Predator-prey shell games: large-scale movement and its implications for decision-making by prey

Many classical models of food patch use under predation risk assume that predators impose patch-specific predation risks independent of prey behavior. These models predict that prey should leave a chosen patch only if and when the food depletes below some critical level. In nature, however, prey individuals may regularly move among food patches, even in the apparent absence of food depletion. We suggest that such prey movement is part of a predator-prey "shell game", in which predators attempt to learn prey location, and the prey attempt to be unpredictable in space. We investigate this shell game using an individual-based model that allows predators to update information about prey location, and permits prey to move with some random component among patches, but with reduced energy intake. Our results show the best prey strategy depends on what the predator does. A non-learning (randomly moving) predator favors non-moving prey – moving prey suffer higher starvation and predation. However, a learning predator favors prey movement. In general, the best prey strategy involves movement biased toward, but not completely committed to, the richer food patch. The strategy of prey movement remains beneficial even in combination with other anti-predator defenses, such as prey vigilance.     

Remarks on the number of persistent states to a fragmented predator–prey model system

We consider a predator–prey model system for spatially distributed species over patches. Each predator species has a unique preferred patch (shelter and reproduction site) and travel for chasing prey. Its individuals are split into resident from the preferred patch and travelers. Further there is at most one resident predator species per patch. Depending on the availability of local anthropized resources not related to local prey on the preferred patch, one distinguishes between well-fed and starving predators. We assume prey species do not disperse at the predator scale.In this study we are interested in the number of persistent stationary states for the resulting ordinary differential equations model system. There exists at most one persistent predator–prey stationary state when there is exactly one starving resident predators per patch provided all functional responses to predation are Lotka–Volterra like or when a single starving resident predators is available. Else multiple persistent predator–prey stationary state are likely to exist. A specific emphasis is put on toy-model systems with 2 or 3 patches. Slow–fast dynamical methodology is also used for locally asymptotically stable purposes.Numerical experiments suggest that several scalings may govern the dynamics at stabilization.     

Predator kairomones change food web structure and function,regardless of cues from consumed prey

Predation risk in aquatic systems is often assessed by prey through chemical cues, either those released by prey or by the predator itself. Many studies on predation risk focus on simple pairwise interactions, with only a few studies examining community‐level and ecosystem responses to predation risk in species‐rich food webs. Further, of these few community‐level studies, most assume that prey primarily assess predation risk through chemical cues from consumed prey, even heterospecific prey, rather than just those released by the predator. Here, we compared the effects of different predation cues (predator presence with or without consumed prey) on the structure and functioning of a speciose aquatic food web housed in tropical bromeliads. We found that the mere presence of the top predator (a damselfly) had a strong cascading effect on the food web, propagating down to nutrient cycling. This predation risk cue had no effect on the identity of colonizing species, but strongly reduced the abundance and biomass of the macroinvertebrate colonists. As a result, bacterial biomass and nitrogen cycling doubled, with a concomitant decrease in bacterial production, but CO₂ flux was unaffected. These community and ecosystem effects of predator presence cues were not amplified by the addition of chemical cues from consumed prey. Our results show that some of the consequences of predation risk observed in controlled experiments with simplified food webs may be observed in a natural, species‐rich food web.     

Patch use in time and space for a meso-predator in a risky world

Predator–prey studies often assume a three trophic level system where predators forage free from any risk of predation. Since meso-predators themselves are also prospective prey, they too need to trade-off between food and safety. We applied foraging theory to study patch use and habitat selection by a meso-predator, the red fox. We present evidence that foxes use a quitting harvest rate rule when deciding whether or not to abandon a foraging patch, and experience diminishing returns when foraging from a depletable food patch. Furthermore, our data suggest that patch use decisions of red foxes are influenced not just by the availability of food, but also by their perceived risk of predation. Fox behavior was affected by moonlight, with foxes depleting food resources more thoroughly (lower giving-up density) on darker nights compared to moonlit nights. Foxes reduced risk from hyenas by being more active where and when hyena activity was low. While hyenas were least active during moon, and most active during full moon nights, the reverse was true for foxes. Foxes showed twice as much activity during new moon compared to full moon nights, suggesting different costs of predation. Interestingly, resources in patches with cues of another predator (scat of wolf) were depleted to significantly lower levels compared to patches without. Our results emphasize the need for considering risk of predation for intermediate predators, and also shows how patch use theory and experimental food patches can be used for a predator. Taken together, these results may help us better understand trophic interactions.     

Behavioural Response of Juvenile Bluegill Sunfish to Variation in Predation Risk and Food Level

The behavioural response of juvenile bluegill sunfish (Lepomis macrochirus) to predation risk when selecting between patches of artificial vegetation differing in food and stem density was investigated. Bluegill foraging activity was significantly affected by all three factors. Regardless of patch stem density or risk of predation bluegills preferred patches with the highest prey number. During each trial bluegill foraging activity was clearly divided into a between- and within-patch component. In the presence of a predator bluegills reduced their between-patch foraging activity by an equivalent amount regardless of patch stem density or food level, apparently showing a risk-adjusting behavioural response to predation risk. Within patches, however, foraging activity was affected by both food level and patch stem density. When foraging in a patch offering a refuge from predation, the presence of a predator had no effect on bluegill foraging activity within this patch. However, if foraging in a patch with only limited refuge potential, bluegill foraging activity was reduced significantly in the presence of a predator. Further, this reduction was significantly greater if the patch contained a low versus a high food level, indicating a risk-balancing response to predation with respect to within-patch foraging activity. Both these responses differ from the risk-avoidance response to predation demonstrated by juvenile bluegills when selecting among habitats. Therefore, our results demonstrate the flexibility of juvenile bluegill foraging behaviour.     

Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

Two common invertebrate predators show varying predation responses to different types of sentinel prey

Sentinel prey (an artificially manipulated patch of prey) are widely used to assess the level of predation provided by natural enemies in agricultural systems. Whilst a number of different methodologies are currently in use, little is known about how arthropod predators respond to artificially manipulated sentinel prey in comparison with predation on free‐living prey populations. We assessed how attack rates on immobilized (aphids stuck to cards) and artificial (plasticine lepidopteran larvae mimics) sentinel prey differed to predation on free‐moving live prey (aphids). Predation was assessed in response to density of the common invertebrate predators, a foliar‐active ladybird Harmonia axyridis (Coleoptera: Coccinellidae), and a ground‐active beetle Pterostichus madidus (Coleoptera: Carabidae). Significant increases in attack rates were found for the immobilized and artificial prey between the low and high predator density treatments. However, an increased predator density did not significantly reduce numbers of free‐living live aphids included in the mesocosms in addition to the alternate prey. We also found no signs of predation on the artificial prey by the predator H. axyridis. These findings suggest that if our assessment of predation had been based solely on the foliar artificial prey, then no increase in predation would have been found in response to increased predator density. Our results demonstrate that predators differentially respond to sentinel prey items which could affect the level of predation recorded where target pest species are not being used.     

Handling time promotes the coevolution of aggregation in predator-prey systems

Predators often have type II functional responses and live in environments where their life history traits as well as those of their prey vary from patch to patch. To understand how spatial heterogeneity and predator handling times influence the coevolution of patch preferences and ecological stability, we perform an ecological and evolutionary analysis of a Nicholson-Bailey type model. We prove that coevolutionarily stable prey and searching predators prefer patches that in isolation support higher prey and searching predator densities, respectively. Using this fact, we determine how environmental variation and predator handling times influence the spatial patterns of patch preferences, population abundances and per-capita predation rates. In particular, long predator handling times are shown to result in the coevolution of predator and prey aggregation. An analytic expression characterizing ecological stability of the coevolved populations is derived. This expression implies that contrary to traditional theoretical expectations, predator handling time can stabilize predator-prey interactions through its coevolutionary influence on patch preferences. These results are shown to have important implications for classical biological control.     

Effect of diffusion and spatially varying predation risk on the dynamics and equilibrium density of a predator-prey system

Starting from natural planktonic systems, we present a new mechanism involving spatial heterogeneity, and develop a new spatial structure model of planktonic predation systems. Firstly, the effect of diffusion on the dynamics of the system is investigated. We find that diffusion of only prey or both prey and predator between different patches with different predation risk may stabilize the dynamics, depending on the flow rate. Only a medium flow rate can lead to the stability of the system. Too large a rate can cause the system to approach the non-spatial limit case of a well-mixed system. Too large a rate can cause the system to approach the non-spatial limit case as a well-mixed system, which is characterized by its strongly oscillatory dynamics. When only prey diffuse, the smaller the parameter f (the proportion of the patchy volume with larger predation risk to the total volume), the more stable the system. If both populations can diffuse, however, only medium and very small f values may stabilize the system. Also, the response of the spatially averaged equilibrium densities of the system to the increasing of the flow rate is examined. With increasing flow rate, the spatial-averaged equilibrium density of prey decreases, while that of predator depends on which species can diffuse. For the case of prey diffusion only, it first remains unchanged and then slightly decreases, while it increases for the case of combinations as the flow rate increases. Our results are, qualitatively, determined by the spatially heterogeneous mechanism that we propose, and further regulated by top-down forces. Of practical importance, the results reported here indicate that which species diffuse plays a key role in the ways in which diffusion influences the dynamics and the spatial-average equilibrium densities of the system responses to the flow rate's increasing.     

Optimal harvesting of prey–predator system with interval biological parameters: A bioeconomic model

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin’s maximum principle. Numerical examples are presented to support the proposed model.     

Dynamics of a intraguild predation model with generalist or specialist predator

Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has “top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: (1) Both IGP models can have multiple attractors with complicated dynamical patterns; (2) Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; (3) IGP model with generalist predator is prone to have coexistence of three species.     

Evolutionary Consequences of Predation for Pathogens in Prey

This article investigates the impact of predation on the coexistence and competitive exclusion of pathogen strains in the prey. Two types of predator are considered—a generalist and a specialist. For each type of predator, we assume that the predator can discriminate among susceptible and infected with each strain prey. The two strains will competitively exclude each other in the absence of predation with the strain with the larger reproduction number persisting. If a generalist predator preys discriminantly and the disease is fatal, then depending on the predation level, a switch in the dominant pathogen may occur. Thus, for some predation levels, the first strain may persist while for other predation levels the second strain may persist. Furthermore, a specialist predator preying discriminantly may mediate the coexistence of the two strains. Although in most cases increasing predation reduces the disease load in the prey, when predation leads to coexistence, it may also lead to increase in the disease load.     

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.     

Interspecific differences in antipredator strategies determine the strength of non‐consumptive predator effects on stream detritivores

Animal species differ considerably in their response to predation risks. Interspecific variability in prey behaviour and morphology can alter cascading effects of predators on ecosystem structure and functioning. We tested whether species‐specific morphological defenses may affect responses of leaf litter consuming invertebrate prey to sit‐and‐wait predators, the odonate Cordulegaster boltonii larvae, in aquatic food webs. Partly or completely blocking the predator mouthparts (mandibles and/or extensible labium), thus eliminating consumptive (i.e. lethal) predator effects, we created a gradient of predator‐prey interaction intensities (no predator < predator – no attack < predator – non‐lethal attacks < lethal predator). A field experiment was first used to assess both consumptive and non‐consumptive predator effects on leaf litter decomposition and prey abundances. Laboratory microcosms were then used to examine behavioural responses of armored and non‐armored prey to predation risk and their consequences on litter decomposition. Results show that armored and non‐armored prey responded to both acute (predator – non‐lethal attacks) and chronic (predator – no attack) predation risks. Acute predation risk had stronger effects on litter decomposition, prey feeding rate and prey habitat use than predator presence alone (chronic predation risk). Predator presence induced a reduction in feeding activity (i.e. resource consumption) of both prey types but a shift to predator‐free habitat patches in non‐armored detritivores only. Non‐consumptive predator effects on prey subsequently decreased litter decomposition rate. Species‐specific prey morphological defenses and behaviour should thus be considered when studying non‐consumptive predator effects on prey community structure and ecosystem functioning.     

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 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.     

Effects of Structural Refuge and Density on Foraging Behaviour and Mortality of Hungry Tadpoles Subject to Predation Risk

Theoretical models of prey behaviour predict that food‐limited prey engage in risk‐prone foraging and thereby succumb to increased mortality from predation. However, predation risk also may be influenced by factors including prey density and structural cover, such that the presumed role of prey hunger on predation risk may be obfuscated in many complex predator–prey systems. Using a tadpole (prey) – dragonfly larva (predator) system, we determined relative risk posed to hungry vs. sated prey when both density and structural cover were varied experimentally. Overall, prey response to perceived predation risk was primarily restricted to increased cover use, and hungry prey did not exhibit risk‐prone foraging. Surprisingly, hungry prey showed lower activity than sated prey when exposed to predation risk, perhaps indicating increased effort in search of refuge or spatial avoidance of predator cues among sated animals. An interaction between hunger level and predation risk treatments indicated that prey state affected sensitivity to perceived risk. We also examined the lethal implications of prey hunger by allowing predators to select directly between hungry and sated prey. Although predators qualitatively favoured hungry prey when density was elevated and structural cover was sparse, the overall low observed variation in mortality risk between hunger treatments suggests that preferential selection of hungry prey was weak. This implies that hunger effects on prey mortality risk may not be readily observed in complex landscapes with additional factors influencing risk. Thus, current starvation‐predation trade‐off theory may need to be broadened to account for other mechanisms through which undernourished prey may cope with predation risk.     

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.