Math 171 - Alexiades
                            HW 1
        Exploring the logistic function, ON PAPER

The logistic function: P(t) = K / (1+e−rt),   −∞ < t < +∞ , with growth rate r and carrying capacity K>0,
arises in many physical situations, such as population dynamics, resource extraction, market share, and many others,
where a quantity increases fast initially but eventually saturates to a maximum value K (called "carrying capacity").
It is the (exact) solution of the logistic differential equation:   dP/dt = r P (1 − P/K) .

For simplicity, consider the case r=1, K=1, so   P(t) = 1 / (1+e−t), solution of the DE   dP/dt = P (1 − P).

Do the following by hand, on paper.

1  (a) Find the limits of P(t) as t tends to ±∞ .
  (b) What is the range of the function P(t) ?
  (c) Sketch P(t) roughly, by hand.

2  (a) Find the derivative Q(t) = dP(t)/dt .
  (b) Show that Q(t) > 0 for all t. What does this tell us about P(t) ?
  (c) Find the limits of Q(t) at t tends to ±∞
  (d) What is its maximum, M = max Q(t) ? When does it occur ?
  (e) What is the range of Q(t) ?
  (f) Show that Q(t) is symmetric about the y-axis, i.e. Q(−t) = Q(t) .
  (g) Sketch Q(t), by hand. It is a nice shape!
  (h) Show that Q = P(1−P) , so in fact P(t) solves the logistic ODE: dP/dt = P(1−P), as mentioned above.

3   Since P(t) is strictly increasing, it is invertible, i.e. for each p in the range, there exists unique t such that P(t) = p.
  The inverse function t(p) of P(t) is known as the logit function.
  (a) Find the formula for the logit function t(p), and specify for what p it exists.

4   For a bell-shaped curve like Q(t), or a pulse, there are various ways of characterizing how "wide" the pulse shape is.
  One widely used way is to specify the Full Width at Half Maximum (FWHM).
  Here, we have the formula for Q(t), so can find it exactly.
  (a) Find the exact FWHM of Q(t), by solving the equation: Q(t) = M/2 for t exactly.
  (b) Evaluate it using a calculator or Matlab.
  But in general we'll only have data values, no formula, so finding FWHM would need to be done numerically.

5   Think about how FWHM can be found if we know only discrete values of a pulse-like quantity like q = Q(t),
  i.e. from the pairs (ti , qi), i=1:N, which are stored in two arrays t and q.
  Here is a strategy, write an algorithm for it, ON PAPER (will be done on matlab in the next lab).
  • Need to find the maximum value M (height of the peak), and the index where it occurs, call it ipeak ,
  • Set the cutoff level at half maximum:  cutoff = M/2.
  • Find the first time ti (before ipeak) at which qi ≥ cutoff. Call it tLeft.
  • Find the first ti (after ipeak) at which qi ≤ cutoff. Call it tRight.
  • Then FWHM = tRight − tLeft .