Population Biology, Nonlinear Systems, and Chaos

A great deal of interest has been sparked by the realization of simple models for the description of population dynamics of organisms with nonoverlapping generations. May (1974), and May and Oster (1976) demonstrated that the use of difference equations such as that for population growth can yield a 

Nt + i = population size at the next time interval K = carrying capacity of the environment r = intrinsic rate of increase over the time interval 

At different sets of initial conditions and with varying r, populations can reach an equilibrium, fluctuate in a stable fashion around the carrying capacity, or exhibit dynamics that have no readily discernible pattern, i.e., they appear chaotic. 

The investigation of chaotic dynamics has also spread to weather forecasting and the physical sciences. An excellent popularization by Gleick (1987) reviews the discovery of the phenomena, from the butterflies of Lorenz in the modeling of weather to complexity theory. What follows is only a brief introduction. 

Comparison of the population dynamics of two systems that begin at the same initial conditions but with different rates of increase. 

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