Population growth models

Exponential population growth model

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

With [math]\displaystyle{ \Delta t\to 0 }[/math] we get [math]\displaystyle{ \frac{d y}{d t} = \alpha\cdot y }[/math], i.e. [math]\displaystyle{ y' = \alpha\cdot y }[/math].

The initial population is [math]\displaystyle{ y(0)= s }[/math].

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

Other models

• Autocatalytic process
• Logistic process

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.Value()*Math.exp(alpha.Value()*x);}],{strokeColor:'red'});

t.hideTurtle();
            
A = 5;
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);
  }
}
</jsxgraph>