From indirect to direct taxis by fast reaction limit

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 $N\leq 2$ the solution to the Lotka-Volterra competition model with repulsive indirect taxis and homogeneous Neumann boundary conditions $$u_t - d_u\Delta u = \chi \nabla \cdot u \nabla w +\mu_1u(1-u-a_1v)\,,$$ $$v_t - d_v\Delta v = \mu_2v(1-v-a_2u)\,,$$ $$\varepsilon ( w_t - d_w\Delta w )= v- w\, ,$$ converges to the solution of repulsive direct-taxis model: $$u_t - d_u\Delta u = \chi \nabla \cdot u \nabla v +\mu_1u(1-u-a_1v)\,,$$ $$v_t - d_v\Delta v = \mu_2v(1-v-a_2u)\,$$ when $\varepsilon\longrightarrow 0$. For space dimension $N\geq 3$ 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.