Coexistence, extinction, and optimal harvesting in discrete-time stochastic population models

TL;DR

This paper studies the long-term behavior of stochastic discrete-time populations under harvesting, providing extinction and coexistence criteria, and deriving optimal harvesting strategies for single and interacting species, including predator-prey and competitive models.

Contribution

It introduces new criteria for extinction and coexistence in stochastic models and characterizes optimal harvesting strategies, including bang-bang and proportional strategies, for complex ecosystems.

Findings

Optimal harvesting for single species is bang-bang with a threshold.

In predator-prey systems, strategies depend on interaction parameters and prices.

In competitive systems, harvesting can lead to coexistence or extinction depending on conditions.

Abstract

We analyze the long term behavior of interacting populations which can be controlled through harvesting. The dynamics is assumed to be discrete in time and stochastic due to the effect of environmental fluctuations. We present extinction and coexistence criteria when there are one or two interacting species. We then use these tools in order to see when harvesting leads to extinction or persistence of species, as well as what the optimal harvesting strategies, which maximize the expected long term yield, look like. For single species systems, we show under certain conditions that the optimal harvesting strategy is of bang-bang type: there is a threshold under which there is no harvesting, while everything above this threshold gets harvested. The second part of the paper is concerned with the analysis of ecosystems that have two interacting species which can be harvested. In particular, we carefully study predator-prey and competitive Ricker models when there are two species. In this setting we show how to find the optimal proportional harvesting strategy. If the system is of predator-prey type the optimal proportional harvesting strategy is, depending on the interaction parameters and the price of predators relative to prey, either to harvest the predator to extinction and maximize the asymptotic yield of the prey or to not harvest the prey and to maximize the asymptotic harvesting yield of the predators. If the system is competitive, in certain instances it is optimal to drive one species extinct and to harvest the other one. In other cases it is best to let the two species coexist and harvest both species while maintaining coexistence. In the setting of the competitive Ricker model we show that if one competitor is dominant and pushes the other species to extinction, the harvesting of the dominant species can lead to coexistence.