A Discrete Model of Population Growth

Difference equations and iterative maps occur naturally in mathematical biology. For example, an important problem is how the population size of a particular species varies from one generation to another. Let $P_{n}$ be the population of a species at time $n$ (corresponding say to the $n$-th generation), and $P_{n+1}$ the population at time $n+1$. Then the change in population size during the time interval is given in the simplest model of population growth by

$$P_{n+1} - P_{n} = k P_{n} ,$$ (3.8)

where $k$ is the growth rate. That is, the population is assumed to increase in one time interval by an amount proportional to its value at the beginning of that interval. However, this would imply a population that increases without bound as time increases. A more realistic model is obtained if the rate of growth is not a constant but depends on $P_{n}$ itself in such a way that it decreases when the population becomes too large and the species runs out of food and/or space. Therefore we put,

$$k = b ( c - P_{n}) ,$$ (3.9)

where $b$ and $c$ are positive constants. Clearly when $P_{n}=c$, $k=0$ and the population has reached a maximum value. Substituting equation (3.9) into (3.8) gives the Verhulst model (also associated with the biologist Robert May),

$$P_{n+1} = P_{n} + b c P_{n} - b P^{2}_{n} .$$ (3.10)

The last term in (3.10) is nonlinear and in addition it 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

$$x_{n+1} = 4 r x_{n} (1 - x_{n}) ,$$ (3.11)

where the symbol $P$ has been replaced with a new variable $x$ that can be interpreted as the fractional population. Equation (3.11) is known as the logistic map and the constant $r$ is called the control parameter ( here interpreted as the reproduction rate). We wish to study the properties of the logistic map as it is evolved forward in time. This evolution is obtained simply by repeated interation of an intial value $x_0$ and the sequence of values taken by the successive $x$ is called an orbit.