From Agent-Based to Population Models: Logistic Growth (cont'd)

We will discover a population model of logistic growth from observing the output of an agent-based simulation. The resulting model should be of the form

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

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 a discrete map, which will be the focus of our future investigations.

Question: Look at the code and make sure it matches the ODD description. Is there anything that is vague or doesn’t match the description?

Question: As grid-size is increased, describe what you see in plot (A) and (B). Explain the patterns that you see.

Answer: In plot (A), we begin to see smoother and smoother versions of an S-shaped curve, which is typical of logistic growth. The curve levels off at what is known as the carrying capacity, which is at grid-size*grid-size.

Question: What type of (simple) curve fits the pattern that you see in plot (B)?

Answer: While the curve is not symmetric, a simple curve that might do reasonably well fitting this curve would be a quadratic function, or more specifically an upside-down parabola.

Question: Write down a function \( F \) for the upside-down parabola from the previous question. Note that the function satisfies the following formula

\[ \Delta P_{t} = P_{t+1}-P_{t} = F(P_{t}) \]
**Hint**: Which points does the curve have to go through?

Based on the hint, the two points that must be on the curve are \( (P_{t}, \Delta P_{t}) = (0,0) \) (no population therefore no growth) and \( (P_{t}, \Delta P_{t}) = (s^2, 0) \) (maximum population therefore no growth), where \( s = \textrm{grid-size} \). The general formula for a quadratic is

\[ F(P_{t}) = aP_{t}^2 + bP_{t} + c \]

First, plug in \( (0,0) \):

\[ \begin{align} 0 & = F(0) \\ & = a\times 0^2 + b\times 0 + c \\ & = c \end{align} \]

Thus, \( c = 0 \).

Based on the hint, the two points that must be on the curve are \( (P_{t}, \Delta P_{t}) = (0,0) \) (no population therefore no growth) and \( (P_{t}, \Delta P_{t}) = (s^2, 0) \) (maximum population therefore no growth), where \( s = \textrm{grid-size} \). The general formula for a quadratic is

\[ F(P_{t}) = aP_{t}^2 + bP_{t} \]

Now, plug in \( (P_{t}, F(P_{t})) = (s^2, 0) \):

\[ \begin{align} & \ 0 = F(s^2) = a\left(s^2\right)^2 + bs^2 \\ \Rightarrow & \ 0 = as^2 + b \ \textrm{(dividing by $s^2$ on both sides)} \\ \Rightarrow & \ b = -as^2. \end{align} \]

Thus, \( b = -as^2 \).

Thus, \( c = 0 \).

Based on the hint, the two points that must be on the curve are \( (P_{t}, \Delta P_{t}) = (0,0) \) (no population therefore no growth) and \( (P_{t}, \Delta P_{t}) = (s^2, 0) \) (maximum population therefore no growth), where \( s = \textrm{grid-size} \). The general formula for a quadratic is

\[ \begin{align} \Delta P_{t} & = F(P_{t}) \\ & = aP_{t}^2 + bP_{t} \\ & = aP_{t}^2 - as^2P_{t} \ \textrm{(since $b = -as^2$)} \\ & = aP_{t}\left(P_{t}-s^2\right) \\ \end{align} \]

In discrete map form, since \( \Delta P_{t} = P_{t+1} - P_{t} \), we arrive at:

\[ P_{t+1} = P_{t} + aP_{t}\left(P_{t}-s^2\right) \]

Note that \( a \) and \( s \) are parameters: \( s \) denotes the grid-size and \( a \) controls the height of the parabola (which is determined by growth rate, in this model given by prob-grow). Also, \( a \) must be negative (downward facing parabola), so in order to deal only with positive numbers, we will replace \( a \) with \( -|a| \) to arrive at

\[ P_{t+1} = P_{t} - |a|P_{t}\left(P_{t} - s^2\right). \]

or after some algebra

\[ P_{t+1} = P_{t} + |a|s^2 P_{t}\left(1 - \frac{P_{t}}{s^2}\right). \]

If we let \( K = s^2 \) and \( r = |a|K \), we arrive at

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