Global bifurcation of solutions for a predator–prey model with prey‐taxis

We study pattern formations in a predator–prey model with prey‐taxis. It is proved that a branch of nonconstant solutions can bifurcate from the positive equilibrium only when the chemotactic is repulsive. Furthermore, we find the stable bifurcating solutions near the bifurcation point under suitable conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.     

Periodic solution and its stability of a delayed Beddington‐DeAngelis type predator‐prey system with discontinuous control strategy

This paper investigates the periodic solution of a delayed Beddington‐DeAngelis (BD) type predator‐prey model with discontinuous control strategy. Firstly, the regularity and visibility analysis of the delayed predator‐prey model is carried out by using the principle of differential inclusion. Secondly, the positiveness and boundeness of the solution is discussed by employing the comparison theorem. Based on the boundary conditions of the model and the Mawhin‐like coincidence theorem, it is shown that the solution of the delayed BD system is asymptotically stable in finite time. Furthermore, it is found that there exists at least one periodic solution of the nonautonomous delayed predator‐prey model by using the principle of topological degree and set value mapping. Specially, when the nonautonomous delayed BD system degenerates into an autonomous system, some criteria are obtained to guarantee the convergence behavior of the harvesting solutions for the corresponding autonomous delayed BD system. Finally, numerical examples are given to demonstrate the applicability and effectiveness of main results. It is worthy to point out that the discontinuous control strategy is superior to the continuous harvesting policies adopted in existing literature.     

具有holling Ⅲ类功能反应的捕食者食系统差分方程周期解的存在性[英文]

本文利用重合度理论研究了一类具有holling Ⅲ类功能反应捕食者食差分方程的正周期解的存在性问题。     

A ratio‐dependent predator–prey model with stage structure and optimal harvesting policy

In this paper, a ratio‐dependent predator–prey model with stage structure and harvesting is investigated. Mathematical analyses of the model equations with regard to boundedness of solutions, nature of equilibria, permanence and stability are performed. By constructing appropriate Lyapunov functions, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. The existence possibilities of bioeconomic equilibria have been examined. An optimal harvesting policy is also given by using Pontryagin's maximal principle. Copyright © 2007 John Wiley & Sons, Ltd.     

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.     

Bogdanov-Takens bifurcation in a delayed Michaelis-Menten type ratio-dependent predator-prey system with prey harvesting

In this paper, we study a delayed Michaelis-Menten Type ratio-dependent predator-prey model with prey harvesting. By considering the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the parameters for the Bogdanov-Takens bifurcation is obtained. The conditions for the characteristic equation having negative real parts are discussed. Using the normal form theory of Bogdanov-Takens bifurcation for retarded functional differential equations, the corresponding normal form restricted to the associated two-dimensional center manifold is calculated and the versal unfolding is considered. The parameter conditions for saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained. Numerical simulations are given to support the analytical results.