Competition is a fundamental force shaping population size and structure as a result of limited availability of resources. In biomathematics, the biological models with competitive interactions exist widely. Furthermore, the nonlinear-diffusion (including self- and cross-diffusions) terms are incorporated to the biological models to better simulate the actual movement of species. Therefore, better compatibility with reality can be achieved by introducing nonlinear-diffusion into biological models with competitive interactions. As a result, a competition system with nonlinear-diffusion and nonlinear functional response is proposed and analyzed in this paper. We first briefly discuss the stability of trivial and semi-trivial solutions by spectrum analysis. Then the boundedness and the non-existence of steady states are studied. Based on the boundedness of the solutions, the existence of the steady states is also investigated by the fixed point index theory in a positive cone. The result shows that the two species can coexist when their diffusion and inter-specific competition pressures are controlled in a certain range.

由于资源有限，竞争是塑造人口规模和结构的根本力量。在生物数学中，具有竞争相互作用的生物模型广泛存在。此外，将非线性扩散（包括自扩散和交叉扩散）项纳入生物学模型，以更好地模拟物种的实际运动。因此，通过将非线性扩散引入具有竞争性相互作用的生物模型中，可以实现与现实的更好兼容性。因此，提出并分析了具有非线性扩散和非线性功能响应的竞赛系统。我们首先通过频谱分析简要讨论平凡和半平凡解的稳定性。然后研究了稳态的有界和不存在。基于解的有界性，还通过定点指标理论研究了正圆锥中稳态的存在。结果表明，当将它们的扩散和种间竞争压力控制在一定范围内时，这两个物种可以共存。