Population growth models

Exponential population growth model

In time [math]\displaystyle{ \Delta y }[/math] the population grows by [math]\displaystyle{ \alpha\cdot y }[/math] elements: [math]\displaystyle{ \Delta y = \alpha\cdot y\cdot \Delta t }[/math], that is [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]

The initial population is $y(0)= s$.