Difference between revisions of "Autocatalytic process"
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Jump to navigationJump to searchA WASSERMANN (talk | contribs) (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>...) |
A WASSERMANN (talk | contribs) |
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− | === | + | ===Autocatalytic population growth model=== |
− | In time <math> \Delta | + | 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 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:'α'}); | var alpha = brd.createElement('slider', [[0,-6], [10,-6],[-1,0.2,2]], {name:'α'}); | ||
− | 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 14:48, 23 April 2009
Autocatalytic population growth model
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 \cdot(A-y)[/math], that is [math] \frac{\Delta y}{\Delta t} = \alpha\cdot y \cdot(A-y)[/math].
With [math]\Delta \to 0[/math] we get [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 blue line is the simulation with [math]\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:'α'});
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>