Population Growth Calculator
Calculate population size using exponential or logistic growth models. Find N(t) at any time, doubling time, or growth rate. Compare exponential (dN/dt = rN) vs logistic (with carrying capacity K).
🌳 Ecology📐 N(t) = N0·e^(rt) or logistic🧬 Biology
Initial population N0
Intrinsic growth rate r (per time unit)
Time t
Carrying capacity K (blank for exponential)
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Population Growth Calculator | — | N(t) = N0·e^(rt) or logistic | individuals |
Step-by-Step Examples
Example 1
Exponential growth
Bacteria: N0=100, r=0.5/hr. Population after 4 hours?
- N(4) = 100 × e^(0.5×4) = 100 × e^2.0
- = 100 × 7.389 = 739
- Doubling time = 0.693/0.5 = 1.39 hr
✓ N(4hr) = 739 bacteria
Example 2
Logistic growth
Deer: N0=50, r=0.3/yr, K=500. After 10 years?
- N(10) = 500/(1+(500-50)/50 × e^(-0.3×10))
- = 500/(1+9×e^-3.0) = 500/(1+0.448) = 344
✓ N(10yr) = 344 deer
Example 3
Doubling time
Human population: r=0.011/yr.
- td = 0.693/0.011 = 63 years
- World population doubles every ~63 years at this rate
✓ Doubling time = 63 years
Real-World Applications
Wildlife Management
Population models guide hunting quotas, conservation targets, and reintroduction planning.
Microbiology
Bacterial growth curves follow logistic model: lag, exponential, stationary phases.
Conservation Biology
Minimum viable population and recovery planning use growth rate models.
Epidemiology
Disease spread models are based on population growth equations (SIR models).
Common Mistakes to Avoid
⚠️
r is per capita rate, not absolute
r = birth rate - death rate per individual. Exponential: dN/dt = rN. Do not confuse with absolute number of births.
⚠️
Logistic assumes constant K
Real carrying capacities fluctuate with resources. Logistic model is a simplification.
⚠️
Doubling time = 0.693/r (rule of 70)
Approximate: doubling time in years ≈ 70/r%. Human pop: r≈1.1%/yr, td≈70/1.1=64 yr.
Frequently Asked Questions
What is intrinsic rate of increase r? ▾
Maximum per-capita growth rate under ideal conditions. r = b - d (birth rate minus death rate). Related to R0 (net reproductive rate) by r = ln(R0)/T.
What is carrying capacity K? ▾
Maximum population size the environment can sustain. Limited by food, space, water, predation. In logistic model: growth rate approaches 0 as N approaches K.
What happens at N = K/2? ▾
Population grows fastest in logistic model at N = K/2 (inflection point). Maximum sustainable yield in fisheries is taken at this point.
What is r-selected vs K-selected? ▾
r-selected: high r, low K, many small offspring (mice, insects). K-selected: low r, high K, few large offspring (elephants, humans). Life history trade-off.
Why is exponential growth unsustainable? ▾
Resources are finite. As population grows, per-capita resources decline, increasing death rate or decreasing birth rate. Growth rate slows toward logistic form.