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Exponential vs. Logistic Growth: Calculating Insect Population Increase Per Unit Time

Population dynamics are central to ecological studies, pest management, and conservation biology. Understanding how populations grow under different conditions—exponential versus logistic—is essential for predicting population trends and managing species effectively. This article demonstrates how to calculate the increase in insect population size per unit time for both exponential and logistic growth, using specific birth and death rates and a known carrying capacity.

Understanding Population Growth Models

Exponential Growth

Exponential growth occurs when a population increases at a rate proportional to its current size, with no limiting factors. The growth rate remains constant, and the population can theoretically grow without bound. The formula for exponential growth is:

dNdt=rNdtdN=rN

where:

• dNdtdtdN: Rate of change in population size per unit time

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

• $N$: Current population size

The intrinsic growth rate ($r$) is calculated as the difference between the per capita birth rate ($b$) and the per capita death rate ($d$):

$$r=b-d$$

Logistic Growth

Logistic growth accounts for environmental limitations, such as finite resources or space, by introducing a carrying capacity ($K$). The growth rate slows as the population approaches $K$, and the population stabilizes at this maximum sustainable size. The logistic growth equation is:

dNdt=rN(K−NK)dtdN=rN(KK−N)

where:

• $K$: Carrying capacity

• $N$: Current population size

Given Data

• Per capita birth rate ($b$): 0.25

• Per capita death rate ($d$): 0.05

• Carrying capacity ($K$): 500

• Current population density ($N$): 100

Calculating the Intrinsic Growth Rate ($r$)

$$r=b-d=0.25-0.05=0.20$$

(a) Exponential Growth

The increase in population size per unit time is:

dNdt=rN=0.20×100=20dtdN=rN=0.20×100=20

Under exponential growth, the insect population increases by 20 individuals per unit time.

(b) Logistic Growth

The increase in population size per unit time is:

dNdt=rN(K−NK)dtdN=rN(KK−N)

Plug in the values:

dNdt=0.20×100×(500−100500)dtdN=0.20×100×(500500−100)=20×(400500)=20×(500400)$$=20\times 0.8$$$$=16$$

Under logistic growth, the insect population increases by 16 individuals per unit time.

Why This Matters

Understanding the difference between exponential and logistic growth is crucial for:

• Pest management: Predicting how quickly insect populations can increase and when control measures are needed.

• Conservation: Estimating how populations recover or stabilize after disturbances.

• Resource planning: Anticipating the demand on resources as populations grow.

Real-World Implications

In nature, most populations experience logistic growth because resources are finite. Exponential growth is typically seen only in the early stages of population expansion or in laboratory settings with unlimited resources. By calculating the actual growth rate under both models, ecologists and managers can make more accurate predictions and decisions.

Summary Table

Conclusion

For an insect population with a birth rate of 0.25, death rate of 0.05, and current density of 100 in a habitat with a carrying capacity of 500:

• Exponential growth: Population increases by 20 individuals per unit time.

• Logistic growth: Population increases by 16 individuals per unit time.

Correct answer:
(3) a: 20, b: 16