Discrete Logistic Growth
April 24, 2023
Discrete Logistic Map
Question: What does it mean to have a solution to this equation, and how many solutions exist?
You might be used to solutions to algebraic equations, such as
\[ax^2 + bx + c = 0\]
Discuss: What does a solution to an algebraic equation look like?
Answer: In this case, the solution to the
quadratic equation is zero, one or two points given by: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
Discrete Logistic Map
Discuss: What does it mean to have a solution to this equation, and how many solutions exist?
Answer: In this case, there are
infinitely many solutions, sometimes referred to as afamily of solutions , because they all share common traits.
You get unique solutions for this equation when you specify a starting value:
\[P_{0} = \mathrm{a \ constant}\]
Note: There are some solutions that don’t make biological sense!
Discrete Logistic Map
A unique solution to a discrete map is an infinite sequence of points:
\[P_{0}, P_{1}, P_{2}, P_{3}, \ldots\]
Let’s simulate the equation using Desmos:
Discrete Logistic Map
Small Population
\[P_{t} \ll K \ \ \textrm{(much smaller than)}\]
\[\frac{P_{t}}{K} \ll 1\]
\[\left(1-\frac{P_{t}}{K}\right) \approx 1\]
\[P_{t+1} \approx (1+r)P_{t} \ \ (\textrm{Exponential growth})\]
Small Population
\[ \begin{align} P_{t+1} & = (1+r)P_{t} \\ & = (1+r)(1+r)P_{t-1}, \ \textrm{since} \ P_{t} = (1+r)P_{t-1} \\ & = (1+r)^2P_{t-1} \\ & = (1+r)^3P_{t-2}, \ \textrm{since} \ P_{t-1} = (1+r)P_{t-2} \\ & = (1+r)^?P_{0} \\ & = (1+r)^{t+1}P_{0} \end{align} \]
We have an explicit solution (exponential function)!!!
\[P_{t} = F(t,P_{0}) = (1+r)^tP_{0}\]
What happens when \(t \rightarrow \infty\)? Depends on \(r\)!
Exponential decay when \(r < 0\) and exponential growth when \(r > 0\). What about \(r = 0\)?
Medium Population
\[P_{t} \approx \frac{K}{2} \Longrightarrow \left(1-\frac{P_{t}}{K}\right) \approx \frac{1}{2}\]
\[P_{t+1} \approx P_{t} + \frac{r}{2}P_{t} = \left(1+\frac{r}{2}\right)P_{t}\]
“Large” Population
\[P_{t} \approx K \Longrightarrow \left(1-\frac{P_{t}}{K}\right) \approx 0\]
\[P_{t+1} \approx P_{t} + rP_{t}\cdot 0 = P_{t} = K\]
\(K\) is called a fixed point.
Discrete Logistic Map
There are a few ways to study a discrete map like this.
- Simulate in time using a computer
- Pro: Easy to do; Con: Not rigorous
- Explicitly solve the equation, i.e. convert to a form \(P_{t} = F(t, P_{0})\)
- Pro: If exists, know everything; Con: Very few models can be explicitly solved.
- Study the behavior of solutions (without solving them!)
For the discrete logistic model, even though it is relatively simple, there does not exist an explicit solution!
We’ve simulated using a computer - what about exploring behavior of solutions?? Dynamical Systems Theory!!
Equilibrium Solutions
In dynamical systems theory, one of the most important questions is:
Question: What happens to solutions as \(t \rightarrow \infty\)?
We saw this in the simulations - there is a transient period (\(t < \infty\)) where the solution seems to converge to an equilibrium (\(t \rightarrow \infty\)).
Fixed Points
The simplest example of equilibrium solutions are known as fixed points. Let’s explore this idea through the simulations.
The Cobweb Plot
All Together Now
Finding Fixed Points
\[ \begin{align} P_{t+1} & = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ P_{t+1} & = P_{t} \end{align} \]
\(\Longrightarrow\)
\[ \begin{align} P_{t} & = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ 0 & = rP_{t}\left(1 - \frac{P_{t}}{K}\right) \\ \end{align} \]
\(\Longrightarrow \ P_{t} = 0\) (or) \(\ 1 - \frac{P_{t}}{K} = 0 \ \Longrightarrow \ P_{t} = 0\) (or) \(\ P_{t} = K\).
Stability of Fixed Points
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 calledunstable if all small deviations from the fixed point DO NOT converge back to the fixed point as \(t\rightarrow\infty\).
Stability of Logistic Map
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
\[ \begin{align} & \left|F^{\prime}(P_{\infty})\right| < 1 \\ \Longrightarrow & \left|1+r-\frac{2r}{K}P_{\infty}\right| < 1 \end{align} \]
Stability of Logistic Map
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
\[ \begin{align} & \ \left|1+r\right| < 1 \\ \Longrightarrow & \ -1 < 1+r < 1 \\ \Longrightarrow & \ -2 < r < 0 \end{align} \]
Stability of Logistic Map
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
\[ \begin{align} & \ \left|1+r-2r\right| < 1 \\ \Longrightarrow & \ \left|1-r\right| < 1 \\ \Longrightarrow & \ \left|r-1\right| < 1 \\ \Longrightarrow & \ -1 < r-1 < 1 \\ \Longrightarrow & \ 0 < r < 2 \\ \end{align} \]
Bifurcation Diagram: Logistic Map
Note: \(r\) values off by 1!!
Bifurcation Diagram: Logistic Map
Note: \(r\) values off by 1!!