Discrete Logistic Growth (cont'd)

Definition: Fixed points of a discrete map are denoted by \( P_{\infty} \) and satisfy the relation

\[ P_{t} = P_{\infty}, \ \textrm{for all $t$} \]

This also means \( P_{t+1} = P_{t} = P_{\infty} \). So how do we find fixed points for a discrete map?

If the discrete map is given by

\[ P_{t+1} = F(P_{t}) \]

then fixed points satisfy the equation

\[ P_{t+1} = {\bf F(P_{t}) = P_{t}} \Rightarrow F(P_{t})-P_{t} = 0 \]

In other words, fixed points simultaneously solve (algebraically) the two equations:

\[ \begin{align} P_{t+1} & = F(P_{t}) \\ P_{t+1} & = P_{t} \end{align} \]

We can visual the dynamics AND these two equations in a cobweb plot.

To do this, we will (graphically) assign \( y = P_{t+1} \) and \( x = P_{t} \) to give

\[ \begin{align} y & = F(x) \\ y & = x \end{align} \]

Let's plot these two equations.

\[ \begin{align} P_{t+1} & = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ P_{t+1} & = P_{t} \end{align} \]

\( \Longrightarrow \)

\[ \begin{align} P_{t} & = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ 0 & = rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ \end{align} \]

\( \Longrightarrow \ P_{t} = 0 \ \) (or) \( \ 1 - \frac{P_{t}}{K} = 0 \ \Longrightarrow \ P_{t} = 0 \ \) (or) \( \ P_{t} = K \).

Definition: A fixed point is called stable (asymptotically stable) if all small deviations from the fixed point converge/limit back to the fixed point as \( t\rightarrow\infty \).

A fixed point is called unstable if all small deviations from the fixed point DO NOT converge back to the fixed point as \( t\rightarrow\infty \).

Question: How do we determine the stability of fixed points?

Definition: A fixed point \( P_{\infty} \) is stable when

\[ \left|F^{\prime}(P_{\infty})\right| < 1 \]
A fixed point \( P_{\infty} \) is unstable when

\[ \left|F^{\prime}(P_{\infty})\right| > 1 \]

Given that \[ F(P_{\infty}) = P_{\infty} + rP_{\infty} - \frac{r}{K}P_{\infty}^2 \] we have \[ \begin{align} F^{\prime}(P_{\infty}) & = 1 + r - \frac{2r}{K}P_{\infty} \end{align} \] Thus, we have a stable fixed point when

\[ \begin{align} & \left|F^{\prime}(P_{\infty})\right| < 1$ \\ \Longrightarrow & \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]

Thus, we have a stable fixed point when

\[ \begin{align} & \ \left|F^{\prime}(P_{\infty})\right| < 1 \\ \Longrightarrow & \ \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]

\( P_{\infty} = 0 \) is stable when

\[ \begin{align} & \ \left|1+r\right| < 1 \\ \Longrightarrow & \ -1 < 1+r < 1 \\ \Longrightarrow & \ -2 < r < 0 \end{align} \]

Thus, we have a stable fixed point when

\[ \begin{align} & \ \left|F^{\prime}(P_{\infty})\right| < 1 \\ \Longrightarrow & \ \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]

\( P_{\infty} = K \) is stable when

\[ \begin{align} & \ \left|1+r-2r\right| < 1 \\ \Longrightarrow & \ \left|1-r\right| < 1 \\ \Longrightarrow & \ \left|r-1\right| < 1 \\ \Longrightarrow & \ -1 < r-1 < 1 \\ \Longrightarrow & \ 0 < r < 2 \\ \end{align} \]

“In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.”
- Wikipedia

Note: \( r \) values off by 1!!

Note: \( r \) values off by 1!!