LONG-TIME PROPERTIES OF PREY-PREDATOR SYSTEM WITH CROSS-DIFFUSION

Abstract

Using calculus inequalities and embedding theorems in <TEX>$R^1$</TEX>, we establish <TEX>$W^1_2$</TEX>-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) <TEX>$d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$</TEX>, and (ii) <TEX>$0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$</TEX>. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when <TEX>$t\;{\to}\;{\infty}$</TEX> via suitable Liapunov functionals.