- #1
squenshl
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Homework Statement
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.
Homework Equations
The Attempt at a Solution
I tried a Taylor series but no luck. Someone please help.
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