Nonlinear Instability Of A Prey-Predator Cross-Diffusion Model With Stage Structure

In this paper, we consider a nonlinear dynamics near an unstable positive constant equilibrium for the prey-predator cross-diffusive model with stage structure where ρ,b1,b2, κ1, κ2,γ1,γ2, m, σ and di(i=1,2,3,4) are positive constant, Td= (0,π)d (d= 1,2,3). The paper is divided into three sections.(1) In Section 1, the locally asymptotical stability of non-negative constant steady states for ODE system is given by using linearization.(2) In Section 2, the locally asymptotical stability of positive constant steady states for reaction-diffusion system is given by applying linearization under the homogeneous Neumann boundary condi-tion, Ω is a bounded domain in Rn.(3) In Section 3, first, the locally asymptotical stability of positive constant steady state for cross diffusion system (1) is given based on linearization.Secondly, the instability condition of the corresponding linearized system is found by any general initial perturbation near the positive constant equilibrium (U1*, U2*, U3*), where F(u) ={k1b1U2*+κ2b2U3*)u1-(A1-γ2)u2+κ2b2U1*3, A1=γ1+γ2+m-k1b1U1* and L2 estimate of the solution to the linearized system (4) is found by using Cramer’s rule, Gronwall inequality and Hadamard inequality.Then, the Bootstrap Lemma is proved by using Sobolev embedding theorem, Poincare inequality, Young inequality and Gagliardo-Nirenberg inequality,last, the main result of this paper is proved. For the prey-predator system with cross diffusive and stage structure in predator, it is proved that given any general perturba-tion of magnitude δ, its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of the order In 1/δ.