Stability and Hopf bifurcation of the stationary solutions to an epidemic model with cross-diffusion

In the previous paper (Cai and Wang, 2015), we investigated the stationary solutions of a cross-diffusion epidemic model with vertical transmission in a spatially heterogeneous environment with Neumann boundary condition and proved that the set of positive stationary solutions forms a bounded bifurcation branch ΓΓ, which is monotone S or fish-hook shaped with respect to the bifurcation parameter δδ. In the present paper, we give some criteria on the stability of solutions on ΓΓ. We prove that the stability of the solutions changes only at every turning point of ΓΓ; while in a different case that a,ka,k and β(x)β(x) are sufficiently large, original stable positive stationary solutions at certain point may lose their stability, and Hopf bifurcation can occur. These results are very different from those of the spatially homogeneous or without cross-diffusion cases.