Autocatalytic process

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Autocatalytic 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].

The blue line is the simulation with [math]\Delta t = 0.1[/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);
  }
}