Abstract
This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.
Highlights
The dynamics of predator-prey systems is one of the fastest developing areas of modern mathematics due to their significant nature in biological fields and other practical fields
One significant component in these systems is the functional response describing the number of prey consumed per predator per unit time for given quantities of prey N and predators P
Many authors have explored the dynamic relationship between predators and their preys
Summary
The dynamics of predator-prey systems is one of the fastest developing areas of modern mathematics due to their significant nature in biological fields and other practical fields. The traditional mathematical model describing the predator-prey interactions consists of the following system of differential equations. Great attention has been paid to the study of the existence of traveling wave solutions in reactiondiffusion system, since they determine the long term behavior of other solutions, and account for phase transitions between different states of physical systems, propagation of patterns, and domain invasion of species in population biology (see, for example, [10]-[12], and the references cited therein). Motivated by the work of Peng and Wang [7] and Wu, Zou [12], in the present paper, we consider the existence of traveling waves of the following predator-prey model with Beddington-DeAngelis functional response and a discrete time delay due to gestation of predator ut − d1= uxx u We give some suitable examples to illustrate our results
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