Operator Semigroup Theory for Abstract Evolution Equations and Applications to Population Dynamics in Semi-Arid Nigeria

This dissertation develops original contributions to the operator semigroup theory for abstract evolution equations in Banach spaces, with applications to structured population dynamics models governing the temporal evolution of age-structured and size-structured populations in the semi-arid ecological zones of North West and North East Nigeria. The theory of strongly continuous operator semigroups provides the natural functional analytic setting for establishing well-posedness, regularity, and long-time behaviour of solutions to abstract Cauchy problems associated with age-structured and size-structured McKendrick-von Foerster partial differential equations. The dissertation proves novel generation theorems for strongly continuous semigroups associated with McKendrick-type operators with non-local birth boundary conditions under weaker regularity assumptions on vital rate functions than those required by existing theorems in the literature, accommodating the discontinuous and empirically estimated vital rates typical of field population studies. Compact perturbation theory and the essential growth bound are used to characterise the spectral-determined growth of the semigroup and to establish exponential stability or instability of the zero steady state in terms of the spectral radius of a generalised net reproductive operator. Positive operator semigroup theory in ordered Banach lattices is applied to prove the existence and stability of positive steady-state solutions. Applications to camel and cattle population management models in Borno and Sokoto states are developed using demographic data from the National Bureau of Statistics agricultural surveys. Keywords: operator semigroups, evolution equations, structured population models, Banach spaces, semi-arid Nigeria