Biology · Organisms and Populations · NEET
Nt is the population density (number) after time t. N0 is the population density at time zero, that is, the starting number. r is the intrinsic rate of natural increase (how fast the population grows per individual). t is the time. e is the base of natural logarithms, a fixed constant equal to 2.71828. Only Nt, N0, r and t change from problem to problem; e is always 2.71828.
e is a constant, not a rate. This is the most common mix-up. The growth rate is r. e is simply a fixed mathematical number (2.71828), the base of natural logarithms. It appears because the equation is the integral (calculus) form of the growth model, and this base makes continuous growth curves smooth. NEET has directly asked what e represents, so remember: e = base of natural logarithms = 2.71828.
When resources are unlimited, every individual keeps reproducing, so the population is added on top of an already larger population each moment. This makes the number rise slowly at first and then very steeply, producing a J-shaped curve when you plot N against time. The equation Nt = N0e^rt is just the maths behind that J-shape.
If r = 0, then e^(0×t) = e^0 = 1, so Nt = N0 × 1 = N0. The population stays exactly the same. r = 0 means birth rate equals death rate (b = d), so there is no net change. ReNEET 2026 tested exactly this idea with b = d = 0.002.
r = b − d, where b is the per capita birth rate and d is the per capita death rate. So the exponential equation is powered by r, and r itself comes from births minus deaths. If b is greater than d, r is positive and the population grows; if b equals d, r is zero and the population is steady.
Only for a short time. Truly unlimited resources do not exist for long, so no population can keep growing exponentially forever. It is seen briefly when a species enters a new area with plenty of food and space. Over time, resources become limited and growth becomes logistic (S-shaped), which is the more realistic model.
In the exponential growth equation Nt = N0e^rt, e represents:
Consider a population of 10 million cells with a per-capita birth rate of 0.002 (per unit time) and a per-capita death rate of 0.002 (per unit time). The expected number of cells after 10 generations is ______.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Nt = N0e^rt. It gives the population size (Nt) after time t, starting from N0, growing at intrinsic rate r, when resources are unlimited.
e = 2.71828. It is the base of natural logarithms and is a fixed constant that never changes.
r is the intrinsic rate of natural increase, the growth rate that changes with the species and conditions. e is a fixed constant (2.71828). r drives the growth; e is just the mathematical base.
A J-shaped curve when population number N is plotted against time, because growth gets steeper and steeper as numbers rise.
It is the integral (calculus) form of dN/dt = rN. NCERT says students familiar with basic calculus can derive Nt = N0e^rt from it.
