Problem 4 - 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. \(R=3, K=120\)

Step-by-Step Solution

Answer

The recursion is \(N_{t+1} = \frac{360N_t}{120 + N_t}\). As \(N_t\) increases, the function \( \frac{360}{120 + N_t} \) decreases.

Step 1: Understand the Beverton-Holt model

The Beverton-Holt recruitment curve describes how a population grows over time, given a growth rate parameter \(R\) and a carrying capacity \(K\). The equation defining this model is \(N_{t+1} = \frac{R \cdot N_t}{1 + \frac{N_t}{K}}\), where \(N_t\) is the population at time \(t\) and \(N_{t+1}\) is the population at time \(t+1\).

Step 2: Simplify the equation for given values

Substitute \(R = 3\) and \(K = 120\) into the Beverton-Holt equation. This gives us \(N_{t+1} = \frac{3 \cdot N_t}{1 + \frac{N_t}{120}}\). Simplifying further, this turns into: \(N_{t+1} = \frac{3N_t}{1 + \frac{N_t}{120}} = \frac{360N_t}{120 + N_t}\).

Step 3: Formulate the recursion

Understand that from the simplified equation, the recursion for the population is \(N_{t+1} = \frac{360N_t}{120 + N_t}\). This implies that the population at the next time step depends on the current population size \(N_t\).

Step 4: Graph the function

To visualize \(\frac{N_{t+1}}{N_t}\) as a function of \(N_t\), plot the graph of \( \frac{\frac{360N_t}{120 + N_t}}{N_t} = \frac{360}{120 + N_t}\). As \(N_t\) increases, \( \frac{360}{120 + N_t} \) will decrease because the denominator increases while the numerator remains constant.

Step 5: Analyze the graph

The graph of \( \frac{360}{120 + N_t} \) will approach zero as \(N_t\) becomes very large, indicating that the population growth rate reduces as you near the carrying capacity. Initially, when \(N_t\) is small, the ratio is high, indicating fast growth.

Key Concepts

Population growth

The Beverton-Holt model is a mathematical way to look at how populations grow over time. Imagine a group of animals living in a region. They start out with a certain number and they reproduce some babies each year. This model uses the growth rate, called parameter \( R \), to explain how quickly the population can increase. If \( R \) is high, the population grows faster. If it's low, the growth is slower. However, growth doesn't go on forever at the same rate. This is where carrying capacity comes in.

Population growth isn't just about multiplying numbers. It also hits limits. Various real-world factors, like space, nutrients, or competition, make it harder for the population to keep growing. - At the start, populations grow quickly because there is less competition and more resources.- As time goes on, these resources get used up and growth starts to slow.The Beverton-Holt model shows this growth using a neat equation that predicts future population sizes based on the current ones.

Carrying capacity

Carrying capacity, indicated by \( K \), is the maximum population size that the environment can sustain indefinitely. Imagine a park with a pond and a limited amount of fish. If there are too many fish, they won't have enough food or space, so some won't survive. That's the concept of carrying capacity in action.

When populations approach this capacity, things change:- They can't grow as fast because the extra individuals don't fit comfortably into the ecosystem.- Resources become limited, leading to competition among individuals, which slows the rate of population increase.Think of carrying capacity like a ceiling. Population growth hits this ceiling when there are just enough resources to support a certain number of individuals. In mathematical terms, as the population \( N_t \) (at time \( t \)) gets closer to the carrying capacity \( K \), the growth of the population slows down, depicted by the formula of the Beverton-Holt model.

Recursion formula

A recursion formula in the Beverton-Holt model connects the population from one year to the next. It uses the current state to figure out what comes next. This formula helps predict future population sizes, showing how the population transitions from time \( t \) to time \( t+1 \).

The recursion formula used here is derived by plugging in the given growth rate and carrying capacity into the original equation:\[N_{t+1} = \frac{360N_t}{120 + N_t}\]- **\( N_t \):** Current population at time \( t \).- **\( N_{t+1} \):** Future population at time \( t+1 \).This equation tells us how the current population affects the future one, taking into account how resources limit growth. When you increase \( N_t \), the growth slows because the denominator (\( 120 + N_t \)) gets larger, reflecting the effects of reaching the carrying capacity.

Understanding this recursion helps explain why populations grow rapidly at first but slow as they near the carrying limits, creating a balance over time.