Harvesting of interacting stochastic populations

Impacts of demographic and environmental stochasticity on population dynamics with cooperative effects

Different types of stochasticity play essential roles in shaping complex population dynamics. This paper presents a novel approach to model demographic and environmental stochasticity in a single-species model with cooperative components that are measured by component Allee effects. Our work provides rigorous mathematical proof on stochastic persistence and extinction, ergodicity (i.e., the existence of a unique stationary distribution) and the existence of a nontrivial periodic solution to study the impacts of demographic and environmental stochasticity on population dynamics. The theoretical and numerical results suggest that stochasticity may affect the population system in a variety of ways, specifically: (i) In the weak Allee effects case (e.g., strong cooperative efforts), the demographic stochasticity from the attack rate contributes to the expansion of the population size, while the demographic stochasticity from the handling rate and the environmental stochasticity have the opposite role, and may even lead to population extinction; (ii) In the strong Allee effects case (cooperative efforts not strong enough), both demographic and environmental stochasticity play a similar role in the survival of population, and are related to the initial population level: if the initial population level is large enough, demographic stochasticity and environmental stochasticity may be detrimental to the survival of population, otherwise if the initial population level is small enough, demographic stochasticity and environmental stochasticity may bring survival opportunities for the population that deterministically would extinct indefinitely; (iii) In the extinction case, demographic and environmental stochasticity can not change the trend of population extinction, but they can delay or promote population extinction.

The effects of random and seasonal environmental fluctuations on optimal harvesting and stocking

We analyze the harvesting and stocking of a population that is affected by random and seasonal environmental fluctuations. The main novelty comes from having three layers of environmental fluctuations. The first layer is due to the environment switching at random times between different environmental states. This is similar to having sudden environmental changes or catastrophes. The second layer is due to seasonal variation, where there is a significant change in the dynamics between seasons. Finally, the third layer is due to the constant presence of environmental stochasticity-between the seasonal or random regime switches, the species is affected by fluctuations which can be modelled by white noise. This framework is more realistic because it can capture both significant random and deterministic environmental shifts as well as small and frequent fluctuations in abiotic factors. Our framework also allows for the price or cost of harvesting to change deterministically and stochastically, something that is more realistic from an economic point of view. The combined effects of seasonal and random fluctuations make it impossible to find the optimal harvesting-stocking strategy analytically. We get around this roadblock by developing rigorous numerical approximations and proving that they converge to the optimal harvesting-stocking strategy. We apply our methods to multiple population models and explore how prices, or costs, and environmental fluctuations influence the optimal harvesting-stocking strategy. We show that in many situations the optimal way of harvesting and stocking is not of threshold type.

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.