In the integral form of the exponential growth equation \( N_t = N_0 e^{rt} \), ‘r’ stands for:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Intrinsic rate of natural increase
When food and space are unlimited, a population grows exponentially. In the integral form of the equation:
\( \displaystyle N_t = N_0 e^{rt} \)
Nt is the population density after time t, N0 is the density at time zero and e is the base of natural logarithms (2.71828). r is the intrinsic rate of natural increase: the birth rate minus the death rate per individual. It measures how fast a population can grow, and it is different for different species.
(1) Extrinsic rate of natural increase — Incorrect. There is no such term. The rate is “intrinsic”, because it is a property of the species itself.
(2) Intrinsic rate of natural increase — Correct. This is r.
(3) Increase in population density — Incorrect. The density itself is N. r is the rate that decides how fast N grows.
(4) Decrease in population density — Incorrect. In exponential growth, the population grows; r is not a decrease in density.
NCERT Extract:
Population Growth – “… If you are familiar with basic calculus, you can derive the integral form of the exponential growth equation as Nt = N0 ert where Nt = Population density after time t N0 = Population density at time zero r = intrinsic rate of natural increase e = the base of natural logarithms (2.71828) Any species growing exponentially under unlimited resource conditions can reach enormous population densities in a short time.”