Growth and Populations by Computer: Quick Revision

A brief note on Growth and Populations by Computer. Read it before a test. Try each question at the end before you open its answer.

Exponential growth

• A loop doubles with N = 2 * N; the formula N0 * 2 ** (t / T) jumps to any time and works on numpy arrays.
• With a growth rate: $N = N_0 e^{rt}$, and $T = \dfrac{\ln 2}{r}$. Use np.log and np.exp, not np.log10.
• On a log axis, plt.yscale("log"), exponential growth is a straight line.

Logistic growth

• $\dfrac{dN}{dt} = rN\left(1 - \dfrac{N}{K}\right)$: nearly exponential while $N$ is small, stopping at the carrying capacity $K$.
• An Euler step: N = N + r * N * (1 - N / K) * dt. Keep dt much smaller than $\dfrac{1}{r}$.
• The S-curve grows fastest at $\dfrac{K}{2}$, at $\dfrac{rK}{4}$, and reaches it at $t = \dfrac{\ln(K/N_0 - 1)}{r}$.

Fitting

• np.polyfit(t, np.log(N), 1) gives the slope $r$ and the intercept $\ln N_0$ of exponential counts.
• curve_fit(model, t, N, p0=guess) returns the best parameters in the order of the model's arguments after t.
• A fit makes the sum of squared misses smallest.

A worked example

What does this print?

1. Hour 1. $40 + 0.5 \times 40 \times 0.9 = 58$.
2. Hour 2. $58 + 0.5 \times 58 \times 0.855 = 82.795$.
3. Hour 3. $82.795 + 0.5 \times 82.795 \times 0.793 = 115.6$.

It prints 116.

Practice with new numbers

A program to try

The program grows the same population exponentially and logistically, side by side, and prints both every two hours.

Mistakes to avoid

• ^ for a power. In Python a power is **.
• Hours and minutes mixed. Keep $t$, $T$ and $dt$ in the same unit as $r$.
• No dt in the step. Without it, each step is one whole unit of time.
• A line fitted to N. Fit the line to $\ln N$.
• The time not first. curve_fit puts the times into the model's first argument.

Check yourself

1. A culture of $25$ cells doubles every $12$ min. How many cells are there after one hour?

2. What is the doubling time for $r = 0.2$ per hour?

3. A logistic population has $K = 600$ cells. At what count is it growing fastest?

4. Why fit a straight line to $\ln N$ and not to $N$?