Problem 3 - 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. \(R=1.5, K=40\)

Step-by-Step Solution

Answer

The recursion is \( \frac{N_t}{N_{t+1}} = \frac{1 + \frac{N_t}{80}}{1.5} \).

Step 1: Understanding the Beverton-Holt Model

The Beverton-Holt recruitment curve is a discrete-time model used to describe population growth. It is represented by the equation: \[ N_{t+1} = \frac{R N_t}{1 + (\frac{R - 1}{K}) N_t} \]where \(N_t\) is the population at time \(t\), \(N_{t+1}\) is the population at time \(t+1\), \(R\) is the growth factor, and \(K\) is the carrying capacity.

Step 2: Substitute Given Values

Insert the given values of \(R = 1.5\) and \(K = 40\) into the equation. The equation becomes:\[ N_{t+1} = \frac{1.5 N_t}{1 + (\frac{1.5 - 1}{40}) N_t} = \frac{1.5 N_t}{1 + (\frac{0.5}{40}) N_t} = \frac{1.5 N_t}{1 + \frac{N_t}{80}} \]

Step 3: Find the Recursion Equation

Rearrange the equation to express it as a function of \(N_t / N_{t+1}\):\[ N_{t+1} = \frac{1.5 N_t}{1 + \frac{N_t}{80}} \]Taking reciprocal on both sides:\[ \frac{N_t}{N_{t+1}} = \frac{1 + \frac{N_t}{80}}{1.5} \]This represents the recursion for the Beverton-Holt recruitment curve.

Step 4: Graph the Function

To graph \( \frac{N_{t}}{N_{t+1}} \) as a function of \(N_t\), plot the equation \[ \frac{N_t}{N_{t+1}} = \frac{1 + \frac{N_t}{80}}{1.5} \].The graph is a straight line with a positive slope, starting at a y-intercept of \(\frac{1}{1.5}\) when \(N_t = 0\) and asymptotically approaching \(\frac{1}{1.5} + \frac{1}{80 \times 1.5}N_t\) as \(N_{t}\) increases.

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population over time. In biology and ecology, understanding how populations grow can be crucial for conserving species and managing ecosystems.
With the Beverton-Holt Model, this growth is analyzed in discrete time steps. This means we look at the population at specific intervals, such as years or generations.
This model is particularly useful when considering populations that have a defined breeding season, since it allows us to predict the size of the population from one time period to the next.
Essentially, the model focuses on how populations can increase given certain conditions, while also considering constraints like limited resources.

Discrete-Time Model

The Beverton-Holt Model operates as a discrete-time model, which means it observes population changes at specific, separate time intervals rather than continuously.
In these types of models, we calculate the size of the population at one time step and then use this value to predict the population size at the next time step.
Discrete-time models are commonly used in ecology when populations breed in

Carrying Capacity

Carrying capacity, denoted by the symbol K , is a crucial concept in population dynamics. It represents the maximum population size that an environment can sustain indefinitely given the available resources like food, habitat, water, and other essentials.
In the Beverton-Holt Model, carrying capacity acts as a limiting factor that balances the growth potential of a population. As population size approaches the carrying capacity, the growth rate slows down because the resources become limited.
Therefore, it prevents the population from growing without bounds. In the provided exercise, the carrying capacity K is set at 40, meaning this is the upper limit of the population size that the environment can support.

Growth Parameter

The growth parameter, often denoted by R in the Beverton-Holt Model, determines how fast a population can grow under ideal conditions, when not limited by resources.
It influences the reproductive potential of a population. A higher growth parameter means the population can potentially increase more rapidly if there are enough resources.
In mathematical terms, R modifies how much the population at the next time point is expected to be. In the exercise, R is set at 1.5, suggesting that the population has a moderately high capacity to grow from one period to the next, although that growth is tempered by the carrying capacity.