Steady State, Oscillation, and Chaos in Population Dynamics

Abstract

Let us return to the simple population models of Chapter 2, in the absence of randomness, and explore the behavior of a simple deterministic model as a parameter value gets pushed outside the realm that is typically considered in these models. Denote the size of the population in time period t as N(t) and the net change in the population size during that period as ΔN. The exogenous parameter influencing the net flow is R. The net flow ΔN updates the stock N:

$$ \Delta {\text{N = N(t + DT) - N(t)}}{\text{.}} $$

(1)

And then so slight, so delicate is death That there’s but the end of a leaf’s fall A moment of no consequence at all. Mark Swann, as quoted by Alfred Lotka, 1956, The Elements of Physical Biology, Dover, NY, p. 376