Asymptotic behaviour of solutions for some competition model with migration effect (Q920055)

This paper is concerned with the study of a couple of hyperbolic equations with a growth term of Volterra-Lotka type: \[ (\partial /\partial t+\partial /\partial x)u_ 1=(1-u_ 1-b_ 1u_ 2)u_ 1,\quad (\partial /\partial t-\partial /\partial x)u_ 2=a(1-b_ 2u_ 1-u_ 2)u_ 2,\quad (x,t)\in {\mathbb{R}}\times {\mathbb{R}}_+, \] with the initial conditions: \[ (u_ 1(x,0)=u_{10}(x),\quad u_ 2(x,0)=u_{20}(x),\quad x\in {\mathbb{R}}, \] where \(u_ i(x,t)\) are the population densities of two competing species. The authors have proved the \(C_ 0\)-stability of a couple of travelling wave solutions and have studied the asymptotic behaviour of the solutions for the Cauchy problem with some class of initial data. The proofs are based on the properties of the lower and upper solutions for the Cauchy problem.