39.A population grows according to the logistic growth equation,

Where, is the rate of population growth, r is the intrinsic rate of increase, N is population size and K is the carrying ‘capacity of the environment. According to this equation, population growth rate is maximum at
(1) K/4 (2) K/2
(3) K (4) 2K

At What Population Size Is the Growth Rate Maximum 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 a cornerstone for describing how populations increase rapidly at first but slow as they approach the environment’s carrying capacity. This article explains when the population growth rate is highest in a logistic growth scenario, using the logistic growth equation as a foundation.

The Logistic Growth Equation

The logistic growth model describes how a population changes over time when resources are limited. The standard form of the equation is:

dNdt=rN(K−NK)dtdN=rN(KK−N)

where:

• dNdtdtdN: Rate of population growth (number of individuals per unit time)

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

• $N$: Current population size

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

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

When Is the Population Growth Rate Maximum?

To find out when the population grows fastest, we look for the point where the growth rate dNdtdtdN is at its maximum. This occurs at the inflection point of the logistic curve, which is the moment when the growth rate transitions from increasing to decreasing.

Mathematical Derivation

The logistic growth rate is:

dNdt=rN(K−NK)dtdN=rN(KK−N)

This can be rewritten as:

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 value with respect to $N$. This is a standard optimization problem in calculus.

Let’s define:

f(N)=rN(1−NK)f(N)=rN(1−KN)

Take the derivative with respect to $N$:

f′(N)=r(1−NK)+rN(−1K)f′(N)=r(1−KN)+rN(−K1)=r−rNK−rNK=r−KrN−KrN=r−2rNK=r−K2rN

Set the derivative equal to zero to find the critical point:

r−2rNK=0r−K2rN=0r=2rNKr=K2rN1=2NK1=K2NN=K2N=2K

So, the growth rate dNdtdtdN is maximum when the population size is half the carrying capacity, N=K2N=2K.

Interpretation

At N=K2N=2K, the population is growing at its fastest rate. Before this point, the growth rate increases as the population grows; after this point, the growth rate decreases as the population approaches the carrying capacity.

Why Is This Important?

Understanding where the growth rate is highest helps ecologists and resource managers:

• Predict population peaks: Knowing when a population will grow fastest allows for better planning and intervention.

• Manage resources: Harvesting or management strategies can be timed to coincide with periods of maximum growth for sustainable yield.

• Model real-world scenarios: Logistic models are used in fisheries, wildlife management, and even epidemiology to predict population trends.

Real-World Example

Consider 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 is not at the start or end: Some might think the population grows fastest when it is smallest or largest, but the maximum growth rate occurs at the midpoint.

• Carrying capacity is not the point of fastest growth: The growth rate is zero at the carrying capacity.

• Inflection point is key: The inflection point of the logistic curve marks the transition from accelerating to decelerating growth.

Summary Table

Conclusion

For a population growing according to the logistic growth equation, the growth rate is maximum when the population size is half the carrying capacity. This is mathematically derived as N=K2N=2K.

Correct answer:
(2) K/2