Search

This paper is concerned with a nonlocal reaction–diffusion prey–predator model with free boundary. The spatial–temporal risk index R 0 p (t) which is threshold value, is introduced for this model. The global existence of positive solutions is proved and the long-time behavior of positive solutions is investigated. The sufficient conditions are found for vanishing of predators and the establishment of themselves in new environment. Furthermore, the spreading–vanishing criteria are established. The results show that the predators will vanish finally and not establish themselves if the sizes of habitant and spreading coefficient are relatively small or the initial value of predators is also relatively small with R 0 p (0) < 1 , and the predators will achieve if initial values of predators and preys are relatively big or R 0 p (t 0) > 1 for t 0 > 0. [ABSTRACT FROM AUTHOR]

This paper is concerned with a cross-diffusion predator–prey system with prey-taxis incorporating Holling type II functional response under homogeneous Neumann boundary condition. By employing global bifurcation theory, it is obtained that a branch of nonconstant solutions can bifurcate from the positive constant solution whenever the chemotactic is attractive or repulsive. Furthermore, by using perturbation of simple eigenvalues it is found that the bifurcating solutions are locally stable near the bifurcation point under suitable conditions. These results imply that cross-diffusion can create coexistence for the predators and preys under the above special case. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a cross-diffusion system arising in a predator–prey population model with Holling type-II functional response in a bounded domain with no flux boundary condition. Employing the energy estimates and Gagliardo–Nirenberg type inequalities to establish -bounds uniform in time, the global existence and uniform boundedness of solutions for a cross-diffusion system are proved. The global asymptotic stability of the constant positive steady state is also investigated. [Copyright &y& Elsevier]