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ODE proof

  1. Jan 18, 2014 #1
    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. jcsd
  3. Jan 18, 2014 #2
    Where does this come from? I don't understand the relation between ##P_p## and ##\tau## in the limit.
     
  4. Jan 18, 2014 #3
    See page 4 of the attached file for the limit and page 6 for the proof.
     

    Attached Files:

  5. Jan 20, 2014 #4
    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.
     
  6. Jan 22, 2014 #5
    I guess no one knows how to do this too :(
     
  7. Jan 25, 2014 #6
    Still got nothing.
     
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