Calculate population growth using exponential and logistic growth models. Determine final population size, doubling time, and intrinsic growth rates with step-by-step solutions for biology and ecology.
Problem: A bacterial colony starts with 500 cells and grows at a rate of 15% per hour. What is the population after 8 hours? What is the doubling time?
Solution: Using the exponential growth model N = N₀ × e^(r × t)
r = 15% = 0.15, t = 8, N₀ = 500
N = 500 × e^(0.15 × 8) = 500 × e^1.2 = 1,660 cells
Doubling time: t₂ = ln(2) / r = 0.6931 / 0.15 = 4.62 hours
The colony nearly doubles every 4.6 hours, reaching over 3 times its original size in 8 hours.
Problem: A deer population of 200 is introduced to an island with a carrying capacity of 2,000 deer. The intrinsic growth rate is 20% per year. What is the population after 10 years?
Solution: Using the logistic growth model
N = K / (1 + ((K - N₀) / N₀) × e^(-r × t))
= 2000 / (1 + ((2000 - 200) / 200) × e^(-0.20 × 10))
= 2000 / (1 + 9 × e^(-2.0)) = 1,239 deer
The population grows rapidly at first but slows as it approaches the carrying capacity of the island.
Problem: A town's population grew from 5,000 to 7,500 over 10 years. What is the annual growth rate and doubling time?
Solution: Using r = (1/t) × ln(N / N₀)
r = (1/10) × ln(7500 / 5000) = 0.1 × ln(1.5)
r = 0.1 × 0.4055 = 0.0405 = 4.05% per year
Doubling time: t₂ = ln(2) / r = 17.1 years
At this growth rate, the town would double in size roughly every 17 years.
Problem: The world population was approximately 8 billion in 2025 with a growth rate of 0.9% per year. If this rate continues exponentially, what will the population be in 2050?
Solution: Using N = N₀ × e^(r × t)
N₀ = 8 × 10⁹, r = 0.009, t = 25
N = 8×10⁹ × e^(0.009 × 25) = 8×10⁹ × e^0.225
N = 10.0 billion
Real-world projections are more complex, factoring in changing fertility rates and logistic constraints.
Where N is the final population size, N₀ is the initial population, r is the intrinsic growth rate (as a decimal), and t is the number of time periods. e is Euler's number (~2.71828).
Where K is the carrying capacity — the maximum sustainable population size. The logistic model accounts for limited resources that slow growth as the population approaches K.
Use this formula to determine the growth rate from known initial and final population sizes. ln is the natural logarithm.
The doubling time is the time required for a population to double in size under exponential growth. For example, with r = 0.05 (5% per year), the doubling time is 0.693 / 0.05 ≈ 13.9 years.
Exponential growth assumes unlimited resources and produces a J-shaped curve. Logistic growth accounts for carrying capacity, producing an S-shaped (sigmoid) curve where growth slows near K.
The intrinsic growth rate r is the per capita rate of increase under ideal conditions. It represents the birth rate minus the death rate. A positive r means the population is growing; negative means declining.
The carrying capacity is the maximum population size an environment can sustain indefinitely. Factors include food availability, habitat space, water, and other resources. When N = K, the population is stable.
Doubling time is a useful metric for understanding how quickly a population grows. Small changes in r significantly affect doubling time — for example, r = 1% yields 69.3 periods to double, while r = 7% yields only 9.9 periods.
⚠️ Important Note: Population growth models are simplifications of complex ecological and biological systems. Exponential growth rarely persists in nature due to resource limitations, predation, and disease. Logistic models assume a constant carrying capacity, which may change over time due to environmental shifts. Always validate model assumptions against real-world data for serious research applications.