Long-time behavior of a cooperative periodic-parabolic system: temporal degeneracy versus spatial degeneracy

In this paper, we continue our study on the cooperative periodic-parabolic system: {partial derivative(t)u - Delta u = mu u + alpha(x, t)v - a(x, t)u(p) partial derivative(t)u - Delta v = mu v + beta(x, t)u - b(x, t)v(q) in Omega x (0, infinity) (partial derivative(v)u, partial derivative(v)v) = (0, 0) on partial derivative Omega x (0, infinity) (u(x, 0), v(x, 0)) = (u(0)(x), v(0)(x)) > (0, 0) in Omega where p, q > 1, Omega subset of R-N (N >= 2) is a bounded smooth domain, alpha, beta > 0 and a, b >= 0 are smooth functions that are T-periodic in t, and mu is a varying parameter. The unknown functions u(x, t) and v(x, t) represent the densities of two cooperative species at location x and time t. In [1], we dealt with the case that and have simultaneous temporal and spatial degeneracies (i.e., vanish), and studied the long-time behavior of (u, v) which resembles that of the scalar periodic-parabolic logistic equation with temporal and spatial degeneracies. The present paper concerns the other two natural situations: simultaneous temporal degeneracy and simultaneous spatial degeneracy. When the species are exposed to such degenerate environments, our investigation reveals new dynamical behaviors, comparing to [1, 2]. The limiting behavior of the principal eigenvalue problem of an associated linear periodic-parabolic system, which may be of independent interest, plays a crucial role in our analysis.