Problem 15 - Calculus for Biology and Medicine

Question

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{3 N_{t}}{1+N_{t} / 30}\)

Step-by-Step Solution

Answer

The fixed points are \(N_t = 0\) and \(N_t = 60\).

Step 1: Understand the Beverton-Holt Model

The Beverton-Holt model describes a type of discrete population growth. The formula given, \[ N_{t+1} = \frac{3N_t}{1 + N_t/30} \], tells us how the population size at time \(t+1\) depends on the size at time \(t\). We want to find fixed points, which occur when the population doesn't change, meaning \(N_{t+1} = N_t\).

Step 2: Set up the Fixed Point Equation

To find the fixed points, set \(N_{t+1} = N_t\). This gives the equation:\[ N_t = \frac{3N_t}{1 + N_t/30} \].Here, we're looking for the \(N_t\) values that satisfy this equation.

Step 3: Simplify the Equation

Multiply both sides by \(1 + N_t/30\) to clear the fraction:\[ N_t (1 + N_t/30) = 3N_t \].This simplifies to:\[ N_t + \frac{N_t^2}{30} = 3N_t \].

Step 4: Rearrange and Factor

Rearrange the equation:\[ \frac{N_t^2}{30} - 2N_t = 0 \].Factor out the common term \(N_t\):\[ N_t \left( \frac{N_t}{30} - 2 \right) = 0 \].

Step 5: Solve for Fixed Points

The equation \(N_t \left( \frac{N_t}{30} - 2 \right) = 0\) gives us two possibilities:1. \(N_t = 0\)2. \(\frac{N_t}{30} - 2 = 0\), which simplifies to \(N_t = 60\).

Step 6: Conclusion

The fixed points of the Beverton-Holt model are \(N_t = 0\) and \(N_t = 60\). These are the population sizes where the population remains constant over time.

Key Concepts

Fixed Points: Stability in Population Models

In the context of population dynamics, a fixed point is a situation where the population size doesn't change over time. For the Beverton-Holt model, this means that at these points, the population for the next generation (\(N_{t+1}\)) equals the current population (\(N_t\)). The step-by-step solution illustrates this process clearly.
To identify fixed points, you equate \(N_{t+1}\) to \(N_t\). This gives an equation that can be simplified algebraically to find the population sizes, \(N_t\), where the population remains unchanged. In this specific exercise, solving the equation gave us fixed points at \(N_t = 0\) and \(N_t = 60\).

• The fixed point \(N_t = 0\) suggests population extinction, where no growth occurs.
• The fixed point \(N_t = 60\) indicates a stable population size that the model maintains over time without external influences.

Understanding fixed points is crucial, as they help us predict and comprehend the behavior of population models under various conditions.

Discrete Population Growth: Understanding the Steps

Discrete population growth models, like the Beverton-Holt model, track changes in a population at distinct time intervals. Unlike continuous models that assume an ever-changing population, discrete models specify population sizes at "steps," usually representing breeding seasons or annual cycles. This specified format reflects more realistic biological processes for many organisms.
For the Beverton-Holt equation, \(N_{t+1} = \frac{3N_t}{1 + N_t/30}\), each iteration or "step" depends heavily on the previous population size (\(N_t\)) and prescribed growth factors. Here, the factor "3" defines the growth rate per time interval, while "30" adjusts the saturation level.

• Every discrete step considers environmental constraints, represented by the denominator in the equation.
• The growth rate, "3," influences how quickly a population potentially increases, assuming resources allow.
• Equilibrium is achieved when values stabilize at specific intervals, leading to fixed points.

Recognizing how these steps interplay helps us grasp complex population behaviors and assess potential strategies for managing real-life populations.

Population Dynamics: The Bigger Picture

Population dynamics examines the factors affecting changes in population sizes over time. A crucial aspect of this study is understanding how populations grow, maintain, or decrease in size, influenced by birth rates, death rates, and environmental factors.
The Beverton-Holt model is specifically tailored to describe logistic growth in a discrete manner. It helps illustrate how populations stabilize rather than increasing infinitely, a process governed by the carrying capacity of the environment. This capacity is depicted in the model through parameters adjusting growth rates and saturation levels.

• Carrying Capacity: The maximum population size that the environment can sustain indefinitely without depleting resources.
• Limiter Factors: Influences like food availability, habitat space, and competition that constrain growth.
• Feedback Mechanisms: As populations grow, these mechanisms reduce birth rates or increase death rates, ensuring stabilization.

Population dynamics, through models like the Beverton-Holt, offer vital insights into forecasting population behavior and aiding ecological and resource management.