Difference between revisions of "Lotka-Volterra equations"

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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:'&epsilon;1',strokeColor:'black',fillColor:'black'});
 
     s = board.createElement('slider', [[20.0,26.0],[25.0,26.0],[0.0,0.3,1.0]],{name:'&epsilon;1',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'});
  
  

Revision as of 10:49, 8 June 2011

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]

Meaning of the variables:

[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', {boundingbox: [-1.5, 28.5, 28.5, -1.5], axis: true, grid: false});

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


    // 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('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];
        }
    }