Problem 1 - 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=15

Step-by-Step Solution

Answer

For \(R = 2\) and \(K = 15\), the recursion is \(N_{t+1} = \frac{2 \cdot N_{t}}{1 + \frac{N_{t}}{15}}\), and the graph plots \(\frac{N_t}{N_{t+1}} = \frac{1 + \frac{N_t}{15}}{2}\).

Step 1: Understanding the Beverton-Holt model

The Beverton-Holt model is described as a difference equation used to model population growth. The relationship is given by the formula: \( N_{t+1} = \frac{R \cdot N_{t}}{1 + \frac{N_{t}}{K}} \), where \( N_t \) is the population at time \( t \), \( R \) is the growth parameter, and \( K \) is the carrying capacity.

Step 2: Substituting the given values

Given the values \( R = 2 \) and \( K = 15 \), substitute these into the Beverton-Holt equation: \( N_{t+1} = \frac{2 \cdot N_{t}}{1 + \frac{N_{t}}{15}} \). This equation will help us graph and find the recursion for population \( N_{t} \).

Step 3: Recursion and deduced expression

The recursion simply refers to referring back to the calculated expression, which in this case is; \( N_{t+1} = \frac{2 \cdot N_{t}}{1 + \frac{N_{t}}{15}} \). It behaves iteratively, meaning that after establishing \( N_{t+1} \), one uses it to calculate \( N_{t+2} \), and so forth.

Step 4: Graph setup

Set up a set of axes with the x-axis representing \( N_{t} \) and the y-axis representing \( \frac{N_{t}}{N_{t+1}} \). We will use the understood equation to plot points. Rearrange the equation \( N_{t+1} = \frac{2 \cdot N_{t}}{1 + \frac{N_{t}}{15}} \) to express \( \frac{N_{t}}{N_{t+1}} \): \( \frac{N_{t}}{N_{t+1}} = \frac{1 + \frac{N_{t}}{15}}{2} \).

Step 5: Plotting the curve

For various values of \( N_{t} \), calculate \( \frac{N_{t}}{N_{t+1}} \) using \( \frac{N_{t}}{N_{t+1}} = \frac{1 + \frac{N_{t}}{15}}{2} \). For example, if \( N_{t}=30 \), \( \frac{N_{t}}{N_{t+1}}=\frac{1 + 2}{2} = 1.5 \). Plot these points on the graph and connect them to visualize the relationship. Expect the curve to increase as \( N_{t} \) increases, reflecting the carrying capacity's limiting effect.

Key Concepts

Population Growth

Population growth is a fundamental concept in ecology, biology, and environmental science. It refers to the change in the number of individuals in a population over time. Understanding how populations grow allows scientists to predict their future sizes and to manage resources effectively.

The Beverton-Holt model is one way to describe population growth. This model is particularly useful for animal populations where each generation replaces itself relatively quickly. It's used to project how a population will grow when a certain number of resources and space are available. The growth parameter, often denoted as \( R \), represents the reproductive rate of the population. If \( R \) is greater than 1, the population is expected to grow.

However, growth doesn't continue indefinitely. A population cannot grow forever without limitation. This is where the concept of carrying capacity (\( K \)) comes into play, and we'll explore that next.

Carrying Capacity

Carrying capacity, symbolized by \( K \), is the maximum population size that an environment can sustain indefinitely. It takes into account the availability of resources such as food, water, habitat, and other necessities needed for survival. Once a population reaches its carrying capacity, its growth slows down and eventually levels off because the resources are fully utilized.

In the context of the Beverton-Holt model, the carrying capacity affects how the population grows towards a stable number. The equation, \( N_{t+1} = \frac{R \cdot N_{t}}{1 + \frac{N_{t}}{K}} \), incorporates this concept by restricting growth as \( N_{t} \) approaches \( K \). As the population size approaches carrying capacity, the fraction \( \frac{N_{t}}{K} \) increases, leading to a smaller denominator in the formula, and thus, smaller increases in population size over each time step. - Population growth is dynamically balanced by the carrying capacity.- As resources become scarcer, growth rates decrease, reaching equilibrium.Understanding carrying capacity helps in predicting how populations interact with their ecosystems and in managing wildlife resources.

Difference Equation

A difference equation is a mathematical tool used to model how a quantity changes over discrete time intervals. Unlike differential equations, which work on continuous change, difference equations are perfect for analyzing systems that evolve step by step, like population growth.

The Beverton-Holt model is expressed as a difference equation: \[ N_{t+1} = \frac{R \cdot N_{t}}{1 + \frac{N_{t}}{K}} \] This equation predicts the population size at the next time point, \( N_{t+1} \), based on the current population size, \( N_{t} \). The difference equation accounts for the growth parameter \( R \) and is tempered by the carrying capacity \( K \).

The iterative nature of difference equations allows for modeling how populations will evolve over time:

• Starting with an initial population, apply the equation to find the next population size.
• Continue the process to assess future population sizes.

This modeling method is straightforward yet powerful for predicting population trends in various ecological and environmental settings. It provides insights into how populations might change under different scenarios, guiding conservation and resource management efforts.