Dynamics of a discrete population model with variable carrying capacity

The carrying capacity is assumed to be constant in population growth models used for resource assessment and management. However, changes to the carrying capacity do occur due to both exogenic and endogenic processes. The need to treat the carrying capacity as a function of time has long been recognised in order to model population dynamics in an environment that undergoes change. Most populations experience fluctuations in their environment due to seasonal change. The simplest approach is to specify some time-dependent function for the carrying capacity that reflects the observed behaviour of the changing environment. To date these models are deterministic with overlapping generations, the kind that are best described using continuous equations. However, the dynamics of some populations may not be appropriately described with continuous equations. Populations with non-overlapping generations are better described by discrete (difference equation) models. In this paper, by considering the carrying capacity as a proxy for the state of the environment, we analyse a population whose growth is governed by a discrete logistic model and whose carrying capacity is modelled by a separate difference equation. The existence of fixed points is established and the stability of fixed points is discussed. Aperiodic behaviour is also shown to exist.