Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.     

Impact of Fear in a Delayed Predator–Prey System with Prey Refuge in Presence of Additional Food

Biophysics - It is observed from field experiments on terrestrial vertebrates that direct predation on predator–prey interaction can not only affect the population dynamics but the indirect...     

Modeling of Multifactor Taxis in a Predator–Prey System

Biophysics - A model of predator–prey dynamics in a spatially heterogeneous range is considered using a system of two nonlinear equations of the diffusion–advection reaction. The...     

Dynamics of a Prey–Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey

Biophysics - This paper mainly deals with the prey?predator dynamics where both the prey and predator exhibit herd behavior. Positivity, boundedness, some extinction criteria, stability of...     

How Predator Food Preference can Change the Destiny of Native Prey in Predator–Prey Systems

The aim of this work is to develop and analyse a mathematical model for a predator-2 preys system arising in insular environments. We are interested in the evolution of a native prey population without behavioural traits to cope with predation or competition, after the introduction of alien species. Here, we consider a long living bird population with low fertility rate. We point out the effects of the preference of the predator for either juvenile or adult stages. In addition, we study the impact of alien prey introduction in such a model. We use a reaction-diffusion system with a singular logistic right hand side. The aim of this work is to bring interesting dynamics to the fore. As a first example, oscillatory behaviour takes place in the model without alien preys and when predators have an average preference coefficient. Introduction of alien preys can lead to species extinction.     

Community Response of Mammalian Predators and Their Prey to Motorways: Implications for Predator–Prey Dynamics

Understanding the interactions between predators and prey is essential for predicting the effects of disturbances to ecosystems. Motorways produce changes in the surrounding biotic and abiotic environment and hence have multiple impacts on wildlife. Some species are known to change their activity patterns in the proximity of motorways but the implications for the structure of food webs are unknown. This study analyzes the activity patterns of both mammalian predators and their prey species near nine motorways in attempt to clarify how motorways affect the mammalian community. Habitat structural variables were also sampled to control the effects of microhabitat on relative prey abundance. Our results revealed different activity patterns of both predators and prey near motorways that are independent of structural differences in microhabitat. Both the red fox and small mammals were found to use the zone close to the motorways more frequently, whereas lagomorphs and mustelids were less active there. These differences suggest that motorways favor the population of the predator that is most tolerant of human activity, the red fox, whose activity could have both direct and indirect effects on that of other members of the predator and prey community. On the one hand, the red fox seems to act as “top predator” and mustelids to follow a “safety match” strategy avoiding the area close to the motorway where fox is more active. On the other hand, abundances of prey species are negatively associated with the activity of their most frequent predators. This study is the first to assess how the proximity to motorways affects the activity of mammals in two levels of the food web and opens the field for research to understand the processes driving the detected patterns. Moreover, such effects at the community scale should be taken into account when evaluating the impacts of motorways on the surrounding ecosystems.     

Adaptive Control Based Harvesting Strategy for a Predator–Prey Dynamical System

This paper deals with designing a harvesting control strategy for a predator–prey dynamical system, with parametric uncertainties and exogenous disturbances. A feedback control law for the harvesting rate of the predator is formulated such that the population dynamics is asymptotically stabilized at a positive operating point, while maintaining a positive, steady state harvesting rate. The hierarchical block strict feedback structure of the dynamics is exploited in designing a backstepping control law, based on Lyapunov theory. In order to account for unknown parameters, an adaptive control strategy has been proposed in which the control law depends on an adaptive variable which tracks the unknown parameter. Further, a switching component has been incorporated to robustify the control performance against bounded disturbances. Proofs have been provided to show that the proposed adaptive control strategy ensures asymptotic stability of the dynamics at a desired operating point, as well as exact parameter learning in the disturbance-free case and learning with bounded error in the disturbance prone case. The dynamics, with uncertainty in the death rate of the predator, subjected to a bounded disturbance has been simulated with the proposed control strategy.     

Biological Conservation of a Prey–Predator System Incorporating Constant Prey Refuge Through Provision of Alternative Food to Predators: A Theoretical Study

We describe a prey–predator system incorporating constant prey refuge through provision of alternative food to predators. The proposed model deals with a problem of non-selective harvesting of a prey–predator system in which both the prey and the predator species obey logistic law of growth. The long-run sustainability of an exploited system is discussed through provision of alternative food to predators. We have analyzed the variability of the system in presence of constant prey refuge and examined the stabilizing effect on predator–prey system. The steady states of the system are derived and dynamical behavior of the system is extensively analyzed around steady states. The optimal harvesting policy is formulated and solved with the help of Pontryagin’s maximal principle. Our objective is to maximize the monetary social benefit through protecting the predator species from extinction, keeping the ecological balance. Results finally illustrated with the help of numerical examples.     

Role of Quantity of Additional Food to Predators as a Control in Predator–Prey Systems with Relevance to Pest Management and Biological Conservation

Necessity to understand the role of additional food as a tool in biological control programs is being increasingly felt, particularly due to its eco-friendly nature. A thorough mathematical analysis in this direction revealed the vital role of quality and quantity of the additional food in the controllability of the predator–prey systems. In this article controllability of the additional food—provided predator–prey system is studied from perspectives of pest eradication and biological conservation. Time optimal paths have been constructed to drive the state of the system to a desired terminal state by choosing quantity of the additional food as control variable. The theory developed in this article has been illustrated by solving problems related to pest eradication and biological conservation.     

The Ideal Free Distribution in a Predator–Prey Model with Multifactor Taxis

Biophysics - The concept of an ideal free distribution (IFD) is analyzed for the predator–prey system in an inhomogeneous ring-shaped habitat. Diffusion–reaction–advection...     

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

This work aims to examine the global behavior of a Gause type predator–prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.     

Effects of Temperature and Turbulence on the Predator–Prey Interactions between a Heterotrophic Flagellate and a Marine Bacterium

Biotic and abiotic factors can influence interactions between microbial grazers and their prey, thus impacting both the cycling of biogenic carbon within the surface layer of the ocean and the export of carbon to the deep ocean and higher trophic levels. In this study, microcosm experiments were used to evaluate the combined effect of temperature and turbulence on the growth rate of a marine bacterium (Vibrio splendidus), a protistan predator (Paraphysomonas sp.), and the community grazing impact of Paraphysomonas sp. on V. splendidus. It was found that the artificial turbulence generated (1.35 × 10⁻¹ cm² s⁻³) significantly increased the rates of growth of Paraphysomonas sp. at high (>10°C), but not low (<5°C) temperatures, and that turbulence had no effect on the growth of V. splendidus. Both flagellate and bacterial growth were temperature dependent and decreased 4- to 6-fold as temperatures decreased from 15 to 0°C. Bacterial grazing mortality by Paraphysomonas sp. was 1.3- to 2.5-fold greater in the turbulent than static treatments among all four temperatures, and the rates of cell-specific ingestion of bacteria by Paraphysomonas sp. was 2-fold greater at 15 and 10°C in the turbulent than in the static treatment. Hence, this study shows that turbulence can influence nanoflagellate grazing at temperatures >5°C and suggests that at low temperatures, increased viscosity may limit the size of organisms that can be affected by small-scale turbulence.Present address (M.P. Delaney): Goddard Earth Sciences & Technology Center, NASA Goddard Space Flight Center, Mail Code 900.1, Greenbelt, MD 20771     

Optimal Culling and Biocontrol in a Predator–Prey Model

Invasive species cause enormous problems in ecosystems around the world. Motivated by introduced feral cats that prey on bird populations and threaten to drive them extinct on remote oceanic islands, we formulate and analyze optimal control problems. Their novelty is that they involve both scalar and time-dependent controls. They represent different forms of control, namely the initial release of infected predators on the one hand and culling as well as trapping, infecting, and returning predators on the other hand. Combinations of different control methods have been proposed to complement their respective strengths in reducing predator numbers and thus protecting endangered prey. Here, we formulate and analyze an eco-epidemiological model, provide analytical results on the optimal control problem, and use a forward–backward sweep method for numerical simulations. By taking into account different ecological scenarios, initial conditions, and control durations, our model allows to gain insight how the different methods interact and in which cases they could be effective.     

A Systematic Overview of Harvesting-Induced Maturation Evolution in Predator–Prey Systems with Three Different Life-History Tradeoffs

There are concerns that anthropogenic harvesting may cause phenotypic adaptive changes in exploited wild populations, in particular maturation at a smaller size and younger age. In this paper, we study the evolutionarily stable size at maturation of prey subjected to size-selective harvesting in a simple predator?Cprey model, taking into account three recognized life-history costs of early maturation, namely reduced fecundity, reduced growth, and increased mortality. Our analysis shows that harvesting large individuals favors maturation at smaller size compared to the unharvested system, independent of life-history tradeoff and the predator??s prey-size preference. In general, however, the evolutionarily stable maturation size can either increase or decrease relative to the unharvested system, depending on the harvesting regime, the life-history tradeoff, and the predator??s preferred size of prey. Furthermore, we examine how the predator population size changes in response to adaptive change in size at maturation of the prey. Surprisingly, in some situations, we find that the evolutionarily stable maturation size under harvesting is associated with an increased predator population size. This occurs, in particular, when early maturation trades off with growth rate. In total, we determine the evolutionarily stable size at maturation and associated predator population size for a total of forty-five different combinations of tradeoff, harvest regime, and predated size class.     

Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in M<Emphasis Type="SmallCaps">ATLAB</Emphasis>

We present two finite-difference algorithms for studying the dynamics of spatially extended predator–prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB. Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator–prey interactions.     

Permanence and Extinction of a Diffusive Predator–Prey Model with Robin Boundary Conditions

The main concern of this paper is to study the dynamic of a predator–prey system with diffusion. It incorporates the Holling-type-II and a modified Leslie–Gower functional responses under Robin boundary conditions. More concretely, we study the dissipativeness of the system by using the comparison principle, and we derive a criteria for permanence and for predator extinction.     

The Effects of Predator Evolution and Genetic Variation on Predator–Prey Population-Level Dynamics

This paper explores how predator evolution and the magnitude of predator genetic variation alter the population-level dynamics of predator–prey systems. We do this by analyzing a general eco-evolutionary predator–prey model using four methods: Method 1 identifies how eco-evolutionary feedbacks alter system stability in the fast and slow evolution limits; Method 2 identifies how the amount of standing predator genetic variation alters system stability; Method 3 identifies how the phase lags in predator–prey cycles depend on the amount of genetic variation; and Method 4 determines conditions for different cycle shapes in the fast and slow evolution limits using geometric singular perturbation theory. With these four methods, we identify the conditions under which predator evolution alters system stability and shapes of predator–prey cycles, and how those effect depend on the amount of genetic variation in the predator population. We discuss the advantages and disadvantages of each method and the relations between the four methods. This work shows how the four methods can be used in tandem to make general predictions about eco-evolutionary dynamics and feedbacks.