A Unified Theoretical Framework for Nonlinear Integro-Differential Equations with Applications to Population Ecology in Nigerian Savanna Ecosystems

This dissertation develops a unified theoretical framework for a class of nonlinear integro-differential equations arising in structured population dynamics, with original applications to ecological systems in the Guinea and Sudan savanna zones of North Central and North West Nigeria. Integro-differential equations that incorporate memory effects and non-local interaction terms have received considerable theoretical attention, yet their rigorous qualitative analysis in the context of ecologically realistic West African population models remains an open research area. The dissertation establishes novel existence, uniqueness, and regularity results for solutions in weighted Sobolev function spaces, using fixed-point theorems in Banach spaces and compactness arguments adapted to the specific kernel structures arising in savanna population models. A global stability analysis framework based on generalised Lyapunov functionals is constructed for the endemic equilibria of coupled predator-prey integro-differential systems, with conditions for stability expressed in terms of kernel decay rates and nonlinear growth function properties. Bifurcation theory is applied to characterise the emergence of periodic solutions through Hopf bifurcation as ecological parameters cross critical thresholds, and Floquet theory is used to assess the stability of bifurcating periodic orbits. Numerical approximation schemes of provable convergence order are developed and applied to simulate ecosystem dynamics under parameterisations derived from published ecological field surveys conducted in Kano and Niger states. The results contribute original theorems and a coherent analytical methodology to the mathematical ecology and functional differential equations literature. Keywords: integro-differential equations, population dynamics, Lyapunov functionals, Hopf bifurcation, savanna ecology