Which Equation Represents Population Growth Under Unlimited Resources? Understanding Exponential Growth

Population growth is a fundamental concept in ecology, shaping how species expand, stabilize, or decline over time. The type of growth a population experiences depends largely on the availability of resources such as food, water, and habitat. When resources are unlimited, populations can grow rapidly and without restraint. This article explains which mathematical equation best represents population growth under unlimited resources, and why this is important for understanding ecological dynamics.

Population Growth Models: An Overview

Ecologists use mathematical models to describe and predict how populations change over time. The two most important models are exponential growth and logistic growth:

• Exponential growth occurs when resources are unlimited and there are no constraints on population expansion.

• Logistic growth occurs when resources are limited, causing the population to slow its growth as it approaches the environment’s carrying capacity.

Understanding these models helps scientists predict future population sizes, manage wildlife, and conserve biodiversity.

What Is Exponential Growth?

Exponential growth describes a situation where a population increases at a rate proportional to its current size. This means that the larger the population becomes, the faster it grows. In nature, this type of growth is rare because resources are usually limited. However, in ideal conditions—such as a laboratory culture with unlimited nutrients—populations can grow exponentially.

The classic example is bacteria in a petri dish: if each bacterium divides into two every hour, the population doubles each hour, leading to rapid, accelerating growth.

Characteristics of Exponential Growth

• Unlimited resources: No constraints on food, space, or other necessities.

• Constant per capita growth rate: Each individual contributes equally to population growth, regardless of population size.

• J-shaped curve: When graphed over time, the population size forms a steeply rising curve that resembles the letter “J”.

Mathematical Representation of Exponential Growth

The standard equation for exponential population growth is:

Nt=N0ertNt=N0ert

where:

• NtNt = population size at time $t$

• N0N0 = initial population size

• $e$ = base of the natural logarithm (~2.718)

• $r$ = intrinsic growth rate (birth rate minus death rate)

• $t$ = time

This equation shows that population size increases exponentially over time when resources are unlimited.

Alternatively, the differential form of the exponential growth equation is:

dNdt=rNdtdN=rN

where:

• dNdtdtdN = rate of change of population size over time

• $r$ = intrinsic growth rate

• $N$ = current population size

Evaluating the Given Options

Let’s examine each of the provided equations to determine which correctly represents population growth under unlimited resources:

1. Nt = NOe –rt

   • Interpretation: The negative exponent (–rt) suggests population decline, not growth. This is incorrect for unlimited resources.

2. dN/dt = rt

   • Interpretation: This implies the population growth rate is proportional to time, not population size. This is not a standard population growth model.

3. Nt/Nt = Nt – NO

   • Interpretation: This equation is mathematically nonsensical and does not describe population growth.

4. dN/dt = Krt

   • Interpretation: This introduces a new variable $K$ (carrying capacity) and is not a standard form for exponential growth. The correct form for unlimited resources is $dN/dt=rN$.

None of the provided options exactly match the standard exponential growth equation. However, if we consider possible typographical errors, option (1) Nt = NOe –rt likely intended to write Nt=N0ertNt=N0ert, which is the correct exponential growth formula. The minus sign is likely a formatting error.

Why Is Exponential Growth Important?

Exponential growth is a foundational concept in ecology because it describes how populations would behave in ideal, resource-rich environments. While this scenario is rare in nature, it helps scientists understand the potential for population explosions and the importance of resource limitations.

Real-World Examples

• Bacteria in a lab: With unlimited nutrients, bacteria can double their population every few hours, leading to explosive growth.

• Invasive species: When introduced to a new environment with few predators and abundant resources, invasive species can grow exponentially until resources become limited.

Limitations of Exponential Growth

Although exponential growth is a useful theoretical model, it is rarely sustained in nature. Eventually, resources become scarce, competition increases, and the population growth rate slows. This leads to logistic growth, where the population stabilizes at the environment’s carrying capacity.

Logistic Growth: The Reality of Limited Resources

When resources are limited, populations follow logistic growth, described by the equation:

dNdt=rN(1−NK)dtdN=rN(1−KN)

where:

• $K$ = carrying capacity (maximum population size the environment can support)

• $N$ = current population size

• $r$ = intrinsic growth rate

This equation produces an S-shaped curve, where growth slows as the population approaches the carrying capacity.

Summary Table

Conclusion

The correct equation for population growth under unlimited resources is the exponential growth equation:

Nt=N0ertNt=N0ert

Among the options provided, none are perfectly correct as written, but option (1) Nt = NOe –rt likely contains a typographical error and should be Nt=N0ertNt=N0ert. This equation describes how populations grow rapidly and without limit when resources are abundant.

Correct answer (assuming typographical error in option 1):
(1) Nt = NOe –rt (should be Nt=N0ertNt=N0ert)

If the equation is written exactly as Nt=N0ertNt=N0ert, it is the correct choice. Otherwise, none of the options are strictly correct, but option (1) is the closest to the standard exponential growth formula.