Q.The population growth curve applicable for a population growing in a geometric fashion, when the resources are not limiting in the habitat will be : (A) [graph: Population density vs Time — horizontal line] (B) [graph: Population density vs Time — straight rising line] (C) [graph: Population density vs Time — exponential (J-shaped) curve] (D) [graph: Population density vs Time — sigmoid (S-shaped) curve]

Concept: Exponential (geometric) growth under unlimited resources

When resources are unlimited and the population grows geometrically, each individual reproduces at a constant rate. The population size at any time is given by:

N_t = N₀ e^{rt}

where r is the intrinsic rate of natural increase. This produces a characteristic J-shaped curve because the absolute number of individuals added per unit time keeps increasing (even though the per capita rate stays constant). …

When a population grows geometrically—meaning each individual produces a constant number of offspring per unit time—and resources are unlimited, we're describing exponential growth. The key insight is that the population doesn't just add a fixed number of individuals each generation; instead, it multiplies by a constant factor. A population of 100 that doubles becomes 200, then 400, then 800—the increment itself grows larger with each step because more parents produce more offspring.

Mathematically, this is captured by:

dN/dt = rN

where N is population size, t is time, and r is the intrinsic rate of increase. The solution is N(t) = N₀ e^{rt}, an exponential function. The hallmark of exponential growth is that the rate of increase is proportional to the current population size—bigger populations grow faster.

Now let's match this behavior to the graph options:

1. Option (A): Horizontal line

   A flat line means zero growth—the population size stays constant. This describes a population at equilibrium or with birth rate exactly matching death rate, not geometric growth.

2. Option (B): Straight rising line

   A linear increase means the population adds the same absolute number of individuals per unit time (arithmetic growth). If you gain 10 individuals per year regardless of population size, that's dN/dt = k (constant), not dN/dt = rN. This doesn't capture the accelerating nature of geometric growth.

3. Option (C): J-shaped (exponential) curve

   This curve starts slowly, then rises steeply, accelerating upward without leveling off. The slope gets steeper as time progresses because the population itself is growing—more individuals mean more reproduction. This is the signature of exponential growth when resources are unlimited and nothing checks the population. …