Lotka-Volterra equations - Revision history

A WASSERMANN at 13:08, 3 March 2021

2021-03-03T13:08:57Z

Michael at 09:26, 19 November 2013

2013-11-19T09:26:30Z

← Older revision		Revision as of 09:26, 19 November 2013
Line 27:		Line 27:

	s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});		s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});
	st = board.createElement('text', [20,25, "Birth rate ~~predators~~"], {fixed:true});		st = board.createElement('text', [20,25, "Birth rate prey"], {fixed:true});
	u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});		u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});
	ut = board.createElement('text', [20,23, "Death rate predators"], {fixed:true});		ut = board.createElement('text', [20,23, "Death rate predators"], {fixed:true});

A WASSERMANN at 10:59, 7 August 2013

2013-08-07T10:59:04Z

Michael at 08:49, 8 June 2011

2011-06-08T08:49:12Z

← Older revision		Revision as of 08:49, 8 June 2011
Line 24:		Line 24:

	<jsxgraph width="600" height="600">		<jsxgraph width="600" height="600">
	board = JXG.JSXGraph.initBoard('jxgbox', {~~originX~~: 40, ~~originY: 560~~, ~~unitX: 20~~, ~~unitY: 20~~, axis: ~~false~~, grid: false});		board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-1.5, 28.5, 28.5, -1.5], axis: true, grid: false});
	~~xax = board.createElement('axis', [[0,0],[1,0]]);~~
	~~yax = board.createElement('axis', [[0,0],[0,1]]);~~
	s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});		s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});
	st = board.createElement('text', [20,25, "Birth rate predators"]);		st = board.createElement('text', [20,25, "Birth rate predators"]);
Line 37:		Line 36:
	pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);		pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);

	startpred = board.createElement('glider', [0, 10, ~~yax~~], {name:'Preys',strokeColor:'red',fillColor:'red'});		startpred = board.createElement('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});
	startprey = board.createElement('glider', [0, 5, ~~yax~~], {name:'Predators',strokeColor:'blue',fillColor:'blue'});		startprey = board.createElement('glider', [0, 5, board.defaultAxes.y], {name:'Predators',strokeColor:'blue',fillColor:'blue'});

	var g3 = null;		var g3 = null;
Line 109:		Line 108:
	<source lang="javascript">		<source lang="javascript">
	// Initialise board		// Initialise board
	board = JXG.JSXGraph.initBoard('jxgbox', {~~originX~~: 40, ~~originY: 560~~, ~~unitX: 20~~, ~~unitY: 20~~, axis: ~~false~~, grid: false});		board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-1.5, 28.5, 28.5, -1.5], axis: true, grid: false});
	~~// Create axis~~
	~~xax = board.createElement('axis', [[0,0],[1,0]]);~~
	~~yax = board.createElement('axis', [[0,0],[0,1]]~~);

	// Define sliders to dynamically change parameters of the equations and create text elements to describe them		// Define sliders to dynamically change parameters of the equations and create text elements to describe them
Line 126:		Line 122:

	// Dynamic initial value as gliders on the y-axis		// Dynamic initial value as gliders on the y-axis
	startpred = board.createElement('glider', [0, 10, ~~yax~~], {name:'Preys',strokeColor:'red',fillColor:'red'});		startpred = board.createElement('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});
	startprey = board.createElement('glider', [0, 5, ~~yax~~], {name:'Predators',strokeColor:'blue',fillColor:'blue'});		startprey = board.createElement('glider', [0, 5, board.defaultAxes.y], {name:'Predators',strokeColor:'blue',fillColor:'blue'});

A WASSERMANN at 20:27, 8 September 2010

2010-09-08T20:27:25Z

← Older revision		Revision as of 20:27, 8 September 2010
Line 61:		Line 61:
	var x0 = [startpred.Y(), startprey.Y()];		var x0 = [startpred.Y(), startprey.Y()];

	var data = JXG.Math.Numerics.rungeKutta(~~JXG.Math.Numerics.predefinedButcher.Euler~~, x0, I, N, f);		var data = JXG.Math.Numerics.rungeKutta('euler', x0, I, N, f);

	var t = [];		var t = [];
Line 159:		Line 159:

	// Solve ode		// Solve ode
	var data = JXG.Math.Numerics.rungeKutta(~~JXG.Math.Numerics.predefinedButcher.Euler~~, x0, I, N, f);		var data = JXG.Math.Numerics.rungeKutta('euler', x0, I, N, f);

	// to plot the data against time we need the times where the equations were solved		// to plot the data against time we need the times where the equations were solved

A WASSERMANN at 07:44, 18 August 2009

2009-08-18T07:44:25Z

← Older revision		Revision as of 07:44, 18 August 2009
Line 213:		Line 213:

	[[Category:Examples]]		[[Category:Examples]]
			[[Category:Calculus]]

A WASSERMANN at 15:01, 30 July 2009

2009-07-30T15:01:53Z

← Older revision		Revision as of 15:01, 30 July 2009
Line 37:		Line 37:
	pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);		pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);

	startpred = board.createElement('glider', [0, 10, yax], {name:'Preys'});		startpred = board.createElement('glider', [0, 10, yax], {name:'Preys',strokeColor:'red',fillColor:'red'});
	startprey = board.createElement('glider', [0, 5, yax], {name:'Predators'});		startprey = board.createElement('glider', [0, 5, yax], {name:'Predators',strokeColor:'blue',fillColor:'blue'});

	var g3 = null;		var g3 = null;
Line 126:		Line 126:

	// Dynamic initial value as gliders on the y-axis		// Dynamic initial value as gliders on the y-axis
	startpred = board.createElement('glider', [0, 10, yax], {name:'Preys'});		startpred = board.createElement('glider', [0, 10, yax], {name:'Preys',strokeColor:'red',fillColor:'red'});
	startprey = board.createElement('glider', [0, 5, yax], {name:'Predators'});		startprey = board.createElement('glider', [0, 5, yax], {name:'Predators',strokeColor:'blue',fillColor:'blue'});

A WASSERMANN at 15:01, 30 July 2009

2009-07-30T15:01:06Z

A WASSERMANN at 07:59, 23 June 2009

2009-06-23T07:59:34Z

← Older revision		Revision as of 07:59, 23 June 2009
Line 4:		Line 4:
	<math>\frac{dN_1}{dt} = N_1(\epsilon_1 - \gamma_1N_2), \quad \frac{dN_2}{dt} = -N_2(\epsilon_2 - \gamma_2N_1)</math>		<math>\frac{dN_1}{dt} = N_1(\epsilon_1 - \gamma_1N_2), \quad \frac{dN_2}{dt} = -N_2(\epsilon_2 - \gamma_2N_1)</math>

	~~Bedeutung der Variablen~~:		Meaning of the variables:
	::{\| border="1" \|\|		::{\| border="1" \|\|
	\|-		\|-

Michael at 18:47, 22 June 2009

2009-06-22T18:47:00Z

← Older revision		Revision as of 18:47, 22 June 2009
Line 1:		Line 1:
	~~{{NotYetImplemented}}~~

	~~This example doesn't work by now because it uses Runge-Kutta methods not implemented in the current release of jsxgraph. With the next release this example should work.~~

	~~--[[User:Michael\|Michael]] 18:12, 29 May 2009 (UTC)~~

	~~----~~

	The Lotka-Volterra equations, a.k.a. the predator-prey equations, are a pair of non-linear differential equations mainly used to describe interaction of two biological species, one a predator and one a prey. The equations were developed independently by Alfred J. Lotka and Vito Volterra.		The Lotka-Volterra equations, a.k.a. the predator-prey equations, are a pair of non-linear differential equations mainly used to describe interaction of two biological species, one a predator and one a prey. The equations were developed independently by Alfred J. Lotka and Vito Volterra.

← Older revision		Revision as of 13:08, 3 March 2021
Line 26:		Line 26:
	board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-1.5, 28.5, 28.5, -1.5], axis: true, grid: false});		board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-1.5, 28.5, 28.5, -1.5], axis: true, grid: false});

	s = board.~~createElement~~('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});		s = board.create('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});
	st = board.~~createElement~~('text', [20,25, "Birth rate prey"], {fixed:true});		st = board.create('text', [20,25, "Birth rate prey"], {fixed:true});
	u = board.~~createElement~~('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});		u = board.create('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});
	ut = board.~~createElement~~('text', [20,23, "Death rate predators"], {fixed:true});		ut = board.create('text', [20,23, "Death rate predators"], {fixed:true});

	o = board.~~createElement~~('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});		o = board.create('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});
	ot = board.~~createElement~~('text', [10,25, "Death rate preys/per predator"], {fixed:true});		ot = board.create('text', [10,25, "Death rate preys/per predator"], {fixed:true});
	p = board.~~createElement~~('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});		p = board.create('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});
	pt = board.~~createElement~~('text', [10,23, "Reproduction rate pred./per prey"], {fixed:true});		pt = board.create('text', [10,23, "Reproduction rate pred./per prey"], {fixed:true});

	startpred = board.~~createElement~~('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});		startpred = board.create('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});
	startprey = board.~~createElement~~('glider', [0, 5, board.defaultAxes.y], {name:'Predators',strokeColor:'blue',fillColor:'blue'});		startprey = board.create('glider', [0, 5, board.defaultAxes.y], {name:'Predators',strokeColor:'blue',fillColor:'blue'});

	var g3 = null;		var g3 = null;
Line 83:		Line 83:
	}		}

	g3 = board.~~createElement~~('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});		g3 = board.create('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});
	g3.updateDataArray = function() {		g3.updateDataArray = function() {
	var data = ode();		var data = ode();
Line 93:		Line 93:
	}		}
	}		}
	g4 = board.~~createElement~~('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});		g4 = board.create('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});
	g4.updateDataArray = function() {		g4.updateDataArray = function() {
	var data = ode();		var data = ode();
Line 111:		Line 111:

	// Define sliders to dynamically change parameters of the equations and create text elements to describe them		// Define sliders to dynamically change parameters of the equations and create text elements to describe them
	s = board.~~createElement~~('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});		s = board.create('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});
	st = board.~~createElement~~('text', [20,25, "Birth rate predators"], {fixed:true});		st = board.create('text', [20,25, "Birth rate predators"], {fixed:true});
	u = board.~~createElement~~('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});		u = board.create('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});
	ut = board.~~createElement~~('text', [20,23, "Death rate predators"], {fixed:true});		ut = board.create('text', [20,23, "Death rate predators"], {fixed:true});

	o = board.~~createElement~~('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});		o = board.create('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});
	ot = board.~~createElement~~('text', [10,25, "Death rate preys/per predator"], {fixed:true});		ot = board.create('text', [10,25, "Death rate preys/per predator"], {fixed:true});
	p = board.~~createElement~~('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});		p = board.create('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});
	pt = board.~~createElement~~('text', [10,23, "Reproduction rate pred./per prey"], {fixed:true});		pt = board.create('text', [10,23, "Reproduction rate pred./per prey"], {fixed:true});

	// Dynamic initial value as gliders on the y-axis		// Dynamic initial value as gliders on the y-axis
	startpred = board.~~createElement~~('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});		startpred = board.create('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});
	startprey = board.~~createElement~~('glider', [0, 5, board.defaultAxes.y], {name:'Predators',strokeColor:'blue',fillColor:'blue'});		startprey = board.create('glider', [0, 5, board.defaultAxes.y], {name:'Predators',strokeColor:'blue',fillColor:'blue'});


Line 183:		Line 183:

	// Plot Predator		// Plot Predator
	g3 = board.~~createElement~~('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});		g3 = board.create('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});
	g3.updateDataArray = function() {		g3.updateDataArray = function() {
	var data = ode();		var data = ode();
Line 195:		Line 195:

	// Plot Prey		// Plot Prey
	g4 = board.~~createElement~~('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});		g4 = board.create('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});
	g4.updateDataArray = function() {		g4.updateDataArray = function() {
	var data = ode();		var data = ode();

← Older revision		Revision as of 10:59, 7 August 2013
Line 27:		Line 27:

	s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});		s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});
	st = board.createElement('text', [20,25, "Birth rate predators"]);		st = board.createElement('text', [20,25, "Birth rate predators"], {fixed:true});
	u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});		u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});
	ut = board.createElement('text', [20,23, "Death rate predators"]);		ut = board.createElement('text', [20,23, "Death rate predators"], {fixed:true});

	o = board.createElement('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});		o = board.createElement('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});
	ot = board.createElement('text', [10,25, "Death rate preys/per predator"]);		ot = board.createElement('text', [10,25, "Death rate preys/per predator"], {fixed:true});
	p = board.createElement('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});		p = board.createElement('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});
	pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);		pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"], {fixed:true});

	startpred = board.createElement('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});		startpred = board.createElement('glider', [0, 10, board.defaultAxes.y], {name:'Preys',strokeColor:'red',fillColor:'red'});
Line 112:		Line 112:
	// Define sliders to dynamically change parameters of the equations and create text elements to describe them		// Define sliders to dynamically change parameters of the equations and create text elements to describe them
	s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});		s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'ε1',strokeColor:'black',fillColor:'black'});
	st = board.createElement('text', [20,25, "Birth rate predators"]);		st = board.createElement('text', [20,25, "Birth rate predators"], {fixed:true});
	u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});		u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'ε2',strokeColor:'black',fillColor:'black'});
	ut = board.createElement('text', [20,23, "Death rate predators"]);		ut = board.createElement('text', [20,23, "Death rate predators"], {fixed:true});

	o = board.createElement('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});		o = board.createElement('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'γ1',strokeColor:'black',fillColor:'black'});
	ot = board.createElement('text', [10,25, "Death rate preys/per predator"]);		ot = board.createElement('text', [10,25, "Death rate preys/per predator"], {fixed:true});
	p = board.createElement('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});		p = board.createElement('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'γ2',strokeColor:'black',fillColor:'black'});
	pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);		pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"], {fixed:true});

	// Dynamic initial value as gliders on the y-axis		// Dynamic initial value as gliders on the y-axis