Problem 6 - 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 \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=2, K=150\)

Step-by-Step Solution

Answer

The recursion is \( N_{t+1} = \frac{300N_t}{150 + N_t} \); graph \( \frac{N_t}{N_{t+1}} = \frac{150+N_t}{300} \).

Step 1: Understanding the Beverton-Holt Recruitment Curve

The Beverton-Holt model is used to describe how a population grows in an environment with a limiting factor, described by a carrying capacity. It is commonly written as \( N_{t+1} = \frac{RN_t}{1 + \frac{N_t}{K}} \), where \( R \) is the growth parameter, \( K \) is the carrying capacity, \( N_t \) is the population at time \( t \), and \( N_{t+1} \) is the population at the next time step.

Step 2: Substitute Given Values into the Formula

We are given \( R = 2 \) and \( K = 150 \). Substitute these values into the Beverton-Holt formula: \[N_{t+1} = \frac{2N_t}{1 + \frac{N_t}{150}}.\]

Step 3: Simplify the Recursion Formula

Simplify the expression: \[N_{t+1} = \frac{2N_t}{1 + \frac{N_t}{150}} = \frac{2N_t}{\frac{150 + N_t}{150}} = \frac{2N_t \times 150}{150 + N_t}.\]Thus, the recursion formula becomes:\[N_{t+1} = \frac{300N_t}{150 + N_t}.\]

Step 4: Plotting the Graph Function \( \frac{N_t}{N_{t+1}} \)

To graph \( \frac{N_t}{N_{t+1}} \), substitute the expression for \( N_{t+1} \) from the Beverton-Holt formula: \[\frac{N_t}{N_{t+1}} = \frac{N_t}{\frac{300N_t}{150 + N_t}} = \frac{150 + N_t}{300}.\]This shows a linear relationship with \( N_t \). Plot this as a line with slope \( \frac{1}{300} \) and y-intercept \( \frac{150}{300} = 0.5 \).

Key Concepts

Population Growth

Population growth in ecological terms refers to the change in population size over time. In mathematical models like the Beverton-Holt recruitment curve, this concept helps us understand how populations evolve in particular environments. This model is essentially used when considering populations with a specific limiting factor causing population dynamics, a factor like resources that the environment can sustain. When a population grows, it typically begins with a rapid, almost exponential increase. However, as resources become scarce or other factors like competition kick in, growth slows down. In the Beverton-Holt model, this is observed through a formula that calculates future populations based on current numbers. The Beverton-Holt curve stems from assumptions that the population grows in discrete steps rather than continuously. Understanding these growth patterns is key to predicting changes in population sizes over time, which also aids in conservation and resource management.

Carrying Capacity

In the context of population dynamics, carrying capacity, denoted as \(K\), is the maximum population size that an environment can sustain indefinitely. This concept illustrates the balance between population growth and the finite nature of resources. As populations approach carrying capacity, the growth rate slows because resources such as food, space, and water become limited, leading to increased competition. In the Beverton-Holt curve, carrying capacity helps limit the growth of the population so it doesn't exceed what the environment can support. Understanding carrying capacity is crucial because it not only provides a ceiling for growth but also informs us about the sustainability of the environment. By knowing \(K\), ecologists and resource managers can make informed decisions about preserving natural habitats and ensuring that wildlife populations remain healthy and stable.

Growth Parameter

The growth parameter, represented as \(R\) in the Beverton-Holt model, is a crucial component of understanding how populations change over time. Think of \(R\) as a representation of the maximum rate at which the population could grow under ideal conditions. In the given exercise, \(R = 2\), meaning the population has the potential to double at each time step, assuming unlimited resources and no environmental constraints. However, since resources are limited, as shown by the carrying capacity \(K\), this growth isn't sustained indefinitely. The growth parameter is foundational because it allows us to determine the potential explosiveness of population growth. High \(R\) values suggest that even small changes in population can lead to large impacts on the future size of the population.

Recursion Formula

The recursion formula is a mathematical expression that helps predict future population sizes based on current values. In the Beverton-Holt model, the recursion formula is given by: \[N_{t+1} = \frac{RN_t}{1 + \frac{N_t}{K}}\] This formula succinctly captures the dynamic relationship between population growth, the growth parameter, and the carrying capacity. It shows how next-step population sizes are computed, making it clear that as the population size \(N_t\) increases, it approaches a ceiling regulated by \(K\). By substituting the values \(R = 2\) and \(K = 150\), we obtain a more specific prediction formula for this particular scenario. The resulting equation, \[N_{t+1} = \frac{300N_t}{150 + N_t},\] reveals how \(N_t\) impacts future populations. Understanding this formula is critical, as it enables the modeling of population sizes over time and aids in resource management and conservation strategies.