Difference between revisions of "Logistic process"

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(New page: ===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\cd...)
 
 
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===Logistic population growth model===
 
===Logistic population growth model===
In time <math> \Delta t</math> the population grows by <math>\alpha\cdot y -\tau\cdot y^2)</math> elements:
+
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> \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>.
 
<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>.
  
 
The initial population is <math>y(0)= s</math>, <math>\tau:=0.3</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>.
 
 
<html>
 
<html>
 
<form><input type="button" value="clear and run" onClick="clearturtle();run()"></form>
 
<form><input type="button" value="clear and run" onClick="clearturtle();run()"></form>
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<jsxgraph height="500" width="600" board="board"  box="box1">
 
<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 brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.5, 11.5, 14.5, -11.5], axis:true});
var t = brd.createElement('turtle',[4,3,70]);
+
var t = brd.create('turtle',[4,3,70]);
           
+
var s = brd.create('slider', [[0,-5], [10,-5],[0,0.5,5]], {name:'s'});
var s = brd.createElement('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
+
var alpha = brd.create('slider', [[0,-6], [10,-6],[-1,0.9,2]], {name:'&alpha;'});
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();
 
t.hideTurtle();
 
              
 
              
A = 5;
+
var A = 5;
tau = 0.3;
+
var tau = 0.3;
 
              
 
              
 
function clearturtle() {
 
function clearturtle() {
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function run() {
 
function run() {
   t.setPos(0,s.X());
+
   t.setPos(0,s.Value());
 
   t.setPenSize(4);
 
   t.setPenSize(4);
   delta = 0.1; // global
+
   dx = 0.1; // global
 
   x = 0.0;  // global
 
   x = 0.0;  // global
 
   loop();
 
   loop();
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function loop() {
 
function loop() {
   //var y = alpha.X()*t.pos[1];  // Exponential growth
+
   var dy = (alpha.Value()*t.Y()-tau*t.Y()*t.Y())*dx; // Logistic process
  //var y = alpha.X()*t.pos[1]*(A-t.pos[1]); // Autocatalytic process
+
   t.moveTo([dx+t.X(),dy+t.Y()]);
  var y = (alpha.X()*t.pos[1]-tau*t.pos[1]*t.pos[1]); // Logistic process
+
   x += dx;
   t.moveTo([1.0+t.pos[0],y+t.pos[1]]);
+
   if (x<20.0) {
   x += delta;
+
     setTimeout(loop,10);
   if (x<10.0) {
 
     setTimeout(loop,50);
 
 
   }
 
   }
 
}
 
}
 
</jsxgraph>
 
</jsxgraph>
 +
 +
===Other models===
  
 
* [[Population growth models]]
 
* [[Population growth models]]
* [[Logistic process]]
+
* [[Autocatalytic process]]
  
 
===The JavaScript code===
 
===The JavaScript code===
<source lang="xml">
+
<source lang="javascript">
<jsxgraph height="500" width="600" board="board"  box="box1">
+
var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.5, 11.5, 14.5, -11.5], axis:true});
brd = JXG.JSXGraph.initBoard('box1', {originX: 10, originY: 250, unitX: 40, unitY: 20, axis:true});
+
var t = brd.create('turtle',[4,3,70]);
var t = brd.createElement('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:'&alpha;'});
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();
 
t.hideTurtle();
 
+
           
A = 5;          
+
var A = 5;
 +
var tau = 0.3;
 +
           
 
function clearturtle() {
 
function clearturtle() {
 
   t.cs();
 
   t.cs();
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function run() {
 
function run() {
   t.setPos(0,s.X());
+
   t.setPos(0,s.Value());
 
   t.setPenSize(4);
 
   t.setPenSize(4);
   delta = 0.1; // global
+
   dx = 0.1; // global
 
   x = 0.0;  // global
 
   x = 0.0;  // global
 
   loop();
 
   loop();
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function loop() {
 
function loop() {
   var y = alpha.X()*t.pos[1]*(A-t.pos[1]); // Autocatalytic process
+
   var dy = (alpha.Value()*t.Y()-tau*t.Y()*t.Y())*dx; // Logistic process
   t.moveTo([1.0+t.pos[0],y+t.pos[1]]);
+
   t.moveTo([dx+t.X(),dy+t.Y()]);
   x += delta;
+
   x += dx;
   if (x<10.0) {
+
   if (x<20.0) {
     setTimeout(loop,50);
+
     setTimeout(loop,10);
 
   }
 
   }
 
}
 
}
</jsxgraph>
 
 
</source>
 
</source>
  
 
[[Category:Examples]]
 
[[Category:Examples]]
 +
[[Category:Calculus]]
 
[[Category:Turtle Graphics]]
 
[[Category:Turtle Graphics]]

Latest revision as of 08:54, 16 July 2019

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].

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:'&alpha;'});

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);
  }
}