Effects of degeneracy and functional response on the bifurcation and positive solutions for a diffusion model

This paper studies a diffusive competition model with degeneracy and Holling‐II functional response in spatially heterogeneous environment. First, we discuss the structures and stability of steady‐state bifurcation solutions. Then, the existence, nonexistence, and multiplicity of steady‐state solutions are established. We conclude that there exist two critical values induced by the spatial degeneracy and the functional response, respectively, such that when the growth rate of one of the competition species is between these two critical values, the model behaves drastically and some qualitative changes occur, which is in sharp contrast to the well‐studied classical models. In addition, it is found that the boundary condition also has important effects on the critical value. These show that not only degeneracy but also the combination of functional response and boundary condition have important influences on the model, especially on the structures of bifurcations and the existence of steady‐state solutions. Finally, the asymptotic behavior and global attractor of positive solutions for the parabolic system are investigated, which enrich the study of dynamical behavior for the model.