Difference between revisions of "Lotka-Volterra equations"

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(New page: 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. --~~~~ ---- <js...)
 
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----
 
----
  
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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.
 +
 +
==Model==
 +
<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:
 +
::{| border="1" ||
 +
|-
 +
| <math>N_1 = N_1(t)</math> || Number of preys
 +
|-
 +
| <math>\epsilon_1>0</math> || Reproduction rate of prey without distortion and with enough food supply
 +
|-
 +
| <math>N_2 = N_2(t)</math> || Number of predators
 +
|-
 +
| <math>\epsilon_2>0</math> || Death rate of predators if no prey available
 +
|-
 +
| <math>\gamma_1>0</math> || Eating rate of predator per prey (equals death rate of prey per predator)
 +
|-
 +
| <math>\gamma_2>0</math> || Reproduction rate of predator per prey
 +
|}
 +
 +
 +
==Plot==
  
 
<jsxgraph width="600" height="600">
 
<jsxgraph width="600" height="600">
Line 57: Line 80:
 
     }
 
     }
 
      
 
      
     function createCurve() {
+
     var data = ode();
 +
    var t = [];
 +
    var dataprey = [];
 +
    var datapred = [];
 +
    for(var i=0; i<data.length; i++) {
 +
        t[i] = data[i][2];
 +
        datapred[i] = data[i][0];
 +
        dataprey[i] = data[i][1];
 +
    }
 +
       
 +
    g3 = board.createElement('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});
 +
    g3.updateDataArray = function() {
 
         var data = ode();
 
         var data = ode();
 +
        this.dataX = [];
 +
        this.dataY = [];
 +
        for(var i=0; i<data.length; i++) {
 +
            this.dataX[i] = t[i];
 +
            this.dataY[i] = data[i][0];
 +
        }
 +
    }
 +
    g4 = board.createElement('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});
 +
    g4.updateDataArray = function() {
 +
        var data = ode();
 +
        this.dataX = [];
 +
        this.dataY = [];
 +
        for(var i=0; i<data.length; i++) {
 +
            this.dataX[i] = t[i];
 +
            this.dataY[i] = data[i][1];
 +
        }
 +
    }
 +
</jsxgraph>
 +
 +
==Source code==
 +
<source lang="javascript">
 +
    // Initialise board
 +
    board = JXG.JSXGraph.initBoard('jxgbox', {originX: 40, originY: 560, unitX: 20, unitY: 20, axis: false, 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
 +
    s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'&epsilon;1'});
 +
    st = board.createElement('text', [20,25, "Birth rate predators"]);
 +
    u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'&epsilon;2'});
 +
    ut = board.createElement('text', [20,23, "Death rate predators"]);
 +
 +
    o = board.createElement('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'&gamma;1'});
 +
    ot = board.createElement('text', [10,25, "Death rate preys/per predator"]);
 +
    p = board.createElement('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'&gamma;2'});
 +
    pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);
 +
 +
    // Dynamic initial value as gliders on the y-axis
 +
    startpred = board.createElement('glider', [0, 10, yax], {name:'Preys'});
 +
    startprey = board.createElement('glider', [0, 5, yax], {name:'Predators'});
 +
 +
 +
    // Variables for the JXG.Curves
 +
    var g3 = null;
 +
    var g4 = null;
 +
 +
    // Initialise ODE and solve it with JXG.Math.Numerics.rungeKutta()
 +
    function ode() {
 +
        // evaluation interval
 +
        var I = [0, 25];
 +
        // Number of steps. 1000 should be enough
 +
        var N = 1000;
 +
 +
        // Right hand side of the ODE dx/dt = f(t, x)
 +
        var f = function(t, x) {
 +
            var bpred = s.Value();//0.3;
 +
            var bprey = u.Value();//0.7;
 +
            var dpred = o.Value();//0.1;
 +
            var dprey = p.Value();//0.3;
 +
 +
            var y = [];
 +
            y[0] = x[0]*(bpred - dpred*x[1]);
 +
            y[1] = -x[1]*(bprey - dprey*x[0]);
 +
 +
            return y;
 +
        }
 +
 +
        // Initial value
 +
        var x0 = [startpred.Y(), startprey.Y()];
 +
 +
        // Solve ode
 +
        var data = JXG.Math.Numerics.rungeKutta(JXG.Math.Numerics.predefinedButcher.Euler, x0, I, N, f);
 +
 +
        // to plot the data against time we need the times where the equations were solved
 
         var t = [];
 
         var t = [];
         var dataprey = [];
+
         var q = I[0];
         var datapred = [];
+
         var h = (I[1]-I[0])/N;
 
         for(var i=0; i<data.length; i++) {
 
         for(var i=0; i<data.length; i++) {
             t[i] = data[i][2];
+
             data[i].push(q);
            datapred[i] = data[i][0];
+
             q += h;
             dataprey[i] = data[i][1];
 
 
         }
 
         }
          
+
 
         g3 = board.createElement('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});
+
        return data;
        g3.updateDataArray = function() {
+
    }
            var data = ode();
+
   
            this.dataX = [];
+
    // get data points
            this.dataY = [];
+
    var data = ode();
            for(var i=0; i<data.length; i++) {
+
 
                this.dataX[i] = t[i];
+
    // copy data to arrays so we can plot it using JXG.Curve
                this.dataY[i] = data[i][0];
+
    var t = [];
            }
+
    var dataprey = [];
 +
    var datapred = [];
 +
    for(var i=0; i<data.length; i++) {
 +
        t[i] = data[i][2];
 +
         datapred[i] = data[i][0];
 +
         dataprey[i] = data[i][1];
 +
    }
 +
   
 +
    // Plot Predator
 +
    g3 = board.createElement('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});
 +
    g3.updateDataArray = function() {
 +
        var data = ode();
 +
        this.dataX = [];
 +
        this.dataY = [];
 +
        for(var i=0; i<data.length; i++) {
 +
            this.dataX[i] = t[i];
 +
            this.dataY[i] = data[i][0];
 
         }
 
         }
        g4 = board.createElement('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});
+
    }
        g4.updateDataArray = function() {
+
 
            var data = ode();
+
    // Plot Prey
            this.dataX = [];
+
    g4 = board.createElement('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});
            this.dataY = [];
+
    g4.updateDataArray = function() {
            for(var i=0; i<data.length; i++) {
+
        var data = ode();
                this.dataX[i] = t[i];
+
        this.dataX = [];
                this.dataY[i] = data[i][1];
+
        this.dataY = [];
            }
+
        for(var i=0; i<data.length; i++) {
 +
            this.dataX[i] = t[i];
 +
            this.dataY[i] = data[i][1];
 
         }
 
         }
 
     }
 
     }
   
+
</source>
    createCurve();
 
</jsxgraph>
 

Revision as of 20:58, 29 May 2009

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.

--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.

Model

[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:

[math]N_1 = N_1(t)[/math] Number of preys
[math]\epsilon_1\gt0[/math] Reproduction rate of prey without distortion and with enough food supply
[math]N_2 = N_2(t)[/math] Number of predators
[math]\epsilon_2\gt0[/math] Death rate of predators if no prey available
[math]\gamma_1\gt0[/math] Eating rate of predator per prey (equals death rate of prey per predator)
[math]\gamma_2\gt0[/math] Reproduction rate of predator per prey


Plot

Source code

    // Initialise board
    board = JXG.JSXGraph.initBoard('jxgbox', {originX: 40, originY: 560, unitX: 20, unitY: 20, axis: false, 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
    s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'&epsilon;1'});
    st = board.createElement('text', [20,25, "Birth rate predators"]);
    u = board.createElement('slider', [[20.0,24.0],[25.0,24.0],[0.0,0.7,1.0]],{name:'&epsilon;2'});
    ut = board.createElement('text', [20,23, "Death rate predators"]);

    o = board.createElement('slider', [[10.0,26.0],[15.0,26.0],[0.0,0.1,1.0]],{name:'&gamma;1'});
    ot = board.createElement('text', [10,25, "Death rate preys/per predator"]);
    p = board.createElement('slider', [[10.0,24.0],[15.0,24.0],[0.0,0.3,1.0]],{name:'&gamma;2'});
    pt = board.createElement('text', [10,23, "Reproduction rate pred./per prey"]);

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


    // Variables for the JXG.Curves
    var g3 = null;
    var g4 = null;

    // Initialise ODE and solve it with JXG.Math.Numerics.rungeKutta()
    function ode() {
        // evaluation interval
        var I = [0, 25];
        // Number of steps. 1000 should be enough
        var N = 1000;

        // Right hand side of the ODE dx/dt = f(t, x)
        var f = function(t, x) {
            var bpred = s.Value();//0.3;
            var bprey = u.Value();//0.7;
            var dpred = o.Value();//0.1;
            var dprey = p.Value();//0.3;

            var y = [];
            y[0] = x[0]*(bpred - dpred*x[1]);
            y[1] = -x[1]*(bprey - dprey*x[0]);

            return y;
        }

        // Initial value
        var x0 = [startpred.Y(), startprey.Y()];

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

        // to plot the data against time we need the times where the equations were solved
        var t = [];
        var q = I[0];
        var h = (I[1]-I[0])/N;
        for(var i=0; i<data.length; i++) {
            data[i].push(q);
            q += h;
        }

        return data;
    }
    
    // get data points
    var data = ode();

    // copy data to arrays so we can plot it using JXG.Curve
    var t = [];
    var dataprey = [];
    var datapred = [];
    for(var i=0; i<data.length; i++) {
        t[i] = data[i][2];
        datapred[i] = data[i][0];
        dataprey[i] = data[i][1];
    }
    
    // Plot Predator
    g3 = board.createElement('curve', [t, datapred], {strokeColor:'red', strokeWidth:'2px'});
    g3.updateDataArray = function() {
        var data = ode();
        this.dataX = [];
        this.dataY = [];
        for(var i=0; i<data.length; i++) {
            this.dataX[i] = t[i];
            this.dataY[i] = data[i][0];
        }
    }

    // Plot Prey
    g4 = board.createElement('curve', [t, dataprey], {strokeColor:'blue', strokeWidth:'2px'});
    g4.updateDataArray = function() {
        var data = ode();
        this.dataX = [];
        this.dataY = [];
        for(var i=0; i<data.length; i++) {
            this.dataX[i] = t[i];
            this.dataY[i] = data[i][1];
        }
    }