Logistic Equation Analytic Solution

Routine Name: Logistic Equation

Author: Kyle Hovey

Language: C++

Description/Purpose:

A logistic equation is a solution \(P\) to the differential equation:

\[ \frac{dP}{dt} = \alpha P - \beta P^2 \]

The solution of which can be massaged to the form:

\[ \frac{\alpha}{(\frac{\alpha}{P_o} - \beta)e^{-\alpha t} + \beta} \ni P_o := P(0)\]

This code takes the free parameters for the logistic equation and returns a function that takes a templated time variable and returns a value of the same type at time t.

Input:

The free parameters that define the logistic equation: \( \{ \alpha, \beta, P_o \} \), as well as a type T that the function should be defined over.

Output:

A function that takes a parameter t of type T and returns the value of the specified logisic equation evaluated at t.

Usage/Example:

To use this function, you have to include it in the file you need to call it from.

#include "logistic/logistic.h"
#include <limits>
#include <iostream>

int main() {
  // Parameters for logistic equation
  const double
    alpha = 2,
    beta = 1,
    Po = 1;

  // Generate the required function
  auto logistic = genLogistic<double>(alpha, beta, Po);

  // Call it for some basic values
  for (int i = -10; i <= 10; i++) {
    std::cout << i << " -> " << logistic(i) << std::endl;
  }

  return EXIT_SUCCESS;
}

Output:

-10 -> 4.12231e-09
-9 -> 3.046e-08
-8 -> 2.2507e-07
-7 -> 1.66306e-06
-6 -> 1.22883e-05
-5 -> 9.07957e-05
-4 -> 0.0006707
-3 -> 0.00494525
-2 -> 0.0359724
-1 -> 0.238406
0 -> 1
1 -> 1.76159
2 -> 1.96403
3 -> 1.99505
4 -> 1.99933
5 -> 1.99991
6 -> 1.99999
7 -> 2
8 -> 2
9 -> 2
10 -> 2

Implementation/Code:

I made this function as abstract as possible. I probably could have gone with partial function application to bind \( \alpha \), \( \beta \), and \( P_o \), but C++ does not have a very clean way of doing this yet. Instead, I created a small factory for the logistic equation that took the free parameters and returns a function that is a logistic map.

template <typename T>
using endomorphism = std::function<T(T)>;

template <typename T>
endomorphism<T> genLogistic(T alpha, T beta, T Po) {
  auto amp = ( alpha / Po - beta);

  return endomorphism<T>(
    [=] (T t) {
      return alpha / (amp * exp(-alpha * t) + beta);
    }
  );
}