The diffusion process

The simplest way to describe the movement of organisms in space is to assume that it is random. If in addition one assumes that the population size $N$ is large enough and makes some specific further hypotheses (which will be better defined below) one can get the so-called equation of transport and diffusion, which is the simplest model for the dispersal of a population of organisms in space (as well as for many other phenomena, such as the release of chemical substances in water and air). Let us see how to derive it by assuming for simplicity that space (indicated hereafter with $x$) is one-dimensional.

Figure 2: The scheme of random walk: an organism moves a step to the right with probability $p$ or to the left with probability $1-p$

First, suppose that both space and time are discrete (Fig. 2). We denote by $p$ the probability that an organism moves to the right with a step of length $\Delta x$ in the time interval $\Delta t$ and by $q = 1-p$ the probability for an organism to move left. If we assume that in every spatial position there is a large number of individuals we can also say that $p$ is the fraction of organisms that moves to the right and $q$ is the fraction of organisms that moves to the left (according to the law of large numbers). Let us introduce the density (or concentration) of organisms in each position $x$ at time $t$ namely

$c(x,t) \Delta x=$ number of organisms that at time $t$ are located between $x-\Delta x/2$ and $x+\Delta x/2$.

Then we can state that

$p c(x,t) \Delta x =$	number of organisms that in the time interval $\Delta t$ move from location $x$ to location $x +\Delta x$;
$(1-p) c(x,t) \Delta x =$	number of organisms that in the time interval $\Delta t$ move from location $x$ to location $x -\Delta x$ .

We can then write the following balance equation

$$c(x,t + \Delta t) \Delta x = p c(x-\Delta x,t) \Delta x + (1-p) c(x+\Delta x,t) \Delta x .$$

Divide now by $\Delta x$ and develop the left- and right-hand side terms in Taylor series, thus getting

$$\begin{array}{rl} c(x,t) + \dfrac{{\partial c}}{{\partial t}}\De... ...\partial {x^2}}}\Delta {x^2} + \mathcal{O}(\Delta {x^3})} \right] \end{array}$$

where with $\mathcal{O}(z)$ we indicate terms of order larger than or equal to $z$. If we simplify some of the terms left and right of the equal sign and divide by $\Delta t$ we obtain

$$\frac{{\partial c}}{{\partial t}} + \frac{{\mathcal{O}(\Delta {t^... ...{{\Delta {x^2}}}{{\Delta t}} + \frac{{\mathcal{O}(\Delta {x^3})}}{{\Delta t}}$$

We must now let the space step $\Delta x$ and the time step $\Delta t$ tend to zero to obtain the appropriate equation in continuous space and time. However, some additional assumptions are needed to obtain the so-called diffusion approximation.More precisely we have to assume that

$\dfrac{{\Delta x}}{{\Delta t}} \to\infty$	absolute movement speed of a single organism
$\dfrac{1}{2}\dfrac{{\Delta {x^2}}}{{\Delta t}} \to D$	finite positive constant
$(2p - 1)\dfrac{{\Delta x}}{{\Delta t}}$	equal to $p\dfrac{{\Delta x}}{{\Delta t}} + (1 - p)\dfrac{{ - \Delta x}}{{\Delta t}} =$ average population speed $\rightarrow v$ finite constant.

In particular, note that the first and the third assumption imply that $2p-1$ must necessarily tend to zero, or, stated otherwise, that the diffusive approximation is valid if the probability to move right or left differ only by a small amount, i.e. differ very little from $1/2$.

Letting $\Delta x$ and $\Delta t$ tend to zero, one obtains

$$\frac{{\partial c}}{{\partial t}} = - v\frac{{\partial c}}{{\partial x}} + D\frac{{{\partial ^2}c}}{{\partial {x^2}}}$$ (5.1)

which is the famous transport (or advection or drift) and diffusion equation. If $v$ is zero one gets the equation of pure diffusion.

Parameter $D$ is termed diffusion coefficient. Note that its measurement unit is a squared distance per unit time. Note also that the transport term is proportional to $\frac{{\partial c}}{{\partial x}}$, i.e. the concentration gradient. It must be remarked that the assumption that every organism is moving at very large speed (infinite in the limit) is crucial to obtaining equation 13. In fact, if we had supposed that the speed of each organism were finite, then $\Delta x$ would tend to zero like $\Delta t$ and then $\Delta x^2$ would tend to zero faster than $\Delta t$. As a result, the diffusive term would vanish and only advection would be operating.

It is easy to understand that the transport term in Eq. 13 has just a translation effect on the solution of the same equation. In other words, the solutions of Eq. 13 can be obtained from solutions of the pure diffusion equation

$$\frac{{\partial c}}{{\partial t}} = D\frac{{{\partial ^2}c}}{{\partial {x^2}}}$$ (5.2)

provided one replaces the space coordinate $x$ with the so-called “moving reference frame coordinate”, or $x + vt$. For this reason, we will from now on focus exclusively on eq. 14.

A solution of eq. 14 is completely determined if one specifies suitable initial conditions and suitable conditions at the space boundary. Providing initial conditions means to specify the organisms density in each location $x$ at the initial instant, conventionally denoted by 0, namely

$$c(x,0) = c_0(x) =$$   a given function of space.$$$$

As regards the boundary conditions, there are several possible cases, depending on the spatial domain considered. In the next sections we will deal with the following two situations only.