Q47. A population growing exponentially can be described by the differential equation
dN/dt = rN, where dN/dt represents the rate of population growth,
N is the size of the population, r is the intrinsic rate of increase, and t is time.
According to this equation, the per capita rate of growth is
The per capita rate of growth in exponential population growth, described by dN/dt = rN, is constant regardless of population size N. This makes option (B) correct.
Equation Breakdown
The differential equation dN/dt = rN models exponential growth, where dN/dt is the population change rate, N is population size, r is the intrinsic growth rate (births minus deaths per individual), and t is time. Rearranging gives (1/N)(dN/dt) = r, showing the per capita growth rate—growth per individual—as the constant r.
Option Analysis
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(A) Highest at large N: Incorrect. Total growth dN/dt accelerates with larger N (since dN/dt = rN), but per capita rate r stays fixed, not higher.
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(B) Constant: Correct. Per capita rate equals r, unchanging with N, as resources are assumed unlimited in this model.
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(C) Lowest at large N: Incorrect. This describes logistic growth (dN/dt = rN(1 – N/K)), where per capita rate drops near carrying capacity K; not exponential.
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(D) Highest at small N: Incorrect. Per capita rate r is steady across all N sizes in exponential growth.
Introduction
In exponential population growth modeled by dN/dt = rN, the per capita rate of growth remains constant, making it a key concept for GATE Life Sciences exams. This intrinsic rate r drives unchecked growth when resources are unlimited, unlike logistic models where it declines. Explore why it’s constant, not highest at large N or lowest at large N.
Exponential Growth Basics
Exponential population growth follows dN/dt = rN, where the population size N increases proportionally to itself over time t. The per capita rate of growth, (1/N)(dN/dt), simplifies to r—a fixed value representing net reproductive success per individual.
This leads to the solution N(t) = N₀ e^{rt}, showing J-shaped curves in plots of N vs. t.
Why Per Capita Rate Is Constant
Each individual’s contribution to growth (r) doesn’t depend on total population size N in this ideal model. As N grows, total growth accelerates, but per capita remains steady at r.
For example, if r = 0.1 per year, a population of 100 adds 10 individuals yearly, while 1,000 adds 100—but per person, it’s always 0.1.
| Model | Equation | Per Capita Rate Behavior |
|---|---|---|
| Exponential | dN/dt = rN | Constant (r) |
| Logistic | dN/dt = rN(1 – N/K) | Declines as N nears K |
Common Exam Confusions
Students often mix this with logistic growth, where per capita rate is lowest at large N due to resource limits. In dN/dt = rN, no such density dependence exists, so options like highest at small N don’t apply.
GATE questions test this distinction for ecology and population dynamics sections.
