Stability of positive stationary solutions to a spatially heterogeneous cooperative system with cross-diffusion

Stability of positive stationary solutions to a spatially heterogeneous cooperative system with cross-diffusion

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In the previous article [Y.-X.Wang and W.-T.

Li, J.Differential Equations, 251 (2011) 1670-1695], the authors have shown that the set of positive stationary solutions of a cross-diffusive Lotka-Volterra cooperative system can form an unbounded fish-hook shaped branch $Gamma_p$.In the present paper, we will show some criteria for the stability of positive stationary solutions tenga air-tech regular on $Gamma_p$.Our results assert that if $d_1/d_2$ is small enough, then unstable positive stationary solutions bifurcate from semitrivial solutions, the stability changes only at every turning point of $Gamma_p$ and no Hopf bifurcation occurs.

While as $d_1/d_2$ becomes large, the stability has a drastic change when $mu<0$ in the supercritical case.Original stable positive stationary solutions at certain point may lose their stability, and Hopf bifurcation can occur.These results nitrile gloves in a bucket are very different from those of the spatially homogeneous case.