Ecological pattern formation in a reaction-diffusion system with hydra and bubbling phenomena

The present investigation focuses on the combined result of the predator’s Allee effect and the constant proportion of prey refuge in a spatiotemporal interacting species model. The bi-stability of equilibria has been analyzed theoretically and numerically. Subject to a feasible set of parameter values, sensitivity analysis has provided a one- and two-parametric bifurcation diagram and identified all potential codimension $1$ and $2$ bifurcations that the system may encounter. Bubbling and hydra impacts, with significant ecological implications, have been examined. The existence conditions and stability attributes under the effect of diffusion are determined using an interacting species model. Several contour images of spatial patterns through diffusion-driven instability have been presented and examined to illustrate the model’s applicability. Additionally, non-Turing patterns outside the Turing space are identified based on the classification of irregular patches in terms of spatiotemporal chaos. The biological implications of the findings have been extensively discussed.