Problem 17 - 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 all fixed points. N_{t+1}=\frac{3 N_{t}}{1+N_{t} / 30}

Step-by-Step Solution

Answer

The fixed points are \( N^* = 0 \) and \( N^* = 60 \).

Step 1: Understand the Beverton-Holt Recruitment

The Beverton-Holt recruitment model is used to describe population dynamics. The equation given is \( N_{t+1} = \frac{3N_t}{1 + \frac{N_t}{30}} \), where \( N_t \) is the population at time \( t \), and \( N_{t+1} \) is the population at time \( t+1 \). The parameters \( R = 3 \) and \( K = 30 \) in this equation represent the growth rate and carrying capacity, respectively.

Step 2: Define a Fixed Point

A fixed point is a value of \( N \) where the population remains constant over time, which means \( N_{t+1} = N_t = N^* \). To find fixed points, set \( N_{t+1} = N_t \) in our given equation.

Step 3: Set up the Fixed Point Equation

Set the equation to find fixed points: \( N^* = \frac{3N^*}{1 + \frac{N^*}{30}} \). Simplifying this will help find the solutions for \( N^* \).

Step 4: Simplify the Equation

Multiply both sides by the denominator \( 1 + \frac{N^*}{30} \) to eliminate the fraction: \[ N^* (1 + \frac{N^*}{30}) = 3N^* \] This simplifies to: \[ N^* + \frac{N^{*2}}{30} = 3N^* \]

Step 5: Solve for Fixed Points

Rearrange the equation: \[ \frac{N^{*2}}{30} = 3N^* - N^* \] \[ \frac{N^{*2}}{30} = 2N^* \] Multiply across by 30: \[ N^{*2} = 60N^* \] Rearrange to: \[ N^{*2} - 60N^* = 0 \].

Step 6: Factor the Equation

Factor out \( N^* \) from the equation \( N^{*2} - 60N^* = 0 \): \[ N^*(N^* - 60) = 0 \]. This gives two solutions: \( N^* = 0 \) and \( N^* = 60 \).

Step 7: Verify Fixed Points

The solutions \( N^* = 0 \) and \( N^* = 60 \) are fixed points since substituting them back into the original equation satisfies the condition \( N_{t+1} = N_t \). Both are potential stable points under the dynamics described.

Key Concepts

Understanding Population Dynamics

The study of population dynamics involves examining how populations of organisms change over time. This includes understanding factors that influence population size and the rates of growth or decline. One important aspect is birth and death rates, which directly affect population numbers.
Resource availability and environmental conditions also play crucial roles. As resources such as food and space become limited, populations may reach a maximum size, known as the carrying capacity.

• Birth rates and death rates affect population size.
• Resource limitations often lead to a carrying capacity.
• Population dynamics explore how these factors interact over time.

This field helps us predict how populations will evolve, which is essential for conservation efforts and resource management.

The Recruitment Curve and its Role in Population Models

The recruitment curve is a concept used to understand how new individuals are added to a population. In the context of the Beverton-Holt model, the recruitment curve helps us express how populations grow with time.
The Beverton-Holt model is described by the equation: \[ N_{t+1} = \frac{RN_t}{1 + \frac{N_t}{K}} \]where \(N_t\) is the population at time \(t\), \(R\) is the growth parameter, and \(K\) is the carrying capacity.

• The recruitment curve shows the relationship between existing population size and new introductions.
• The shape of the curve helps determine how quickly a population can grow before reaching its limit.
• It is fundamental in understanding population management.

The recruitment curve is significant in predicting how populations adjust to changes in environment or resource levels.

Exploring Fixed Points in Population Dynamics

A fixed point in population dynamics is a state where the population size remains unchanged over time. This means that if a population reaches a fixed point, it is stable and will not increase or decrease under the current conditions.
In mathematical terms, a fixed point occurs when \(N_{t+1} = N_t = N^*\), which implies there is no net growth.

• Fixed points indicate stability in population size.
• These are crucial for understanding long-term population trends.
• In the Beverton-Holt model, fixed points can show where a population will settle over time.

Finding these points involves setting the equation for population growth equal to itself and solving for \(N^*\). The exercise finds solutions for \(N^* = 0\) and \(N^* = 60\), indicating potential equilibrium states.

What is Carrying Capacity?

Carrying capacity is a key concept in population dynamics, indicating the maximum number of individuals an environment can sustain indefinitely. It is determined by the amount of available resources, such as food, habitat, and water.

• Carrying capacity limits population growth.
• Environmental factors such as climate and resource availability directly affect it.
• It shapes how populations adjust to their surroundings.

In the Beverton-Holt model, the carrying capacity, \(K\), is a central parameter. It shapes the recruitment curve, determining how population growth slows as numbers approach \(K\). For example, in the exercise, \(K = 30\) suggests the environment can support a maximum of 30 individuals.
This understanding helps manage populations sustainably, keeping them in balance with their environment.