Optimal harvesting in a two-species model under critical depensation: The case of optimal harvesting in semi-arid grazing systems

Abstract

Ecological systems in general often exhibit the mechanism of critical depensation, i.e. the system may collapse due to decreasing population densities caused by, for example, an increasing predation pressure on the prey that causes both populations to collapse to extinction. In this study a semi-arid predator–prey grazing system is taken as an example and optimal, model based, harvesting rates for the herbivore population are presented that allow survival of both herbivores and grasses during long dry periods without precipitation. Recovery to a maximum sustainable yield is achieved in the next rain season under the assumption of model validity. Optimal control theory is utilized in this example as the principal method of solution. Both analytical and numerical issues of the solution method and obtained solutions are discussed. Survival of the system is possible if the pastoralist is willing to decrease his population considerably once the dry period has set in. The social willingness, necessary to implement these management strategies, can be interpreted in terms of a discount rate in the current value Hamiltonian associated with the dynamical grazing system. The example is worked out in more detail and dynamical solutions for two discount rates are presented.