Logistic Population Growth: Videos & Practice Problems

The logistic population growth model accounts for environmental limitations, contrasting with the exponential model, which assumes unlimited resources. The carrying capacity (k) represents the maximum sustainable population size, creating a sigmoidal growth curve. As population size (n) approaches k, growth rate decreases, eventually reaching zero. The equation for instantaneous growth rate incorporates a term, 1-nk, differentiating it from exponential growth. Understanding these dynamics is crucial for studying population ecology and resource management.

Logistic Growth in a Limited Environment

The logistic population growth model describes how populations grow in environments with limited resources, contrasting with the rare occurrence of unlimited exponential growth. In nature, exponential growth is often short-lived due to environmental constraints, which the logistic model accounts for by incorporating density-dependent factors that influence carrying capacity.

The carrying capacity, denoted as k, represents the maximum population size that an environment can sustain. In graphical representations, this is illustrated as a horizontal line, serving as an asymptote that limits population growth. The logistic growth model's equation for instantaneous population growth rate includes a term that modifies the exponential growth equation, specifically 1 - \frac{n}{k}, where n is the current population size. This term reflects the impact of environmental limitations, resulting in a sigmoidal (S-shaped) growth curve.

Initially, when the population size n is small, the logistic growth approximates exponential growth. However, as n approaches half of the carrying capacity k, the growth rate begins to slow down, indicating a unique characteristic of the logistic model. As the population nears the carrying capacity, the growth rate approaches zero, leading to a stabilization around k.

While populations can temporarily exceed their carrying capacity, this is usually followed by a decline back to or below k. Over time, populations may fluctuate around the carrying capacity before stabilizing. Understanding these dynamics is crucial for applying logistic growth concepts to real-world ecological problems.

Logistic Population Growth Example 1

In this example, we explore the logistic population growth model using barn owls as a case study. The intrinsic growth rate (r) is given as 0.15 owls per year per owl, and the carrying capacity (K) is 500 owls. To find the instantaneous population growth rate (dN/dt) at different population sizes, we utilize the logistic growth equation:

$$ \frac{dN}{dt} = r \cdot N \cdot \left(1 - \frac{N}{K}\right) $$

Here, N represents the current population size. We will calculate dN/dt for three different population sizes: 50, 250, and 450 owls.

1. **For a population size of 50 owls:**

Substituting the values into the equation:

$$ \frac{dN}{dt} = 0.15 \cdot 50 \cdot \left(1 - \frac{50}{500}\right) $$

Calculating this gives:

$$ \frac{dN}{dt} = 0.15 \cdot 50 \cdot \left(1 - 0.1\right) = 0.15 \cdot 50 \cdot 0.9 = 6.75 $$

Thus, the instantaneous population growth rate is 6.75 owls per year.

2. **For a population size of 250 owls:**

Substituting N = 250 into the equation:

$$ \frac{dN}{dt} = 0.15 \cdot 250 \cdot \left(1 - \frac{250}{500}\right) $$

Calculating this gives:

$$ \frac{dN}{dt} = 0.15 \cdot 250 \cdot \left(1 - 0.5\right) = 0.15 \cdot 250 \cdot 0.5 = 18.75 $$

Thus, the instantaneous population growth rate is 18.75 owls per year.

3. **For a population size of 450 owls:**

Substituting N = 450 into the equation:

$$ \frac{dN}{dt} = 0.15 \cdot 450 \cdot \left(1 - \frac{450}{500}\right) $$

Calculating this gives:

$$ \frac{dN}{dt} = 0.15 \cdot 450 \cdot \left(1 - 0.9\right) = 0.15 \cdot 450 \cdot 0.1 = 6.75 $$

Thus, the instantaneous population growth rate is again 6.75 owls per year.

From these calculations, we observe that the growth rate increases as the population approaches half of the carrying capacity (250 owls), reaching a peak at this point. Beyond this, as the population nears the carrying capacity, the growth rate begins to decline. Notably, both the 50 and 450 owl populations yield the same growth rate due to their equidistance from the half carrying capacity, illustrating the dynamics of logistic growth.

In some cases, a population can exceed its carrying capacity for a brief period of time. In a population of dragonflies that approaches its carrying capacity, many females use the last of their energy reserves to reproduce. As a result, the population size reaches 675 dragonflies, but the estimated carrying capacity ($K$) of the area is 550 dragonflies. The r_max value for this population is 0.5. Calculate the rate of population growth (dN/dt) & the per capita rate of growth, r(1 – N/k).

• A. dN/dt = –97, r(1 – N/k) = –0.14
• B. dN/dt = –77, r(1 – N/k) = –0.11
• C. dN/dt = –153, r(1 – N/k) = –0.23
• D. dN/dt = 828, r(1 – N/k) = 0.23

Answer: (B)

A population of spotted hyenas in a South African National Park currently has 87 individuals. The carrying capacity for spotted hyenas in the park is estimated to be 125 hyenas & the intrinsic growth rate ($r$) is 0.25 hyenas/year/hyena. What is the instantaneous rate of population growth of the spotted hyena population?

• A. 22 hyenas/year.
• B. 7 hyenas/year.
• C. 15 hyenas/year.
• D. 13 hyenas/year.

Answer: (B)

Exponential Growth vs. Logistic Growth

In studying population dynamics, two primary models are often compared: exponential growth and logistic growth. Both models describe how populations increase over time, but they do so under different assumptions regarding environmental limitations.

The exponential growth model assumes that a population grows without any constraints, leading to a J-shaped curve when graphed. This model is characterized by a constant intrinsic growth rate, denoted as r. For instance, if we set the initial population size (n) to 50 and the growth rate (r) to 1.0, the population will increase continuously without any signs of slowing down. The formula for instantaneous population growth in this model is given by:

$$\frac{d n}{d t} = r \cdot n$$

In contrast, the logistic growth model incorporates environmental limitations, represented by a carrying capacity (K). This model produces an S-shaped curve, where growth is rapid at first but slows as the population approaches the carrying capacity. The carrying capacity acts as a cap on population growth, and once reached, the growth rate declines to zero. The logistic growth model modifies the growth rate to account for these limitations, using the equation:

$$\frac{d n}{d t} = r \cdot n \cdot \left(1 - \frac{n}{K}\right)$$

Here, the term 1 - n/K reduces the growth rate as the population size increases, reflecting the impact of limited resources. For example, if the carrying capacity is set at 3,000, the growth rate will decrease as the population approaches this limit. Initially, both models may show similar growth patterns, but as the population nears half of the carrying capacity, the differences become more pronounced. The logistic model will start to slow down, while the exponential model continues to grow unchecked.

As the population reaches the carrying capacity, the logistic growth model's growth rate approaches zero, indicating that the population has stabilized. If the population exceeds the carrying capacity, the growth rate becomes negative, leading to a decline in population size. This reflects the reality that populations cannot sustain growth indefinitely without facing environmental constraints.

In summary, while both exponential and logistic growth models describe population dynamics, they differ significantly in their treatment of environmental limitations. Understanding these differences is crucial for predicting population behavior in various ecological contexts.

Summary of Key Features & Assumptions for Each Population Growth Model

Understanding population growth models is essential for studying ecological dynamics. There are three primary models: linear, exponential, and logistic, each with distinct characteristics and assumptions that influence how populations grow over time.

The linear population growth model is defined by a constant population growth rate, represented as \(\Delta n / \Delta t\), which remains unchanged regardless of the population size (n). This means that the growth is steady and predictable, making it unique among the models.

In contrast, the exponential growth model features a growth rate that is directly proportional to the population size. This relationship indicates that as the population increases, the growth rate accelerates, leading to rapid population expansion. The mathematical representation of this relationship can be expressed as \(\Delta n / \Delta t = r \cdot n\), where r is the constant per capita growth rate.

The logistic growth model introduces a more complex dynamic, where the growth rate initially increases but eventually decreases as the population approaches its carrying capacity (k). This model accounts for environmental limitations, making it realistic for many natural populations. The logistic growth can be described by the equation \(\Delta n / \Delta t = r \cdot n \cdot (1 - \frac{n}{k})\), where the term (1 - \frac{n}{k}) modifies the growth rate as the population nears its carrying capacity.

Key features of these models include:

• The linear model has a constant growth rate, while the exponential model's growth rate is proportional to the population size.
• The logistic model uniquely incorporates a carrying capacity and density-dependent factors, reflecting resource limitations.
• All three models assume a closed population, where changes in size are solely due to births and deaths, ignoring immigration and emigration.
• They also assume a homogeneous environment, disregarding age structure, sex ratio, and external factors that could influence population dynamics.

In summary, the linear model is characterized by constant growth, the exponential model by proportional growth, and the logistic model by growth that is limited by environmental factors. Understanding these distinctions is crucial for applying the appropriate model to ecological studies and predicting population trends.

Here’s what students ask on this topic:

What is the logistic population growth model?

The logistic population growth model describes how populations grow in environments with limited resources. Unlike the exponential growth model, which assumes unlimited resources and continuous growth, the logistic model incorporates environmental limitations. The key feature of this model is the carrying capacity (k), which represents the maximum population size that an environment can sustain. As the population size (n) approaches the carrying capacity, the growth rate decreases and eventually reaches zero, resulting in a sigmoidal or S-shaped growth curve. The equation for the logistic growth model includes a term, (1 - n/k), which accounts for these limitations.

How does the logistic growth model differ from the exponential growth model?

The logistic growth model differs from the exponential growth model in that it accounts for environmental limitations on population growth. While the exponential model assumes unlimited resources and continuous growth, the logistic model incorporates the concept of carrying capacity (k), which is the maximum population size an environment can sustain. As the population size (n) approaches k, the growth rate decreases and eventually reaches zero, creating a sigmoidal or S-shaped curve. The logistic growth equation includes a term, (1 - n/k), which modifies the growth rate based on the population size relative to the carrying capacity.

What is carrying capacity in the context of logistic population growth?

In the context of logistic population growth, carrying capacity (k) refers to the maximum population size that an environment can sustain over time. It is determined by the availability of resources such as food, water, and shelter. As the population size (n) approaches the carrying capacity, the growth rate slows down and eventually reaches zero, resulting in a stable population size. The carrying capacity acts as a limiting factor, preventing the population from growing indefinitely and ensuring that the population remains within the bounds of the available resources.

What factors influence the carrying capacity of an environment?

The carrying capacity of an environment is influenced by several factors, including the availability of resources such as food, water, and shelter. Other factors include the presence of predators, disease, and competition among individuals for resources. Environmental conditions such as climate, weather patterns, and habitat quality also play a role in determining the carrying capacity. Human activities, such as deforestation, pollution, and urbanization, can further impact the carrying capacity by altering the availability and quality of resources.

How does the logistic growth model account for environmental limitations?

The logistic growth model accounts for environmental limitations by incorporating the concept of carrying capacity (k) into its equation. The model includes a term, (1 - n/k), which modifies the growth rate based on the population size (n) relative to the carrying capacity. As the population size approaches the carrying capacity, the growth rate decreases and eventually reaches zero, resulting in a stable population size. This term ensures that the population growth slows down as resources become limited, preventing the population from exceeding the carrying capacity and depleting the available resources.