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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 purported to investigate a ratio-dependent invasion-diffusion predator-prey epidemic system in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without invasion-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are found for this invasion-diffusion epidemic system, which implies the existence of spatiotemporal pattern formation. The results show that in wide space for the predator and the prey to diffuse by the self pressure and with a little tendency of predator to catch prey, they can't coexist and there isn't endemic disease extensively, but they can coexist and endemic disease exists extensively when the coefficient of invasion-diffusion for predator is big enough with other random diffusion coefficients being fixed and satisfying curtain conditions. [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]

This paper is concerned with a diffusive Holling–Tanner prey–predator model in a bounded domain with Dirichlet boundary condition and a free boundary. The global existence of the unique solution is proved. Moreover, the criteria governing spreading–vanishing are derived by mainly using the comparison principle. The results show that if the length of the occupying line is bigger than a threshold value (spreading barrier), then the spreading of predators will make an achievement, and, if the length of the occupying line is smaller than this spreading barrier and the spreading coefficient is relatively small depending on initial size of predators, then the predators will fail in establishing themselves and eventually die out. [ABSTRACT FROM AUTHOR]

A ratio-dependent diffusion predator-prey system with free boundary is investigated to understand the impact of free boundary on spreading-vanishing dichotomy and a long time behavior of species. The existence and uniqueness of solutions are verified and the behavior of positive solutions is considered for this system. Moreover, the criteria for spreading-vanishing dichotomy are also derived. The results show that if the length of the initial occupying area is longer than a critical size for the predators or the length of the initial occupying area is shorter than a critical size, but the moving coefficient of free boundary is relatively big, then the spreading of predators always happens under relatively small rate of death for the predator. On the other hand, it is found that if the initial value of free boundary is smaller than a threshold value and the moving coefficient of free boundary is relatively small depending on initial size of predator or the rate of death is relatively big, the predators fail in spreading to new environment. [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]