Given a continuous exponential growth model, say P(t)=P0 ert, is it immediate that we have an exponential distribution? An example I have in mind is the division of cells: Assuming continuous division with exponential growth, do we automatically know that the probability of division between t=a and t=b is era−e−rb or is the exponential distribution of cell division an additional assumption one must impose?

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Given a continuous exponential growth model, say P(t)=P0 ert, is it immediate that we have an exponential distribution? An example I have in mind is the division of cells: Assuming continuous division with exponential growth, do we automatically know that the probability of division between t=a and t=b is era−e−rb or is the exponential distribution of cell division an additional assumption one must impose?

missygouldzl · Accepted Answer

The division rate at time t being λ(t) means that P(t+dt)=P(t)+(λ(t)dt)P(t). Hence, for every t≥0,P′(t)=λ(t)P(t) and P(t)=eA(t)P0 with A(t)=∫t0λ(s)ds. In your case, A(t)=rt hence λ(t)=r for every t. In the general case, the probability that an individual cell present at time t stays undivided until time t+s is e−A(t+s)+A(t). Thus, assuming that λ(t)=r for every t, the probability that an individual cell present at time t divides before time t+s is 1−e−sr

Frauental91 · Accepted Answer

The assumption that division times are exponentially distributed results from disregarding the population&#039;s age distribution. For models of cell division that are age-structured, see Glenn Webb&#039;s publications. This models go from an ODE to a PDE, fortunately there is a characteristic line that still describes it which simplifies the solution. However the mathematics are more complicated, and do not fit in a simple&amp;nbsp;ex&amp;nbsp;type model.

Given a continuous exponential growth model, say P(t)=P0ert, is it immediate that we have an exponen

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