Optimal Control of a Threatened Wildebeest-Lion Prey-Predator System in the Serengeti Ecosystem

Abstract

We develop a two-species prey-predator model in which prey is wildebeest and predator is lion. The threats to wildebeest are poaching and drought while to lion are retaliatory killing and drought. The system is found in the Serengeti ecosystem. Optimal control theory is applied to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for poaching, construction of strong bomas for retaliatory killing and construction of dams for drought control. The possible impact of using a combination of the three controls either one at a time or two at a time on the threats facing the system is also examined. We observe that the best result is achieved by using all controls at the same time, where a combined approach in tackling threats to yield optimal results is a good approach in the management of wildlife populations.

Effects of Over-Harvesting and Drought on a Predator-Prey System with Optimal Control

“…The technique uses Runge-Kutta 4th order numerical method (RK4) in solving the optimal control problem. The following scholars [5] [18] [23],…”

“…The dynamics of predator-prey interactions is affected by factors such as over-harvesting, over predation, environmental pollution, mismanagement of the habitat, and hazards fire and drought. It has been recognized that many species in the ecosystem, have been pushed to extinction and many others are at the point of extinction due to interaction with the above mentioned factors [5]. From the bottom-up theory of community ecology, prey population density determines the predator's population density since predators depend on prey as their only source of food.…”

“…From the bottom-up theory of community ecology, prey population density determines the predator's population density since predators depend on prey as their only source of food. Thus, any change in prey population density affects the predator population density which means that a decrease in prey population density results in a decrease in predator population density and increase in prey population density results in increase in predator population density [5] [6]. Therefore, in order to sustain the prey and predator species in the ecosystem, harvesting rate of prey population should not be either greater or equal or close to intrinsic growth rate of prey but should be at a rate at which the prey species will still survive in the system.…”

“…For example, in the 1993 Serengeti ecosystem drought, about 30% of the wildebeest and about 40% to 50% of the total park population of large mammals died [5]. These external forces cause a rapid decrease of the population densities and probably the species can go to extinction if control measures are not taken into account.…”

“…proof for the existence of optimal control provided by[5] [21][22] is also appropriate in here.1) Model Equation (3.1) with control are linear in control variables.…”