Problem 18 - 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{5 N_{t}}{1+N_{t} / 120}

Step-by-Step Solution

Answer

The fixed points are \(N = 0\) and \(N = 480\).

Step 1: Identify the Beverton-Holt equation

The Beverton-Holt model is given as \( N_{t+1} = \frac{R N_t}{1 + N_t / K} \). In this equation, \(R\) is the growth parameter and \(K\) is the carrying capacity.

Step 2: Compare to the given population model

The given equation is \( N_{t+1} = \frac{5 N_t}{1 + N_t/120} \). Thus, we identify \(R = 5\) and \(K = 120\) from comparison with the general form of the Beverton-Holt model.

Step 3: Set up the fixed point equation

A fixed point is a value of \(N_t\) such that the population does not change over time, meaning \(N_{t+1} = N_t\). Set the equation \(N_{t+1} = N_t = x\) to find the fixed points: \[ x = \frac{5x}{1 + x/120} \].

Step 4: Solve the fixed point equation

Multiply both sides by \(1 + x/120\) to eliminate the fraction: \[ x(1 + x/120) = 5x \].This simplifies to: \[ x + \frac{x^2}{120} = 5x \].Rearrange to obtain: \[ \frac{x^2}{120} = 4x \].

Step 5: Simplify and solve the quadratic equation

Multiply everything by 120 to get rid of the fraction: \[ x^2 = 480x \].Rearrange to obtain a standard quadratic form: \[ x^2 - 480x = 0 \].Factor out \(x\): \[ x(x - 480) = 0 \].

Step 6: Find the solutions for x

Setting each factor in the equation \(x(x - 480) = 0\) to zero gives the solutions: \(x = 0\) and \(x = 480\).

Key Concepts

Beverton-Holt recruitment curve

The Beverton-Holt recruitment curve is a simple model used to describe population dynamics over time. It captures how a population grows each period based on factors like individual reproduction rates and environmental constraints. This model is formulated as:

• \( N_{t+1} = \frac{R N_t}{1 + N_t / K} \)

Here, \(N_{t+1}\) signifies the population size in the next time period, and \(N_t\) represents the current population size. The Beverton-Holt equation artfully balances growth potential with limitations caused by resource availability.

This equation is often used in ecology to predict how a population might change, considering limitations like food, space, or other critical resources that act as a cap on exponential growth. Such models help policymakers and scientists understand and manage wildlife populations and fish stocks effectively.

Population growth

Population growth implicates how populations expand or decline over time. In ecological models like the Beverton-Holt model, population growth is determined not by unlimited exponential growth. Instead, it accounts for negative feedback mechanisms.

• Initially, a population may grow rapidly when resources appear abundant.
• As the population number approaches the carrying capacity, growth slows.

This nonlinear growth ensures that the population doesn't over-exploit its environment, potentially leading to dramatic declines. Such controlled growth models are vital for sustainable management of ecosystems and understanding broader biological processes.

Carrying capacity

Carrying capacity, denoted as \(K\) in population models, defines the maximum population size an environment can sustain indefinitely. Beyond this threshold, the environment's resources are insufficient to support additional individuals.

The carrying capacity is influenced by several factors:

• Available food and water resources
• Living space
• Competition with other species

In the Beverton-Holt model, the carrying capacity directly affects how population growth slows as **\(N_t\)** approaches \(K\). This demonstrates a realistic cap on population growth, reflecting real-world limitations on resources.

It's important to understand that carrying capacities can change due to environmental shifts, policy changes, or technological advances, making dynamic models crucial for accurate forecasting.

Growth parameter

The growth parameter, abbreviated as \(R\) in the Beverton-Holt model, reflects the potential rate at which a population can grow in an ideal scenario without limiting factors. A larger \(R\) means that, in the absence of constraints, the population can grow more quickly.

In practical terms, \(R\) represents factors like the reproductive rate of the population and vitality of the environment.

• An \(R\) value less than 1 can lead to a declining population.
• An \(R\) value greater than 1 suggests a population can grow if not constrained by resources.

In our example, \(R = 5\) implies a strong potential for growth, affecting how rapidly the population expands until it nears the carrying capacity \(K\). Balancing \(R\) with \(K\) ensures ecological stability and sustainability.