Global dynamics of a ratio-dependent commensalism model with non-selective harvesting and partial closure for the populations

Abstract

Populations of species in ecosystems subjected to harvesting often face the problem of extinction of one or more species. We propose and analyze the global dynamics of a ratio-dependent commensalism model with non-selective harvesting and partial closure of populations. In addition to positivity and uniform boundedness, we discuss the persistence and permanence of the system. The conditions for the existence, local, and global stability of all four equilibrium points were studied. Additionally, the analytical results showed a negative correlation between the fraction of stock available for harvesting and the equilibrium densities of the species. This implies a positive correlation between the partial closure for the populations and equilibrium densities of the species. We performed numerical simulations to validate the theoretical results. It was found that for at least 80% of the stock available for harvesting (or a partial closure of at most 20%), both species could be driven to extinction within a finite time. On the other hand, for a sustainable ecosystem where both species coexist in a stable equilibrium, at most 35% of stock available for harvesting (or a partial closure of at least 65%) should be implemented.