Harvesting and seeding of stochastic populations: analysis and numerical approximation

Abstract

It is well known that excessive harvesting or hunting has driven species to extinction both on local and global scales. This leads to one of the fundamental problems of conservation ecology: how should we harvest a population so that economic gain is maximized, while also ensuring that the species is safe from extinction? We study an ecosystem of interacting species that are influenced by random environmental fluctuations. At any point in time, we can either harvest or seed (repopulate) species. Harvesting brings an economic gain while seeding incurs a cost. The problem is to find the optimal harvesting-seeding strategy that maximizes the expected total income from harvesting minus the cost one has to pay for the seeding of various species. We consider what happens when one, or both, of the seeding and harvesting rates are bounded. The focus of this paper is the analysis of these three novel settings: bounded seeding and infinite harvesting, bounded seeding and bounded harvesting, and infinite seeding and bounded harvesting. We prove analytical results and develop numerical approximation methods. By implementing these approximations, we are able to gain qualitative information about how to best harvest and seed species. We are able to show that in the single species setting there are thresholds $0<L_1<L_2<\infty$ such that: 1) if the population size is `low', so that it lies in $(0, L_1]$, there is seeding using the maximal seeding rate; 2) if the population size `moderate', so that it lies in $(L_1,L_2)$, there is no harvesting or seeding; 3) if the population size is `high', so that it lies in the interval $[L_2, \infty)$, there is harvesting using the maximal harvesting rate. Once we have a system with at least two species, numerical experiments show that constant threshold strategies are not optimal anymore.