Dispersal- and harvesting-induced dynamics of single-species inhabited in minimal ring-shaped patches

Abstract

We investigate two discrete-time models of a single-species dispersed between three patches located on a ring. The dynamic models are formulated by identical logistic maps with linear coupling. The co-existing equilibrium completely depends on the intrinsic growth rate and carrying capacity. However, stability depends only on intrinsic growth rate and dispersal rate. We shall analytically present the stability analysis of both the trivial and coexisting equilibrium in a two parameter plane. Our main focus is to explore the bifurcations at the coexisting equilibrium and their consequences in ecology. Increasing dispersal rate leads to a period-doubling bifurcation in the bi-directional dispersal model followed by a Neimark-Sacker bifurcation arising from each periodic branch. Our analysis reports the existence of either three stable 2-cycles or three distinct quasi-periodic modes resulting in either periodic–periodic–periodic multistability or quasiperiodic–quasiperiodic–quasiperiodic multistability. In contrast to the bi-directional model, only a Neimark-Sacker occurs in the uni-directional dispersal model for increasing dispersal rate. This uni-directional dispersal strategy does not exhibit any multistability. The co-existing equilibrium may experience an instability switching in both models while introducing harvesting in one of the patches. Under harvesting, the bi-directional model could induce a Neimark-Sacker bifurcation which is impossible to occur for increasing dispersal. We shall estimate the effort levels to achieve the same amount of harvested yield in both uni- and bi-directional dispersal models. These results might be interesting from biological conservation and fishery management viewpoints.