Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition

A diffusive Rosenzweig-MacArthur model involving nonlocal prey competition is studied. Via considering joint effects of prey's carrying capacity and predator's diffusion rate, the first Turing (Hopf) bifurcation curve is precisely described, which can help to determine the parameter region where coexistence equilibrium is stable. Particularly, coexistence equilibrium can lose its stability through not only codimension one Turing (Hopf) bifurcation, but also codimension two Bogdanov-Takens, Turing-Hopf and Hopf-Hopf bifurcations, even codimension three Bogdanov-Takens-Hopf bifurcation, etc., thus the concept of Turing (Hopf) instability is extended to high codimension bifurcation instability, such as Bogdanov-Takens instability. To meticulously describe spatiotemporal patterns resulting from \begin{document}$ Z_2 $\end{document} symmetric Bogdanov-Takens bifurcation, the corresponding third-order normal form for partial functional differential equations (PFDEs) involving nonlocal interactions is derived, which is expressed concisely by original PFDEs' parameters, making it convenient to analyze effects of original parameters on dynamics and also to calculate normal form on computer. With the aid of these formulas, complex spatiotemporal patterns are theoretically predicted and numerically shown, including tri-stable nonuniform patterns with the shape of \begin{document}$ \cos \omega t\cos \frac{x}{l}- $\end{document} like or \begin{document}$ \cos \frac{x}{l}- $\end{document} like, which reflects the effects of nonlocal interactions, such as stabilizing spatiotemporal nonuniform patterns.