Bogdanov-Takens bifurcation and multi-peak spatiotemporal staggered periodic patterns in a nonlocal Holling-Tanner predator–prey model

A reaction-diffusion Holling-Tanner predator-prey model with nonlocal prey competition involving purely spatial heat kernel is investigated. The first bifurcation curve is mathematically described, that is a piecewise smooth parameter curve of dividing the stability and instability of the coexistence equilibrium. The concepts of Turing/Hopf instability are extended to the higher codimension bifurcation instability, because the non-smooth points of the first bifurcation curve can be Bogdanov-Takens/Turing-Hopf/Hopf-Hopf instability point. Utilizing normal form method, spatiotemporal dynamics near \(Z_2\) $Z_2$ symmetric Bogdanov-Takens singularity are theoretically and numerically studied, including the stable coexistence of a pair of steady states with the shape of \(\cos \frac{2x}{l}\) $cos\frac{2x}{l}$ and a spatiotemporal staggered periodic solution with the shape of \(\cos \omega t\cos \frac{2x}{l}\) $cos\omega tcos\frac{2x}{l}$ . It is found that the larger the spatial size of a habitat is, the more complex the distributions of a species can be, while too narrow or wide range of nonlocal interactions inhibit the formations of complex spatiotemporal patterns.