Modeling and analysis of two prey-one predator system with harvesting, Holling type II and ratio-dependent responses

Abstract

A Mathematical model is proposed and analysed to study the dynamics of a system of two prey and one predator in which the predator shows a Holling Type II response to one prey that is also harvested, and a ratio-dependent response to the other prey. The model is used to study the ecological dynamics of the lion-buffalo-Uganda Kob prey-predator system of Queen Elizabeth National Park, Western Uganda.

Results of analysis of the model showed that the 3 species would co-exist if the Uganda Kobs were not harvested beyond their intrinsic growth rate. Another important result of analysis was that the lion should convert the biomass of the Uganda Kobs into fertility at a rate greater than its natural mortality rate and the time it took to handle the Uganda Kobs. Also, the rate at which the lion captures the buffalo should be greater than the product of the buffalo’s intrinsic growth rate and its anti-predator behaviour.

One of the major observations from results of numerical simulation is that the predator population density increased significantly when the intrinsic growth rate of both prey increased. This can imply that a high intrinsic growth rate of the prey initially increases their population density which increases the predator’s chance of capturing the prey and so the predator’s population density increases.

Numerical simulation of the model also revealed that the dynamical behaviour of the system changes mostly from a limit cycle to a stable spiral and vice - versa when values of some parameters such as the harvesting rate, natural death rate of the predator and food conversion rate of predator are varied. This implied that these parameters can be controlled so that the dynamical behaviour of the steady state is a stable spiral which implies that the steady state is globally asymptotically stable. However, varying some parameters such as the inter-specific competition among prey does not change the dynamical behaviour of the system.