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In population biology, the following equation is the Pielou Logistic Equation, is used to model population with non-overlapping generations

$$x_{n+1} = \frac{\alpha x_{n}}{1+\beta x_{n}}$$

Show that

$$\lim_{n \to\infty} x_n = \begin{cases} \frac{\alpha -1}{\beta},& |\alpha| > 1,\\ 0,& \alpha = 1 \text{ or } |\alpha|<1,\\ \big\{ x_0, \frac{-x_0}{1+\beta x_0} \big\},&\alpha = -1\quad(\text{note: a two cycle}) \end{cases} $$

To be honest, I am not sure where to begin to return. I tried to use the substitution $z_n = \frac{1}{x_n}$ to transform this logistic equation into a linear equation.

Can you please give me some hints or pointers in order for me to understand what I need to do. I am sorry the notation is a little bit confusing.

I appreciate all of the help.

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  • $\begingroup$ "I tried to use the substitution $z_n = \frac{1}{x_n}$ to transform this logistic equation into a linear equation. " Did you actually try this? This is an excellent idea. And what did you find then? More precisely: what prevented you to find a full solution once you had started along this path? $\endgroup$
    – Did
    Jun 18, 2015 at 6:32

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This is a fractional linear transformation (a Möbius transformation, when considered as a map on the complex numbers). Iterating these can be done using the representation of these using $2 \times 2$ matrices. If $f(x) = \dfrac{ax+b}{cx+d}$ corresponds to the matrix $A = \pmatrix{a & b\cr c & d\cr}$, then the $n$'th iterate $f^{n}$ corresponds to the $n$'th power of the matrix: $A^n$.

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