Demographic increase and diffusion

From the ecological viewpoint things get much more interesting when, in addition to movement, we introduce the population demography. In other words, we consider that population density at each spatial location changes over time not only because of movement, but also because of occurring births and deaths. Suppose that the birthrate $\nu$ and mortality rate $\mu$ that operate in a certain spatial location depend only upon local density $c(x,t)$ (see section ). If we indicate the per capita rate of increase by $R$ we can write

$$R(c) = \nu (c) - \mu (c) .$$

Then the time variation of density at each location $x$ is given by the following equation

$$\frac{{\partial c}}{{\partial t}} = D\frac{{{\partial ^2}c}}{{\partial {x^2}}} + R(c)c.$$ (5.12)

Eq. 24 is called reaction-diffusion equation because this same equation is used to describe the kinetics of a chemical reactor in which $R$ is the speed at which a chemical component with concentration $c$ is formed. The analysis of the reaction-diffusion equation is not so simple in the general case of any $R(c)$, but is quite simple when the demography is Malthusian, namely when the per capita growth rate $R$ is independent of density :

$$R(c) = r$$    constant$$.$$

In fact, Eq. 24 reduces to

$$\frac{{\partial c}}{{\partial t}} = D\frac{{{\partial ^2}c}}{{\partial {x^2}}} + rc$$

and can be easily solved via a change of variables. By introducing the new variable

$$z(x,t) = \exp (-rt) c(x,t)$$

it is easy to derive that

$$\frac{{\partial z}}{{\partial t}} = - r\exp ( - rt)c(x,t) + \exp ... ...partial {x^2}}} + rc} \right] = D\frac{{{\partial ^2}z}}{{\partial {x^2}}}.$$

Therefore, $z$ satisfies a pure diffusion equation whose solutions we have already learnt how to derive in the previous section. Let us separately consider the two Cauchy and Dirichlet problems previously defined.