Malthusian growth and dispersal in an infinite habitat

Suppose we introduce a new species or a reintroduce one that was once present in the environment by releasing a number $N_0$ of individuals, in the appropriate sex ratio, at time $t = 0$ at location $x = 0$ . These organisms will disperse without barriers in space, will reproduce and will die. How does the organisms' density vary in locations far away from that of release? If $r > 0$, it is interesting to wonder whether the density at every location is going to increase or whether dispersal prevents the local growth of the population. From eq. 15 we know that

$$z(x,t) = {N_0}\frac{1}{{\sqrt {4\pi Dt} }}\exp \left( { - \frac{1}{2}\frac{{{x^2}}}{{2Dt}}} \right)$$

and thus we easily derive that

$$c(x,t) = {N_0}\frac{1}{{\sqrt {4\pi Dt} }}\exp \left( {rt - \frac{1}{2}\frac{{{x^2}}}{{2Dt}}} \right).$$ (5.13)

Fig. 9 shows the evolution of density in successive time instants. One can remark for example that at the release point $x = 0$ the density initially decreases because diffusion is prevailing, but in the long term population growth prevails because the term $\exp(rt)$ has the most influence on the population evolution for large $t$. In spatial sites that were initially unpopulated, density is always growing: the locations are reached via diffusion and then basically population keeps growing because of demography. It is obvious that the total number of organisms grows exponentially because

$$N(t) = \int\limits_{ - \infty }^{ + \infty } {c(x,t)dx} = \exp(rt)\int\limits_{ - \infty }^{ + \infty } {z(x,t)dx = } {N_0}\exp (rt)$$

Figure 9: Time evolution of the solution to the problem of Malthusian growth and diffusion in an unbounded habitat. The figure shows various snapshots of the solution in successive instants.

If we consider the problem from a more realistic viewpoint, i.e. the organisms are released and can move in a two-dimensional environment, the solution is simply given by

$$c(x,y,t) = {N_0}\frac{1}{{4\pi Dt}}\exp \left( {rt - \frac{1}{2}\frac{{{x^2} + {y^2}}}{{2Dt}}} \right).$$

From this formula we can gather how the population colonizes new space. It is possible to prove (Pielou, 1977) that the number $N_{out}$ of individuals in the population that at time $t$ are outside a contour line of radius $R_t$ is given by

$$N_{out} = {N_0}\exp \left( {rt - \frac{{R_t^2}}{{4Dt}}} \right).$$

It is reasonable to assume that the presence of individuals is no longer detectable when $N_{out}$ is below a certain threshold fraction $f_{\min}$ (for example 1%) of the number $N_0$ of initially released organisms. Therefore the radius within which the entire population is in practice contained satisfies the equation

$$f_{\min} = \exp \left( {rt - \frac{1}{2}\frac{{R_t^2}}{{2Dt}}} \right)$$

and thus

$$R_t^2 = 4Dr{t^2} - 4Dt\ln {f_{\min }}.$$ (5.14)

Figure: Expansion radius as a function of time for a Malthusian population dispersing via diffusion (blue curve). $f_{\min}$ is set to 1%. Asymptotically the radius increases linearly. The dashed red straight line is the asymptote which is given by $2\sqrt {Dr} t - \frac{{D\ln ({f_{\min }})}}{{\sqrt {Dr} }}$.

With the passing of time the first term of Eq. 26 becomes much larger than the second, hence we can conclude that, if we include Malthusian growth in addition to diffusion, the radius marking population expansion, after an initial transient, grows approximately linearly with time, not with the root of the time (see Fig. 10). The colonization speed is thus approximately equal to $2\sqrt {Dr}$. Equivalently, we can say that the area which virtually contains all the population increases with the square of time.

Figure 11: The expansion of muskrat (Ondatra zibethica) in Europe following the accidental release of five muskrats in the location shown as a white dot in panel (A), near Prague. Maps are redrawn after the original reported in Elton (1958). (B) If we approximate the areas occupied by the muskrats with circles, their root increases linearly with time, as reported in the seminal paper by Skellam (1951). By dividing the slope of the fitting line by $\sqrt {\pi }$ one obtains the average radius increase as kilometres per year. The estimated value is $11.8$ km/year.

There are several examples that confirm the validity of this simple model. Many refer to the introduction of alien species. The classical example is that of the muskrat (see Fig. 11), a rodent native to North America that was imported to Bohemia at the beginning of 1900's to exploit its fur. It looks like five animals managed to escape from a breeding farm near Prague in 1905. The muskrat population in the wild began then to increase colonizing space at a speed of about 12 km per year. Another example, that of the Argentine ant Iridomyrmex humilis introduced in some areas of western United States, is summarized in Figure 12. Table 4.1 shows some values ​​of expansion velocity as observed in both terrestrial and marine invasive species.

Figure 12: The expansion of Argentine ant Iridomyrmex humilis in a meadow close to San Diego (California), initially occupied by the Californian ant Pogonomyrmex californicus. The experiment took place between October 1963 and October 1968. (A) Solid lines represent the invasion wave fronts of the Argentine ant (left to right). Elapsed times (months) between one front and the next are indicated below each solid line. (B) Distance from release location averaged along the vertical transect as a function of time. Redrawn after Figure 1 by Erickson (1971).

Table 1: Various expansion speeds for a few terrestrial and marine species (Grosholz, 1996).

TERRESTRIAL SPECIES
Species name (common name)	Observed speed(km/year)
Impatiens glandulifera (Himalayan balsam)	9.4 – 32.9
Lymantria dispar (Asian gypsy moth)	9.6
Pieris rapae (small white butterfly)	14.7 – 170
Oulema melanopus (cereal leaf beetle)	26.5 – 89.5
Ondatra zibethica (muskrat)	0.9 – 25.4
Sciurus carolinensis (grey squirrel)	7.66
Streptopelia decaocto (collared dove)	43.7
Sturnus vulgaris (European starling)	200
Yersinia pestis (Bubonic plague bacterium)	400
MARINE SPECIES
Species name (common name)	Observed speed(km/year)
Botrylloides leachi (colonial tunicate)	16
Membranipora membranacea (lacy crust bryozoan)	20
Carcinus maenas (green crab)	55
Hemigraspus sanguineus (Asian shore crab)	12
Elminius modestus (acorn barnacle)	30
Littorina littorea (common periwinkle)	34
Mytilus galloprovincialis (Mediterranean mussel)	115
Perna perna ( brown mussel)	95