Volterra-Lotka Fractals

Nice Image


In The Beauty of Fractals, these images are offered as an example of "how an apparently innocent system of differential equations gives rise to unimaginably rich and complex behavior after discretization." The Volterra-Lotka equations are a refinement of attempts to model predator- prey systems.

If x represents the prey population and y represents the predator population, their relationship can be expressed as:

    dx/dt =  Ax - Bxy = f(x,y)
    dy/dt = -Cy + Dxy = g(x,y)

According to Peitgen and Richter, "Hence, x grows at a constant rate in the absence of y, and y decays at a constant rate in the absence of x. The prey is consumed in proportion to y, and the predators expand in proportion to x." They proceed to "discretize" this system, by "mating" the Euler and Heun methods. For purposes of image computation, their formula (Equation 8.3 on page 125) can be interpreted as:

    x(new) = x + h/2 * [ f(x,y) + f[x + pf(x,y), y + pg(x,y)] ]
    y(new) = y + h/2 * [ g(x,y) + g[x + pf(x,y), y + pg(x,y)] ]

This formula can be used to plot or connect single points, starting with arbitrary values of x(0) and y(0), to produce typical "strange attractor" images such as the ones commonly derived from the Henon or Lorenz formulae. But to produce an escape-time fractal, we iterate this formula for all (x, y) pairs that we can associate with pixels on our monitor screen. The standard window is: 0.0 < x < 6.0; 0.0 < y < 4.5. Since the "unimaginably rich and complex behavior" occurs with the points that do NOT escape, the inside coloring method assumes considerable importance.

The parameters h and p can be selected between 0.0 and 1.0, and this determines the types of attractors that will result. Infinity and (1, 1) are predictable attractors. For certain combinations, an "invariant circle" (which is not strictly circular) and/or an orbit of period 9 also are attractive.

The invariant circle and periodic orbit change with each (h, p) pair, and they must be determined empirically. That process would be thoroughly impractical to implement through any kind of fixed formula. This is especially true because even when these attractors are chosen, the threshold for determining when a point is "close enough" is quite arbitrary, and yet it affects the image considerably. The best compromise in the context of a generalized formula is to use either the "zmag" or "bof60" inside coloring options. See Inside=bof60|bof61|zmag|fmod|period|atan for details. This formula performs best with a relatively high bailout value; the default is set at 256, rather than the standard default of 4.

Reference: Peitgen, H.-O. and Richter, P.H. The Beauty of Fractals, Springer-Verlag, 1986; Section 8, pp. 125-7.

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