2025/04/02 by Tello, J. Ignacio, Wrzosek, Dariusz
#35B30 #35K59 #35Q92 #92D25 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.01546
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⊂ ℝN with regular boundary and prove that for the space dimension N≤ 2 the solution to the Lotka-Volterra competition model with repulsive indirect taxis and homogeneous Neumann boundary conditions ut - duΔu = χ∇ ⋅ u ∇ w +μ1u(1-u-a1v) , vt - dvΔv = μ2v(1-v-a2u) , ε ( wt - dwΔw )= v- w , converges to the solution of repulsive direct-taxis model: ut - duΔu = χ∇ ⋅ u ∇ v +μ1u(1-u-a1v) , vt - dvΔv = μ2v(1-v-a2u) when ε\longrightarrow 0. For space dimension N≥ 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.