Population growth models

Exponential population growth model

In time [math]\displaystyle{ \Delta y\lt /math the population grows by \lt math\gt \alpha\cdot y }[/math] elements:

[math]\displaystyle{ \Delta y = \alpha\cdot y\cdot \Delta t }[/math]

It follows:

[math]\displaystyle{ \frac{\Delta y}{\Delta t} = \alpha\cdot y }[/math]

With $\Delta \to 0$ we get:

[math]\displaystyle{ \frac{d y}{d t} = \alpha\cdot y }[/math]

i.e.

[math]\displaystyle{ y' = \alpha\cdot y }[/math]