# ODE proof

1. Jan 18, 2014

### squenshl

1. The problem statement, all variables and given/known data
Given that we have the equation pi which is the probability that the centre of cell i is not in A (i and A not is not important), given that cells 1 to i-1 are not in A, then we have
$$p_i = \frac{1-E_{i-1}-A}{1-E_{i-1}},$$
where Ei is the proportion of the total area excluded by the first i cells. The probability Ps that a proliferation attempt will be successful is then the probability that the centres of all N(t) cells lie outside A,
$$P_s = \prod_{i=1}^{N(t)-1} \frac{1-E_i-A}{1-E_i}.$$
Each agent excludes an area $$\pi\Delta^2$$ although the area excluded by different agents can overlap. Hence, we may write a recurrence relation for Ei as
$$E_{i+1} = E_i+\pi d^2\left(1-q_i\right),$$
where
$$d = \frac{\Delta}{\Omega}$$
and qi is the expected proportion of cells i's excluded area that overlaps with the area already excluded by the first i-1 cells. After some work we get the recurrence relation
$$E_i = 1-\left(1-\pi d^2\right)^i.$$
Provided that the domain size is large
$$\left(d<<1\right),$$
we can treat the spatially averaged agent density as a continuous variable. Combining equations Ps, Ei+1 and Ei gives
$$\frac{dC_m}{dt} = \lambda C_m\prod_{i=1}^{c_m/d^2-1}\frac{\left(1-\pi d^2\right)^i-2d^2}{\left(1-\pi d^2\right)^i},$$
where
$$\lambda = \lim_{\tau \to 0} \left(P_p/\tau\right).$$
and Pp is the probability the a cell attempts to proliferate. My question is how do we combine equations Ps, Ei+1 and Ei to get dCm/dt. In other words, how do we go from a difference/recurrence equation to a first order ODE. I have never come across this before.
2. Relevant equations

3. The attempt at a solution
I tried a Taylor series but no luck. Someone please help.

Last edited: Jan 18, 2014
2. Jan 18, 2014

### Mandelbroth

Where does this come from? I don't understand the relation between $P_p$ and $\tau$ in the limit.

3. Jan 18, 2014

### squenshl

See page 4 of the attached file for the limit and page 6 for the proof.

#### Attached Files:

• ###### SimpsonCells.pdf
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1.2 MB
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4. Jan 20, 2014

### squenshl

Still struggling on how to get equation 3.9. I just don't know where the 2d^2 term and the lambdaC_m come from. I know where everything does.

5. Jan 22, 2014

### squenshl

I guess no one knows how to do this too :(

6. Jan 25, 2014

### squenshl

Still got nothing.