The Dynamics of a Population in a Markovian Environment

A model of population growth in a partially predictable environment is developed and analyzed. The essential features of the model are as follows: (1) There are only 2 possible environmental states, and the transitions between the 2 states are the same as for a Poisson process, so that time spent in a state before a transition follows an exponential distribution. (2) Arbitrary rules for population growth in each of the states are assumed. (3) The model is formulated in terms of the joint distribution function of environmental states and population sizes. An equilibrium solution is found, from which it can be determined if there is a stationary joint distribution by whether or not the equilibrium solution exists and can be normalized. Several specific growth models are analyzed, including the effect of variations in the intrinsic rate of increase and in the carrying capacity for the logistic and related models. The main result is that it is possible to have a stationary distribution of populations sizes indicating persistent fluctuations driven by environmental changes only when the correlation time of environmental changes is roughly comparable to the average response time of the population to those changes. Also, in many models of growth the dominant parameter that determines the tendency of the population to go extinct is the mean of the intrinsic rates of increase in different environments. There is no support for earlier conjecture that the relative values of the mean and variance of a parameter determine the extinction tendencies.