# Population growth models

### Exponential population growth model

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

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

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

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

### 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);
}
}