Logistic equation in The Azimuth Project

The logistic equation is a simple model of population growth in conditions where there are limited resources. When the population is low it grows in an approximately exponential way. Then, as the effects of limited resources become important, the growth slows, and approaches a limiting value, the equilibrium population or carrying capacity . The logistic growth model is. describing exponential population growth: The logistic model can be normalised by rescaling the units of population and time. Define y : = x / K y\colon = x/K and s : = rt s\colon = rt . The result is. It is easy to find the explicit solution of the logistic equation, since it is a first-order separable differential equation. The growing solutions are all time-translated versions of the logistic function (See Wikipedia) which looks like this: Chaos in the logistic model. Discrete-time version. When the logistic equation is discretised it displays chaos. It is, in fact, the canonical ground for the study of the period-doubling cascade. Assume that the Euler formula is used to discretise the logistic growth model, that is, This can be normalised by letting λ = 1 + r Δ t \lambda = 1+r\Delta t : When the logistic map is interpreted as a numerical approximation to the continuous logistic growth model, the time step Δ t \Delta t is largely arbitrary and can be chosen to be less than 1 / r 1/r , so that 1 λ 2 1 \lt \lambda \lt 2 ensuring numerical stability. Continuous version. Here chaos arises as a consequence of delayed feedback. Rewrite the normalised logistic growth model as. and add a delay to the right-hand side, obtaining. To see this, write x = e y x = e^y : Assuming now that y ≈ 0 y\approx 0 , An analysis of this equation is carried out in the article on delayed feedback. Complementary approaches. More details on the populations age structure has been added by several research groups. Also the Individual based ecology approach has bee used both for complementary and comparative research. See Law,Murrell and Dieckmann. References. Richard Law David J. Murrell Ulf Dieckmann, 2003 Population Growth in Space and Time: The Spatial Logistic Equation. How great an effect does self-generated spatial structure have on logistic population growth? Results are described from an individual based model (IBM) with spatially localised dispersal and competition, and from a deterministic approximation to the IBM describing the dynamics of the first and second spatial moments. The dynamical system incorporates a novel closure that gives a close approximation to the IBM in the presence of strong spatial structure. Population growth given by the spatial logistic equation can differ greatly from that of the non-spatial logistic model. Numerical simulations show that populations may grow more slowly or more rapidly than would be expected from the non-spatial model, and may reach their maximum rate of increase at densities other than half of the carrying capacity. Populations can achieve asymptotic densities substantially greater than or less than the carrying capacity of the non-spatial logistic model, and can even tend towards extinction. These properties of the spatial logistic equation are caused by local dispersal and competition which affect spatial structure, which in turn affects population growth. Accounting for these local spatial processes brings the theory of single-species population growth a step closer to the growth of real spatially structured populations. See also Allee effect for a generalization of the logistic equation where the per capita population growth rate becomes smaller for very small populations.