Coexistence of Three Competing Species with Non-negative Cross-diffusion rate
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In this paper, we report on our investigation of the existence and non-existence of positive solutions to a 3 × 3 competition interaction system with non-negative cross-diffusion under Dirichlet boundary conditions. First, it is shown that the system with constant diffusions can have a positive solution under suitable assumptions even though each of three 2 × 2 sub-systems coupled from the three equations of the system does not have a positive solution. Second, we show that the emergence of cross-diffusion in one equation of the system may generate a positive solution in case that the corresponding competition interaction system without cross-diffusion does not have a positive solution.
KeywordsPositive solutions Competition model Non-negative cross-diffusion Fixed point index
Mathematics Subject Classifications (2010)35J60 35Q80
The authors thank the referee for the careful reading and valuable comments which have helped to improve the presentation of this paper. This work was supported by a Korea University Grant.
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