Global existence and asymptotic stability in a predator–prey chemotaxis model

Publication date: August 2020Source: Nonlinear Analysis: Real World Applications, Volume 54Author(s): Shengmao Fu, Liangying MiaoAbstractIn this paper, we consider the global behavior of the fully parabolic predator–prey chemotaxis model u1t=d1Δu1+χ∇⋅(u1∇v)+μ1u1(1−u1−e1u2),x∈Ω,t>0,u2t=d2Δu2−ξ∇⋅(u2∇v)+μ2u2(1+e2u1−u2),x∈Ω,t>0,vt=d3Δv+αu1+βu2−γv,x∈Ω,t>0,∂u1∂ν=∂u2∂ν=∂v∂ν=0,x∈∂Ω,t>0,u1(x,0)=u1,0(x),u2(x,0)=u2,0(x),v(x,0)=v0(x),x∈Ωin a smooth bounded domain Ω⊂Rn, where d1,d2,d3,χ,ξ,μ1,μ2,e1,e2,β,γ are positive constants, α∈R. It is proved that if n≤2 and the parameters μ1, μ2, e1, e2 satisfy some suitable conditions, then for all appropriate regular nonnegative initial data, the model has a unique global classical solution (u1,u2,v). Furthermore, the following criteria on the global asymptotic stability of the equilibria to the model are given by constructing Lyapunov functions. (i) If e1<1 and both μ1χ2 and μ2ξ2 are sufficiently large, then the solution (u1,u2,v) satisfying u1,0,u2,0≥(⁄≡)0 converges to a unique positive equilibrium point of the model. (ii) If e1≥1 and μ2ξ2 is sufficiently large, then the solution (u1,u2,v) with u2,0≥(⁄≡)0 converges to the semi-trivial equilibrium point (0,1,βγ).In particular, the respective convergence rates are at least exponential if e1≠1, and algebraic if e1=1.

Source: Nonlinear Analysis: Real World Applications - January 7, 2020 Category: Research Source Type: research