Difference between revisions of "Population growth models"

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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);
Line 57: Line 56:
  
 
===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() {
Line 74: Line 74:
 
              
 
              
 
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 13:50, 8 June 2011

Exponential population growth model

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] \frac{\Delta y}{\Delta t} = \alpha\cdot y [/math].

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].

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 t}[/math]. The blue line is the simulation with [math]\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);
  }
}