Problem 13 - Calculus for Biology and Medicine

Question

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{2 N_{t}}{1+N_{t} / 90}\)

Step-by-Step Solution

Answer

Fixed points are 0 and 90.

Step 1: Understanding the Beverton-Holt Model

The Beverton-Holt model is a discrete-time population model. It is generally written in the form: \[ N_{t+1} = \frac{r N_{t}}{1+ \frac{N_{t}}{K}} \]where \( r \) is the growth rate and \( K \) is the carrying capacity. In this problem, the model is given by\[ N_{t+1} = \frac{2 N_{t}}{1+ \frac{N_{t}}{90}} \]which implies that \( r = 2 \) and \( K = 90 \).

Key Concepts

Discrete-Time Population Model

The Beverton-Holt model represents a type of discrete-time population model. Unlike continuous models, which update populations continuously over time, discrete models operate in phases or specific time steps, often reflecting changes season by season or year by year. In this case, we see population changes from one time step, denoted as \( N_t \), to the next, \( N_{t+1} \), according to a specific formula. This type of modeling is beneficial for understanding population dynamics in scenarios like wildlife management, where changes are more evident over distinct periods.
One advantage of using a discrete-time model is its simplicity and ease of application in simulations and calculations. It allows scientists and researchers to make predictions about the future population sizes based on current data, while also accounting for factors like initial population and environmental limitations. The Beverton-Holt model, in particular, is useful for predicting carrying capacity and population stabilization over time.

Fixed Points

In the context of the Beverton-Holt model, a fixed point refers to a population size where there is no change from one step to the next. Mathematically, this is when \( N_{t+1} = N_t \). Finding fixed points helps us understand stable population sizes where growth ceases naturally, either due to resource limitations or other environmental factors.
To find a fixed point in the given Beverton-Holt model \( N_{t+1} = \frac{2N_t}{1 + \frac{N_t}{90}} \), you set \( N_{t+1} = N_t \) and solve for \( N_t \). Fixed points often represent equilibrium states where population growth is balanced by available resources, offering insight into long-term sustainability and the effects of parameters like growth rate \( r \) and carrying capacity \( K \). In this exercise, solving results in \( N_t = 0 \) and \( N_t = 90 \), indicating the potential stable populations under given conditions.

Carrying Capacity

Carrying capacity, denoted as \( K \), is a key element in the Beverton-Holt model, and it represents the maximum population size an environment can support indefinitely. This concept is crucial in ecological studies as it reflects the balance between resource availability and population growth.
In the equation \( N_{t+1} = \frac{2N_t}{1 + \frac{N_t}{90}} \), the carrying capacity is 90. Essentially, this means that when the population size reaches 90, the rate of growth will stabilize, and the population will neither increase nor decrease significantly. At this point, resources are optimally utilized, preventing further growth without the risk of degradation to the environment.

• High growth rates or sudden increases in population size might lead to overshooting carrying capacity, potentially resulting in population crashes.
• If resources improve, the carrying capacity can increase, allowing for a larger sustainable population.

Understanding carrying capacity helps in designing conservation strategies, ensuring sustainable resource use, and anticipating challenges in changing environments.