Harvesting and seeding of stochastic populations: analysis and numerical approximation

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. In Hening et al. (J Math Biol 79(2):533-570, 2019b) we considered this problem when one has absolute control of the population (infinite harvesting and seeding rates are possible). In many cases, these approximations do not make biological sense and one must 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. Even one dimensional harvesting problems can be hard to tackle. Once one looks at an ecosystem with more than one species analytical results usually become intractable. In order to gain information regarding the qualitative behavior of the system we develop rigorous numerical approximation methods. This is done by approximating the continuous time dynamics by Markov chains and then showing that the approximations converge to the correct optimal strategy as the mesh size goes to zero. By implementing these numerical approximations, we are able to gain qualitative information about how to best harvest and seed species in specific key examples. We are able to show through numerical experiments that in the single species setting the optimal seeding-harvesting strategy is always of threshold type. This means there are thresholds [Formula: see text] such that: (1) if the population size is 'low', so that it lies in [Formula: see text], there is seeding using the maximal seeding rate; (2) if the population size 'moderate', so that it lies in [Formula: see text], there is no harvesting or seeding; (3) if the population size is 'high', so that it lies in the interval [Formula: see text], 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. Suppose there are two competing species and we are only allowed to harvest or seed species 1. The optimal strategy of seeding and harvesting will involve lower and upper thresholds [Formula: see text] which depend on the density [Formula: see text] of species 2.