Permanence of periodic predator–prey system with two predators and stage structure for prey

A stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response is proposed and analyzed. Based on the comparison theorem, some sufficient and necessary conditions are derived for permanence of the system. Finally, two examples are presented to illustrate the application of our main results.     

The spreading speed for a predator–prey model with one predator and two preys

We study a predator–prey model with two preys and one predator. Our main concern is the invasion process of the predator into the habitat of two aborigine preys. We consider the case when the two preys are weak competitors in the absence of predator. Under certain conditions, we are able to characterize the asymptotic spreading speed by the parameters of the model.     

Weak-renormalized solutions for predator–prey system

In this article, we consider the predator–prey system with Dirichlet boundary conditions which is used in the modelling of ecology. Under the assumptions of no growth conditions and integrable data, we prove the existence of weak-renormalized solutions to the predator–prey system.     

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.     

Traveling wavefronts for a two-species predator–prey system with diffusion terms and stage structure

In this paper, we considered an important model describing a two-species predator–prey system with diffusion terms and stage structure. By using the linearized method, we investigated the locally asymptotical stability of the nonnegative equilibria of the system and obtained the locally stable conditions. And by using the approach introduced by Canosa [J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev. 17 (1973) 307–313] and the method of upper and lower solutions, we studied the existence of traveling wavefronts, connecting the zero solution with the positive equilibrium of the system. Our results show that the traveling wavefronts exist and appear to be monotone. Finally, we given a conclusion to summarize the overall achievements of the work presented in the paper.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.     

Traveling wave solutions of a diffusive predator–prey model with modified Leslie–Gower and Holling-type II schemes

We study a diffusive predator–prey model with modified Leslie–Gower and Holling-II schemes with \(D=0\). We establish the existence of traveling wave solutions connecting a positive equilibrium and a boundary equilibrium via the ‘shooting method’, and the non-existence by the ‘eigenvalue method’. It should be emphasized that a threshold value \(c^*=\sqrt{4\alpha }\) is found in our paper.     

Permanence and periodic solutions for a diffusive ratio-dependent predator–prey system

This paper considers a diffusive predator–prey model, in which there is a ratio-dependent functional response with Holling III type. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. The existence of a unique globally stable periodic solution is also presented.     

Traveling wavefronts for a two-species ratio-dependent predator–prey system with diffusion terms and stage structure

In this paper, we considered an important nonlinear reaction-diffusion equations describing a two-species ratio-dependent predator–prey system with diffusion terms and stage structure. By using the linearized method, we investigated the locally asymptotical stability of the nonnegative equilibria of the above mentioned system and obtained the locally stable conditions. And by combining the approach introduced by J. Canosa (see [J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Deve. 17 (1973) 307–313]) with the method of upper and lower solutions, we proved that the traveling wavefronts which connect the zero solution with the positive constant equilibrium of the system exist and appear to be monotone. Finally, we gave a conclusion to summarize the achievements of the work.     

Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion

In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.     

Multiple periodic solutions of a delayed predator–prey system with stage structure for the predator

By using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a delayed predator–prey system with stage structure for the predator is established.     

Periodic solutions for a semi-ratio-dependent predator–prey system with nonmonotonic functional response and time delay

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for the semi-ratio-dependent predator–prey system with nonmonotonic functional responses and time delay is established. Further, by constructing a Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the uniqueness and global stability of positive periodic solutions to the system. Finally, some numerical simulations are carried out to support the theoretical analysis of the research.     

Multiple positive periodic solutions for a generalized predator–prey system with exploited terms

In this paper, we consider a generalized predator–prey system with exploited terms and prove the existence of eight positive periodic solutions by employing the continuation theorem of coincidence degree theory.     

Periodic traveling wave train and point-to-periodic traveling wave for a diffusive predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type scheme is investigated in this article. The existences of a small amplitude periodic traveling wave train _Γp${\Gamma }_p$ and the traveling wave solution connecting the boundary equilibrium _Eu(1,0)$E_u(1,0)$ to the periodic traveling wave _Γp${\Gamma }_p$ are obtained. The existence of this point-to-periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain, and thus plays a mild role in the evolution of predator–prey communities. The techniques used here are the Hopf bifurcation theorem, the improved shooting method combining with the geometric singular perturbation method.     

A predator–prey system with stage-structure for predator and nonlocal delay

This paper deals with the behavior of solutions to the reaction–diffusion system under homogeneous Neumann boundary condition, which describes a prey–predator model with nonlocal delay. Sufficient conditions for the global stability of each equilibrium are derived by the Lyapunov functional and the results show that the introduction of stage-structure into predator positively affects the coexistence of prey and predator. Numerical simulations are performed to illustrate the results.     

Stability and Hopf bifurcation in a delayed predator–prey system with stage structure for prey

We consider a predator–prey system of Lotka–Volterra type with time delays and stage structure for prey. By analyzing the corresponding characteristic equations, the local stability of the equilibria is investigated and Hopf bifurcations occurring at the positive equilibrium under some conditions are demonstrated. The mathematical tools which enable us to obtain the sufficient conditions, guaranteeing the global asymptotical stability of the equilibria, are the well-known Kamke comparison theorem and an iteration technique. Numerical simulations are carried out to illustrate our theoretical results.     

Stability and instability analysis for a ratio-dependent predator–prey system with diffusion effect

In this paper, we consider a ratio-dependent predator–prey system with diffusion. And we mainly discuss the following problems: (1) stability and Hopf bifurcation analysis of the positive equilibrium for the reduced ODE system; (2) Diffusion-driven instability of the equilibrium solution; (3) Hopf bifurcations for the corresponding diffusion system with homogeneous Neumann boundary conditions. In order to verify our theoretical results, some numerical simulations are also included, respectively.     

Uniqueness and stability of positive steady state solutions for a ratio-dependent predator–prey system with a crowding term in the prey equation

This paper deals with a ratio-dependent predator–prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrate some special behaviors of solutions to the system which the coexistence states of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we demonstrate that under some conditions, the positive steady state solution of the predator–prey system with a crowding term in the prey equation is unique and stable. Our result is different from those ones of the predator–prey systems without the crowding terms.