Lotka-Volterra equations

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]\displaystyle{ \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]\displaystyle{ N_1 = N_1(t) }[/math]	Number of preys
[math]\displaystyle{ \epsilon_1\gt 0 }[/math]	Reproduction rate of prey without distortion and with enough food supply
[math]\displaystyle{ N_2 = N_2(t) }[/math]	Number of predators
[math]\displaystyle{ \epsilon_2\gt 0 }[/math]	Death rate of predators if no prey available
[math]\displaystyle{ \gamma_1\gt 0 }[/math]	Eating rate of predator per prey (equals death rate of prey per predator)
[math]\displaystyle{ \gamma_2\gt 0 }[/math]	Reproduction rate of predator per prey

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:'ε1',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:'ε2',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:'γ1',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:'γ2',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];
        }
    }