Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

We extend the discrete-time mathematical models developed in (Ackleh et al., 2019) and (Ackleh et al., 2024) to account for seasonal prey reproduction and build a class of discrete-time predator-prey seasonal models. Each model distinguishes between breeding and non-breeding seasons, representing prey reproduction as a periodic function of period 2. Altogether, three different models are analyzed. In the first part, we extend the predator-prey model from (Ackleh et al., 2019) to incorporate seasonality. We study the resulting dynamics and show that when the inherent reproduction number of the prey and the invasion reproduction number of the predator are larger than one, the system is permanent and an interior 2-cycle is locally asymptotically stable. Comparing the impact of seasonality on prey density, we show that while seasonality is always deleterious (resulting in lower average prey density) when the predator is absent, it can be advantageous for certain parameter ranges when the predator is present. Similarly, in the second part, we extend the evolutionary predator-prey model from (Ackleh et al., 2019) to incorporate seasonality. We establish conditions for population persistence, the stability of periodic solutions, and the impact of seasonal reproduction under toxicant stress. Our analysis reveals how seasonal fluctuations in reproduction influence the evolutionary adaptation of prey to toxicants and how this affects the persistence and stability of both populations. Comparisons assess how seasonality modifies the effects of toxicant exposure, identifying parameter regimes where it may either increase or decrease population densities. These findings contribute to our understanding of the interplay between environmental stressors, evolutionary processes, and temporal dynamics in shaping ecosystem stability. Lastly, in the third part, we extend the model from (Ackleh et al., 2024). We establish the global stability of the interior equilibrium for the non-seasonal case using a novel nullcline approach. Upon incorporating seasonality, we show that both species persist when the inherent and invasion reproduction numbers are greater than one. We establish local stability conditions for the interior 2-cycle and show numerically that the system may exhibit rich dynamical behavior. Finally, we demonstrate that seasonality shrinks the region where dynamics were stable under certain parameter conditions.

Date

2-6-2026

DOI

https://proquest.com/docview/3347845044

First Committee Chair

Azmy S. Ackleh

First Committee Member

Amy Veprauskas

Second Committee Member

Hayriye Gulbudak

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Mathematics Commons

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