Naderi Yeganeh, Hamid; (2025) Invariant Sets and Stability in Dynamical Systems Applied to Theoretical Ecology and Population Genetics. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Dynamical systems play a crucial role in studying various aspects of ecology and population genetics. Among discrete dynamical systems, different instances of Kolmogorov maps have been widely used to model populations of organisms and predict their long-term behavior. Two well-known Kolmogorov maps in theoretical ecology are the Leslie-Gower model and the Ricker model, while the Stein-Ulam spiral map has been applied in population genetics. In this thesis, I establish a theoretical framework that resolves a conjecture stating that the local stability of the interior fixed point of the Ricker map implies the global stability of the system, within a specific range of parameters. As a second topic, I prove the convexity of carrying simplices in the logarithmically scaled Leslie-Gower and Ricker models for a range of parameters. Finally, I develop a method that provides deeper insight into the behavior of the Stein-Ulam spiral map near its boundary.

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