Problem 9 - 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 \(R\) and \(K .\) N_{t+1}=\frac{1.5 N_{t}}{1+0.5 N_{t} / 30}

Step-by-Step Solution

Answer

Growth parameter \( R = 1.5 \) and carrying capacity \( K = 60 \).

Step 1: Identify the Beverton-Holt Formula

The Beverton-Holt recruitment model is given by \[ N_{t+1} = \frac{R N_t}{1 + \frac{N_t}{K}} \], where \(R\) is the growth parameter and \(K\) is the carrying capacity.

Step 2: Match Given Formula with Beverton-Holt Formula

We need to match the given equation \( N_{t+1} = \frac{1.5 N_t}{1 + 0.5 N_t / 30} \) to the Beverton-Holt form. Comparing the numerators, we identify \( R = 1.5 \) as the growth parameter.

Step 3: Identify Denominator Structure

In the denominator of the given formula, the term is \( 1 + 0.5 N_t / 30 \). In the Beverton-Holt form, this is \( 1 + \frac{N_t}{K} \). Set \( \frac{N_t}{K} = 0.5 \frac{N_t}{30} \) to find \( K \).

Step 4: Solve for Carrying Capacity \( K \)

Solving for \( K \) in \( \frac{1}{K} = \frac{0.5}{30} \) yields \( K = \frac{30}{0.5} = 60 \). Thus, the carrying capacity \( K = 60 \).

Step 5: Verify Solution

With \( R = 1.5 \) and \( K = 60 \), the given equation \( N_{t+1} = \frac{1.5 N_t}{1 + 0.5 N_t / 30} \) corresponds correctly to the Beverton-Holt model \( N_{t+1} = \frac{R N_t}{1 + \frac{N_t}{K}} \).

Key Concepts

Population Growth in the Beverton-Holt Model

The Beverton-Holt model is a fundamental concept used to describe population growth. It is particularly useful in ecological studies where a population approaches an upper limit, also known as the carrying capacity. The formula for this model is expressed as:

• \[ N_{t+1} = \frac{R N_t}{1 + \frac{N_t}{K}} \]

In this model, \(N_t\) represents the current population size and \(N_{t+1}\) the population size at the next time interval. This recursive relationship helps us understand how populations grow over time.
Unlike simple exponential models where populations can grow infinitely, the Beverton-Holt model accounts for environmental limits. Populations are seen to grow rapidly when small and slow down as they approach the carrying capacity \(K\). This is crucial in understanding real-world populations where resources such as space, food, and other essentials are limited. These natural limits ensure we model more realistic scenarios.
Through this model, researchers can predict how quickly a population might reach a stable state or how it might respond to changes in the environment.

Understanding the Growth Parameter

The growth parameter, denoted as \(R\) in the Beverton-Holt model, plays a crucial role in determining how a population increases relative to its current size. Specifically, it quantifies the rate at which a small population can grow when environmental conditions are ideal.

• The growth parameter is directly related to the numerator of the Beverton-Holt equation.
• Higher \(R\) values suggest rapid population growth, while lower values indicate slower growth rates.

In practical applications, determining \(R\) involves analyzing ecological or biological data to fit the model. For example, if a population census over several years shows a steady rise in numbers, this information helps in estimating \(R\).
In the given exercise, comparing the given formula \( N_{t+1} = \frac{1.5 N_t}{1 + 0.5 N_t / 30} \) with the standard Beverton-Holt formula reveals that \(R = 1.5\). This implies that the present population is expected to increase by 1.5 times under optimal conditions. Such rapid initial growth is typical in newly established populations where competition for resources is initially minimal.

Carrying Capacity: The Ecological Ceiling

Carrying capacity, represented as \(K\) in the Beverton-Holt model, is crucial as it defines the upper limit to the size of a population that an environment can sustain. It effectively acts as the ecological ceiling.

• It stabilizes population growth when resources become scarce or competition increases.
• Environmental factors like food supply, habitat space, and climate conditions influence \(K\).

What's fascinating about \(K\) is that it doesn't denote a fixed point but can fluctuate due to changes in the environment or human interventions. It's a dynamic threshold that adjusts to external conditions, which is why it's essential for sustainability studies.
In solving for \(K\) in the exercise, where \(N_{t+1} = \frac{1.5 N_t}{1 + 0.5 N_t / 30}\), we found that \(K = 60\). This implies that when the population size approaches 60, the growth rate will slow, and the population will stabilize around this number, assuming no changes in the environment. Understanding \(K\) helps in managing resources and anticipating future population needs.