Global Stability for a Binary Reaction-Diffusion Lotka-Volterra Model with Ratio-Dependent Functional Response
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Global Stability for a Binary Reaction-Diffusion Lotka-Volterra Model with Ratio-Dependent Functional Response Florinda Capone · Roberta De Luca
Received: 13 December 2013 / Accepted: 9 January 2014 © Springer Science+Business Media Dordrecht 2014
Abstract A reaction-diffusion system modeling the predation between two species is analyzed in the case in which the predators have to search, share and compete for food. The boundedness and uniqueness of the solutions is proved and conditions guaranteeing the global nonlinear asymptotic stability of the positive equilibrium point have been found. These conditions improve those ones present in the existing literature. Keywords Predator-prey · Ratio-dependent · Global stability
1 Introduction The predation between two species has been deeply developed in literature (see, for instance, [1–12] and the references therein). When predators have to search, share and compete for food, it is useful to take into account of predator’s rate of feeding upon prey, the functional response that, in this case, is a function of the ratio between prey and predator. Numerous papers are devoted to the qualitative analysis of the ratio-dependent predator-prey system (see, for instance, [7–12]). When the population is non-homogeneously mixed, the diffusion has to be taken into account and reaction-diffusion dynamical systems are more appropriate to describe the problem (see, for example, [4–6, 9–20]). In [10, 11] a reaction-diffusion dynamical model governing the interaction between predator and prey, is considered. The boundary conditions are of Neumann type, meaning that the population has no flux across the boundary of the domain. By using the Liapunov Direct Method, conditions guaranteeing the global asymptotic stability of the biologically meaningful equilibrium point, have been
In honor of Professor Salvatore Rionero, on the occasion of his 80th birthday.
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F. Capone · R. De Luca ( ) Department of Mathematics and Applications ‘R. Caccioppoli’, University of Naples Federico II, Complesso Universitario Monte S. Angelo, Via Cinzia, 80126 Naples, Italy e-mail: [email protected] F. Capone e-mail: [email protected]
F. Capone, R. De Luca
obtained. In particular, the results in [11] improve those ones in [10] in the case of equal diffusion coefficients. In this paper we want to reconsider the problem in [10, 11], under the more general Robin boundary conditions, aimed to find the best conditions ensuring the global nonlinear stability of the positive meaningful equilibrium point. The plan of the paper is as follows. Section 2 is devoted to the preliminaries in which the existence, boundedness and uniqueness of solutions are investigated and the results obtained in [10, 11] are recalled. Preliminaries to the stability are furnished in Sect. 3 where suitable scalings are introduced together with a peculiar Liapunov functional which allows to obtain necessary and sufficient conditions for the linear stability of the positive equilibrium point (Sect. 4). The global, nonlinear stability
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