Population Persistence and Density Dependence

This paper reformulates the notion of density dependence and shows how this notion plays an important role in constructing appropriate models for data analysis. The regulation and persistence of population processes are interpreted as a close resemblance to the behavior of a series of random variables in which the second moments are bounded. On this basis the formal criteria of persistence are deduced. General structural models of population processes are set up and translated into discrete single—variable difference equations, ranging from the simplest linear first—order process to more complex nonlinear second—order processes. The discussion includes the derivation of general conditions for the second—order limit cycles, a reanalysis of the Canadian lynx 10—yr cycle, and models for population outbreaks. Based on the results of the preceding study of models, the notion of density dependence is reformulated. First, the meaning of the word 'dependence' is discussed. In the context of 'density dependence,' the word has two meanings; the causal dependence of a factor on density, and the statistical dependence. Statistical dependence is defined as a converse of statistical independence, the latter being a process in which the rate of change in density has zero correlation with density; this is a very special class of processes and is unlikely to occur in natural population processes. Therefore, the test of density dependence against the null hypothesis of statistical independence will not provide much insight. It is also argued that a deduction from the persistence criteria shows that a negative correlation between density and its rate of change is a necessary outcome of regulation and hence that the notion of 'density—dependent regulation' in statistical dependence is an uninspiring tautology. As opposed to statistical density independence, which necessarily generates an unbounded population process, causal density independence may satisfy the persistence conditions and hence may regulate populations. However, such a causally 'density—independent regulation' tends to be 'fragile' against perturbations by random exogenous factors. It is a particular class of causally density—dependent processes that can ensure regulation more durable against such perturbations. The inference of generating mechanism from observation is discussed. Although regression analysis is an essential method of inference, simple regression analysis will not work unless the observed processes are known to be a simple Markov chain. Statistical inference of generating mechanisms in observed systems depends largely on the choice of appropriate models, and it is in the construction of such models that the notion of causal density dependence plays an important role.