Ultimate boundedness for nonautonomous diffusive Lotka-Volterra patches (Q1115374)

The asymptotic behavior of n biological species living in two patches is investigated. The governing mathematical model reads as a system of ordinary differential equations of the form \[ \dot x_ i=x_ i[e_ x(t)+A_ x(t)x]_ i+D_ i(t)(y_ i-x_ i),\quad \dot y_ i=y_ i[e_ y(t)+A_ y(t)y]_ i+D_ i(t)(x_ i-y_ i), \] i\(=1,2,...,n\), where \(e_ x,e_ y: {\mathbb{R}}\to {\mathbb{R}}^ n\), \(A_ x,A_ y: {\mathbb{R}}\to R^{n\times n}\), \(x=col(x_ 1,x_ 2,..,x_ n)\), \(y=col(y_ 1,y_ 2,...,y_ n)\). Sufficient conditions which guarantee the ultimate boundedness property of the (positive) solutions are given.