Population growth models: Difference between revisions

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===Exponential population growth model===
===Exponential population growth model===
In time <math> \Delta y</math the population grows by <math>\alpha\cdot y </math> elements:
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>
<math> \Delta y = \alpha\cdot y\cdot \Delta t </math>, that is
It follows:
<math> \frac{\Delta y}{\Delta t} = \alpha\cdot y </math>.
:<math> \frac{\Delta y}{\Delta t} = \alpha\cdot y </math>
With $\Delta \to 0$ we get:
With $\Delta \to 0$ we get:
:<math> \frac{d y}{d t} = \alpha\cdot y </math>
:<math> \frac{d y}{d t} = \alpha\cdot y </math>
i.e.
i.e.
:<math> y' = \alpha\cdot y </math>
:<math> y' = \alpha\cdot y </math>
The initial population is $y(0)= s$.
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Revision as of 16:06, 22 April 2009

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 $y(0)= s$.