Effect of prey refuge on a harvested predator–prey model with generalized functional response

A predator–prey model with generalized response function incorporating a prey refuge and independent harvesting in each species are studied by using the analytical approach. A constant proportion of prey using refuges is considered. We will evaluate the effects with regard to the local stability of equilibria, the equilibrium density values and the long-term dynamics of the interacting populations. Some numerical simulations are carried out.     

Analysis of a diffusive Leslie–Gower predator–prey model with nonmonotonic functional response

In this paper, a diffusive Leslie–Gower predator–prey system with nonmonotonic functional respond is studied. We obtain the persistence of this model and show the local asymptotic stability of positive constant equilibrium by linearized analysis and the global stability by constructing Liapunov function. Besides, Turing instability of this equilibrium is obtained. The existence and nonexistence of positive nonconstant steady states of this model are established. Furthermore, by numerical simulations we illustrate the patterns of prey and predator.     

Codimension-two bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response

In this paper, we investigate dynamical behaviours of a discrete predator–prey model with nonmonotonic functional response. Codimension-2 bifurcations associated with 1:2, 1:3 and 1:4 resonances are analyzed by using bifurcation theory. Codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams and phase portraits, which not only illustrate the validity of the theoretical results but also display some interesting complex dynamical behaviours, are obtained by numerical simulations.     

A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

Periodic solutions to a delayed predator–prey model with Hassell–Varley type functional response

In this paper, a delayed predator–prey model with Hassell–Varley type functional responses is studied. Some sufficient conditions are obtained for the existence of positive periodic solutions to it by applying the coincidence degree theorem. It is interesting that the result is based on the delay, which is different from the previous work (the results are delay-independent). Furthermore, the simulation shows that some conditions are sharp.     

Steady state in a cross-diffusion predator–prey model with the Beddington–DeAngelis functional response

The purpose of this paper is to study the existence of steady state in a linear cross-diffusion predator–prey model with Beddington–DeAngelis functional response. The proofs mainly rely on Fixed point index theory and analytical techniques.     

Impulsive periodic oscillation for a predator–prey model with Hassell–Varley–Holling functional response

This paper considers impulsive periodic oscillation in a predator–prey model with Hassell–Varley–Holling functional response. Based on classical continuous theorem of coincidence degree and skilful analysis techniques, some sufficient and necessary criteria for the existence of positive periodic solutions are derived, which are used to generate impulsive control strategy to excite or remove the periodic oscillation in this system. We give some numerical examples to illustrate main results and present some interesting problems.     

Positive solutions of a diffusive Leslie–Gower predator–prey model with Bazykin functional response

In this paper, we consider a diffusive Leslie–Gower predator–prey model with Bazykin functional response and zero Dirichlet boundary condition. We show the existence, multiplicity and uniqueness of positive solutions when parameters are in different regions. Results are proved by using bifurcation theory, fixed point index theory, energy estimate and asymptotical behavior analysis.     

Multiple periodic solutions of an impulsive predator–prey model with Holling-type IV functional response

An impulsive periodic predator–prey model with Holling-type IV functional response is considered. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. It is important to point out that we establish a better estimation on the difference between the supremum and infimum of a differentiable piecewise continuous periodic function. As illustrated in this paper, with the help of this estimation, many existing results can be improved.     

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.     

Delay-induced Bogdanov–Takens bifurcation in a Leslie–Gower predator–prey model with nonmonotonic functional response

This work is concerned with the dynamics of a Leslie–Gower predator–prey model with nonmonotonic functional response near the Bogdanov–Takens bifurcation point. By analyzing the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the delay inducing the Bogdanov–Takens bifurcation is obtained. In this case, the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. The bifurcation diagram near the Bogdanov–Takens bifurcation point is drawn according to the obtained normal form. We show that the change of delay can result in heteroclinic orbit, homoclinic orbit and unstable limit cycle.     

Predator–prey model of Beddington–DeAngelis type with maturation and gestation delays

A predator–prey model of Beddington–DeAngelis type with maturation and gestation delays is formulated and analyzed. This two-delay model is similar to the stage-structured model by Liu and Beretta [S. Liu, E. Beretta, Stage-structured predator–prey Model with the Beddington–DeAngelis functional response, SIAM J. Appl. Math. 66 (2006) 1101–1129] but contains an extra gestation delay term. Criteria for permanence and for predator extinction as well as the global attractiveness of the interior equilibrium are derived. The combined effects of the two delays and the degree of predator interference on the dynamical behaviors of the coexistence equilibrium are also studied both analytically and numerically. It is shown that complicated behaviors including chaotic and multi-periodic solutions may occur with the introduction of gestation delay, and that the predator interference can stabilize the system by simplifying the dynamical behaviors and enlarging the stability parameter fields.     

Periodic solution of a delayed ratio-dependent predator–prey model with monotonic functional response and impulse

In this paper, a delayed ratio-dependent predator–prey model with monotonic functional response and impulse is investigated. By using the continuation theorem of coincidence degree theory, an easily verifiable sufficient condition for the existence of at least one positive periodic solution is established. In particular, our results generalize some known criteria.     

Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment

We consider a predator–prey model with prey-taxis and Holling-type II functional responses in a spatially heterogeneous environment to analyze the effects of prey-taxis and the heterogeneity of an environment on predator invasion. To achieve our goal, we investigate the stability of semi-trivial solution in which the predator is absent. It is known that both the predator diffusion and the death rate contribute to the predator invasion in a heterogeneous habitat when there is no prey-taxis. In this paper, we show that predator invasion is affected by the prey-taxis and diffusions of the prey-taxis model for a certain range of predator death rates in a heterogeneous environment. Furthermore, in cases where predator invasion by predator diffusion does not occur in a particular death rate range of the predator, predator invasion can occur by prey-taxis in a spatially heterogeneous habitat. In addition, we compare this phenomenon to the corresponding predator–prey model with ratio-dependent functional responses. It is observed that none of the predator’s diffusion and prey-taxis affect the predator’s invasion, and that only the predator’s death rate contributes to predator invasion for the model with ratio-dependent functional responses.     

Dynamical behavior of a delayed diffusive predator–prey model with competition and type III functional response

In this paper, a delayed diffusive predator–prey model with competition and type III functional response is investigated. By using inequality analytical technique, some sufficient conditions which ensure the permanence of the model have been derived. By Lyapunov functional method, a series of sufficient conditions which assure the global asymptotic stability of the system are established. The paper ends with some numerical simulations that illustrate our analytical predictions.