We study a predator–prey system with Holling–Tanner interaction terms. We show that if the saturation rate m is large, spatially inhomogeneous steady-state solutions arise, contrasting sharply with the small-m case, where no such solution could exist. Furthermore, for large m, we give sharp estimates on the ranges of other parameters where spatially inhomogeneous solutions can exist. We also determine the asymptotic behaviour of the spatially inhomogeneous solutions as m → ∞, and an interesting relation between this population model and free boundary problems is revealed.
<p style='text-indent:20px;'>We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [<xref ref-type="bibr" rid="b7">7</xref>], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.</p>
We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin [2], with its origin tracing back to Capasso et al. [5,6]. We prove that the problem has a unique solution defined for all t > 0, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local diffusion model in [2]. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the model of [2], where the spreading has finite speed whenever spreading happens [33].proposed by Capasso and Paveri-Fontana [6] to describe the cholera epidemic which spread in the European Mediterranean regions in 1973. Here a, b, c are all positive constants, u(t) and v(t), respectively, stand for the average population concentration of the infectious agents and the infective humans in the infected area at time t.
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