Problem 36 - Calculus for Biology and Medicine

Question

Investigate the advantage of dimensionless variables. You are studying a population that obeys the discrete logistic equation. You know that \(R_{0}=2 .\) One year you measure \(N_{t}=10\). The next year you measure that \(N_{t+1}=15 .\) What value of \(b\) is needed in the model to fit these data?

Step-by-Step Solution

Answer

The value of \(b\) needed is \(b = \frac{3}{40}\).

Step 1: Understand the Discrete Logistic Equation

The discrete logistic equation is given by \(N_{t+1} = N_t + R_0 N_t \left(1 - \frac{N_t}{K}\right)\), where \(R_0\) is the intrinsic growth rate and \(K\) is the carrying capacity. The parameter \(b\) relates to these variables as \(b = \frac{1}{K}\). We need to determine \(b\) such that the model fits the given data.

Step 2: Solve the Model for Known Values

Given \(R_0 = 2\), \(N_t = 10\), and \(N_{t+1} = 15\), we can substitute these into the equation: \(15 = 10 + 2 \times 10 \times \left(1 - \frac{10}{K}\right)\). This simplifies to \(15 = 10 + 20 - \frac{20 \times 10}{K}\).

Step 3: Simplify the Equation

Combine like terms in the equation: \(15 = 30 - \frac{200}{K}\). Rearrange to solve for the carrying capacity \(K\): \(15 - 30 = -\frac{200}{K}\) which simplifies to \(-15 = -\frac{200}{K}\).

Step 4: Solve for Carrying Capacity \(K\)

Manipulating \(-15 = -\frac{200}{K}\), we get \(K = \frac{200}{15}\) or \(K = \frac{40}{3}\).

Step 5: Calculate \(b\)

Since \(b = \frac{1}{K}\), substitute \(K = \frac{40}{3}\) to find \(b\). Thus, \(b = \frac{3}{40}\).

Key Concepts

Discrete Logistic Equation

The discrete logistic equation is a mathematical model that describes how populations grow over discrete time steps. This model accounts for various factors like reproduction and environmental resistance. It generally represents populations that have a limited carrying capacity, rather than those that grow indefinitely. This makes it realistic for studying real-world biological populations.
The equation is expressed as: \[ N_{t+1} = N_t + R_0 N_t \left(1 - \frac{N_t}{K}\right) \]Where:

• \(N_{t}\) is the population size at time \(t\).
• \(N_{t+1}\) is the population size at time \(t + 1\).
• \(R_0\) is the intrinsic growth rate.
• \(K\) is the carrying capacity.

This formula helps track changes in a population over time, accommodating for growth limitations due to resources and space.

Intrinsic Growth Rate

The intrinsic growth rate, denoted as \(R_0\), is a fundamental parameter in population dynamics models. It reflects how fast a population can grow without any limitations. A higher \(R_0\) means the population can potentially increase more rapidly.
In the discrete logistic equation:

• A high intrinsic growth rate indicates rapid birth rates.
• However, it does not consider environmental constraints, hence is not sustainable indefinitely if the carrying capacity is exceeded.

For example, in our exercise, an \(R_0\) value of 2 implies the population could theoretically double each time step if the carrying capacity \(K\) weren't a factor.

Carrying Capacity

Carrying capacity, represented by \(K\), is a critical concept in ecology and population dynamics. It refers to the maximum population size that an environment can sustainably support. Limited resources and other ecological constraints determine this limit.
In the context of the discrete logistic equation, \(K\) restricts population growth as it approaches its maximum sustainable size.

• Once a population nears its carrying capacity, its growth rate slows down to zero.
• When exceeded, negative effects like decreased food availability may reduce the population.

In this exercise, we've calculated that \(K = \frac{40}{3}\) based on given parameters and data, which feeds into the calculation of a dimensionless parameter \(b = \frac{1}{K}\).

Population Dynamics

Population dynamics deals with the patterns and processes of change within populations. By studying population dynamics, scientists gain insight into how populations grow, shrink, and maintain equilibrium over time.
The discrete logistic equation is one tool for understanding these dynamics. It incorporates factors that can help predict future population changes:

• Reproduction rates, captured by the intrinsic growth rate.
• Environmental limitations, represented by the carrying capacity.

Population dynamics can be affected by various factors, including predation, disease, and changes in the environment. Studying these changes enables ecologists to make informed decisions about conservation and resource management.