Difference between revisions of "Logistic process"

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

Revision as of 19:14, 12 May 2009

Logistic population growth model

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

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

The initial population is [math]y(0)= s[/math], [math]\tau:=0.3[/math].

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

Other models

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();

tau = 0.3;           
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]-tau*t.pos[1]*t.pos[1])*delta; // Logistic process
  t.moveTo([delta+t.pos[0],y+t.pos[1]]);
  x += delta;
  if (x<20.0) {
     setTimeout(loop,10);
  }
}
</jsxgraph>