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Logistic Equation — Definition, Formula & Examples

The logistic equation is a differential equation that models population growth which starts exponentially but slows and levels off as the population approaches a maximum sustainable size called the carrying capacity.

The logistic equation is the first-order autonomous ODE dPdt=rP(1−PK)\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right), where rr is the intrinsic growth rate and KK is the carrying capacity. Its solution is a sigmoidal (S-shaped) curve bounded between 00 and KK.

Key Formula

P(t)=K1+(K−P0P0)e−rtP(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}
Where:
  • P(t)P(t) = Population at time $t$
  • KK = Carrying capacity (maximum sustainable population)
  • P0P_0 = Initial population at $t = 0$
  • rr = Intrinsic growth rate
  • tt = Time

How It Works

When the population PP is small relative to KK, the factor (1−PK)\left(1 - \frac{P}{K}\right) is close to 1, so growth is approximately exponential at rate rr. As PP approaches KK, this factor shrinks toward 0, decelerating growth. At P=KP = K, growth stops entirely because dPdt=0\frac{dP}{dt} = 0. The equation is separable: you can solve it by partial fractions to obtain the explicit solution for P(t)P(t).

Worked Example

Problem: A population follows the logistic equation with carrying capacity K = 1000, growth rate r = 0.5, and initial population P₀ = 100. Find P(6).
Compute the constant: Find the ratio that appears in the solution formula.
K−P0P0=1000−100100=9\frac{K - P_0}{P_0} = \frac{1000 - 100}{100} = 9
Write the solution: Substitute all known values into the logistic solution.
P(t)=10001+9e−0.5tP(t) = \frac{1000}{1 + 9e^{-0.5t}}
Evaluate at t = 6: Compute the exponential term and then the population. Note that e−3≈0.04979e^{-3} \approx 0.04979.
P(6)=10001+9(0.04979)=10001.44811≈690.6P(6) = \frac{1000}{1 + 9(0.04979)} = \frac{1000}{1.44811} \approx 690.6
Answer: At t=6t = 6, the population is approximately 691.

Why It Matters

The logistic equation is the standard model for resource-limited growth in ecology, epidemiology (modeling disease spread), and chemistry (autocatalytic reactions). In a differential equations course, it serves as a key example of a separable, nonlinear ODE solved by partial fractions.

Common Mistakes

Mistake: Forgetting the negative sign in the exponent and writing erte^{rt} instead of e−rte^{-rt} in the solution.
Correction: The exponent must be −rt-rt so that the denominator grows over time, causing P(t)P(t) to increase toward KK rather than diverge.