Autocatalytic process: Difference between revisions

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The initial population is <math>y(0)= s</math>, <math>A := 5</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>.
<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>
</html>
</html>


<jsxgraph height="500" width="600" board="board"  box="box1">
<jsxgraph height="500" width="600" 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, 12.5, 14.5, -12.5], keepaspectratio: false, axis:true});
var t = brd.createElement('turtle',[4,3,70]);
var t = brd.create('turtle',[4,3,70]);
              
              
var s = brd.createElement('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
var s = brd.create('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 alpha = brd.create('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'});
//var e = brd.create('functiongraph', [function(x){return s.Value()*Math.exp(alpha.Value()*x);}],{strokeColor:'red'});


t.hideTurtle();
t.hideTurtle();
              
              
A = 5;
var A = 5;
tau = 0.3;
var tau = 0.3;
              
              
function clearturtle() {
function clearturtle() {
Line 41: Line 40:
              
              
function loop() {
function loop() {
   var dy = alpha.Value()*t.pos[1]*(A-t.pos[1])*dx; // Autocatalytic process
   var dy = alpha.Value()*t.Y()*(A-t.Y())*dx; // Autocatalytic process
   t.moveTo([dx+t.pos[0],dy+t.pos[1]]);
   t.moveTo([dx+t.X(),dy+t.Y()]);
   x += dx;
   x += dx;
   if (x<20.0) {
   if (x<20.0) {
Line 55: Line 54:


===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, 12.5, 14.5, -12.5], keepaspectratio: false, 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.createElement('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
var s = brd.create('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 alpha = brd.create('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'});
//var e = brd.create('functiongraph', [function(x){return s.Value()*Math.exp(alpha.Value()*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 loop() {
function loop() {
   var dy = alpha.Value()*t.pos[1]*(A-t.pos[1])*dx; // Autocatalytic process
   var dy = alpha.Value()*t.Y()*(A-t.Y())*dx; // Autocatalytic process
   t.moveTo([dx+t.pos[0],dy+t.pos[1]]);
   t.moveTo([dx+t.X(),dy+t.Y()]);
   x += dx;
   x += dx;
   if (x<20.0) {
   if (x<20.0) {
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   }
   }
}
}
</jsxgraph>
</source>
</source>


[[Category:Examples]]
[[Category:Examples]]
[[Category:Calculus]]
[[Category:Turtle Graphics]]
[[Category:Turtle Graphics]]

Latest revision as of 07:55, 16 July 2019

Autocatalytic population growth model

Here, in time [math]\displaystyle{ \Delta t }[/math] the population grows by [math]\displaystyle{ \alpha\cdot y \cdot(A-y) }[/math] elements: [math]\displaystyle{ \Delta y = \alpha\cdot y\cdot \Delta t \cdot(A-y) }[/math], that is [math]\displaystyle{ \frac{\Delta y}{\Delta t} = \alpha\cdot y \cdot(A-y) }[/math].

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

The initial population is [math]\displaystyle{ y(0)= s }[/math], [math]\displaystyle{ A := 5 }[/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:'&alpha;'});
//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);
  }
}