Optimality and Sustainability of Delayed Impulsive Harvesting

Date
2022-07-18
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Abstract
Optimal and sustainable management of natural resources requires knowledge about the behaviour of mathematical models of harvesting under many different types of conditions. In this thesis, the effects of delays on the optimality and sustainability of harvesting models are studied, with a particular focus on delayed impulsive harvesting models. We begin by considering delays within a continuous harvesting model and derive sufficient conditions for stability of harvesting models with general growth and harvesting rate. We also derive maximum sustainable yields for models with both logistic and Gompertz growth, and show that they are delay dependent. Then we consider the main object of the thesis, a logistic differential equation subject to impulsive delayed harvesting, where the deduction information is a function of the population size at the time of one of the previous impulses. A close connection to the dynamics of high-order difference equations is used to conclude that while the inclusion of a delay in the impulsive condition does not impact the optimality of the yield, sustainability may be highly affected and is once again delay-dependent. Maximum and other types of yields are explored, and sharp stability tests are obtained for the model, as well as explicit sufficient conditions. It is also shown that persistence of the solution is not guaranteed for all positive initial conditions, and extinction in finite time is possible, as is illustrated in the simulations. The results of this thesis imply that delays within harvesting should be kept short to maintain the sustainability of resources.
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Keywords
Delays, Impulsive Harvesting, Optimal Harvesting, Logistic Equation, Impulsive Delayed Harvesting, Population Dynamics
Citation
Lawson, J. L. (2022). Optimality and Sustainability of Delayed Impulsive Harvesting (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.