, this delay was reduced in the presence of additional predators. The interdependence of pre-and post-encounter behaviors revealed by the evolutionary algorithm suggests an intricate co-evolution of multi-behavioral predator–prey behavioral strategies.     

Effects of predator-specific defence on biodiversity and community complexity in two-trophic-level communities

Summary Antipredator strategies employed by prey may be specific (effective against only one type of predator) or non-specific (effective against all predators). To examine the effects of the specificity of antipredator behaviour on biodiversity and community complexity, we analyse mathematical models including both evolutionary and population dynamics of a system including multiple prey species and multiple predator species. The models assume that all predator species change in their prey choice and all prey species have evolutionary change in their antipredator effort in evolution. The traits of each species change in an adaptive manner, whose rate is proportional to the slope of their fitness function. We calculate community complexity, resource-overlap between predators, an index of biodiversity and other properties of the coevolutionarily stable community for two cases: (1) all prey species have non-specific antipredator behaviour and (2) all prey species have predator-specific defence. Predator-specificity in defence increases community complexity, resource-overlap between predators, the total abundance of predators and the ratio of predator to prey abundance. Specific defence also decreases the number of isolated subwebs within the entire foodweb.     

Alternative food, switching predators, and the persistence of predator-prey systems

Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to alternative food sources. It has long been known that sigmoid (Holling's Type III) functional responses may stabilize an otherwise unstable equilibrium of prey and predators in Lotka-Volterra models. This poses the question of under what conditions such switching-mediated stability is likely to occur. A more complete understanding of the effect of predator switching would therefore require the analysis of one-predator/two-prey models, but these are difficult to analyze. We studied a model based on the simplifying assumption that the alternative food source has a fixed density. A well-known result from optimal foraging theory is that when prey density drops below a threshold density, optimally foraging predators will switch to alternative food, either by including the alternative food in their diet (in a fine-grained environment) or by moving to the alternative food source (in a coarse-grained environment). Analyzing the population dynamical consequences of such stepwise switches, we found that equilibria will not be stable at all. For suboptimal predators, a more gradual change will occur, resulting in stable equilibria for a limited range of alternative food types. This range is notably narrow in a fine-grained environment. Yet, even if switching to alternative food does not stabilize the equilibrium, it may prevent unbounded oscillations and thus promote persistence. These dynamics can well be understood from the occurrence of an abrupt (or at least steep) change in the prey isocline. Whereas local stability is favored only by specific types of alternative food, persistence of prey and predators is promoted by a much wider range of food types.