Nonlinear diffusion effect on bifurcation structures for a predator-prey model. (Q636951)

The authors consider stationary solutions to a nonlinear diffusive predator-prey model which incorporates a modified version of Leslie-Gower and Holling-type II functional responses. More precisly, the system \[ -\Delta [(1+\alpha v)u]=(a-u-\frac {cv}{1+mu})u,\;-\Delta [(\mu +\frac {1}{1+\beta u})v]=(b-\frac {v}{1+mu})v \] is studied in a bounded domain in \(\mathbb {R}^n\) with zero Dirichlet boundary data, where \(\alpha ,\;\mu ,\;\beta ,\;a,\;b,\;c\) and \(m\) are positive constants. Existence of positive solutions bifurcating from semi-trivial solutions is proved. Global behaviour of branches of bifurcating solutions is shown, the coexistence region is described. This region spreads as \(\beta \) increases and narrows for large \(\alpha \). In the case \(\alpha =0\), only two types of positive solutions exist for large \(\beta \). The main tools are the global bifurcation theory and a priori estimates.