International Encyclopedia of Systems and Cybernetics

Curiously, the logistic equation has been studied during more than one century and put to considerable uses, before anybody noted its strange properties for higher values of the coefficient a. (See for example A.J. LOTKA, 1956, and V. VOLTERRA's, 1931 works).

However, some aspects of populations dynamics remained unexplained by the equation and this fact was more than once attributed to some insufficient data or statistical errors.

More recently, it has been discovered that, as stated by R.V. JENSEN: "The time-evolution of x_n, generated by this single algebraic equation exhibits an extraordinary transformation from order to chaos as the parameter a, which measures the strength of the nonlinearity, is increased… In fact, the review article on the logistic map by the biologist Robert MAY (1976) is a historical milestone in the modern development of nonlinear dynamics" (1987, p.170).

This chaotic randomness of the logistic equation, once a specific threshold is crossed, is explained by J. CASTI as follows: "When we put the birth and overcrowding effects together into a single expression, what pops out is the formula for randomness" (1990, p.94).

Moreover, as over croowding results of an excess in inputs, this feature of systems becomes related to PRIGOGINE's thermodynamics of systems far-from-equilibrium, submitted to giant fluctuations.

Grounding his analysis on the quadratic form (2) of the logistic equation, JENSEN writes: "When the original population X₀ is small (much less than 1 on a normalized scale, where 1 might stand for any number, such a 1 million individuals), the nonlinear term can initially be neglected. Then the population at time-step (year) n = 1 will be approximately equal to aX₀. If a>1, the population increases. If a<1, the population decreases. Therefore, the linear term in equation (1) can be interpreted as a linear growth or death rate which by itself would lead to exponential population growth or decay. If a>1, the population will eventually grow to a value large enough for the nonlinear term –aX_n to become important. Since this term is negative, it represents a nonlinear death rate which dominates when the population becomes too large" (1987, p.170).

So far, so good.

However, while the "… graphic analysis tells us that if the normalized population starts out larger than 1, then it immediately goes negative, becoming extinct in one time-step" (Ibid.); for values of a between 0 and 4, "Conventional perturbation theory gives no hint of the existence of a nonvanishing steady state population" (p.171).

"For values of a between 1 and 3 almost all initial populations evolve to this equilibrium population" (See asymptotic stability). "Then, as a is increased between 3 and 4, the dynamics change in remarquable ways. First, the fixed point becomes unstable and the population evolves to a dynamic steady state inwhich it alternates between a large and a small population… (a period-2 cycle). For somewhat larger values of a this period-2 cycle becomes unstable and is replaced by a period-4 cycle in which the population alternates high-low, returning to its original value every four time steps. As a is increased, the long-time motion converges to period -8, -16, -32, -64 cycles, finally accumulating to a cycle of infinite period for a = a₈ 3,57" (Ibid.) In other words, at this value, chaotic stability is achieved and for higher values of a the evolution of populations "… is indistinguishable from a random process, even though the logistic map is fully deterministic in the sense that there are no "random" forces and that the future is completely determined by the initial condition, X₀•(p•171).

This is an absolutely general property of the logistic equation, for any kind of populations and is thus a general systemic property.