Logistic Map in Ecology

Pierre Verhulst (1840s) first proposed the logistic equation for describing population growth.

How does the population size of a particular species varies from one generation to another ?

If the generations are non-overlapping (e.g., as in many insect species) this can be modeled using a difference equation or map.

Logistic map is the simplest such map which gives non-trivial behavior.

Let Pn be the population of a species at time n (corresponding to the n-th generation).
The change in population size during the time interval n to n+1 is given in the simplest model of population growth by

Pn+1 - Pn  =  kPn

where k is the growth rate.
==> The population is assumed to increase in one time interval by an amount proportional to its value at the beginning of that interval.

BUT, this implies a population that increases without bound as time increases.

In reality, this is checked by Malthusian effects (famine, war, natural disasters).

A more realistic model: if the growth rate is not a constant but depends on Pn itself such that it decreases when the population becomes too large and the species runs out of food and/or space.
Let,

kb ( c - Pn )

(b,c are positive constants).

When Pn= c (carrying capacity), k =0 : the population has reached a maximum value.

This gives the Verhulst-May model (the physicist turned ecologist Robert May),

Pn+1=Pn + b c Pn - b Pn2.

The last term is nonlinear: provides negative feedback as compared to the positive feedback due to the second term.

By a change of variables the equation can be written in the simplified and conventional form (the logistic map)

xn+1 = a xn (1-xn)

where the new variable x can be interpreted as the normalised population density and the control parameter a interpreted as the reproduction rate.

Other maps have been proposed as more accurate descriptions of populations density variation over generations. For example, the Ricker map proposed by W. Ricker in1954

xn+1 = a xn exp (-xn)

But these "improved" maps have qualitatively the same dynamics as the logistic map.