Linear Population Models with Stochastic Time Delays

Abstract

Previously it was shown that reproductive—cycle parameters such as time to maturity, ovulation interval, gestation period, duration of regression, duration of nonreproductive lactation period, and the like, can be incorporated into population models rather easily through the use of a simple network approach. In this paper, the network approach is extended to include the same types of reproductive parameters when their values are not necessarily fixed, but may vary randomly from one member of a population to the next and/or for a given member from one time to the next. It is shown that linear transforms of the parameter distribution functions can be incorporated directly into the network models and that analysis of the resulting dynamics follows in a straightforward manner, the characteristic dynamical equation being obtainable by inspection with Mason's algorithm and the roots of the equation being obtainable by direct analysis in simple cases or by well—established numerical methods in complicated cases. The roots themselves can be interpreted directly in terms of dominant patterns of population growth and deduced propensity of the population to sustain oscillations triggered by external stimuli. In the case of a simple natality cycle with gamma, negative binomial, and binomial distributions of maturation times, it is shown that the dominant growth pattern approximates rather closely that expected for a nonrandom maturation time equal to the mean of the distribution, and that the propensity to sustain population oscillations decreases markedly both with increasing standard deviation and with increasing (positive) skewness in the distribution.