Problem 33 - Calculus for Biology and Medicine

Question

Investigate the advantage of dimensionless variables. A population obeys the discrete logistic equation: $$ N_{t+1}=R_{0} \cdot N_{t}-b N_{t}^{2} $$ Find the possible fixed points of the population size (one fixed point will depend on the unknown parameters \(R_{0}\) and \(b\) ).

Step-by-Step Solution

Answer

The fixed points are \(N_t = 0\) and \(N_t = \frac{R_0 - 1}{b}\).

Step 1: Understanding Fixed Points

Fixed points in a dynamical system occur when the system remains constant over time. In this equation, it means the population size does not change from one generation to the next. Mathematically, this is expressed as \(N_{t+1} = N_{t}\).

Step 2: Setting Up the Equation

To find the fixed points, set \(N_{t+1} = N_{t}\) in the given equation: \(N_{t} = R_{0} \cdot N_{t} - b \cdot N_{t}^{2}\).

Step 3: Simplifying the Equation

Rearrange the equation to bring all terms involving \(N_{t}\) to one side: \(0 = (R_{0} - 1) \cdot N_{t} - b \cdot N_{t}^{2}\).

Step 4: Factoring the Equation

Factor the equation: \(0 = N_{t}((R_{0} - 1) - b \cdot N_{t})\). This gives us two possible solutions: \(N_{t} = 0\) or \((R_{0} - 1) = b \cdot N_{t}\).

Step 5: Solving for the Non-trivial Fixed Point

Solve the second equation \((R_{0} - 1) = b \cdot N_{t}\) for \(N_{t}\), resulting in \(N_{t} = \frac{R_{0} - 1}{b}\). This is the non-trivial fixed point that depends on \(R_{0}\) and \(b\).

Key Concepts

Logistic Equation

The logistic equation is a fundamental concept in biological and ecological modeling. It describes how a population evolves over time, considering limited resources. In essence, it models the population's growth rate, which decreases as the population size approaches its maximum sustainable size, or the carrying capacity.

For the discrete logistic equation given:

• \(N_{t+1} = R_0 \cdot N_t - b \cdot N_t^2\)
• \(N_t\) represents the population size at time \(t\)
• \(R_0\) is the intrinsic growth rate
• \(b\) is a measure of the intensity of competition for resources

As the population grows, the quadratic term \(-b \cdot N_t^2\) becomes significant, modeling the competition effect. When this term balances the growth term \(R_0 \cdot N_t\), the population reaches a fixed point.

Dimensionless Variables

Dimensionless variables help simplify complex equations, allowing for a clearer analysis by removing units. They are particularly advantageous in scaling problems, highlighting the relationships between different parameters.

By introducing dimensionless variables in the logistic equation, one can reduce the number of parameters, often simplifying the analysis significantly:

• You can rescale \(N_t\) relative to a typical population size or carrying capacity.
• Use \( ilde{N}_t = \frac{N_t}{K} \) where \(K\) is the carrying capacity.
• This can lead to new equations that are easier to solve or interpret.

This simplification can also reveal universal behavior or dynamics intrinsic to the system that might not be obvious with the original equation's parameters.

Dynamical System

In mathematics and physics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Dynamical systems are a key concept in modeling how populations evolve over time, such as with our logistic equation.

Characteristics of a dynamical system include:

• A defined rule, often a differential or discrete equation, like the logistic equation, that dictates how the state of the system evolves.
• Fixed points, which are states where the system remains unchanging over time.
• The behavior of nearby points can determine stability, showing how small changes can grow, shrink, or remain constant over time.

Understanding these systems can illustrate long-term behaviors of populations and help determine strategies for sustainability and management. Fixed points, like those found in the logistic equation, provide key insights into stable states, critical for population management and prediction.