From indirect to direct taxis by fast reaction limit

Abstract

Many ecological population models consider taxis as the directed movement of animals in response to a stimulus. The taxis is named direct if the animals are guided by the density gradient of some other population or indirect if they are guided by the density of a chemical secreted by individuals of the other population. Let $u$ and $v$ denote the densities of two populations and $w$ the density of the chemical secreted by individuals in the $v$ population. We consider a bounded, open set $\Omega \subset {\mathbb{R}}^N$ with regular boundary and prove that for the space dimension $N\le 2$ the solution to the Lotka-Volterra competition model with repulsive indirect taxis and homogeneous Neumann boundary conditions $$u_t-d_u\Delta u=\chi \nabla \cdot u\nabla w+{\mu }_{1}u(1-u-a_{1}v),$$ $$v_t-d_v\Delta v={\mu }_{2}v(1-v-a_{2}u),$$ $$\epsilon (w_t-d_w\Delta w)=v-w,$$ converges to the solution of repulsive direct-taxis model: $$u_t-d_u\Delta u=\chi \nabla \cdot u\nabla v+{\mu }_{1}u(1-u-a_{1}v),$$ $$v_t-d_v\Delta v={\mu }_{2}v(1-v-a_{2}u)$$ when $\epsilon ⟶0$. For space dimension $N\ge 3$ we use the compactness argument to show that the result holds in some weak sense. A similar result is also proved for a typical prey-predator model with prey taxis and logistic growth of predators.