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Logistic sequences are a class of sequences that possess a very simple recursive definition, yet exhibit a wide variety of end behaviors. Furthermore, they can be applied to population studies. The recursive definition of a logistic sequence is:
\begin{equation*} a_n=ra_{n-1}(1-a_{n-1}) \end{equation*}Let us begin with the logistic sequence $a_n=2a_{n-1}(1-a_{n-1})$. A recursive equation does not completely define a sequence, since a starting value is needed. Looking at the first few terms for a few different initial values, we get:
Although it is not always apparent what particular end behavior will occur from just the first few terms, we should already begin to see patterns developing. We might be able to reinforce our hunches by looking at the first 100 terms for each sequence. But to confirm our conclusions, another approach is needed. The terms at the beginning of a sequence (or even in the middle) are never sufficient to determine the end behavior of the sequence. Other means are needed. But once we have those other approaches, if we examined all possible initial values and their end behaviors, we would find that we could classify the possible end behaviors of this particular logistic sequence into three categories:
In this particular example, two of the three categories involve convergence to a particular limit point, and the third category diverges. But these are only two of the four different possible end behaviors for a logistic sequence.
Let us now choose a specific initial value, and look at the effect on changing the parameter $r$ in the equation. Mathematically, $r$ could be any real number. In population studies, $r$ is typically restricted so as to satisfy the inequality $0\le r\le 4$. We shall begin by using the initial value $a_1=0.4$. We get:
Just a brief look at these examples suggests other end behaviors are possible. When $r$ is 0, 0.8, 1.6, or 2.4, the terms of the sequence do appear to be converging, though not to the same limit point. When $r=3.2$, the sequence appears to be jumping between two values which are slowly receding from one another. And there is no apparent pattern in the sequence when $r=4$, in spite of the fact that the values were produced by a very specific rule. These preliminary observations will, in fact, turn out to be accurate.
When used in studies of populations, the logistic sequence gives the percentage of saturation of a population in a particular habitat. In other words, if $a_n=0.76$ at some point in time, then the population has reached 76% of the carrying capacity of that habitat. By this definition, the population can never exceed 100% (and if it did, the equation would predict that a negative population would follow, which would be immediate extinction). Adjusting the parameter $r$ yields different population growth scenarios. As long as the parameter remains in the interval $0 < r < 4$, the predictions made by the equation are at least possible values in a population. When the parameter $r$ leaves that interval, the equation will predict negative populations at some point in time.
The end behaviors of the logistic sequence can be classified into four types.
It is a much harder problem to classify the end behaviors of all logistic sequences. We have already seen that a single parameter value can produce more than one type of end behavior, depending on the initial value $a_1$. And that a single initial value can produce more than one type of end behavior, depending on the parameter $r$. But there are some easy cases.
When any sequence is converging to a particular value, then the difference in the terms of the sequence, $a_n-a_{n-1}$, must be going to zero. In other words, $a_n \approx a_{n-1}$. If we assumed that they did eventually become equal, so $a_n=a_{n-1}$, then we could solve the resulting equation for $a_n$.
When a sequence is defined recursively, this turns out to be a very simple situation. In a recursive definition, $a_{n-1}$ is used as the input into the formula, and $a_n$ is the output. If we wrote this in function notation, then $a_n=f(a_{n-1})$. But since $a_n=a_{n-1}$, we actually have $a_n=f(a_n)$, or equivalently $x=f(x)$. Solutions of the equation $x=f(x)$ are called fixed points of the function, and they will be the limit points of a convergent sequence.
So what are the fixed points of a logistic sequence? Setting up the fixed point equation $x=f(x)$, we get $x=rx(1-x)$. Expanding this will produce the quadratic equation $rx^2+(1-r)x=0$, which has two solutions, $x=0$ and $x=\dfrac{r-1}{r}$.
The case $x=0$ is for a specific term $a_n=0$. A quick look at the recursive definition of a logistic sequence confirms that any sequence having zero as a term will have zero for every successive term, and thus the sequence would converge to zero. Also, any logistic sequence with initial value $a_1=0$ will also be constantly zero.
The case $x=\dfrac{r-1}{r}$ is more interesting. It says that if a logistic sequence has parameter $r$, and at some point in the sequence there is a term $a_n=\dfrac{r-1}{r}$, then every subsequent term of the sequence will have the same value. In the examples we previously examined, most of the sequences only approached their limit point, they never reached it. But if we could begin the sequence with the initial value $a_1=\dfrac{r-1}{r}$, and then every subsequent term would have the same value. In other words, for every parameter $r$, there are at least two initial values which produce convergent (and actually constant) sequences.
Using the fixed point as the initial value of a logistic sequence will always produce a convergent sequence. But what if we choose another value as the initial value? Will the sequence converge if the initial value is close enough to the fixed point? Maybe. If we investigate logistic sequences whose initial values are only close to the fixed point, we will discover two different cases. Some fixed points will attract sequences, and some will repel them. For example:
Whether a fixed point is attractive or repelling depends on the slope of the function $f(x)=rx(1-x)$ when $x$ is the fixed point. Jumping directly to the conclusion (without proof) for logistic sequences, $x=0$ will be an attractive fixed point for $-1 \le r < 1$, and $x=\dfrac{r-1}{r}$ will be an attractive fixed point for $1 < x \le 3$. Logistic sequences with any other parameter values will not have an attractive fixed point, so the only convergent behavior they can exhibit is if the sequence is constant, which occurs when the initial value is equal to the fixed point itself.
When the parameter satisfies the restriction $3 < r < 1+\sqrt{6}$, with an appropriate initial value the logistic sequence will produce periodic end behavior, cycling between two numbers. These two values are two of the four solutions of the equation $x=f(f(x))$ (the other two solutions are the fixed points). Algebraically, these values are given by $x=\dfrac{r^2+r\pm\sqrt{r^2-2r-3}}{2r^2}$.
When the parameter $r$ increases from about 3.45 to about 3.57, periodic end behavior is still exhibited, but the length of the period repeatedly doubles over this interval. Most values of the parameter $r$ between about 3.57 and 4 will produce chaotic end behavior, although there are a few isolated regions of periodicity (such as a period-three interval near $r=3.83$).
When a logistic sequence exhibits converging end behavior, or even cyclic end behavior, it will approach its limit point (or points) over a range of initial values. But the same is not true when chaotic behavior takes over. Consider the logistic sequence with parameter $r=4$. Here are the terms of the sequence for two very similar initial values.
Although the two sequences begin quite similarly, after twenty iterations they can be seen to be quite different. In other words, a small perturbation in an initial value can lead to some large differences as the sequence progresses, and one sequence may be increasing while the other is decreasing. A chaotic environment is so sensitive to its initial conditions that when Edward Lorenz, both mathematican and meteorologist, gave a presentation to scientific colleagues titled "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?", the term "butterfly effect" took root.