A nonautonomous predator–prey system with stage structure and double time delays

In the present paper we study a nonautonomous predator–prey model with stage structure and double time delays due to maturation time for both prey and predator. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the immature prey. Based on some comparison arguments we discuss the permanence of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of positive periodic solution. By means of constructing an appropriate Lyapunov functional, we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution. Two examples are given to illustrate the feasibility of our main results.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

Permanence and stability of a predator–prey system with stage structure for predator

A stage-structured predator–prey system with delays for prey and predator, respectively, is proposed and analyzed. Mathematical analysis of the model equations with regard to boundedness of solutions, permanence and stability are analyzed. Some sufficient conditions which guarantee the permanence of the system and the global asymptotic stability of the boundary and positive equilibrium, respectively, are obtained.     

Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses

A delayed stage-structured predator–prey model with non-monotone functional responses is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age, and that immature predators do not have the ability to attack prey. Some new and interesting sufficient conditions are obtained for the global existence of multiple positive periodic solutions of the stage-structured predator–prey model. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions to Lx = λNx. An example is given to illustrate the feasibility of our main result.     

A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting

In this paper, we propose a bioeconomic differential algebraic predator–prey model with Holling type II functional response and nonlinear prey harvesting. As the nonlinear prey harvesting is introduced, the proposed model displays a complex dynamics in the predator–prey plane. Taking into account of the economic factor, our predator–prey system is established by bioeconomic differential algebraic equations. The effect of economic profit on the proposed model is analyzed by viewing it as a bifurcation parameter. By jointly using the normal form of differential algebraic models and the bifurcation theory, the stability and bifurcations (singularity induced bifurcation, Hopf bifurcation) are discussed. These results obtained here reveal richer dynamics of the bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting, and suggest a guidance for harvesting in the practical word. Finally, numerical simulations are given to demonstrate the results.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

VOLE POPULATION DYNAMICS UNDER THE INFLUENCE OF SPECIALIST AND GENERALIST PREDATION

Abstract To understand the impact of predation by different types of predators on the vole population dynamics, we formulate a three differential equation model describing the population dynamics of voles, the “specialist predator” and the “generalist predator.” First we perform a local stability study of the different steady states of the basic model and deduce that the predation rates of the “specialist” as well as the “generalist” predator are the main parameters controlling the existence/extinction criteria of the concerned populations. Next we analyze the model from a thermodynamic perspective and study the thermodynamic stability of the different equilibria. Finally using stochastic driving forces, we incorporate the exogenous factor of environmental forcing and investigate the stochastic stability of the system. We compare the stability criteria of the different steady states under deterministic, thermodynamic and stochastic situations. The analysis reveals that when the “specialist” and the “generalist” predator are modeled separately, the system exhibits rich dynamics and the predation rates of both types of predators play a major role in controlling vole oscillation and/or stability. These findings are also seen to resemble closely with the observed behavior of voles in the natural setting. Numerical simulations are carried out to illustrate analytical findings.     

Rich dynamical behaviours for predator–prey model with weak Allee effect

The weak Allee effect on the predator is introduced into the classic predator–prey model of Lotka–Volterra type. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. It is shown that the weak Allee effect can bring rich and complicated dynamics to the previous simple model, such as the saddle–node bifurcation, subcritical and supercritical Hopf bifurcations, and Bogdanov–Takens bifurcations, implying that weak Allee effect can be one of the simple reasons for many complicated behaviours in the predator–prey communities.     

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.     

A stage-structured food chain model with stage dependent predation: Existence of codimension one and codimension two bifurcations

Stage-structured predator–prey models exhibit rich and interesting dynamics compared to homogeneous population models. The objective of this paper is to study the bifurcation behavior of stage-structured prey–predator models that admit stage-restricted predation. It is shown that the model with juvenile-only predation exhibits Hopf bifurcation with the growth rate of the adult prey as the bifurcation parameter; also, depending on parameter values, a stable limit cycle will emerge, that is, the bifurcation will be of supercritical nature. On the other hand, the analysis of the model with adult-stage predation shows that the system admits a fold-Hopf bifurcation with the adult growth rate and the predator mortality rate as the two bifurcation parameters. We also demonstrate the existence of a unique limit cycle arising from this codimension-2 bifurcation. These results reveal far richer dynamics compared to models without stage-structure. Numerical simulations are done to support analytical results.     

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.     

Chaos in delay-induced Leslie–Gower prey–predator–parasite model and its control through prey harvesting

In this paper we analyze a delay-induced predator–prey–parasite model with prey harvesting, where the predator–prey interaction is represented by Leslie–Gower type model with type II functional response. Infection is assumed to spread horizontally from one infected prey to another susceptible prey following mass action law. Spreading of disease is not instantaneous but mediated by a time lag to take into account the time required for incubation process. Both the susceptible and infected preys are subjected to linear harvesting. The analysis is accomplished in two phases. First we analyze the delay-induced predator–prey–parasite system in absence of harvesting and proved the local & global dynamics of different (six) equilibrium points. It is proved that the delay has no influence on the stability of different equilibrium points except the interior one. Delay may cause instability in an otherwise stable interior equilibrium point of the system and larger delay may even produce chaos if the infection rate is also high. In the second phase, we explored the dynamics of the delay-induced harvested system. It is shown that harvesting of prey population can suppress the abrupt fluctuations in the population densities and can stabilize the system when it exceeds some threshold value.     

Spatiotemporal complexity of a discrete space-time predator−prey system with self- and cross-diffusion

In this research, we investigate the spatiotemporal dynamics of a discrete space-time predator−prey system with self- and cross-diffusion. Through stability analysis and bifurcation analysis, Turing pattern formation conditions are derived and two nonlinear mechanisms of pattern formation are found, i.e., pure Turing instability and Hopf-Turing instability. Numerical simulations reveal rich dynamics of the discrete predator−prey system. In spatially homogeneous case, stable homogeneous stationary states, homogeneous periodic, quasiperiodic and chaotic oscillating states are exhibited; in spatially heterogeneous case, a surprising variety of prey and predator patterns are described, including spotted, striped, labyrinth, gapped, spiral, circled patterns and many intermediate patterns. Moreover, sensitivity of spatiotemporal pattern formation to initial conditions is predicted along with Hopf-Turing instability, suggesting the self-organization of diverse patterns under identical parametric conditions. In comparison with former results in literature, the discrete version of reaction-diffusion model developed in this research capture more complicated and richer nonlinear dynamical behaviors, contributing to a new comprehending on the complex pattern formation of spatially extended discrete predator−prey systems.     

Dispersal permanence of periodic predator–prey model with Ivlev-type functional response and impulsive effects

In this paper, we study the permanence of a periodic Ivlev-type predator–prey system where the prey disperses in patchy environment with two patches. We assume the Ivlev-type functional response within-patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent. Furthermore, we give numerical analysis to confirm our theoretical results. It will be useful to ecosystem control.     

A cannibalistic eco-epidemiological model with disease in predator population

In this present article, we propose and analyze a cannibalistic predator–prey model with disease in the predator population. We consider two important factors for the dynamics of predator population. The first one is governed through cannibalistic interaction, and the second one is governed through the disease in the predator population via cannibalism. The local stability analysis of the model system around the biologically feasible equilibria are investigated. We perform global dynamics of the model using Lyapunov functions. We analyze and compare the community structure of the system in terms of ecological and disease basic reproduction numbers. The existence of Hopf bifurcation around the interior steady state is investigated. We also derive the sufficient conditions for the permanence and impermanence of the system. The study reveals that the cannibalism acts as a self-regulatory mechanism and controls the disease transmission among the predators by stabilizing the predator–prey oscillations.     

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.     

A Lyapunov functional for a stage-structured predator–prey model with nonlinear predation rate

We consider the dynamics of a general stage-structured predator–prey model which generalizes several known predator–prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.     

A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.