From indirect to direct taxis by fast reaction limit
J. Ignacio Tello, Dariusz Wrzosek
TL;DR
This work analyzes the fast-reaction (small $\varepsilon$) limit that connects indirect taxis, modeled by a signaling chemical, to direct taxis in two-population reaction–diffusion systems. By establishing $\varepsilon$-independent a priori bounds and employing compactness arguments, the authors prove that for space dimension $N\le 2$ the full solution sequence converges to the direct-taxis limit, with explicit convergence rates under a compatibility condition, while for $N\ge 3$ convergence holds in a weak sense via subsequences. They also extend the analysis to a prey–predator system with prey taxis, showing the same limit behavior under similar assumptions. The results provide a rigorous derivation of direct-taxis models from indirect taxis and clarify how indirect taxis can qualitatively alter dynamics, depending on dimension and regularity.
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 $Ω\subset \mathbb{R}^N$ 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 $$u_t - d_uΔu = χ\nabla \cdot u \nabla w +μ_1u(1-u-a_1v)\,,$$ $$ v_t - d_vΔv = μ_2v(1-v-a_2u)\,,$$ $$\varepsilon ( w_t - d_wΔw )= v- w\, , $$ converges to the solution of repulsive direct-taxis model: $$ u_t - d_uΔu = χ\nabla \cdot u \nabla v +μ_1u(1-u-a_1v)\,,$$ $$ v_t - d_vΔv = μ_2v(1-v-a_2u)\,$$ 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.