Multi-patch integrodifference models and their eigenvalue problems in spatial ecology
Loading...
Date
2024-10
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of New Brunswick
Abstract
In the realm of spatial ecology, we grapple with fundamental questions: How can we design effective nature reserves to safeguard the survival of species? In the context of fisheries, how wide can a fishing zone be without compromising the stability of fish populations? These inquiries have fueled my motivation to delve into the subject matter of this thesis. While we acknowledge that precise answers to such questions remain elusive, I have endeavored to contribute to our understanding of these critical issues.
Our journey begins with an exploration of integrodifference equations (IDEs) in spatial ecology in Chapter 1. These mathematical models serve as powerful tools for unraveling the intricate spatial and temporal dynamics of populations characterized by discrete generations and continuous spatial domains.
Imagine a population confined to a single isolated patch — a scenario akin to a lake surrounded by hard boundaries. Within this patch, there exists a gap devoid of reproduction, effectively separating the population. Consider, for instance, a protected fishing zone within a lake. Our focus in Chapter 2 lies on understanding the persistence of such populations. We model their life cycles using IDEs and present a method to calculate the maximum allowable gap size that ensures population persistence.
The concept of critical patch size takes center stage in Chapter 3. It refers to the minimum favorable area below which a population faces the risk of extinction. Our investigation accounts for the demographic and dispersal traits of individuals, recognizing that these traits may vary across patches. Surprisingly, we find that the smallest critical patch size occurs when individuals exhibit a propensity to leave the patch. Conversely, the largest critical patch size arises when boundaries are more restrictive, limiting the chances of individuals leaving the patch.
In the patchy landscape, Chapter 4 introduces an approximation method that simplifies equilibrium population calculations. Our approach involves a form of the redistribution approximation tailored for piecewise continuous kernels. The accuracy of our estimate improves as movement biases near patch boundaries intensify. Key factors influencing our estimate include the growth term’s derivative and the deviation of the equilibrium solution from its average across patches.