Autocatalytic process

Autocatalytic population growth model

Here, in time $\Delta t$ the population grows by $\alpha\cdot y \cdot(A-y)$ elements: $\Delta y = \alpha\cdot y\cdot \Delta t \cdot(A-y)$, that is $\frac{\Delta y}{\Delta t} = \alpha\cdot y \cdot(A-y)$.

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

The initial population is $y(0)= s$, $A := 5$.

The blue line is the simulation with $\Delta t = 0.1$.

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();

A = 5;
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]*(A-t.pos[1])*delta; // Autocatalytic process
t.moveTo([delta+t.pos[0],y+t.pos[1]]);
x += delta;
if (x<20.0) {
setTimeout(loop,10);
}
}
</jsxgraph>