# Convert ODE into discrete probabilistic model

how can I turn an ODE equation into a discrete probabilistic model?

I take for example the Verhulst equation for the growth of a population.

$$\frac{dP}{dt} = rP(1-P/K)$$

I was thinking to simulate the population dynamic simply as a loop (for the time) and at every time step, for each member of the population evaluate the probability of procreation (in which case I add 1 to the population counter) and the probability that a member has died, in which case I remove 1 from the population counter.

I don't know though how to use r and K to evaluate such probabilities. Can you give me some hints or link me to some good resource that explains how to do this?

In general, which is the strategy to convert an ODE (or system of ODEs) into it probabilistic equivalent???

• What kind of distribution do you expect the model to have? Uniform? Normal? Other? – Paul Mar 28 '12 at 14:53
• What are your constraints on $r$ and $K$? – Deathbreath Mar 28 '12 at 16:58
• @Paul my idea is that each "animal" has its own reproduction and death rates that are normally distributed and . However initially I can assume that the reproduction rate is constant. – lucacerone Mar 28 '12 at 19:38
• @Deathbreath as I told Paul, I'd like to assume the rates are normally distributed – lucacerone Mar 28 '12 at 19:44