Differential equation population models in a slowly varying environment

Many investigations have analysed population dynamics using differential equation models in which the governing parameters are assumed to be constant. However, in reality, this assumption of constancy is not always true, since the dynamics of such models can be influenced by regular environmental changes, through such parameters varying with time. Moreover, this time variation can occur on one time scale or on a number of different time scales. In general, for such cases, an exact solution of population evolution problems involving the model cannot be obtained and numerical solution methods must be used. However, when the variation of the model parameters is slow relative to other quantities, analytic multi-scaling techniques can be applied to obtain approximate solutions which can successfully represent the variation of the population over time. This technique is widely applied in the fields of physics and engineering and has begun to become more established as a technique in recent times in population dynamics.<br><br>This thesis considers single species differential equations population models in which the model parameters are functions of time that vary slowly compared to the intrinsic time variation of the population itself and applies multi-timing methods to obtain closed-form (explicit or implicit ) approximate expressions representing the evolving populations represented by these models. These expressions are then applied to analyse the ongoing population behaviour . They are compared with solutions obtained by numerical solution methods in particular cases and are shown to give very satisfactory approximations over a range of (variable) parameters. <br><br>Several models of interest are considered. For a harvested logistic model, the multi-timimg approximations are used to show how, for some starting populations and under suitable harvesting, the population survives to a slowly varying limiting state, while for others, it is driven to extinction. For a model displaying an Allee effect, the approximations yield similar results. Finally, for a logistic model subject to a saturating harvesting, clear criteria for survival and extinction are established.<br><br>In many situations, the survival/extinction potential of a population changes, through a change in the (slowly varying) model parameters, termed a transition. Such transitions are analysed through the approximations developed, to give a clear picture of the ultimate fate of a population as it moves through a transition state.<br>