Exponential Growth Calculator
Calculate exponential population growth: N(t) = N0 * e^(rt). Find population at any time, doubling time, or growth rate from two population censuses. Applies to bacteria, populations below K.
🌳 Ecology📐 N(t) = N0·e^(rt)🧬 Biology
Initial population N0
Growth rate r per time unit
Time t
OR find r: N at time t
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Exponential Growth Calculator | — | N(t) = N0·e^(rt) | individuals |
Step-by-Step Examples
Example 1
Bacteria growth
E. coli: N0=100, r=1.386/hr (doubles every 30 min). N after 2 hours?
- N(2) = 100 × e^(1.386×2) = 100 × e^2.772
- = 100 × 15.99 = 1599
- Doubling time = 0.693/1.386 = 0.5 hr
✓ N(2hr) = 1600 bacteria; 4 doublings
Example 2
Find r from census data
N0=500 in 2010, N=750 in 2020. Find r.
- r = ln(750/500)/10 = ln(1.5)/10 = 0.405/10 = 0.0405/yr
- Doubling time = 0.693/0.0405 = 17.1 years
✓ r = 0.0405/yr; doubling time 17 years
Example 3
Rule of 70
Population growing at r=2%/yr. Doubling time?
- td = 0.693/0.02 = 34.7 years
- Rule of 70: 70/2 = 35 years (approximate)
✓ Doubling time ≈ 35 years
Real-World Applications
Microbiology
Bacterial growth in log phase follows exponential model precisely.
Demography
Human population grew exponentially until ~1960. Now logistic due to demographic transition.
Finance
Compound interest follows same exponential equation: A = P×e^(rt).
Astrobiology
Exponential growth is one criterion used in SETI models for civilization spread.
Common Mistakes to Avoid
⚠️
r can be negative (population decline)
If deaths > births: r < 0, population declines exponentially. Same formula, N(t) decreases.
⚠️
e^r vs (1+r) convention
Continuous: N = N0×e^rt. Discrete: N = N0×(1+r)^t. For small r, essentially identical. Use continuous for biological problems.
⚠️
Exponential is only valid below K
Real populations eventually reach limits. Use logistic model when approaching carrying capacity.
Frequently Asked Questions
What is the difference between r and R0? ▾
r is intrinsic rate of increase (continuous). R0 = net reproductive rate (offspring per lifetime). Related by r = ln(R0)/T where T = generation time.
When does exponential growth apply? ▾
Early colonization, introduced species, populations far below K, pathogen growth early in infection, laboratory cultures in log phase.
What is the J-curve? ▾
Graph of exponential population growth over time. J-shaped because growth is slow initially then accelerates. Contrast with S-curve (logistic) which levels off.
Does any population grow exponentially forever? ▾
No. Resources are finite. Exponential growth always eventually gives way to logistic growth, crash, or cyclical dynamics.
What is Malthusian growth? ▾
Thomas Malthus (1798) proposed human populations grow geometrically (exponentially) while food supply grows arithmetically, predicting eventual famine and population checks.