The Stability of Nontrivial Positive Steady States for the SKT Model with Large Cross Diffusion
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Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
The Stability of Nontrivial Positive Steady States for the SKT Model with Large Cross Diffusion Qing LI1 , Qian XU2,† 1 College
of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China of Fundamental and Interdisciplinary Sciences, Beijing Union University, Beijing 100101, China
2 Institute
(E-mail: [email protected])
Abstract
This paper is concerned with the existence and stability of steady state solutions for the SKT
biological competition model with cross-diffusion. By applying the detailed spectral analysis and in virtue of the bifurcating direction to the limiting system as the cross diffusion rate tends to infinity, it is proved the stability/instability of the nontrivial positive steady states with some special bifurcating structure. Further, the existence and stability/instability of the corresponding nontrivial positive steady states for the original cross-diffusion system are proved by applying perturbation argument. Keywords
spectral analysis; stability; cross-diffusion system
2000 MR Subject Classification
1
35B32; 35B35
Introduction and Statement of Main Results
In an attempt to model segregation phenomena between two competing species, Shigesada et al.[21] proposed the following cross-diffusion model in 1979 (abbreviated as S-K-T model): ut = ∆[(d1 + ρ11 u + ρ12 v)u] + u(a1 − b1 u − c1 v), x ∈ Ω, t > 0, v = ∆[(d + ρ u + ρ v)v] + v(a − b u − c v), x ∈ Ω, t > 0, t 2 21 22 2 2 2 Bu = Bv = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0 (x) ≥ 0, v(x, 0) = v0 (x) ≥ 0, x ∈ Ω,
(1.1)
where Ω is a bounded domain in RN with smooth boundary ∂Ω; Bu = u (Dirichlet boundary condition) or Bu = ∂u ∂ν with outward normal ν on ∂Ω (Neumann boundary condition); di , ai , bi , ci are positive constants throughout this paper. In case Bu = u for (1.1), the boundary condition means that the habitat Ω is surrounded by a hostile environment. In case Bu = ∂u ∂ν , the boundary condition ∂u = 0 implies that there exists no migration across the boundary. ∂ν u(x, t) and v(x, t) represent the densities of two competing species at location x and time t. The coefficients ρ11 and ρ22 are the self-diffusion rates which represent intra-specific population pressures, ρ12 and ρ21 denote the cross diffusion coefficients which measure the population pressure from the competing species. One can see the details for the biological background in Okubo and Levin[19] . Manuscript received December 17, 2018. Accepted on January 17, 2020. This paper is supported by the National Natural Science Foundation of China (No.11871048, No.11501031, No.11471221, No.11501016),Premium Funding Project for Academic Human Resources Development in Beijing Union University(BPHR2019CZ07, BPHR2020EZ01) and Beijing Municipal Education Commission (KZ201310028030,KM202011417010). † Corresponding author.
Q. LI, Q. XU
658
In the case when ρij = 0, the system (1.1) becomes the classical Lotka-Volterra model while if ρij ̸= 0 for so
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