Team:Valencia Biocampus/Demonstration/Diffusion3

Proof of a Group Behavior Diffusion Model from a Random Walk Model

For proving that, we will make the assumption that the worm only moves in one dimension ($x$), but it can be extrapolated to $\;n\;$ dimensions.

At each time step $\;q\;$ it either moves a distance $\;h\;$ to the left with probability $\;l\;$, a distance $\;h\;$ to the right with probability $\;r\;$, or stays in the same position with probability $\;1−r−l\;$ (the isotropic random walk has $\;r\;=\;l\;=\;1/2$, so it cannot rest motionless). We also define the probability that a worm is at a position $\;x\;$ at time $\;t\;$ by $\;P(x,t)\;$. One time step earlier, at time $\;t − q\;$, the walker must have been at position $\;x − δ\;$ and then moved to the right, or at position $\;x + δ\;$ and then moved to the left, or at position $\;x\;$ and then not moved at all. Thus:

$$ P(x,t)\;=\;P(x,t-q)\;\left(1 - l - q\right) + P(x-h,t-q)\;r + P(x+h,t-q)\;l $$
Assuming that $\;q\;$ and $\;h\;$ are so small, that are negligible compared to $\;t\;$ and $\;x\;$ respectively, we can expand de function as a Taylor series, around $\;t\;$ and $\;x\;$. Notice that higher terms than $\;q^2\;$ and than $\;h^3\;$ have been included in $\;O(q^2)\;$ and $\;O(h^3)\;$, respectively:

$$ P\;=\;\left(P - q\;\frac{\partial P}{\partial t}\right)\;\left(1 - l - r\right) + \left(P - q\;\frac{\partial P}{\partial t} - h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;r + \left(P - q\;\frac{\partial P}{\partial t} + h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;l + O(h^3) + O(q^2)$$
Rearranging this gives:

$$ \frac{\partial P}{\partial t}\;=\;\frac{\alpha\;h^2}{2\;q}\;\frac{\partial^2 P}{\partial x^2} - \frac{\beta\;h}{q}\;\frac{\partial P}{\partial x} + O(h^3) + O(q^2)$$
Where $\;\alpha\;=\;r + l\;$ and $\;\beta\;=\;r - l\;$. We now let $\;h,\;q,\;\beta\;\rightarrow\;0\;$ in such a way that the following limits are finite:

$$ D\;=\;\alpha\;\lim\limits_{h,\;q,\;\beta\rightarrow 0}\frac{h^2}{2\;q} $$ $$ v\;=\;\lim\limits_{h,\;q,\;\beta\rightarrow 0}\frac{h\;\beta}{q} $$
So we can neglect $\;O(h^3)\;$ and $\;O(q^2)\;$, resting:

$$ \frac{\partial P}{\partial t}\;=\;D\;\frac{\partial^2 P}{\partial x^2} - v\;\frac{\partial P}{\partial x} $$
Considerations:

• If we set $\;r\;=\;l\;=\;1/2\;$ as in the isotropic random walk, then $\;\beta\;=\;0\;$, so $\;u\;=\;0\;$, giving as a result the non-biased Diffusion Equation:
  
  $$ \frac{\partial P}{\partial t}\;=\;D\;\frac{\partial^2 P}{\partial x^2}$$
  
• In this case, $\;v\;$ is constant for all the space, not as in the case that concerns us, where $\;v\;$ depends on the gradient of the attractant, normally distributed (with Gaussian Distributions) in the space of interest. So, with $\;v\;$ constant, it is possible to obtain an analytical solution, given by Montroll & Shlesinger (1984), with initial condition $\;P(x,0)\;=\;\delta(x)\;$, that is:
  
  $$ P(x,t)\;=\;\frac{1}{\sqrt{4 \pi D t}}\;e^{-\left(x - v t\right)^2/\left(4 D t\right)}$$

Proof of a Group Behavior Diffusion Model from a Random Walk Model

In addition to last proof, we will make the assumption that the worm can now move in two dimensions ($x,y$).

As before, time step is $\;q\;$, and space step (either in $x$ or in $y$ coordinate) is $\;h\;$. Each time step $\;q\;$, the worm can move a distance $\;h\;$ either up, down, left or right with probabilities $\;u\;$, $\;d\;$, $\;l\;$ and $\;r\;$, respectively. Nevertheless, now we will consider that this probabilities will be dependent on position, so we must rename them as $\;u(x,y)\;$, $\;d(x,y)\;$, $\;l(x,y)\;$ and $\;r(x,y)\;$. The probability of remaining at the same position is, obviously, $\;1 - u(x,y) - d(x,y) - l(x,y) - r(x,y)\;$. The probability of being in position ($x,y$) at time $\;t\;$, $\;P(x,y,t)\;$ is:

$$ P(x,y,t)\;=\;P(x,y,t-q)\;\left(1 - l(x,y) - r(x,y) - u(x,y) - d(x,y)\right) + P(x-h,y,t-q)\;r(x-h,y) + \\ P(x+h,y,t-q)\;l(x+h,y) + P(x,y-h,t-q)\;u(x,y-h) + P(x,y+h,t-q)\;d(x,y+h)$$
Then, expanding each term as a Taylor series, centered in $\;t\;$, $\;x\;$ and $\;y\;$, including all terms $\;h^3\;$ and $\;q^2\;$ and higher, into $\;O(h^3)\;$ and $\;O(q^2)\;$, respectively:

$$ P\;=\;\left(P - q\;\frac{\partial P}{\partial t}\right)\;\left(1 - l - r - u - d\right) + \left(P - q\;\frac{\partial P}{\partial t} - h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;\left(r - h\;\frac{\partial r}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 r}{\partial x^2}\right) + \\ \left(P - q\;\frac{\partial P}{\partial t} + h\;\frac{\partial P}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial x^2}\right)\;\left(l + h\;\frac{\partial l}{\partial x} + \frac{h^2}{2}\;\frac{\partial^2 l}{\partial x^2}\right) + \left(P - q\;\frac{\partial P}{\partial t} - h\;\frac{\partial P}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial y^2}\right)\;\left(u - h\;\frac{\partial u}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 u}{\partial y^2}\right) + \\ \left(P - q\;\frac{\partial P}{\partial t} + h\;\frac{\partial P}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 P}{\partial y^2}\right)\;\left(d - h\;\frac{\partial d}{\partial y} + \frac{h^2}{2}\;\frac{\partial^2 d}{\partial y^2}\right) + O(h^3) + O(q^2) + O(h\;q)$$
That, after operating, results:

$$ q\;\frac{\partial P}{\partial t}\; = \;h\;\left(\frac{\partial l}{\partial x} - \frac{\partial r}{\partial x}\right)\;f + h\;\left(\frac{\partial d}{\partial y} - \frac{\partial u}{\partial y}\right)\;f + \frac{h^2}{2}\left(\frac{\partial^2 r}{\partial x^2} + \frac{\partial^2 l}{\partial x^2}\right)\;f + \frac{h^2}{2}\left(\frac{\partial^2 d}{\partial y^2} + \frac{\partial^2 u}{\partial y^2}\right)\;f + h\;\left(l - r\right)\;\frac{\partial f}{\partial x} + h\;\left(d - u\right)\;\frac{\partial f}{\partial y} + \\ h^2\;\left(\frac{\partial l}{\partial x} + \frac{\partial r}{\partial x}\right)\;\frac{\partial f}{\partial x} + h^2\;\left(\frac{\partial d}{\partial y} + \frac{\partial u}{\partial y}\right)\;\frac{\partial f}{\partial y} + \frac{h^2}{2}\;\left(l + r\right)\;\frac{\partial^2 f}{\partial x^2} + \frac{h^2}{2}\;\left(d + u\right)\;\frac{\partial^2 f}{\partial y^2} + O(h^3) + O(q^2) + O(h\;q)$$
Now we define:

$$ b_1\;=\;\lim\limits_{h,\;q,\;\beta_1\rightarrow 0}\frac{\beta_1\;h}{q} $$ $$ b_2\;=\;\lim\limits_{h,\;q,\;\beta_2\rightarrow 0}\frac{\beta_2\;h}{q} $$ $$ a_11\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_1\;h^2}{2q} $$ $$ a_22\;=\;\lim\limits_{h,\;q\rightarrow 0}\frac{\alpha_2\;h^2}{2q} $$
Where $\;\alpha_1 \;=\; r + l\;$, $\;\alpha_2 \;=\; u + d\;$, $\;\beta_1 \;=\; r - l\;$ and $\;\beta_2 \;=\; u - d\;$.

We now take equation, divide through by $\;q\;$ and take the limit as $\;h,q,\beta_1,\beta_2\rightarrow 0\;$ such that $\;\beta_1 h/q\;$, $\;\beta_2 h/q\;$, $\;\alpha_1 h^2/2q\;$ and $\;\alpha_2 h^2/2q\;$ all tend to a constant, giving:

Okopinska A. (2002) Fokker-Planck equation for bistable potential in the optimized expansion. Physical review E, Volume 65, 062101