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On a predator-prey system with cross diffusion representing the tendency of predators in the presence of prey species. (English) Zbl 1160.35021

The present paper is concerned with the existence of positive solution of the following Lotka-Volterra predator-prey system with cross-diffusion

$-{\Delta }u=u\left({a}_{1}-u-{b}_{1,2}v\right),\phantom{\rule{1.em}{0ex}}-D{\Delta }u-{\Delta }v=v\left({a}_{2}+{b}_{2,1}u-v\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },$

$\left(u,v\right)=\left(0,0\right)\text{on}\phantom{\rule{0.277778em}{0ex}}\partial {\Omega }$. The system represents the tendency of predators to avoid the group defense by a large number of prey or diffuse in the direction of higher concentration of the prey species. The approach uses the method of upper and lower solutions. Sufficient conditions for the existence of positive solutions are provided and the non-existence of positive solutions is also investigated.

##### MSC:
 35Q92 PDEs in connection with biology and other natural sciences 92D25 Population dynamics (general)
##### References:
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