Abstract
Time delays are an important aspect of ecological systems. Delays are accurate ways of modelling phenomena such as maturation and gestation and can result in more complicated dynamics than in the absence of delay. Delays often appear in the model equations because of age-structure in the population. Inclusion of diffusion results in nonlocal delays. Additionally, the equations can have delay-dependent coefficients. The equilibria of such equations usually depend on the time delays. We propose and analyse some mathematical models of stage-structured populations where the individuals have two stages; immature and mature. The time from birth to maturity can be modeled by either a discrete or a distributed time delay. We study both ordinary differential delay equations and reaction-diffusion equations with delay, the latter to incorporate spatial effects. The possibility of individuals not all maturing at the same age can be modeled by using a distributed delay formulation. We start by deriving a stage-structured reaction-diffusion model with discrete delay. Existence and monotonicity of travelling front solutions of the resulting system are investigated. To address the possibility of individuals maturing at different ages we then develop a distributed delay ODE model and examine its dynamics. A reaction-diffusion extension is then developed and attention paid to travelling front solutions. A two-species competition system is then proposed, in which only the adults are in competition. In the absence of competition, each species evolves according to the distributed delay model proposed earlier. Global stability of the equilibria of the competition model is investigated. In the situation when there is no coexistence equilibrium, travelling fronts are shown to exist connecting the two boundary equilibria. These correspond to the weaker competitor being driven to extinction by the stronger in a travelling wave of invasion. We then propose a general stage-structured population model with a state- dependent time delay, in which the time delay is an increasing differentiable function of the total population (immature plus mature). This type of equation is motivated by previous research on whale and seal populations. Finally, we examine a three species food chain model with delay, in which the predator is assumed to have a stage-structure. A combination of analytical and numerical methods are used here. Results are proved on positivity, boundedness, linear and nonlinear stability. Existence of periodic solutions is demonstrated numerically.