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http://dx.doi.org/10.4134/CKMS.2006.21.2.293

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

Shim Seong-A (Department of Mathematics Sungshin Women's University)
Publication Information
Communications of the Korean Mathematical Society / v.21, no.2, 2006 , pp. 293-320 More about this Journal
Abstract
Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$, and (ii) $0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$. 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 $t\;{\to}\;{\infty}$ via suitable Liapunov functionals.
Keywords
prey-predator system; cross-diffusion; self-diffusion; calculus inequalities; uniform bound; Liapunov functional; convergence;
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