# Use of differential equations for modeling population which is a discrete variable

Population dynamics is often modeled using ODE. For example one common model is logistic growth model:

$$\frac {dx}{dt} = kx\left(1-\frac{x}{C}\right)$$ where $x$ is population size, $k$ is rate constant for growth, $C$ is carrying capacity.

But population is a discrete variable. It is not continuous. It always takes whole numbers. You can have a population of 3000 fishes, but not 3001.2 fishes.

Then how can one use population as a dependent variable in a differential equation?

Integration of the ODE, given above, will give me a function to calculate size of population, $x_t$, at time $t$, when size of the population at $t = 0$, was $x_0$. We can specify a whole number for $x_0$. But, $x_t$ can be a real number with fraction. But a population size is always a whole number.

How does one tackle this anomaly?

• Well for small populations one should consider using a difference equation instead of a differential equation. But for large populations the error of not being an integer is small. That being said its interesting that the descrete version of logistic growth can exhibit chaotic behaviour, whereas the differential equation cannot (so there are subtle issues) – Martin Aug 10 '16 at 11:27
• Take a million people as unit. Then the population is for all purposes a continuous variable. – Christian Blatter Aug 10 '16 at 11:54

## 1 Answer

There is no anomaly. If the assumption that "the equation so-and-so describes the population size at time $t$" seems to be arbitrary for you, you should start with underlying stochastic system. In particular, assume that $X$ is a discrete random variable that takes only integer values. Specify the rules this variable follows. In this way you get a stochastic process. However, if this process is nonlinear it is usually very difficult to analyze. There is a significant body of work that allows to deduce a system of ordinary differential equations to approximate some characteristics of this process, in particular, the mean of $X$ (not always though). Hence, very often you should actually replace the words "let $N(t)$ be the population size at time $t$" with "let $N(t)$ be the expectation of the population size at time $t$" In this way there is no internal conflict.