Emergence and robustness of solitary states and cluster states in prey–predator model

Abstract

We investigate the possible scenarios for the emergence and robustness of solitary states (SS) in an identically coupled Rosenzweig–MacArthur (RM) prey–predator model, where the coupling is limited to the predator. In SS, a small subset of patches splits off and behaves differently from other patches. We also explored cluster state (CS) and synchronized oscillatory state (SOS) in an identically coupled Rosenzweig–MacArthur (RM) prey–predator model. In CS, the patches split into groups, with high (exhibits periodic oscillations) and zero prey density. In contrast, the predator population has a low density and shows periodic oscillations. Additionally, we have analyzed two distinct types of solitary states (SS1 and SS2) and evaluated the basins of attraction and probability of attaining different dynamical states using basin stability measures. Moreover, we demonstrate the emergence of two inhomogeneous steady states (IHSS), namely solitary death (SD) and cluster death (CLD), alongside solitary and cluster states. Further, we show the transition of SS1 to SD and CS to CLD state with varying the number of patches. In addition, we show the robustness and critical value of the dispersal rate required for the occurrence of SS, CS, SD, and CLD states with varying network size. Finally, we also observed that CS and CLD states were obtained in different ratios, including high and zero prey density patches. Our analysis also reveals that CS is a prerequisite for the emergence of a solitary state. We have also determined the stability conditions for solitary and cluster states.