Diffusion and density-dependent population growth

The Malthusian demographic model is not very realistic if the intrinsic rate of population increase $r$ is positive. In fact, if we wait for a sufficiently long time the organisms' density in any point can become very large. Instead, we know that intraspecific competition acts to limit the growth rate for high densities and even makes it vanish at the carrying capacity.

The mathematical treatment of the reaction-diffusion equation in the case of non-Malthusian demographics is a bit complicated and will not be illustrated in detail. We just report the main results, which anyway are quite intuitive in the light of what we learned for the Malthusian case . The fundamental assumption that we will make is that the growth rate $R(c)$ is a decreasing function of density $c$, thus excluding the case of Allee effect (or depensation, see section 2 of chapter ), which implies more complicated phenomena. For example demographics may be logistic

$$R(c) = r\left(1-\frac{c}{K}\right),$$

with $K$ being the carrying capacity.

Figure 16: Behaviour of the solution to the diffusion-reaction equation with density-dependent demography in an unbounded one-dimensional environment. The figure displays the snapshots of $c(x,t)$ in successive time instants. The carrying capacity $K$ in any site is 100.

Let us first analyse what happens in the case of an environment without barriers. For the Malthusian demographics described in the previous section, the density at any point tends to exponentially increase with time. If there is logistic dependence, instead, density in the long run tends to the carrying capacity $K$ at any point in space. Therefore, if organisms are released in a given location (with coordinate $x = 0$), the solution behaviour is the one shown in Fig. 16. Initially, diffusion prevails, but then the demographic growth leads to saturation the zone close to the release site, and from that moment on there substantially occurs propagation of two wave fronts: one to the right and one to the left. One can show that the front speed is $2\sqrt {DR(0)}$. Note that this formula is the exact analogue of the asymptotic expansion velocity for the radius of the area containing a Malthusian population: in fact that radius was shown to grow as $2\sqrt {Dr} t$. We should not wonder that the velocity of the front waves is influenced only by the value of $R(0)$ in the density-dependent growth rate, because the density in close proximity to the wave front in the direction of propagation is indeed very low, close to zero.

Figure: Time behaviour of the solution to the reaction-diffusion equation with logistic demography in a finite one-dimensional domain ($L=30$). The figure shows the stationary profile of density that is reached in the long run. The carrying capacity is $450$ and is reached only in the center of the domain while in the other locations diffusion through the boundary hinders the attainment of carrying capacity. Density obviously vanishes at the boundary.

Even the solution of the case with finite habitat has a close analogy to what we learned for the Malthusian demographics. In fact, even with density dependence there exists a critical habitat size under which the population is doomed to extinction, because dispersal through the boundary prevails over population growth. To obtain expressions that provide the critical sizes it is sufficient to replace the Malthusian growth rate r with $R(0)$. Therefore, we obtain

$$L_{cr} = \pi \sqrt {D/R(0)}$$

$$A_{cr} = 2{\pi ^2}\frac{D}{{R(0)}}$$

$$A_{cr} = 1.84{\pi ^2}\frac{D}{{R(0)}}$$

Because of density-dependence $c(x,t)$ tends over the course of time towards a stationary profile of symmetrical shape, like that shown in Fig. .