Population growth models: Difference between revisions

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===Exponential population growth model===
===Exponential population growth model===
In time <math> \Delta t</math> the population grows by <math>\alpha\cdot y </math> elements:
In time <math> \Delta t</math> the population consisting of <math>y</math> elements grows by <math>\alpha\cdot y </math> elements:
<math> \Delta y = \alpha\cdot y\cdot \Delta t </math>, that is  
<math> \Delta y = \alpha\cdot y\cdot \Delta t </math>, that is  
<math> \frac{\Delta y}{\Delta t} = \alpha\cdot y </math>.
<math> \frac{\Delta y}{\Delta t} = \alpha\cdot y </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 </math>, i.e. <math> y' = \alpha\cdot y </math>.
<math> \frac{d y}{d t} = \alpha\cdot y </math>, i.e. <math> y' = \alpha\cdot 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 red line shows the exact solution of the differential equation <math>y(t)=s\cdot e^{\alpha t}</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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<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: [-0.25, 12.5, 14.75, -12.5], 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],[-5,0.5,5]], {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],[-1,0.2,2]], {name:'&alpha;'});
var alpha = brd.createElement('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'});
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]*delta;  // Exponential growth
   var dy = alpha.Value()*t.Y()*dx;  // Exponential growth
   t.moveTo([delta+t.pos[0],y+t.pos[1]]);
   t.moveTo([dx+t.X(),dy+t.Y()]);
   x += delta;
   x += dx;
   if (x<20.0) {
   if (x<20.0) {
     setTimeout(loop,10);
     setTimeout(loop,10);
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}
}
</jsxgraph>
</jsxgraph>
===Other models===


* [[Autocatalytic process]]
* [[Autocatalytic process]]
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===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: [-0.25, 12.5, 14.75, -12.5], 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],[-5,0.5,5]], {name:'s'});
           
var alpha = brd.create('slider', [[0,-6], [10,-6],[-1,0.2,2]], {name:'&alpha;'});
var s = brd.createElement('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'});
var e = brd.create('functiongraph', [function(x){return s.Value()*Math.exp(alpha.Value()*x);}],{strokeColor:'red'});
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();
           
var A = 5;
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]*delta;  // Exponential growth
   var dy = alpha.Value()*t.Y()*dx;  // Exponential growth
   t.moveTo([delta+t.pos[0],y+t.pos[1]]);
   t.moveTo([dx+t.X(),dy+t.Y()]);
   x += delta;
   x += dx;
   if (x<20.0) {
   if (x<20.0) {
     setTimeout(loop,10);
     setTimeout(loop,10);
   }
   }
}
}
</jsxgraph>
</source>
</source>


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

Latest revision as of 11:50, 8 June 2011

Exponential population growth model

In time [math]\displaystyle{ \Delta t }[/math] the population consisting of [math]\displaystyle{ y }[/math] elements grows by [math]\displaystyle{ \alpha\cdot y }[/math] elements: [math]\displaystyle{ \Delta y = \alpha\cdot y\cdot \Delta t }[/math], that is [math]\displaystyle{ \frac{\Delta y}{\Delta t} = \alpha\cdot y }[/math].

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

The initial population is [math]\displaystyle{ y(0)= s }[/math].

The red line shows the exact solution of the differential equation [math]\displaystyle{ y(t)=s\cdot e^{\alpha t} }[/math]. The blue line is the simulation with [math]\displaystyle{ \Delta t = 0.1 }[/math].

Other models

The JavaScript code

var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.25, 12.5, 14.75, -12.5], 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()*dx;   // Exponential growth
  t.moveTo([dx+t.X(),dy+t.Y()]);
  x += dx;
  if (x<20.0) {
     setTimeout(loop,10);
  }
}