# Population growth models

### Exponential population growth model

In time $\displaystyle{ \Delta t }$ the population consisting of $\displaystyle{ y }$ elements grows by $\displaystyle{ \alpha\cdot y }$ elements: $\displaystyle{ \Delta y = \alpha\cdot y\cdot \Delta t }$, that is $\displaystyle{ \frac{\Delta y}{\Delta t} = \alpha\cdot y }$.

With $\displaystyle{ \Delta t\to 0 }$ we get $\displaystyle{ \frac{d y}{d t} = \alpha\cdot y }$, i.e. $\displaystyle{ y' = \alpha\cdot y }$.

The initial population is $\displaystyle{ y(0)= s }$.

The red line shows the exact solution of the differential equation $\displaystyle{ y(t)=s\cdot e^{\alpha t} }$. The blue line is the simulation with $\displaystyle{ \Delta t = 0.1 }$.

### The JavaScript code

var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.25, 12.5, 14.75, -12.5], axis:true});
var t = brd.create('turtle',[4,3,70]);
var s = brd.create('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
var alpha = brd.create('slider', [[0,-6], [10,-6],[-1,0.2,2]], {name:'&alpha;'});
var e = brd.create('functiongraph', [function(x){return s.Value()*Math.exp(alpha.Value()*x);}],{strokeColor:'red'});

t.hideTurtle();

var A = 5;
var tau = 0.3;

function clearturtle() {
t.cs();
t.ht();
}

function run() {
t.setPos(0,s.Value());
t.setPenSize(4);
dx = 0.1; // global
x = 0.0;  // global
loop();
}

function loop() {
var dy = alpha.Value()*t.Y()*dx;   // Exponential growth
t.moveTo([dx+t.X(),dy+t.Y()]);
x += dx;
if (x<20.0) {
setTimeout(loop,10);
}
}