# What does the population growth model equation mean? dN/dt=rN

Feb 10, 2016

The equation $\frac{\mathrm{dN}}{\mathrm{dt}} = r N$means that rate change of the population is proportional to the size of the population, where r is the proportionality constant.

#### Explanation:

This is a rather simple and impractical equation because it signifies an Exponential Population Growth. If you are familiar to the Future Value of a compounded interest rate, $F V = P V {\left(1 + r\right)}^{n}$.
dN/dt = rN : a differential equation describing the population growth
where N is the population size, r is the growth rate, and t is time.
$N \left(t\right) = {N}_{0} {e}^{r t}$ : the solution of the differential equation for exponential growth. The equation grows exponential and you know population does not grow exponentially, as a result we have have a more reasonable model called "The Logistic Equation". The Logistic model sets limit to the growth. Why? Well a control space like a nation, a savanna, or the plane carry a finite amount of resources and cannot support exponential populations growth in perpetuity.
$\left(\frac{\mathrm{dN}}{\mathrm{dt}}\right) = r N \left(1 - \frac{N}{K}\right)$ : The logistic differential equation, has
N as the population size, r is growth rate, K is carrying capacity.
This equation forces, populations to converge to the carrying capacity. The speed at which the populations approach K is related to the growth rate r.