Problem 12 - Calculus for Biology and Medicine

Question

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{2 N_{t}}{1+N_{t} / 60}\)

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 is a discrete-time population model with density dependence. It describes the population size in the next time period, \(N_{t+1}\), as a function of the current population size, \(N_t\). In this case, the model is given by \(N_{t+1} = \frac{2N_t}{1+\frac{N_t}{60}}\). We need to find the fixed points, which are values of \(N_t\) where the population remains constant over time.

Step 2: Define the Fixed Point Condition

A fixed point in this context occurs when the population size does not change between periods, meaning \(N_{t+1} = N_t\). Therefore, we must set up the equation \(N_{t+1} = N_t\) using the given model: \(\frac{2N_t}{1+\frac{N_t}{60}} = N_t\).

Step 3: Solve the Fixed Point Equation

To find \(N_t\) that satisfies the fixed point condition, solve the equation \(\frac{2N_t}{1+\frac{N_t}{60}} = N_t\). Multiplying both sides by \(1 + \frac{N_t}{60}\) gives \(2N_t = N_t + \frac{N_t^2}{60}\). Rearranging terms gives the quadratic equation \(\frac{N_t^2}{60} - N_t = 0\).

Step 4: Factor the Quadratic Equation

The equation \(\frac{N_t^2}{60} - N_t = 0\) can be factored by taking \(N_t\) common: \(N_t(\frac{N_t}{60} - 1) = 0\). This gives us two solutions: \(N_t = 0\) and \(\frac{N_t}{60} - 1 = 0\).

Step 5: Solve for Non-zero Fixed Point

For the non-zero fixed point, solve the equation \(\frac{N_t}{60} - 1 = 0\). This simplifies to \(\frac{N_t}{60} = 1\), which means \(N_t = 60\).

Step 6: Identify All Fixed Points

The solutions to the equation are \(N_t = 0\) and \(N_t = 60\). Thus, the fixed points of the Beverton-Holt model in this case are \(N_t = 0\) and \(N_t = 60\).

Key Concepts

Fixed Points

In the context of the Beverton-Holt model, fixed points are particular population sizes at which the population remains constant from one time period to the next. When we say fixed points, imagine a plateau in the population growth where its size stabilizes.

This occurs when the population size in the next period, \(N_{t+1}\), is equal to the current population size, \(N_t\).
This can be thought of as a state of equilibrium. To find these points, we substitute into the model's equation and solve for \(N_t\):

• Set \(N_{t+1} = N_t\) in the original equation \(\frac{2N_t}{1+\frac{N_t}{60}} = N_t\)
• Rearrange and solve this equation to find the values of \(N_t\)

In this problem, we found two fixed points: \(N_t = 0\) and \(N_t = 60\).
These points suggest that if the current population is at either zero or sixty, it won't change in subsequent generations. Understanding fixed points in population growth helps predict sustainable population sizes.

Population Growth

The Beverton-Holt model helps to grasp how populations grow in an environment with limited resources. Unlike exponential growth models, which can predict unbounded growth, the Beverton-Holt model introduces the concept of density dependence. This means the growth rate decreases as population size increases, due to factors like limited resources or increased competition.

The model suggests that population grows more rapidly when it's small but slows down as it approaches a certain threshold. This threshold is often determined by resource availability, and the fixed points (like \(N_t = 60\) in our example) reflect the maximum sustainable population size.

• Rapid growth when population is small.
• Slowed growth as population nears resource limits.

Understanding this model can give insight into real-world scenarios where maintaining balance with resources is vital for sustainability.

Discrete-Time Model

The concept of a discrete-time model is to analyze processes at distinct time intervals. Unlike continuous models, which observe changes smoothly over time, discrete models view changes in steps.

In the Beverton-Holt model, each time step represents a new generation of the population. This approach is useful when trying to predict future population sizes over regular intervals, such as yearly or seasonally.

• Population changes are calculated for each time step.
• This model focuses on change over intervals rather than continuous monitoring.

Using a discrete-time frame allows scientists and ecologists to better predict and manage population dynamics, especially when resources are allocated or policies are implemented in fixed cycles.