Chaotic dynamics in mathematical models of predator-prey systems

Abstract

This thesis explores mathematical predator-prey models and methods for identifying system characteristics, with a focus on chaotic dynamics. The study begins by investigating the relationship between the Hurst exponent and the largest Lyapunov exponent across various one-dimensional maps, which are then integrated into a Coupled Map Lattice (CML). Our analysis reveals small variations in Hurst exponents within each CML, suggesting a single Hurst exponent may represent the system. It is found that a strong positive correlation between the Hurst exponent and chaos, with higher Hurst exponents indicating a higher Lyapunov exponent in 85% of cases. Given the quicker computation of the Hurst exponent, it is proposed as a more efficient measure for detecting chaos. Machine learning techniques are then employed to predict the largest Lyapunov exponent across diverse scenarios.

Next, two discrete-time predator-prey models, one linear and one non-linear, featuring a herbivore feeding on a basal food source are examined. Our findings show species coexistence and chaotic behaviour as parameters change, with fertilization expanding the coexistence parameter space.

The models are further developed into four three-dimensional models, each incorporating different predator behaviours. These models exhibit various behaviours, including chaotic and hyper-chaotic patterns, with increased basal consumption affecting species survival. Mutations, particularly omnivorous ones, significantly influence stability by reducing chaotic behaviour.

Finally, coupled maps with multiple prey and predator phenotypes are presented. Adjusting consumption and kill rates reveals fluctuations in survival and population levels. Mutation enhances system robustness, and the introduction of an apex predator impacts species survival without compromising ecosystem stability or chaos.

Date of Award	Sept 2024
Original language	English
Supervisor	David Glass (Supervisor) & Mark Mc Cartney (Supervisor)