Contrasting Discrete and Continuous Density Dependent Population Models
0James Sandefur, Georgetown University, Washington DC USA
https://qubeshub.org/community/groups/simiode/expo
Abstract: Density dependent population models are of the form:
p′ = r(p)p or p(n+1) = (1+r(p(n))p(n),
where the per capita growth rate, r, is a function of the population size. Under the assumptions: 1) there is an intrinsic per capita growth rate, 2) there is a carrying capacity, 3) r is decreasing, the most common functions for r are linear, rational, or exponential.
The well-known logistic model, in which r is linearly decreasing, gives clear contrasts between the discrete and continuous: The continuous model can always be solved and has a stable positive equilibrium population size, while the discrete model a) cannot be solved for most r’s, b) exhibits period doubling to chaos, and c) solutions go to negative infinity if the slope of r or the initial population size are too large.
The discrete model is more realistic if we also assume, 4) r decreases to negative one. This leads to four of the classic discrete models. Solutions for these models cannot, in general, be found. Solutions for the rational r in the continuous case can theoretically be found, but in practice, cannot realistically be found nor do the solutions give useful information.
Analysis of the continuous models can be done easily using phase-line analysis, resulting in stable and unstable equilibrium population sizes while for the discrete case, we must also use r′ at equilibrium to determine stability. In fact, many discrete models, like the logistic model, exhibit period doubling to chaos.