Optimal harvesting strategies for stochastic competitive Lotka–Volterra ecosystems

Abstract

This work develops optimal harvest strategies for Lotka–Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. To better reflect reality, a continuous-time Markov chain is used to model the random environment. The underlying systems are thus controlled regime-switching diffusions that belong to the class of singular control problems. Starting with a model having multiple species, we construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. Moreover, in some cases, the lower bounds provide a good approximation of the value functions.