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Qualitative analysis of a ratio-dependent predator-prey system with diffusion. (English) Zbl 1059.92056
The authors consider a predator-prey model based on the ratio-dependent theory in which the predator growth rate is a function of the ratio of prey to predator with attached diffusion over a domain with closed boundary. This type of interaction is believed to be a more faithful realisation of the natural state when the predators have to search, share and complete for the prey than the standard Lotka-Volterra model in which the predator growth rate is proportional to the number of prey present. In the model function used in this paper there is no sense of the limit as the search, share and complete requirement disappears being that of Lotka-Volterra. One could consider this a little curious.
The major part of the paper is concerned with persistence, stability and dissipation of nonnegative constant and nonconstant solutions of the diffusion system, i.e., the evolutionary aspect of the system is not explored in detail. This does make for something of a departure from the original dynamical system.
The paper contains very many theorems and is dense with results. There is no suggestion of an application to any system based upon reality.

MSC:
92D40 Ecology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
92D25 Population dynamics (general)
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