Questions: Population Growth Models: Exponential and Logistic

Question 1

A population currently at N = K/2 is growing logistically. At what population size is the instantaneous growth rate dN/dt at its maximum?

• A. N = 0 (growth is fastest when the population is smallest)
• B. N = K/4
• C. N = K/2 (the population is already at the maximum growth rate)
• D. N = K (growth is fastest just before the ceiling)

Answer: (C)

The logistic growth rate dN/dt = rN(K−N)/K is a quadratic function of N with a maximum at N = K/2. At this midpoint, N is large enough to contribute many births but (K−N)/K is still large enough that density-dependent limitation hasn't strongly kicked in. Below K/2, growth is limited by low N; above K/2, it is limited by resource scarcity. The population in this question is already at its peak growth rate.

Question 2

Once a logistically growing population reaches carrying capacity K, it will stabilize exactly at K and remain there.

• T. True
• F. False

Answer: False

The logistic equation predicts that dN/dt = 0 at N = K, making K a theoretical equilibrium. But real populations overshoot K due to time lags between resource depletion and reduced reproduction, then undershoot as mortality catches up. Environmental stochasticity (variable rainfall, disease outbreaks) also continuously shifts the effective K. Real populations fluctuate around K rather than settling at it.

Question 3

Why does exponential growth (dN/dt = rN) eventually become an unrealistic model for real biological populations?

Model answer: Exponential growth assumes unlimited resources — that every individual always finds food, space, and mates. In reality, as population density increases, resources become limiting, competition intensifies, and disease spreads more easily. These density-dependent factors cause birth rates to fall and death rates to rise, slowing growth below the exponential rate. No environment has truly unlimited resources.

The exponential model captures the biology accurately only at low densities when resources genuinely aren't limiting — early bacterial colonization, introduced species before competitors arrive. As density rises, density-dependent regulation kicks in, and the logistic model's (K−N)/K term captures this deceleration. The switch from J-shaped to S-shaped growth reflects the shift from resource-unlimited to resource-limited conditions.