The Fokker–Planck law of diffusion and pattern formation in heterogeneous environments

Abstract

We analyze the influence of spatially inhomogeneous diffusion on several common ecological problems. Diffusion is modeled with Fick’s law and the Fokker–Planck law of diffusion. We discuss the differences between the two formalisms and when to use either the one or the other. In doing so, we start with a pure diffusion equation, then turn to a reaction–diffusion system with one logistically growing component which invades the spatial domain. We also look at systems of two reacting components, namely a trimolecular oscillating chemical model system and an excitable predator–prey model. Contrary to Fickian diffusion, spatial inhomogeneities promote spatial and spatiotemporal pattern formation in case of Fokker–Planck diffusion.

Appendix

1.1 Dimensionless diffusion equation

We transform a reaction–diffusion–equation with the Fokker–Planck equation into a dimensionless form. The original equation reads

$$\begin{aligned} \frac{\partial U}{\partial \tau }=f(U)-\frac{\partial }{\partial X}\left( \alpha \frac{\partial \delta (X)}{\partial X}U\right) +\frac{\partial ^2}{\partial X^2}(\delta (X) U), \end{aligned}$$

(22)

where U has the dimension of a concentration, X is the spatial variable, $\tau$ has the dimension of time, and $\delta (X)$ is a diffusion coefficient with dimension $\text {space}^2/\text {time}$.

We introduce dimensionless variables: $u=U/U_0, t=\tau /T_0$, and $x=X/X_0$. $U_0, T_0$, and $X_0$ are constants with the dimension of U, T, or X, respectively.

Using these variables gives

$$\begin{aligned} \frac{\partial u}{\partial t}=f(uU_0)\frac{T_0}{U_0}-\frac{T_0}{X_0^2}\frac{\partial }{\partial x}\left( \alpha \frac{\partial \delta (X)}{\partial x}u\right) +\frac{T_0}{X_0^2}\frac{\partial ^2}{\partial x^2}(\delta (X) u).\end{aligned}$$

(23)

Now we can define a dimensionless coefficient of diffusion $D(x)\equiv \delta (xX_0) \frac{T_0}{X_0^2}$. This coefficient scales with the timescale $T_0$ and the scaling factor of space $X_0$. The final, dimensionless version of the Fokker–Planck equation with a reaction term then reads

$$\begin{aligned} \frac{\partial u}{\partial t}=F(u)-\frac{\partial }{\partial x}\left( \alpha \frac{\partial D(x)}{\partial x}u\right) +\frac{\partial ^2}{\partial x^2}(D(x) u), \end{aligned}$$

(24)

where F(u) is a dimensionless formulation of the reaction term f(U).

1.2 Stationary solution of the Fokker–Planck equation

A stationary solution to the Fokker–Planck equation (6) with $\varLambda =\alpha \frac{\partial D}{\partial x}$ has to fulfil the following condition:

$$\begin{aligned} 0&=-\frac{\partial }{\partial x}\left( \alpha \left( \frac{\partial D(x)}{\partial x}\right) u(x)\right) +\frac{\partial ^2}{\partial x^2}\left( D(x)u(x)\right) \\ c&=(1-\alpha )\left( \frac{\partial D(x)}{\partial x}\right) u(x)+D(x)\frac{\partial u(x)}{\partial x}, \end{aligned}$$

where x is a constant. Using the ansatz $u(x)=k\exp (f(x))>0$ with an unknown function f(x) we get

$$\begin{aligned} c&=(1-\alpha )u(x)\frac{\partial D(x)}{\partial x}+D(x)u(x)\frac{\text {d} f(x)}{\text {d} x}\\&\quad \Rightarrow \frac{\text {d} f(x)}{\text {d} x}=\frac{c}{D(x)}-(1-\alpha )\frac{1}{D(x)}\frac{\partial D(x)}{\partial x} \text { if }D(x)\ne 0\text { and }u(x)\ne 0\\&\quad \Rightarrow f(x)= c\int _0^x\frac{1}{D(x^{\,'})}\text {d}x^{\,'}-(1-\alpha )\left( \ln |D(x)|+c^*\right) , \end{aligned}$$

where $c^*$ is another constant. Substituting f(x) into the ansatz we made for u(x) gives us the steady state solution for the Fokker–Planck equation:

$$\begin{aligned} u(x)=\exp \left( c\int _0^x\frac{1}{D(x^{\,'})} \text {d}x^{\,'}\right) u_0D(x)^{-(1-\alpha )}\quad \text { with } u_0=k\cdot \exp (-(1-\alpha )c^*). \end{aligned}$$

We used periodic boundary conditions. In our case with $D(x)=A+B\cos (2\pi x), D(x)$ is a periodic function with the same values at $x=0$ and $x=L$, where L is the length of the spatial domain. The same must hold for u(x). Therefore the constant c must be set to zero. The solution reads

$$\begin{aligned} u(x)=u_0D(x)^{-(1-\alpha )}, \end{aligned}$$

where the constant $u_0$ depends on the initial conditions,

$$\begin{aligned} u_0=\dfrac{\int _0^Lu(x,t=0) \text {d}x}{\int _0^LD(x)^{-(1-\alpha )}\text {d}x}. \end{aligned}$$

Fig. 8

Spatial and temporal periodic blooms in an excitable predator–prey system with periodic boundary conditions. Only prey concentrations are shown. Turbulent and ecological diffusion vary in phase. $D_{turb}(x)=D_{eco}^{(2)}(x)=(1+0.9\cos (2\pi x))10^{-3}$. The initial condition is a homogeneous distribution of the stationary solution of system (19)

Fig. 9

Numerically analyzed values of $D_{eco}^{(2)min}$ and $\varDelta D_{eco}^{(2)}$ where excitation occur in a spatially diffusive system with $D_{eco}^{(2)}= (D_{eco}^{(2)min}+\varDelta \frac{D_{eco}^{(2)}}{2}(1+\cos (2\pi x)))$. $D_{turb}(x)=10^{-3}\cdot (1+0.9\cos (2\pi x))$. We tested whether the value of the prey concentration exceeded a threshold of 0.5 for at least two times or not, to define if there are periodic blooms in the system. Homogeneous initial conditions are the stationary solutions of system (19). Note that the values of $\varDelta D_{eco}^{(2)}$ are much larger than in the case in which $D_{eco}$ and $D_{turb}$ vary in anti-phase (Fig. 7)

1.3 Plankton model with turbulent and ecological diffusion in phase

To study the effect of ecological diffusion, which varies in phase with the turbulent diffusion $D_{turb}$, we set $D_{eco}^{(2)}(x)=D_{turb}(x)$. In this case, the zooplankton is drifted to areas with small $D_{turb}$, and blooms occur with the same mechanism as in the previous simulations (Fig. 8). The drifting effect caused by the Fokker–Planck law of diffusion has to be stronger than in the former case (where $D_{turb}$ and $D_{eco}$ are in anti-phase) to create a bloom (Fig. 9). This is because the smoothing effect of $D_{turb}$ prevails in areas where the predators are diminished by the drift.