Malthusian growth and diffusion in a finite habitat with an absorbing barrier

Suppose that in a wood, suited to the ecological requirements of an insect, there are initially $N_0$ individuals distributed with density $c_0(x)$ (for example as in Fig. 3A). What will happen in the long run? Will diffusion prevail, which tends to bring insects out of the wood, or population growth, which tends to let population increase inside the wood? From equations 21 and 23 we know that

$$z(x,t) = \sum\limits_{k = 1}^\infty {{b_k}\exp \left( { - D\frac{{{\pi ^2}{k^2}}}{{{L^2}}}t} \right)\sin } \left( {\frac{{\pi k}}{L}x} \right)$$

where $b_k$ are the weights of the modal components of $c_0(x)$. Then we easily derive that the solution to the problem of growth and diffusion in a finite interval of length $L$ with an absorbing barrier at the interval extremes is given by

$$c(x,t) = \exp (rt)z(x,t) = \sum\limits_{k = 1}^\infty {{b_k}\exp ... ...}{k^2}}}{{{L^2}}}} \right)t} \right]\sin } \left( {\frac{{\pi k}}{L}x} \right).$$ (5.15)

This relationship is extremely important because it tells us that $c(x,t)$ is the sum of modal components that might dampen or amplify over time according to the exponential coefficient being positive or negative, that is depending on

$$r - D\frac{{{\pi ^2}{k^2}}}{{{L^2}}} > 0$$    or $$\qquad r - D\frac{{{\pi ^2}{k^2}}}{{{L^2}}} < 0.$$

The crucial remark is that the coefficient $r - D\frac{{{\pi ^2}{k^2}}}{{{L^2}}}$ is a decreasing function of $k^2$ and thus the mode with the highest growth rate over time is the fundamental mode (i.e. with wave number $k = 1$). Therefore if

$$r - D\frac{{{\pi ^2}}}{{{L^2}}} < 0$$

namely if

$$L < \pi \sqrt {\frac{D}{r}} = {L_{cr}}$$ (5.16)

then all the modal components will vanish in the long time and the function $c(x, t) \rightarrow 0$ for $t \rightarrow \infty$. Eq. 28 is one of the most important results of reaction-diffusion theory. It tells us that if the size of the habitat suitable for a species is smaller than a critical value ${L_{cr}} = \pi \sqrt {\frac{D}{r}}$ then the population is surely doomed to extinction. It is worthwhile to note the fundamental difference with the case of growth and diffusion in a virtually unbounded domain: in that case the only condition for population extinction was that the per capita growth rate $r$ be negative, whereas here extinction is possible even with $r > 0$ if the habitat size is smaller than the critical threshold.

In addition we note that the damping coefficient of each mode increases with the square of the wave number $k$. Therefore, the modal components of higher frequency will dampen much more rapidly (see example in Fig. 13).

Figure 13: Behaviour of the solution to the problem of Malthusian growth and diffusion in a finite one-dimensional habitat with a length that is smaller than the critical size. The figure shows various snapshots of the solution in successive time instants. Parameters are $D = 1$, $r = 2$, $L = 2$. The critical habitat length is $2.221$. The solid line displays the initial density profile $c_0(x)$.

On the contrary, if the habitat size $L$ is larger than $L_{cr}$, the damping regards only the modal components with wave number $k$ large enough, precisely those for which

$$k > \frac{L}{\pi }\sqrt {\frac{r}{D}}$$

Instead, the low-frequency modes with wave number such that $r - D\frac{{{\pi ^2}{k^2}}}{{{L^2}}}>0$ grow exponentially. The fundamental mode ($k = 1$, period = $2L$) is the one growing most rapidly with a rate of increase given by

$$r - D\frac{{{\pi ^2}}}{{{L^2}}}.$$

Thus we can conclude that if $L> L_{cr}$ population growth prevails over dispersal and the population can grow even within a confined habitat. Fig. 14 shows an example of the solution behaviour in this case.

Figure 14: Behaviour of the solution to the problem of Malthusian growth and diffusion in a finite one-dimensional habitat with a length that is larger than the critical size. The figure shows various snapshots of the solution in successive time instants. Parameters are $D = 1$, $r = 2$, $L = 2.6$. The critical habitat length is $2.221$. The fundamental mode is the only one with a positive rate of increase. The solid line displays the initial density profile $c_0(x)$.

One can easily extend the analysis to the more realistic case in which diffusion occurs on a surface; then the diffusion equation in two dimensions must be employed. Of course, in this case the bounded habitat where the population is located can have different shapes: the two simplest are square and circular. In the first case (square with side $L$) the condition of absorbing barrier is

$$\begin{array}{l} c(0,y,t) = c(L,y,t) = 0\\ c(x,0,t) = c(x,L,t) = 0 \end{array}$$

while in the second case the condition is that $c(x,y,t) = 0$ for all the points satisfying the relationships ${x^2} + {y^2} = {R^2}$ (circle of radius $R$).

In a way quite similar to the one-dimensional case one can show that the population is doomed to extinction if the suitable habitat area $A$ is smaller than a critical value $A_{cr}$. In particular, for a square habitat the critical size is

$${A_{cr}} = 2{\pi ^2}\frac{D}{r}$$ (5.17)

and for a circular habitat it is

$${A_{cr}} = 1.84{\pi ^2}\frac{D}{r}.$$ (5.18)

Suppose you want to rescue a species threatened with extinction by placing a number of individuals within a reserve. Formulas 29 and 30 are fundamental to decide the size and shape of the reserve. Of course, to determine the critical area it is necessary to have at least rough estimates of the per capita growth rate and the diffusion coefficient.

Fig. 15 is an indirect confirmation of the theory of critical habitat size: William Newmark (1986) analyzed several national parks in the West of United States and found that the species extinction probability significantly decreases with the park size.

Figure 15: Number of originally present species that became extinct after setting up a park as a function of each park area. The analysis has been conducted for national parks in the western part of United States. Redrawn after Newmark (1986).