Question

Write an equation or function that models the number of bacteria, y, in terms of time, x.

Answer 1

I can give several standard models; which one you use depends on how the bacteria reproduce (continuously vs in discrete steps) and whether growth is unlimited or limited by resources. If you give an initial amount and a growth rate (or doubling time or a later measurement), I can make a specific formula. Meanwhile, here are the common options:

1) Continuous exponential growth (unlimited resources):
y = y0 e^{k x}
- y0 = number at time x = 0
- k = continuous growth rate (per unit time)
- If you know a measurement y1 at time x1, then k = (1/x1) ln(y1/y0).
- If you know doubling time T_d, then y = y0 2^{x/T_d} or equivalently k = ln 2 / T_d.

2) Discrete (reproduces in generations/intervals):
y = y0 r^{n} or y = y0 (1 + p)^{x}
- r (or 1 + p) is the factor per time interval (n or x is number of intervals).
- Example: doubles every 3 hours → y = y0 2^{x/3} (x in hours).

3) Logistic growth (limited carrying capacity L):
y = L / (1 + A e^{-k x}), where A = (L - y0)/y0
- Use when resources eventually limit growth; L is the carrying capacity.

If you give initial bacteria y0 and either k, r, doubling time, or a second data point, I will write the explicit function for you.