Rabbits killing birds revisited

We formulate and study a three-species population model consisting of an endemic prey (bird), an alien prey (rabbit) and an alien predator (cat). Our model overcomes several model construction problems in existing models. Moreover, our model generates richer, more reasonable and realistic dynamics. We explore the possible control strategies to save or restore the bird by controlling or eliminating the rabbit or the cat when the bird is endangered. We confirm the existence of the hyperpredation phenomenon, which is a big potential threat to most endemic prey. Specifically, we show that, in an endemic prey-alien prey-alien predator system, eradication of introduced predators such as the cat alone is not always the best solution to protect endemic insular prey since predator control may fail to protect the indigenous prey when the control of the introduced prey is not carried out simultaneously.     

Should "handled" prey be considered? Some consequences for functional response, predator-prey dynamics and optimal foraging theory

Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered.     

Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

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.     

An impulsively controlled pest management model with n predator species and a common prey

This paper investigates the dynamics of a competitive single-prey n-predators model of integrated pest management, which is subject to periodic and impulsive controls, from the viewpoint of finding sufficient conditions for the extinction of prey and for prey and predator permanence. The per capita death rates of prey due to predation are given in abstract, unspecified forms, which encompass large classes of death rates arising from usual predator functional responses, both prey-dependent and predator-dependent. The stability and permanence conditions are then expressed as balance conditions between the cumulative death rate of prey in a period, due to predation from all predator species and to the use of control, and to the cumulative birth rate of prey in the same amount of time. These results are then specialized for the case of prey-dependent functional responses, their biological significance being also discussed.     

Not attackable or not crackable—How pre‐ and post‐attack defenses with different competition costs affect prey coexistence and population dynamics

It is well‐known that prey species often face trade‐offs between defense against predation and competitiveness, enabling predator‐mediated coexistence. However, we lack an understanding of how the large variety of different defense traits with different competition costs affects coexistence and population dynamics. Our study focusses on two general defense mechanisms, that is, pre‐attack (e.g., camouflage) and post‐attack defenses (e.g., weaponry) that act at different phases of the predator—prey interaction. We consider a food web model with one predator, two prey types and one resource. One prey type is undefended, while the other one is pre‐ or post‐attack defended paying costs either by a higher half‐saturation constant for resource uptake or a lower maximum growth rate. We show that post‐attack defenses promote prey coexistence and stabilize the population dynamics more strongly than pre‐attack defenses by interfering with the predator's functional response: Because the predator spends time handling “noncrackable” prey, the undefended prey is indirectly facilitated. A high half‐saturation constant as defense costs promotes coexistence more and stabilizes the dynamics less than a low maximum growth rate. The former imposes high costs at low resource concentrations but allows for temporally high growth rates at predator‐induced resource peaks preventing the extinction of the defended prey. We evaluate the effects of the different defense mechanisms and costs on coexistence under different enrichment levels in order to vary the importance of bottom‐up and top‐down control of the prey community.     

具有脉冲投放捕食者阶段结构时滞的捕食-食饵模型

讨论了具有阶段结构脉冲时滞的HollinglI功能反应的捕食模型,其中天敌（益虫）进行人工脉冲周期投放,害虫具有阶段结构及成熟期的时滞现象,并进行了系统的数学及生物方面的研究．利用脉冲及时滞微分方程的基本知识证明了该害虫根除周期解的唯一性和全局吸引性．进一步证明了当天敌的投放量或者投放周期在一定的范围内,能够控制害虫在作物的经济危害水平（EIL）运行的情况下使天敌与害虫可以共存．得出的结论为害虫治理提供了策略基础．     

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.     

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.     

Individual base model of predator-prey system shows predator dominant dynamics

Individual base model of predator-prey system is constructed. Both predator and prey species have age structure and cohorts of early reproductive age have competitive advantage. The model has linear functional response in predation behavior and includes the effect of interference among predators and delay of population growth from resource intake, not by functional response but by calculation procedure. Each foraging action is calculated successively and surplus or scarce of acquired resources is interpreted into population size through individual birth and death. This model shows that biomass of prey killed by predator is dependent on demand of predator and that heterogeneity in predator population is essential in persistency and stability of predator-prey system. Heterogeneity of predator makes predator individuals of less competing ability die rapidly. Rapid death of weak individuals causes rapid decrease of total demand of predator and that makes enough room for survived predators. Therefore, the biomass of killed prey is dependent on predator's demand. As young or infant population of predator are the more vulnerable to shortage of prey, and when many of them cannot survive to reproductive age, they can stabilize the system by wasting excessive prey with only temporal numerical increase of predator population.     

Individual variation in prey choice in a predator-prey community

One predator-two prey community models are studied with an emphasis on individual variation in predator behavior. The predator behaves according to a well-known prey choice model. The behavioral model predicts that predators should always attack the primary prey (more profitable prey of the two), but only attack the alternative prey (less profitable prey of the two) when the density of the primary prey is below a threshold density. The predator that accepts the alternative prey does not discriminate between the primary and alternative prey (all-or-nothing preference for the alternative prey). However, empirical studies do not result in clear all-or-nothing responses. Previous models examined the relaxation of the all-or-nothing response by assuming partial preference (e.g., predators preferentially forage on the primary prey even when they also attack the alternative prey). In this study, I consider individual variation in two predator traits (prey density perception and handling time) as the sources of the variation in the threshold density, which can make empirical data appear deviated from the expectation. I examine how community models with partial preference and individual variation differ in their dynamics and show that the differences can be substantial. For example, the dynamics of a model based on individual variation can be more stable (e.g., stable in a wider parameter region) than that of a model based on partial preference. As the general statistical property (Jensen’s inequality) is a main factor that causes the differences, the results of the study have general implications to the interpretation of models based on average per-capita rates.     

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

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

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.     

Stochastic modelling of prey depletion processes

The modelling of prey-predator interactions is of major importance for the understanding of population dynamics. Classically, these interactions are modelled using ordinary differential equations, but this approach has the drawbacks of assuming continuous population variables and of being deterministic. We propose a general approach to stochastic modelling based on the concept of functional response for a prey depletion process with a constant number of predators. Our model could involve any kind of functional response, and permits a likelihood-based approach to statistical modelling and stable computation using matrix exponentials. To illustrate the method we use the Holling-Juliano functional response and compare the outcomes of our model with a deterministic counterpart considered by Schenk and Bacher [2002. Functional response of a generalist insect predator to one of its prey species in the field. Journal of Animal Ecology 71 (3), 524-531], who observed the depletion of Cassida rubiginosa due to its exclusive predator, Polistes dominulus. The predation was found to be Holling type III, reflecting the ability of the predator to regulate its prey. Our approach corroborates this result, but suggests that the prey depletion census should have been performed more often, and that predation features were significantly different between the two years for which data are available.     

Predator functional response changed by induced defenses in prey

Functional responses play a central role in the nature and stability of predator-prey population dynamics. Here we investigate how induced defenses affect predator functional responses. In experimental communities, prey (Paramecium) expressed two previously undocumented inducible defenses--a speed reduction and a width increase--in response to nonlethal exposure to predatory Stenostomum. Nonlethal exposure also changed the shape of the predator's functional response from Type II to Type III, consistent with changes in the density dependence of attack rates. Handling times were also affected by prey defenses, increasing at least sixfold. These changes show that induced changes in prey have a real defensive function. At low prey densities, induction led to lower attack success; at high prey densities, attack rates were actually higher for induced prey. However, induction increased handling times sufficiently that consumption rates of defended prey were lower than those of undefended prey. Modification of attack rate and handling time has important potential consequences for population dynamics; Type III functional responses can increase the stability of population dynamics and persistence because predation on small populations is low, allowing a relict population to survive. Simulations of a predator-prey population dynamic model revealed the stabilizing potential of the Type III response.     

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.     

Predator–prey relationships in a Mediterranean vertebrate system: Bonelli’s eagles,rabbits and partridges

How predators impact on prey population dynamics is still an unsolved issue for most wild predator–prey communities. When considering vertebrates, important concerns constrain a comprehensive understanding of the functioning of predator–prey relationships worldwide; e.g. studies simultaneously quantifying ‘functional’ and ‘numerical responses’ (i.e., the ‘total response’) are rare. The functional, the numerical, and the resulting total response (i.e., how the predator per capita intake, the population of predators and the total of prey eaten by the total predators vary with prey densities) are fundamental as they reveal the predator’s ability to regulate prey population dynamics. Here, we used a multi-spatio-temporal scale approach to simultaneously explore the functional and numerical responses of a territorial predator (Bonelli’s eagle Hieraaetus fasciatus) to its two main prey species (the rabbit Oryctolagus cuniculus and the red-legged partridge Alectoris rufa) during the breeding period in a Mediterranean system of south Spain. Bonelli’s eagle responded functionally, but not numerically, to rabbit/partridge density changes. Type II, non-regulatory, functional responses (typical of specialist predators) offered the best fitting models for both prey. In the absence of a numerical response, Bonelli’s eagle role as a regulating factor of rabbit and partridge populations seems to be weak in our study area. Simple (prey density-dependent) functional response models may well describe the short-term variation in a territorial predator’s consumption rate in complex ecosystems.     

Oscillations in a size-structured prey-predator model

This article introduces a predator–prey model with the prey structured by body size, based on reports in the literature that predation rates are prey-size specific. The model is built on the foundation of the one-species physiologically structured models studied earlier. Three types of equilibria are found: extinction, multiple prey-only equilibria and possibly multiple predator–prey coexistence equilibria. The stabilities of the equilibria are investigated. Comparison is made with the underlying ODE Lotka–Volterra model. It turns out that the ODE model can exhibit sustain oscillations if there is an Allee effect in the net reproduction rate, that is the net reproduction rate grows for some range of the prey’s population size. In contrast, it is shown that the structured PDE model can exhibit sustain oscillations even if the net reproductive rate is strictly declining with prey population size. We find that predation, even size-non-specific linear predation can destabilize a stable prey-only equilibrium, if reproduction is size specific and limited to individuals of large enough size. Furthermore, we show that size-specific predation can also destabilize the predator–prey equilibrium in the PDE model. We surmise that size-specific predation allows for temporary prey escape which is responsible for destabilization in the predator–prey dynamics.