Chemotaxis-Induced Turing Instability and Steady-State Bifurcation in a Ross–Lotka Epidemic Model on Compact Metric Graphs

We study a chemotactic Ross–Lotka malaria model on a compact metric graph, where spatial movement is described by graph diffusion together with mosquito chemotaxis toward infectious humans. Using the Crandall-Rabinowitz theorem together with a higher-order expansion, we derive a second-order Lyapunov-type coefficient, which determines the local stability of the bifurcating branch. Near the bifurcation point, the stable nonconstant steady state is locally unique and dominated by the critical Laplacian eigenmode. Numerical simulations are presented on a diamond-shaped graph, which illustrate that the chemotaxis can induce the occurrence of Turing patterns, and the emerging pattern is selected by the first nonconstant Laplacian mode.