Invasive-Invaded Interaction Incorporating a Bramson Model with Density-Dependent Diffusion and a Non-Lipschitz Reaction

Abstract

The primary objective of the presented study is to investigate the pairwise interaction dynamics between invasive and invaded species, considering a model characterized by a non-regular, non-Lipschitz type reaction, as well as non-homogeneous diffusion. To achieve this, we begin with the foundational model proposed by Bramson in 1988 and tailor it to account for density-dependent diffusion and the non-Lipschitz type reaction, rendering it more applicable to our specific ecological scenario. Subsequently, our newly developed model is subjected to different analyses to ascertain the existence and uniqueness of positive weak solutions. It is noteworthy that density-dependent diffusive operators exhibit a property known as "finite propagation", which manifests as the existence of a propagating front in the ecological system. Furthermore, we delve into the problem domain by employing the concept of traveling waves to identify specific solutions. A key outcome of our investigation is as follows: When both species propagate at significantly different speeds in the context of traveling waves, the interaction between them is deemed unstable, resulting in oscillations in the concentration of the invaded species. Conversely, when both species propagate within a similar range of speeds, the dynamics of the system are predominantly governed by the invasive species.