Large time behavior in a predator-prey system with indirect pursuit-evasion interaction

Abstract

In a bounded domain Ω ⊂ R n with smooth boundary, this work considers the indirect pursuit-evasion modelwith positive parameters χ, ξ, λ, µ, a and b.It is firstly asserted that when n ≤ 3, for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if bλ < µ and under some explicit smallness conditions on χ and ξ, any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if bλ > µ, however, then under an explicit smallness assumption on χ but without any restriction on ξ, any bounded classical solution (u, v) with u ≡ 0 stabilizes to (λ, 0) as t → ∞.1. Introduction. Lotka-Volterra type interplay, forming a class of paradigmatic interaction mechanisms in population dynamics, occurs in various biological and ecological processes, and the investigation of possible effects has been stimulated both theoretical biology and applied mathematics during the past century. A significant potential with regard to structure formation, preferably in contexts involving non-negligible migration of individuals, is indicated by numerous findings on colorful solution behavior already in simple reaction-diffusion systems combining Lotka-Volterra kinetics with undirected random diffusion ([15]).

Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model

“…Next, we plan to establish L ∞ estimates for u, v. Firstly, we derive the L p estimates for u, v when p ∈ I, where I is defined in Lemma 2.3. In fact, the following proof is based on the ideas from [23,42]. Since the elliptic equation differs from the above cites, the procedure for the estimates vary from them slightly.…”

Convergence rates of solutions in apredator-preysystem withindirect pursuit-evasion interaction in domains of arbitrary dimension

“…In drastic contrast to (3) with logistic term, the system (1) with the indirect pursuit-evasion exhibits a new property concerning the competition between the chemotactic crossdiffusion and logistic dampening in the N -dimensional case (N ≥ 3), which is different from (3) with logistic term. In particular, an arbitrarily small quadratic degradation term is sufficient to suppress any blow-up phenomenon in (1) when N = 3 (see [20]). In fact, if f = u(λ − u + av) and g = v(µ − v − bu), Li-Tao-Winkler ( [20]) proved that the solutions of parabolic-elliptic model ( 5) are global and bounded provided that N ≤ 3 and b > 0.…”

Large time behavior in a predator-prey system with pursuit-evasion interaction

“…In 2019, Amorim, Telch and Villada ( [2]) provided uniform estimates in Lebesgue spaces, which lead to boundedness and the global well-posedness for the system. Recently, Li, Tao and Winkler ( [14]) obtained a global and bounded smooth solution when N ≤ 3, f (u, v) = u(µ−u+av) and g(u, v) = v(λ − v − bu). Moreover, under some exact assumptions on χ, ξ, µ, and λ, the solution converges to a spatially homogeneous coexistence state or a prey-extinction state in the large time limit.…”