Traveling Wavefronts in a Delayed Food-limited Population Model

In this paper we develop a new method to establish the existence of traveling wavefronts for a food-limited population model with nonmonotone delayed nonlocal effects. Our approach is based on a combination of perturbation methods, the Fredholm theory, and the Banach fixed point theorem. We also develop and theoretically justify Canosa's asymptotic method for the wavefronts with large wave speeds. Numerical simulations are provided to show that there exists a prominent hump when the delay is large. 1. Introduction. There has been some success in establishing the existence of traveling wavefronts for the reaction-diffusion equation with nonlocal delayed nonlin- earity. When the nonlinearity is monotone, the existence of traveling wavefronts can be obtained by extension of the methods of the super/subsolution pair (1), (7), (29), homotopy (3), and Leray-Schauder degree (28). Unfortunately, when the delayed non- linearity is no longer monotone, very little has been achieved (except for the work in (9)). While one suspects that the method developed by Wu and Zou (29) and based on a nonstandard ordering could be applicable, the construction of a supersolution and subsolution pair is nontrivial, and it is almost as difficult as solving the original given equations. In this paper we develop a new approach to establish the existence of trav- eling wavefronts in the case when the delayed nonlinearity is nonmonotone. We shall demonstrate this approach by considering the following food-limited reaction-diffusion equation

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