一类带有交叉扩散的捕食模型的共存问题

Coexistence of a prey-predator model with cross-difusion

研究带有齐次Dirichlet边界条件的捕食-食饵模型,得到了平凡解存在的条件,并给出半平凡解存在的充分条件以及解的先验估计,最后利用Shauder不动点定理,得到问题至少有一个正解存在的充分条件.该结果说明只要捕获率足够小,物种的交叉扩散相对弱,问题就至少存在一个正解.

A predator-prey model with homogeneous Dirichlet boundary conditions of is studied. The sufficent conditions of trivial solution and semi-trivial solutions are given, estimates of the solutions are given also. Finally by using the Shauder fixed point theorem, sufficient condition for the existence of positive solutions is obtained. The results show that as long as the capture rate is small enough, cross-diffusion species are relatively weak, the problem on the existence of at least one positive solution.