Complex Discrete Dynamics from Simple Continuous Population Models

The concept of chaos in ecological populations is widely known for non-overlapping generations since the early theoretical works of May [1974, 1976], and successively applied to laboratory and field studies [Hassell et al., 1976]. A classical approach using very simple models consists of using discrete, first-order non-linear difference equations for populations with $N_i$ individuals at time $i$ of the form $N_{t+1}+f(N_t)$, where $F(N_t)=aN_t g(N_t,...N_{t-j})$ and $g(N_t,...,N_{t-j})$ is some nonlinear function describing some degree of density-dependence with time delay $j$. In fact, a well-known equation describing a full range of dynamic behaviors was developed by Ricker (1954): $N_{t+1}=\mu N_t e^{-bNt}$, where $\mu$ stands for the discrete initial growth rate, and the initial population $N_t$ is exponentially reduced as a function of some mortality rate $b>0$. The use of discrete models, although very popular due to their simplicity, contains serious drawbacks if some biological within-generation properties are to be taken into account. A given population may indeed reproduce at certain fixed time steps; however, its mortality might not be constant, but conditioned by the starvation rate, which in turn depends on the quantity of resources available for the population and its consumption along time between successive generations. How this resource-consumer interaction affects the behavior of the population and how it is related to classical discrete models is crucial if a well-defined dynamic scenario is required.