Bifurcation analysis of a piecewise-smooth Ricker map with proportional threshold harvesting

Abstract

Proportional threshold harvesting (PTH) refers to some control rules employed in fishing policies, which specify a biomass level below which no fishing is permitted (the threshold), and a fraction of the surplus above the threshold is removed every year. When these rules are applied to a discrete population model, the resulting map governing the harvesting model is piecewise smooth, so border-collision bifurcations play an essential role in the dynamics. In this paper, we carry out a bifurcation analysis of a PTH model, providing a thorough picture of the 2-parameter bifurcation diagram in the plane for a case study. Here, T is the threshold and q is the harvest proportion. Our results explain some numerical bifurcation diagrams in previous work for PTH, and uncover new features of the dynamics with interesting consequences for population management.