When Is the Change in Population Density Greatest in Logistic Growth?

Understanding how populations grow and what limits their expansion is fundamental in ecology, conservation, and resource management. The logistic growth model is widely used to describe how populations increase rapidly at first but slow as they approach the environment’s carrying capacity. This article explains at what population size the change in density per unit time is greatest, and why this point is important for sustainable management.

The Logistic Growth Model

The logistic growth model describes how a population grows when resources are limited. The standard equation for logistic growth is:

dNdt=rN(1−NK)dtdN=rN(1−KN)

where:

• $N$: Current population size (density)

• $K$: Carrying capacity (maximum population size the environment can support)

• $r$: Intrinsic growth rate (per capita)

• dNdtdtdN: Change in population density per unit time

This equation shows that as the population ($N$) approaches the carrying capacity ($K$), the growth rate slows and eventually stops when $N=K$.

The S-Shaped Curve and the Inflection Point

A graph of logistic growth forms an S-shaped (sigmoid) curve. The curve has three main phases:

1. Initial slow growth: The population is small, and growth is slow.

2. Rapid growth: As the population increases, resources are abundant, and the growth rate accelerates.

3. Plateau at carrying capacity: As the population nears $K$, resources become scarce, and the growth rate slows until the population stabilizes.

The inflection point of the curve is where the growth rate transitions from accelerating to decelerating. This is the point at which the change in population density per unit time (dNdtdtdN) is at its maximum.

Finding the Maximum Growth Rate

To find where dNdtdtdN is maximum, we look for the inflection point of the logistic curve. Mathematically, this occurs when the population is half the carrying capacity ($N=K/2$).

Here’s why:

The logistic growth equation is:

dNdt=rN(1−NK)dtdN=rN(1−KN)

To find the maximum, we can treat dNdtdtdN as a function of $N$ and find its maximum with respect to $N$. Taking the derivative and setting it to zero:

ddN(dNdt)=r−2rNK=0dNd(dtdN)=r−K2rN=0

Solving for $N$:

r=2rNK1=2NKN=K2r=K2rN1=K2NN=2K

So, the change in population density per unit time (dNdtdtdN) is maximum when $N=K/2$.

Why Is This Point Important?

• Sustainable Harvesting: The point of maximum growth rate is where the population can sustain the highest harvest without declining.

• Conservation Planning: Knowing when the population is growing fastest helps in planning conservation and management strategies.

• Resource Management: It allows for efficient use of resources by targeting the period of maximum productivity.

Real-World Example

Imagine a fish population in a lake with a carrying capacity of 1,000. The population will grow fastest when there are 500 fish. At this point, the number of new fish added per unit time is at its maximum.

Common Misconceptions

• Maximum growth rate at carrying capacity: Some think the population grows fastest at $K$, but in reality, the growth rate is zero at $K$.

• Maximum growth rate at very low population sizes: At low population sizes, the growth rate is increasing but not yet at its maximum.

• Inflection point is at $K/4$ or $K/2$: The inflection point is always at $K/2$, not $K/4$ or any other fraction.

Summary Table

Conclusion

In a population growing logistically and approaching carrying capacity ($K$), the change in density per unit time (dNdtdtdN) is maximum when the population size is half the carrying capacity ($N=K/2$). This is the inflection point of the logistic curve, where the population grows at its fastest rate.

Correct answer:
(2) K/2