57. The following graph is for a logistically growing population, with N_tplotted on the X-axis. What is the parameter plotted on the Y-axis?

(1) dN/dt (2) N_t+1
(3) dN/dt. 1/N (4) K

Introduction to Logistic Growth Graphs

Logistic growth describes how populations expand rapidly under ideal conditions but slow as they approach environmental limits, stabilizing at a maximum called the carrying capacity (K). When studying this pattern, ecologists often plot population size (Nt) on the X-axis. But what appears on the Y-axis? Understanding this relationship is crucial for interpreting real-world population dynamics, from invasive species to conservation efforts.

The Key Parameter: dN/dt

For a logistically growing population, the parameter plotted on the Y-axis is the population growth rate, dN/dt. This represents the instantaneous change in population size over time. When graphed against Nt, this relationship produces a distinctive bell-shaped curve (Figure 1), which peaks when the population reaches half the carrying capacity (N = K/2) and tapers to zero as N approaches K.

Why dN/dt?

The logistic growth equation explains this relationship:

dNdt=rN(1−NK)dtdN=rN(1−KN)

• N = population size (X-axis)

• dN/dt = growth rate (Y-axis)

• r = intrinsic growth rate

• K = carrying capacity

As population size (N) increases, competition for resources intensifies, causing dN/dt to rise, peak, and fall symmetrically. This curve is a hallmark of density-dependent growth.

Why Other Options Are Incorrect

• (2) Nt+1: This represents the population size at the next time step. Plotting Nt+1 against Nt produces a parabolic curve in discrete models, not the bell shape unique to dN/dt vs. N.

• (3) (dN/dt) · (1/N): This is the per capita growth rate, which decreases linearly with N in logistic growth. It would produce a straight line, not the bell curve.

• (4) K: Carrying capacity is a constant. Plotting K against Nt would yield a horizontal line, failing to capture growth dynamics.

Real-World Example: Insect Invasion

Consider crickets invading a grassland:

• Low Nt: dN/dt is low as the population establishes.

• Moderate Nt: dN/dt peaks (rapid expansion).

• High Nt: dN/dt declines as resources deplete, stabilizing near K.

Why This Graph Matters

• Predicting Population Peaks: The bell curve’s maximum at N = K/2 signals when growth is fastest.

• Conservation Insights: Declining dN/dt at high N warns of overpopulation risks, guiding habitat management.

Conclusion

For a logistically growing population with Nt on the X-axis, the Y-axis represents dN/dt. This curve captures how growth accelerates, peaks, and slows under resource constraints—a universal pattern in ecology. Understanding this graph helps scientists predict booms, crashes, and sustainable population thresholds.