Exploring quasi-periodic shrimps in the parameter space of a discrete-time food chain model

Shrimps are islands of regularity within chaotic regimes in bi-parameter spaces of nonlinear dynamical systems. While the presence of periodic shrimps has been extensively reported, recent research has uncovered the existence of quasi-periodic shrimps. Compared to their periodic counterparts, quasi-periodic shrimps require a relatively higher-dimensional phase-space to come into existence and are also quite uncommon to observe. This Focus Issue contribution delves into the existence and intricate dynamics of quasi-periodic shrimps within the parameter space of a discrete-time, three-species food chain model. Through high-resolution stability charts, we unveil the prevalence of quasi-periodic shrimps in the system's unsteady regime. We extensively study the bifurcation characteristics along the two borders of the quasi-periodic shrimp. Our analysis reveals that along the outer border, the system exhibits transition to chaos via intermittency, whereas along the inner border, torus-doubling and torus-bubbling phenomena, accompanied by finite doubling and bubbling cascades, are observed. Another salient aspect of this work is the identification of quasi-periodic accumulation horizon and different quasi-periodic (torus) adding sequences for the self-distribution of infinite cascades of self-similar quasi-periodic shrimps along the horizon in certain parameter space of the system.