Autocatalytic process: Difference between revisions

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(New page: ===Exponential population growth model=== In time <math> \Delta y</math> the population grows by <math>\alpha\cdot y </math> elements: <math> \Delta y = \alpha\cdot y\cdot \Delta t </math>...)
 
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===Exponential population growth model===
===Autocatalytic population growth model===
In time <math> \Delta y</math> the population grows by <math>\alpha\cdot y </math> elements:
In time <math> \Delta t</math> the population grows by <math>\alpha\cdot y \cdot(A-y)</math> elements:
<math> \Delta y = \alpha\cdot y\cdot \Delta t </math>, that is  
<math> \Delta y = \alpha\cdot y\cdot \Delta t \cdot(A-y)</math>, that is  
<math> \frac{\Delta y}{\Delta t} = \alpha\cdot y </math>.
<math> \frac{\Delta y}{\Delta t} = \alpha\cdot y \cdot(A-y)</math>.


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


The initial population is <math>y(0)= s</math>.
The initial population is <math>y(0)= s</math>.


The red line shows the exact solution of the differential equation <math>y(t)=s\cdot e^{\alpha x}</math>.
The blue line is the simulation with <math>\Delta t = 0.1</math>.
The blue line is the simulation with <math>\Delta t = 0.1</math>.
<html>
<html>
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var s = brd.createElement('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
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 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'});
//var e = brd.createElement('functiongraph', [function(x){return s.X()*Math.exp(alpha.X()*x);}],{strokeColor:'red'});


t.hideTurtle();
t.hideTurtle();

Revision as of 12:48, 23 April 2009

Autocatalytic population growth model

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 \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].

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

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();
            
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]*(A-t.pos[1]); // Autocatalytic process
  t.moveTo([1.0+t.pos[0],y+t.pos[1]]);
  x += delta;
  if (x<10.0) {
     setTimeout(loop,50);
  }
}
</jsxgraph>