In the integral form of the exponential growth equation, Nt = N0 ert,…

The correct answer is Option (4) → N_t – Population density after time t

For a population growing without limit, the exponential equation dN/dt = rN can be written in its integral form, N_t = N₀ e^rt. Each symbol stands for one quantity: N₀ is the population density at the start (time zero), N_t is the density after time t, r is the intrinsic rate of natural increase and e is the base of natural logarithms.

Option (1) N_t – Population density at time zero — Incorrect. N_t is the density after time t, not at time zero. The density at time zero is N₀.

Option (2) N₀ – Population density after time t — Incorrect. N₀ is the starting density, at time zero. The density after time t is N_t.

Option (3) e – Carrying capacity of the habitat — Incorrect. e is the base of natural logarithms (about 2.718), a mathematical constant. Carrying capacity is K, which belongs to the logistic equation.

Option (4) N_t – Population density after time t — Correct. N_t is the population density reached after time t.

NCERT Extract:
Growth Models – “… If you are familiar with basic calculus, you can derive the integral form of the exponential growth equation as N_t = N₀ e^rt where N_t = 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)”