Diffusion-driven instabilities in a tri-trophic food web model: From Turing to non-Turing patterns and waves

This work uses a reaction–diffusion system to explore self-organized pattern-forming phenomena through species distribution in a tri-trophic food web system. The interacting non-diffusive system involves a bottom prey, one specialist intermediate predator dependent on the bottom prey, and a generalist top predator, having its food preference regulation on both the bottom prey and intermediate predator. The system encounters temporal instability through Hopf-bifurcation and chaotic oscillations about the coexistence equilibrium with over-dependency on a particular food source. We provide analytical conditions for diffusion-driven Turing and wave instabilities. A weakly nonlinear analysis (WNA) is performed to examine the patterns of Turing instability close to the critical threshold. Numerical simulations show spatiotemporal patterns like spots, stripes, a mixture of spots & stripes and spatiotemporal chaos. The numerical investigations highlight the non-Turing instabilities consisting of Hopf, wave, Hopf-Turing, Hopf-wave. The diffusion of species suppresses regular and irregular spatiotemporal oscillations, giving stability to the system. Using a heat map of the Lyapunov exponents of the time series for different pairs of parameter values in a bi-parametric plane, it is demonstrated that the Turing pattern dominates the oscillatory Hopf pattern if the critical parameter value is from the Hopf-Turing region and is close to the Hopf bifurcation threshold, but the Hopf pattern dominates if the parameter pair is significantly away from it. The qualitative comparison of the non-Turing instabilities is provided with the corresponding spatiotemporal distribution of the species. It is observed that the high-amplitude oscillations of Hopf and Hopf-dominated non-Turing oscillations are vicious for the spatially distributed population.