Difference between revisions of "Autocatalytic process"
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===Autocatalytic population growth model=== | ===Autocatalytic population growth model=== | ||
− | + | Here, 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> \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>. | <math> \frac{\Delta y}{\Delta t} = \alpha\cdot y \cdot(A-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 \cdot (A-y) </math>, i.e. <math> y' = \alpha\cdot y \cdot (A-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>, <math>A := 5</math>. | The initial population is <math>y(0)= s</math>, <math>A := 5</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" | + | <jsxgraph height="500" width="600" box="box1"> |
− | brd = JXG.JSXGraph.initBoard('box1', { | + | var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.5, 12.5, 14.5, -12.5], keepaspectratio: false, axis:true}); |
− | var t = brd. | + | var t = brd.create('turtle',[4,3,70]); |
− | var s = brd. | + | var s = brd.create('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'}); |
− | var alpha = brd. | + | var alpha = brd.create('slider', [[0,-6], [10,-6],[-1,0.2,2]], {name:'α'}); |
− | //var e = brd. | + | //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 33: | Line 32: | ||
function run() { | function run() { | ||
− | t.setPos(0,s. | + | t.setPos(0,s.Value()); |
t.setPenSize(4); | t.setPenSize(4); | ||
− | + | dx = 0.1; // global | |
x = 0.0; // global | x = 0.0; // global | ||
loop(); | loop(); | ||
Line 41: | Line 40: | ||
function loop() { | function loop() { | ||
− | var | + | var dy = alpha.Value()*t.Y()*(A-t.Y())*dx; // Autocatalytic process |
− | t.moveTo([ | + | t.moveTo([dx+t.X(),dy+t.Y()]); |
− | x += | + | x += dx; |
if (x<20.0) { | if (x<20.0) { | ||
setTimeout(loop,10); | setTimeout(loop,10); | ||
Line 50: | Line 49: | ||
</jsxgraph> | </jsxgraph> | ||
+ | ===Other models=== | ||
* [[Population growth models]] | * [[Population growth models]] | ||
* [[Logistic process]] | * [[Logistic process]] | ||
===The JavaScript code=== | ===The JavaScript code=== | ||
− | <source lang=" | + | <source lang="javascript"> |
− | + | var brd = JXG.JSXGraph.initBoard('box1', {boundingbox: [-0.5, 12.5, 14.5, -12.5], keepaspectratio: false, axis:true}); | |
− | brd = JXG.JSXGraph.initBoard('box1', { | + | var t = brd.create('turtle',[4,3,70]); |
− | var t = brd. | ||
− | var s = brd. | + | var s = brd.create('slider', [[0,-5], [10,-5],[-5,0.5,5]], {name:'s'}); |
− | var alpha = brd. | + | var alpha = brd.create('slider', [[0,-6], [10,-6],[-1,0.2,2]], {name:'α'}); |
− | var e = brd. | + | //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; |
+ | var tau = 0.3; | ||
+ | |||
function clearturtle() { | function clearturtle() { | ||
t.cs(); | t.cs(); | ||
Line 72: | Line 73: | ||
function run() { | function run() { | ||
− | t.setPos(0,s. | + | t.setPos(0,s.Value()); |
t.setPenSize(4); | t.setPenSize(4); | ||
− | + | dx = 0.1; // global | |
x = 0.0; // global | x = 0.0; // global | ||
loop(); | loop(); | ||
Line 80: | Line 81: | ||
function loop() { | function loop() { | ||
− | var | + | var dy = alpha.Value()*t.Y()*(A-t.Y())*dx; // Autocatalytic process |
− | t.moveTo([ | + | t.moveTo([dx+t.X(),dy+t.Y()]); |
− | x += | + | x += dx; |
if (x<20.0) { | if (x<20.0) { | ||
setTimeout(loop,10); | setTimeout(loop,10); | ||
} | } | ||
} | } | ||
− | |||
</source> | </source> | ||
[[Category:Examples]] | [[Category:Examples]] | ||
+ | [[Category:Calculus]] | ||
[[Category:Turtle Graphics]] | [[Category:Turtle Graphics]] |
Latest revision as of 08:55, 16 July 2019
Autocatalytic population growth model
Here, 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 t\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], [math]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:'α'});
//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);
}
}