Global existence and asymptotic stability in a predator–prey chemotaxis model. (August 2020) - Fu, Shengmao; Miao, Liangying
Abstract: In this paper, we consider the global behavior of the fully parabolic predator–prey chemotaxis model u 1 t = d 1 Δ u 1 + χ ∇ ⋅ ( u 1 ∇ v ) + μ 1 u 1 ( 1 − u 1 − e 1 u 2 ), x ∈ Ω, t > 0, u 2 t = d 2 Δ u 2 − ξ ∇ ⋅ ( u 2 ∇ v ) + μ 2 u 2 ( 1 + e 2 u 1 − u 2 ), x ∈ Ω, t > 0, v t = d 3 Δ v + α u 1 + β u 2 − γ v, x ∈ Ω, t > 0, ∂ u 1 ∂ ν = ∂ u 2 ∂ ν = ∂ v ∂ ν = 0, x ∈ ∂ Ω, t > 0, u 1 ( x, 0 ) = u 1, 0 ( x ), u 2 ( x, 0 ) = u 2, 0 ( x ), v ( x, 0 ) = v 0 ( x ), x ∈ Ω in a smooth bounded domain Ω ⊂ R n, where d 1, d 2, d 3, χ, ξ, μ 1, μ 2, e 1, e 2, β, γ are positive constants, α ∈ R . It is proved that if n ≤ 2 and the parameters μ 1, μ 2, e 1, e 2 satisfy some suitable conditions, then for all appropriate regular nonnegative initial data, the model has a unique global classical solution ( u 1, u 2, v ) . Furthermore, the following criteria on the global asymptotic stability of the equilibria to the model are given by constructing Lyapunov functions. (i) If e 1 < 1 and both μ 1 χ 2 and μ 2 ξ 2 are sufficiently large, then the solution ( u 1, u 2, v ) satisfying u 1, 0, u 2, 0 ≥ ( ⁄ ≡ ) 0 converges to a unique positive equilibrium point of the model. (ii) If e 1 ≥ 1 and μ 2 ξ 2 is sufficiently large, then the solution ( u 1, u 2, v ) with u 2, 0 ≥ ( ⁄ ≡ ) 0 converges to the semi-trivial equilibrium point ( 0, 1, β γ ) . In particular, the respective convergence rates are at least exponential if e 1 ≠ 1, and algebraic if e 1 = 1 .