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Applications of the logistic equation in science

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From: veritasium

The logistic map, a seemingly simple equation, demonstrates profound connections between various phenomena, including a dripping faucet, the Mandelbrot set, a population of rabbits, thermal convection in a fluid, and the firing of neurons in the brain [00:00:01].

The Logistic Map Explained

The logistic map is defined by the equation: Xn+1 = R * Xn * (1 - Xn) [00:01:16]. This equation models a population (X) as a percentage of its theoretical maximum (ranging from 0 to 1), where Xn is the current population and Xn+1 is the population in the next time period [00:01:00], [00:01:16]. The variable ‘R’ represents the growth rate [00:00:41].

Initially, a simple model might multiply the population by a growth rate ‘R’, leading to exponential growth forever [00:00:37], [00:00:49]. The term (1 - X) is added to introduce a negative feedback loop, representing environmental constraints that limit growth as the population approaches its maximum [00:00:52], [00:01:08], [00:01:32]. Graphing the next year’s population versus the current year’s population reveals an inverted parabola [00:01:21], [00:01:27].

For certain values of ‘R’, the population stabilizes at an equilibrium value, largely independent of the initial population [00:02:37], [00:02:50], [00:03:04]. If R is below 1, the population eventually goes extinct [00:03:27], [00:03:31]. As R increases above 1, the equilibrium population also increases [00:03:57], [00:04:02].

Period Doubling and Chaos

A fascinating phenomenon occurs when ‘R’ surpasses 3: the equilibrium value splits into two [00:04:11], [00:04:15]. Instead of stabilizing at a single constant value, the population oscillates between two values (a “period two” cycle) [00:04:20], [00:04:27]. As ‘R’ continues to increase, each branch splits again, leading to cycles of four values, then eight, sixteen, thirty-two, and so on [00:04:45], [00:04:49], [00:05:08]. These are known as “period doubling bifurcations” [00:05:00].

At approximately R = 3.57, the system enters a chaotic state [00:05:11], [00:05:15]. The population never settles down, bouncing around as if at random [00:05:19]. This equation was one of the first methods for generating pseudo-random numbers on computers, producing unpredictable results from a deterministic machine [00:05:21], [00:05:39], [00:05:41].

Surprisingly, as ‘R’ further increases, “windows of stable periodic behavior” emerge from the chaos [00:05:46], [00:05:52]. For instance, at R = 3.83, a stable cycle with a period of three years appears [00:05:54], [00:05:57]. This then undergoes its own period doubling, leading to cycles of 6, 12, 24, and so on, before returning to chaos [00:06:03], [00:06:07]. The logistic map contains periods of every possible length [00:06:10].

Connection to the Mandelbrot Set

The bifurcation diagram, which plots the long-term behavior of the logistic map as ‘R’ varies, exhibits fractal properties, with large-scale features repeating on smaller scales [00:06:23], [00:06:25]. Intriguingly, this diagram is actually part of the Mandelbrot set [00:06:40], [00:06:43].

The Mandelbrot set is based on the iterated equation Zn+1 = Zn2 + C, where ‘C’ is a complex number [00:06:52], [00:06:55], [00:06:57]. A number ‘C’ is part of the Mandelbrot set if the value of Z remains finite after unlimited iterations, starting with Z=0 [00:07:00], [00:07:04], [00:07:06]. When the Mandelbrot set’s iterated equation is plotted in 3D, showing the iterated values on the Z-axis, looking at it from the side reveals the bifurcation diagram [00:08:25], [00:08:31], [00:08:38], [00:08:42].

This connection reveals that:

Real-World Applications

Beyond population modeling, the logistic map applies to a vast range of scientific fields:

The Feigenbaum Constant and Universality

Physicist Mitchell Feigenbaum discovered a universal constant by examining the ratio of the widths of successive bifurcation sections in the logistic map [00:14:32], [00:14:35]. This ratio converges to approximately 4.669, now known as the Feigenbaum constant [00:14:38], [00:14:40], [00:14:44], [00:14:46]. This constant is considered a fundamental constant of nature, unrelated to other known physical constants [00:14:59], [00:15:00], [00:15:03], [00:15:06].

Even more remarkably, this universality means that any iterated equation with a single “hump” (like Xn+1 = sin(Xn)) will exhibit the same period-doubling bifurcations, and the ratio of when these bifurcations occur will approach the same Feigenbaum constant [00:15:10], [00:15:12], [00:15:16], [00:15:20], [00:15:22], [00:15:28], [00:15:33], [00:15:40]. This universality suggests a deep, fundamental process underlying the emergence of complex behavior from simple systems [00:15:51], [00:15:55].

Educational Significance

Biologist Robert May’s 1976 paper in Nature about the logistic equation sparked a revolution in understanding complex systems [00:16:04], [00:16:07], [00:16:11]. May advocated for teaching students about this simple equation to develop an intuition for how simple rules can generate very complex behaviors [00:16:18], [00:16:20], [00:16:24], [00:16:26], [00:16:29]. This approach challenges traditional teaching methods that often focus solely on simple equations leading to simple outcomes [00:16:36], [00:16:39], [00:16:42].

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