Population growth models

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Exponential population growth model

In time [math] \Delta y[/math] the population grows by [math]\alpha\cdot y [/math] elements: [math] \Delta y = \alpha\cdot y\cdot \Delta t [/math], that is [math] \frac{\Delta y}{\Delta t} = \alpha\cdot y [/math]. With $\Delta \to 0$ we get:

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

i.e.

[math] y' = \alpha\cdot y [/math]

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