Understanding Verhulst-Pearl Logistic Growth

The Verhulst-Pearl Logistic Growth model is a fundamental concept in population ecology. It describes how a population grows when its resources are limited, leading to a carrying capacity for the environment. Unlike exponential growth, which assumes unlimited resources, logistic growth incorporates environmental resistance.

The Logistic Growth Equation Explained

The rate of change in population size over time in the logistic growth model is represented by a differential equation. This equation considers the intrinsic rate of increase and the effect of the population size approaching the environment's carrying capacity.

The standard form of the Verhulst-Pearl Logistic Growth equation is:

\( \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \)

This equation can also be written as:

\( \frac{dN}{dt} = rN \left(\frac{K-N}{K}\right) \)

Let's break down the components of this equation:

• \( \frac{dN}{dt} \): This represents the rate of change in the population size (\(N\)) over time (\(t\)). It's the population growth rate.
• \( r \): This is the intrinsic rate of increase, also known as the biotic potential. It's the rate at which the population would grow if there were unlimited resources. It is often calculated as the birth rate (\(h\)) minus the death rate (\(d\)), i.e., \(r = h - d\).
• \( N \): This is the current population size.
• \( K \): This is the carrying capacity of the environment. It's the maximum population size that the environment can sustainably support given the available resources.
• \( \left(\frac{K-N}{K}\right) \): This term represents the environmental resistance. As the population size \(N\) approaches the carrying capacity \(K\), this term gets closer to zero, slowing down the population growth rate. When \(N=K\), this term is zero, and the growth rate becomes zero. When \(N\) is much smaller than \(K\), this term is close to 1, and the growth rate is close to exponential growth (\(rN\)).

Analyzing the Given Options

Let's compare the provided options with the standard logistic growth equation \( \frac{dN}{dt} = rN \frac{K-N}{K} \):

Option 1: \( \frac{dN}{dt} = (h - d) \frac{N(K-N)}{K} \)

Since \(r = h - d\), this equation is \( \frac{dN}{dt} = r \frac{N(K-N)}{K} \). This can be rewritten as \( \frac{dN}{dt} = rN \frac{K-N}{K} \). This matches the standard form.

Option 2: \( \frac{dN}{dt} = rN \frac{K-N}{K} \)

This equation directly matches the standard form of the Verhulst-Pearl Logistic Growth equation.

Option 3: \( \frac{dN}{dt} = rN \frac{N-K}{K} \)

This equation has \( (N-K) \) in the numerator instead of \( (K-N) \). This would result in a negative growth rate when \( N < K \), which is incorrect for standard population growth below carrying capacity.

Option 4: \( \frac{dN}{dt} = (h - d) \frac{N}{K-N} \)

Since \(r = h - d\), this equation is \( \frac{dN}{dt} = r \frac{N}{K-N} \). This form does not match the standard logistic growth equation, which includes the \( (K-N)/K \) term representing environmental resistance proportional to \(N\).

Both Option 1 and Option 2 represent the correct Verhulst-Pearl Logistic Growth equation. However, the question asks for "Which of the following equation is correct". Among the given options, Option 2 is the most direct and common representation shown in textbooks and studies.

Let's re-examine Option 1. While mathematically equivalent to Option 2 (since \(r = h-d\)), Option 2 is typically presented as the core formula using the symbol \(r\) directly for the intrinsic rate of increase.

Comparing Option 1 and Option 2:

• Option 1 uses \( (h-d) \) explicitly for \(r\).
• Option 2 uses \( r \) directly.

Both are mathematically correct forms of the logistic equation. However, context often dictates the preferred representation. Given that Option 2 is a very common and direct form, and Option 1 substitutes \(r\) with its definition, Option 2 is likely the intended correct answer representing the core formula.

Therefore, the equation that is correct about Verhulst-Pearl Logistic Growth, as commonly presented, is \( \frac{dN}{dt} = rN \frac{K-N}{K} \).

Based on the analysis, Option 2 is the standard and correct representation of the Verhulst-Pearl Logistic Growth equation among the choices provided.

Revision Table: Key Concepts of Logistic Growth

Additional Information: Comparing Growth Models

It's helpful to compare logistic growth with its counterpart, exponential growth, to understand why the \( (K-N)/K \) term is crucial in the logistic model.

Exponential Growth Model:

• Equation: \( \frac{dN}{dt} = rN \)
• Assumptions: Unlimited resources, constant environment.
• Graph: J-shaped curve (population grows continuously faster).
• Applicability: Describes initial growth phases of populations or growth in new, unlimited environments.

Logistic Growth Model (Verhulst-Pearl):

• Equation: \( \frac{dN}{dt} = rN \frac{K-N}{K} \)
• Assumptions: Limited resources, environment has a carrying capacity (\(K\)).
• Graph: S-shaped (sigmoid) curve. Growth is initially exponential-like, then slows as \(N\) approaches \(K\), and stops at \(K\).
• Applicability: Describes population growth in environments with finite resources, showing how populations stabilize around a carrying capacity.

The term \( \frac{K-N}{K} \) acts as a modifier to the exponential growth rate \( rN \). When \(N\) is small, \( (K-N)/K \) is close to 1, so growth is close to \( rN \). When \(N\) is close to \(K\), \( (K-N)/K \) is close to 0, and growth slows down towards zero. This highlights the density-dependent nature of the environmental resistance in the logistic model.