Autocatalytic process
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Jump to navigationJump to searchAutocatalytic population growth model
Here, in time [math] \Delta t[/math] the population grows by [math]\alpha\cdot y \cdot(A-y)[/math] elements: [math] \Delta y = \alpha\cdot y\cdot \Delta t \cdot(A-y)[/math], that is [math] \frac{\Delta y}{\Delta t} = \alpha\cdot y \cdot(A-y)[/math].
With [math]\Delta t\to 0[/math] we get [math] \frac{d y}{d t} = \alpha\cdot y \cdot (A-y) [/math], i.e. [math] y' = \alpha\cdot y \cdot (A-y) [/math].
The initial population is [math]y(0)= s[/math], [math]A := 5[/math].
Other models
The JavaScript code
var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.5, 12.5, 14.5, -12.5], keepaspectratio: false, 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:'α'});
//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()*(A-t.Y())*dx; // Autocatalytic process
t.moveTo([dx+t.X(),dy+t.Y()]);
x += dx;
if (x<20.0) {
setTimeout(loop,10);
}
}