# Population growth models

### Exponential population growth model

In time $\displaystyle{ \Delta y }$ the population 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 $\displaystyle{ \Delta \to 0 }$ 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 $\displaystyle{ y(t)=s\cdot e^{\alpha x} }$. The blue line is the simulation with $\displaystyle{ \Delta t = 0.1 }$.

### The JavaScript code

<jsxgraph height="500" width="600" board="board"  box="box1">
brd = JXG.JSXGraph.initBoard('box1', {originX: 10, originY: 250, unitX: 40, unitY: 20, axis:true});
var t = brd.createElement('turtle',[4,3,70]);

var s = brd.createElement('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
var alpha = brd.createElement('slider', [[0,-6], [10,-6],[-1,0.2,2]], {name:'&alpha;'});
var e = brd.createElement('functiongraph', [function(x){return s.X()*Math.exp(alpha.X()*x);}],{strokeColor:'red'});

t.hideTurtle();

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

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

function loop() {
var y = alpha.X()*t.pos[1];   // Exponential growth
t.moveTo([1.0+t.pos[0],y+t.pos[1]]);
x += delta;
if (x<10.0) {
setTimeout(loop,50);
}
}
</jsxgraph>