# Differential equations involving random variables

First, consider a logistic decay model

$$\frac{dN}{dt} = -N(1-r+rN)$$

Where $$r>0$$. Depending on $$r$$, $$N(t) \rightarrow 0$$ or $$N(t) \rightarrow 1-1/r$$ are the possible behaviors.

Second, consider a pure birth process with Poisson distribution $$p_N(t)$$ and birth rate $$\lambda$$: $$p_N'(t) = (1-\lambda) p_{N}(t) + \lambda p_{N+1}(t)$$

The distribution function is $$p_N(t) = e^{-\lambda t} \frac{(\lambda t)^N}{N!}$$

Now my question:

How can I mix these two concepts together?

I want to model a population which is a logistic decay from the initial condition as the birth rate goes to zero, but it is a pure birth process as the death rate goes to zero.

I'd like to obtain the probability distribution of a logistically decaying population with random birth. Any input is appreciated. Thanks!

• Stochastic differential equation? – R zu Nov 8 '18 at 3:41
• probably, but which would produce the desired limiting behavior? Where in the logistic equations would the birth driving terms go, and how would they relate to $\lambda$? – kevinkayaks Nov 8 '18 at 3:48
• You want to model a combination of a deterministic death process with a stochastic birth process, yes? – aghostinthefigures Nov 8 '18 at 4:06
• Yep, I think that's the situation. It's almost like a standard birth death process but the death is not random without a random population: it is governed by a logistic equation – kevinkayaks Nov 8 '18 at 4:11

## 1 Answer

Recall the solution to the deterministic logistic equation:

$$N(t) = \frac{1}{1 + (\frac{1}{N_0} - 1)e^{-rt}}$$

for initial population size $$N_0$$. Because the birth rate $$\lambda$$ is (extraordinarily luckily) independent of the population size, a realization of this process will consist of chained intervals of these solutions for initial conditions $$N_i$$ over time intervals $$W_i$$ where the size of each time interval $$W_i$$ is distributed exponentially with parameter $$\lambda$$ and the initial conditions at each time interval $$N_{i>0}$$ are defined recursively as

$$N_{i+1} = \frac{1}{1 + (\frac{1}{N_i} - 1)e^{-rW_i}}+1$$

In this sense, the only source of randomness is $$W_i$$ which then feeds into the value of $$N(t)$$ through the definition of $$N_{i+1}$$. All of this is conditional on the fact that the logistic decay process "resets" at every birth event.

Finding information about the specific distribution this system follows is very difficult, since this is a continuous-time continuous-space system, but it can be easily found computationally.

Hope this helps!