Traveling waves for a diffusive predator–prey system with general functional response

By combining the simplified shooting method with a sandwich method, the existence and nonexistence of two types of traveling wave solutions for a class of diffusive predator–prey systems with general functional response are investigated. These two types of point-to-point traveling waves include the connections of the zero equilibrium to the positive equilibrium and the boundary equilibrium to the positive equilibrium. Furthermore, we also give a discussion about the parameter threshold value whether any of the traveling waves approaches the positive equilibrium monotonically or has exponentially damped oscillations about the positive equilibrium. Some applications with different functional response functions are given to illustrate the application of our results.     

Multiple periodic solutions in generalized Gause-type predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple positive periodic solutions in periodic Gause-type predator–prey systems with non-monotonic numerical responses and time delays. As corollaries, some applications are listed. In particular, our results improve and supplement those obtained by Chen [Y. Chen, Multiple periodic solutions of delayed predator–prey systems with type IV functional responses, Nonlinear Anal. Real World Appl. 5 (2004) 45–53].     

Global stability of stage-structured predator–prey models with Beddington–DeAngelis functional response

Two stage-structured predator–prey systems with Beddington–DeAngelis functional response are proposed. The first one is deterministic. The Second one takes the random perturbation into account. For each system, sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.     

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.     

An optimal control problem for a prey–predator system with a general functional response

An optimal control problem is studied for a prey–predator system with a general functional response. The control functions represent the rate of mixture of the populations and the cost functional is of Mayer type. The number of switching points of the optimal control is discussed in terms of the sign of a specific constant.     

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.     

Discrete-time analogues of predator–prey models with monotonic or nonmonotonic functional responses

Discrete-time analogues of predator–prey models with monotonic and nonmonotonic functional responses are introduced, respectively. The discrete-time analogues are considered to be numerical discretizations of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models with monotonic functional responses. Moreover, it is the first time that multiplicity of periodic solutions are studied when modeled with nonmonotonic functional responses. Unlike other types of functional responses, nonmonotone functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation $\mathit{Lx}=\lambda \mathit{Nx}$ are inapplicable to our case and some new arguments are employed for the first time.     

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.     

Predator–prey model with sigmoid functional response

The sigmoid functional response in the predator–prey model was posed in 1977. But its dynamics has not been completely characterized. This paper completes the classification of the global dynamics for the classical predator–prey model with the sigmoid functional response, whose denominator has two different zeros. The dynamical phenomena we obtain here include global stability, the existence of the heteroclinic and homoclinic loops, the consecutive canard explosions via relaxation oscillation, and the canard explosion to a homoclinic loop among others. As we know, the last one is a new dynamical phenomenon, which has never been reported previously. In addition, with the help of geometric singular perturbation theory, we solve the problem of connection between stable and unstable manifolds from different singularities, which has not been well settled in the published literature.     

Asymptotic properties of a stochastic predator–prey system with Holling II functional response

A stochastic predator–prey system with Holling II functional response is proposed and investigated. We show that there is a unique positive solution to the model for any positive initial value. And we show that the positive solution to the stochastic system is stochastically bounded. Moreover, under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean.     

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.     

The impact of Allee effect on a predator–prey system with Holling type II functional response

In this paper, the Allee effect is incorporated into a predator–prey model with Holling type II functional response. Compared with the predator–prey model without Allee effect, we find that the Allee effect of prey species increases the extinction risk of both predators and prey. When the handling time of predators is relatively short and the Allee effect of prey species becomes strong, both predators and prey may become extinct. Moreover, it is shown that the model with Allee effect undergoes the Hopf bifurcation and heteroclinic bifurcation. The Allee effect of prey species can lead to unstable periodical oscillation. It is also found that the positive equilibrium of the model could change from stable to unstable, and then to stable when the strength of Allee effect or the handling time of predators increases continuously from zero, that is, the model admits stability switches as a parameter changes. When the Allee effect of prey species becomes strong, longer handling time of predators may stabilize the coexistent steady state.     

Stochastic predator–prey model with Allee effect on prey

We study a predator–prey model with the Allee effect on prey and whose dynamics is described by a system of stochastic differential equations assuming that environmental randomness is represented by noise terms affecting each population. More specifically, we consider a term that expresses the variability of the growth rate of both species due to external, unpredictable events. We assume that the intensities of these perturbations are proportional to the population size of each species. With this approach, we prove that the solutions of the system have sample pathwise uniqueness and bounded moments. Moreover, using an Euler–Maruyama-type numerical method we obtain approximated solutions of the system with different intensities for the random noise and parameters of the model. In the presence of a weak Allee effect, we show that long-term survival of both populations can occur. On the other hand, when a strong Allee effect is considered, we show that the random perturbations may induce the non-trivial attracting-type invariant objects to disappear, leading to the extinction of both species. Furthermore, we also find the Maximum Likelihood estimators for the parameters involved in the model.     

Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

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.     

Dynamics of cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response

In this paper, cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response is studied. By using comparison theorem and some analysis techniques as well as the coincidence degree theory, sufficient conditions are obtained for the permanence, extinction and the existence of positive periodic solution.