Discrete Logistic Growth

Last time, we discovered a population model of logistic growth from observing the output of an agent-based simulation. The resulting model is of the form

\[ P_{t+1} = P_{t} + rP_{t}\left(1 - \frac{P_{t}}{K}\right), \]

which deterministically determines the population size in the next generation (\( P_{t+1} \)) as a function of the current population size (\( P_{t} \)). This is known as the discrete logistic map where \( r \) is the intrinsic growth rate and \( K \) is the carrying capacity.

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: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]

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 a family 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!

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:

\[ P_{t} \ll K \ \ \textrm{(much smaller than)} \]

\[ \frac{P_{t}}{K} \ll 1 \]

\[ \left(1-\frac{P_{t}}{K}\right) \approx 1 \]

\[ P_{t+1} \approx (1+r)P_{t} \ \ (\textrm{Exponential growth}) \]

\[ \begin{align} P_{t+1} & = (1+r)P_{t} \\ & = (1+r)(1+r)P_{t-1}, \ \textrm{since} \ P_{t} = (1+r)P_{t-1} \\ & = (1+r)^2P_{t-1} \\ & = (1+r)^3P_{t-2}, \ \textrm{since} \ P_{t-1} = (1+r)P_{t-2} \\ & = (1+r)^?P_{0} \\ & = (1+r)^{t+1}P_{0} \end{align} \]

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 \( r = 0 \)?

\[ P_{t} \approx \frac{K}{2} \Longrightarrow \left(1-\frac{P_{t}}{K}\right) \approx \frac{1}{2} \]

\[ P_{t+1} \approx P_{t} + \frac{r}{2}P_{t} = \left(1+\frac{r}{2}\right)P_{t} \]

\[ P_{t} \approx K \Longrightarrow \left(1-\frac{P_{t}}{K}\right) \approx 0 \]

\[ P_{t+1} \approx P_{t} + rP_{t}\cdot 0 = P_{t} = K \]

\( K \) is called a fixed point.

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!!