Logistic process

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Logistic population growth model

In time Δt the population grows by \alpha\cdot y -\tau\cdot y^2 elements:  \Delta y = (\alpha\cdot y- \tau\cdot y^2)\cdot \Delta t, that is  \frac{\Delta y}{\Delta t} = \alpha\cdot y -\tau\cdot y^2.

With \Delta t\to 0 we get  \frac{d y}{d t} = \alpha\cdot y  -\tau\cdot y^2 , i.e.  y' = \alpha\cdot y  -\tau\cdot y^2 .

The initial population is y(0) = s, τ: = 0.3.

The blue line is the simulation with Δt = 0.1.

Other models

The JavaScript code

var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.5, 11.5, 14.5, -11.5], axis:true});
var t = brd.create('turtle',[4,3,70]);
var s = brd.create('slider', [[0,-5], [10,-5],[0,0.5,5]], {name:'s'});
var alpha = brd.create('slider', [[0,-6], [10,-6],[-1,0.9,2]], {name:'α'});
 
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()-tau*t.Y()*t.Y())*dx; // Logistic process
t.moveTo([dx+t.X(),dy+t.Y()]);
x += dx;
if (x<20.0) {
setTimeout(loop,10);
}
}
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