Problem 24 - Calculus for Biology and Medicine

Question

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find the population sizes for \(t=1,2, \ldots\), 5 and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0}\). R=4, K=20, N_{0}=10

Step-by-Step Solution

Answer

The population sizes are approximately 16, 18.82, 19.70, 19.92, and 19.98 for \( t=1 \) to 5, with a limit of 20 as \( t \rightarrow \infty \).

Step 1: Understand the Beverton-Holt Model

The Beverton-Holt recruitment curve is a discrete-time population model described by the recursive equation \( N_{t+1} = \frac{R N_t}{1 + \left(\frac{R-1}{K}\right) N_t} \), where \( N_t \) is the population at time \( t \). Here, \( R = 4 \) is the growth factor, \( K = 20 \) is the carrying capacity, and \( N_0 = 10 \) is the initial population size.

Step 2: Calculate Population at t=1

Using the Beverton-Holt equation, substitute \( N_0 = 10 \), \( R = 4 \), and \( K = 20 \) to find \( N_1 \):\[ N_1 = \frac{4 \times 10}{1 + \left(\frac{4-1}{20}\right) \times 10} \]Calculate the denominator:\( 1 + \left(\frac{3}{20}\right) \times 10 = 1 + 1.5 = 2.5 \).Substitute back into the equation:\( N_1 = \frac{40}{2.5} = 16 \).

Step 3: Calculate Population at t=2

Use \( N_1 = 16 \) to find \( N_2 \):\[ N_2 = \frac{4 \times 16}{1 + \left(\frac{3}{20}\right) \times 16} \]Calculate the denominator:\( 1 + \left(\frac{3}{20}\right) \times 16 = 1 + 2.4 = 3.4 \).Substitute back:\( N_2 = \frac{64}{3.4} \approx 18.82 \).

Step 4: Calculate Population at t=3

Use \( N_2 \approx 18.82 \) to find \( N_3 \):\[ N_3 = \frac{4 \times 18.82}{1 + \left(\frac{3}{20}\right) \times 18.82} \]Calculate the denominator:\( 1 + \left(\frac{3}{20}\right) \times 18.82 \approx 3.822 \).Substitute back:\( N_3 \approx \frac{75.28}{3.822} \approx 19.70 \).

Step 5: Calculate Population at t=4

Use \( N_3 \approx 19.70 \) to find \( N_4 \):\[ N_4 = \frac{4 \times 19.70}{1 + \left(\frac{3}{20}\right) \times 19.70} \]Calculate the denominator:\( 1 + \left(\frac{3}{20}\right) \times 19.70 \approx 3.955 \).Substitute back:\( N_4 \approx \frac{78.8}{3.955} \approx 19.92 \).

Step 6: Calculate Population at t=5

Use \( N_4 \approx 19.92 \) to find \( N_5 \):\[ N_5 = \frac{4 \times 19.92}{1 + \left(\frac{3}{20}\right) \times 19.92} \]Calculate the denominator:\( 1 + \left(\frac{3}{20}\right) \times 19.92 \approx 3.992 \).Substitute back:\( N_5 \approx \frac{79.68}{3.992} \approx 19.98 \).

Step 7: Determine Long-Term Behavior

To find \( \lim_{t \rightarrow \infty} N_t \), observe that as \( t \) increases, the population approaches the carrying capacity \( K = 20 \). Thus, the long-term population limit is:\( \lim_{t \rightarrow \infty} N_t = K = 20 \).

Key Concepts

Population Dynamics

Population dynamics refers to the study of how and why the numbers of individuals in populations change over time. This change is influenced by various factors such as birth rates, death rates, and migration patterns. In the study of population dynamics, we are interested in understanding these factors and predicting future population sizes.

For example, in the case of the Beverton-Holt recruitment model, which we are examining here, the dynamics are determined by both growth parameters and constraints like carrying capacity. The model gives us insight into how a population grows each period and stabilizes over time.

Studying population dynamics is essential for managing wildlife, conserving endangered species, and understanding the impacts of human activity on different ecosystems.

Carrying Capacity

Carrying capacity is the maximum number of individuals of a particular species that an environment can sustainably support, given available resources such as food, habitat, and other essentials. It acts as a controlling element in population dynamics to ensure that a population does not grow unchecked.

In the Beverton-Holt model, the carrying capacity is represented by the variable \( K \). This value dictates the upper limit the population can reach as it grows. In this specific example, we have a carrying capacity \( K = 20 \), meaning that, in the long-term, our population will level off and not exceed this number.

Recognizing carrying capacity allows us to measure the sustainability of populations within ecosystems, ensuring that they thrive without depleting resources.

Discrete-time Population Model

A discrete-time population model is a mathematical approach where changes in population size are calculated at discrete, separate time intervals. This contrasts with continuous models which calculate changes continuously over time.

The Beverton-Holt model is a well-known discrete-time model. It calculates population size at specific times, like \( t = 1, 2, 3 \), by using a defined equation that considers the current population, growth rate, and carrying capacity.

Discrete-time models are particularly useful when data is collected at regular intervals, such as yearly censuses or quarterly wildlife surveys. They help in capturing changes over each period and identifying trends over time.

Population Growth Models

Population growth models help describe how populations increase or decrease over time. These models can be based on various assumptions and factors, such as resources, reproduction rates, and external influences.

The Beverton-Holt recruitment model is one example of a population growth model that factors in both growth capacity, represented by \( R \), and environmental constraint \( K \). In our case, with \( R = 4 \) and \( K = 20 \), the population grows initially due to the high growth factor but then stabilizes as it approaches the carrying capacity.

Different types of growth models, such as exponential, logistic, and recruitment models like Beverton-Holt, are used to predict how populations behave under various conditions. These models aid in ecological management, conservation efforts, and resource planning.