34. In a population growing according to the logistic growth model

(A) individuals reproduce according to their physiological capacity

(B) the per capita rate of increase approaches zero as the population nears the carrying capacity

(D) the birth-to-death ratio is NOT influenced by the carrying capacity

Logistic Growth Model: What Happens to the Per Capita Rate of Increase Near Carrying Capacity?

Understanding the Logistic Growth Model

The logistic growth model describes the growth of a population in an environment where resources are limited. Unlike the exponential growth model, which assumes unlimited food, space, and other essential resources, logistic growth recognizes that every environment has a maximum population size that it can support sustainably.

This maximum sustainable population size is known as the carrying capacity, represented by K. When the population size is small, resources are relatively abundant and the population can grow rapidly. However, as population size increases, competition for food, space, nutrients, shelter, and other resources becomes stronger. Consequently, the rate of population increase gradually slows down.

The correct answer is Option (B) because, in the logistic growth model, the per capita rate of increase approaches zero as the population approaches the carrying capacity.

Logistic Growth Equation and Its Biological Meaning

The logistic growth model is mathematically expressed as:

dN/dt = rN [(K − N)/K]

dN/dt = rN (1 − N/K)

Here, N represents the population size, r represents the intrinsic rate of natural increase, K represents the carrying capacity of the environment, and dN/dt represents the rate of change in population size with time.

The most important component of this equation is:

(1 − N/K)

This term represents the effect of environmental limitation on population growth. When the population is very small compared with the carrying capacity, the value of N/K is close to zero. Therefore, the term (1 − N/K) is close to one, and the population can grow rapidly.

As the population size increases and approaches the carrying capacity, the value of N/K approaches one. Therefore:

1 − N/K → 0

As a result, population growth gradually slows down.

When:

N = K

then:

1 − N/K = 1 − 1 = 0

Therefore:

dN/dt = 0

At carrying capacity, there is no net increase in population size.

Why Option (B) Is Correct

The Per Capita Rate of Increase Approaches Zero Near Carrying Capacity

Option (B) the per capita rate of increase approaches zero as the population nears the carrying capacity is the correct answer.

To understand this statement properly, we need to examine the per capita rate of population increase. Starting with the logistic growth equation:

dN/dt = rN (1 − N/K)

Dividing both sides by N gives:

(1/N)(dN/dt) = r(1 − N/K)

Therefore, the per capita rate of increase is:

r(1 − N/K)

When the population size N is much smaller than the carrying capacity K, the term N/K is very small. Therefore, the per capita rate of increase remains relatively high.

However, as N approaches K:

N/K → 1

and therefore:

1 − N/K → 0

Thus, the per capita rate of increase approaches zero.

Biologically, this happens because increasing population density intensifies competition for limited resources. Individuals may obtain less food, experience reduced reproductive success, face increased disease transmission, and suffer higher mortality. These density-dependent effects reduce the net contribution of each individual to future population growth.

At the carrying capacity, births may still occur and deaths may still occur, but their rates become approximately equal. Therefore, the population shows no net increase.

Hence, Option (B) is correct.

Why Option (A) Is Incorrect

Individuals Reproduce According to Their Physiological Capacity

Option (A) individuals reproduce according to their physiological capacity is incorrect in the context of logistic population growth.

Reproduction according to the maximum physiological capacity of individuals is more closely associated with the exponential growth model, particularly under ideal environmental conditions. If food, space, nutrients, and other resources are unlimited, individuals can reproduce close to their maximum biological potential.

This maximum potential is related to the intrinsic rate of natural increase, represented by r.

However, the logistic growth model assumes that environmental resources become increasingly limited as population density rises. Therefore, individuals cannot continue reproducing indefinitely according to their maximum physiological capacity.

As the population approaches carrying capacity, several environmental limitations become important. Competition for food increases, available space decreases, disease transmission may become more frequent, predation pressure may change, and reproductive success may decline. These factors collectively create environmental resistance.

Therefore, reproduction in a logistic population is influenced by population density and resource availability rather than being determined only by physiological capacity.

Hence, Option (A) is incorrect.

Why Option (C) Is Incorrect

The Number of Births Is Always More Than the Number of Deaths

Option (C) the number of births is always more than the number of deaths is incorrect.

During the early and rapidly growing phases of logistic population growth, the number of births may indeed exceed the number of deaths. This produces a positive population growth rate.

However, the word “always” makes the statement incorrect.

As the population approaches the carrying capacity, the difference between births and deaths becomes progressively smaller. At carrying capacity, the number of births is approximately equal to the number of deaths.

Therefore:

Birth rate ≈ Death rate

and consequently:

Net population growth ≈ 0

A population at carrying capacity is not necessarily biologically inactive. Individuals continue to be born and individuals continue to die. The important point is that these processes approximately balance each other, resulting in little or no net change in population size.

Therefore, births are not always greater than deaths in the logistic growth model.

Hence, Option (C) is incorrect.

Why Option (D) Is Incorrect

The Birth-to-Death Ratio Is Not Influenced by Carrying Capacity

Option (D) the birth-to-death ratio is NOT influenced by the carrying capacity is incorrect.

Carrying capacity has a major influence on population dynamics. As a population approaches K, increasing population density affects both birth rates and death rates.

Limited food availability can reduce reproductive output. Competition for resources can decrease survival. Overcrowding may increase disease transmission. Shortage of nesting sites, territories, or shelter can reduce reproductive success. These factors can lower the birth rate, increase the death rate, or produce both effects simultaneously.

Thus, the relationship between births and deaths is strongly influenced by the population’s proximity to carrying capacity.

When the population is far below carrying capacity, resources are relatively abundant and births generally exceed deaths. As the population approaches carrying capacity, environmental resistance becomes stronger and the difference between births and deaths decreases. At or near carrying capacity, births and deaths become approximately balanced.

Therefore, the birth-to-death relationship is influenced by carrying capacity.

Hence, Option (D) is incorrect.

Phases of the Logistic Growth Curve

The logistic growth curve is commonly described as an S-shaped or sigmoid growth curve. Population growth changes as the population passes through different stages.

Initial Slow Growth Phase

At the beginning, the population is small. Although resources are abundant, the total number of reproductive individuals is also small. Therefore, total population growth may initially be relatively slow.

Rapid Growth Phase

As the population increases, more individuals contribute to reproduction. Resources are still sufficiently available, so the population enters a phase of rapid growth.

Decelerating Growth Phase

As population density rises, resources become increasingly limited. Competition becomes stronger and the population growth rate begins to slow down.

Stationary Phase Near Carrying Capacity

When the population approaches the carrying capacity, the per capita rate of increase approaches zero. Births and deaths become approximately balanced, and the population size stabilizes around K.

This gradual slowing of population growth is the central feature that distinguishes logistic growth from unlimited exponential growth.

Relationship Between Population Size and Per Capita Growth Rate

One of the most important concepts in the logistic growth model is that the per capita growth rate decreases linearly as population size increases.

The relationship is:

Per capita growth rate = r(1 − N/K)

When N is close to zero, the per capita growth rate is close to r.

When N = K/2, the per capita growth rate becomes:

r(1 − 1/2) = r/2

When N approaches K, the per capita growth rate approaches zero.

When N = K, the per capita growth rate is exactly zero in the ideal logistic model.

This density-dependent decline in per capita population growth is one of the defining characteristics of the logistic growth model.

Logistic Growth Versus Exponential Growth

The difference between exponential and logistic growth is essential for understanding this question.

In exponential growth, resources are assumed to be unlimited and the per capita rate of increase remains constant. The population grows according to a J-shaped curve.

In logistic growth, resources are limited and the per capita rate of increase decreases as population density increases. The population eventually approaches carrying capacity and produces an S-shaped or sigmoid growth curve.

Therefore, the statement that individuals reproduce according to their full physiological capacity is more suitable for ideal exponential growth, whereas the decline in per capita growth rate near carrying capacity is characteristic of logistic growth.

Role of Density Dependence in Logistic Population Growth

The logistic growth model is a classic example of density-dependent population regulation.

Density dependence means that the effect of certain environmental factors becomes stronger as population density increases. Competition for food, competition for space, disease transmission, accumulation of waste products, and competition for reproductive opportunities can all intensify in crowded populations.

These factors reduce the average contribution of each individual to population growth. As a result, the per capita rate of increase declines progressively with increasing population size.

This is why a population cannot continue growing indefinitely in a limited environment.

Final Answer

Correct Option: (B) The per capita rate of increase approaches zero as the population nears the carrying capacity.

In the logistic growth model, increasing population density strengthens the effects of limited resources and environmental resistance. The per capita rate of increase is represented by r(1 − N/K). As population size N approaches carrying capacity K, the term (1 − N/K) approaches zero. Consequently, the per capita rate of increase also approaches zero. Therefore, Option (B) is the most accurate statement.