Problem 30 - Calculus for Biology and Medicine

Question

Assume that the discrete logistic equation is used with parameters \(R\) and \(K .\) Write the equation in the canonical form \(x_{t+1}=r x_{t}\left(1-x_{t}\right)\), and determine \(r\) and \(x_{t}\) in terms of \(R, K\), and \(R=2.5, K=50\)

Step-by-Step Solution

Answer

In terms of \\(R = 2.5\\) and \\(K = 50\\), \\(r = 3.5\\) and \\(x_t = \frac{x_t}{50}\\). Equation: \\[x_{t+1} = 3.5 \left(\frac{x_t}{50}\right) (1 - \frac{x_t}{50})\].\\

Step 1: Understand the Logistic Equation

The discrete logistic equation given is typically used to model population growth where resources are limited. The equation is: \[x_{t+1} = x_t + R x_t \left(1 - \frac{x_t}{K}\right)\] where \(x_t\) is the current population, \(R\) is the growth rate, and \(K\) is the carrying capacity.

Step 2: Rearrange to Canonical Form

We need to express the logistic equation in the canonical form \(x_{t+1} = r x_t (1 - x_t)\). To do this, factor out \(x_t\): \[x_{t+1} = x_t + R x_t - \frac{R x_t^2}{K}\]. Simplifying gives \[x_{t+1} = x_t(1 + R - \frac{R x_t}{K})\].

Step 3: Define Canonical Parameters

Compare the simplified equation \(x_{t+1} = x_t(1 + R - \frac{R x_t}{K})\) to the canonical form \(x_{t+1} = r x_t (1 - x_t)\) and identify: \[r = 1 + R\] and \[x_t = \frac{x_t}{K}\]. Thus, \(r\) is the effective growth rate and \(x_t\) is scaled by the carrying capacity \(K\).

Step 4: Substitute Given Values

Given \(R = 2.5\) and \(K = 50\), substitute these values into the expressions for \(r\) and scaled \(x_t\): \[r = 1 + 2.5 = 3.5\] and \[x_t = \frac{x_t}{50}\].

Step 5: State the Resulting Equation

Replace \(r\) and scaled \(x_t\) in the canonical form: \[x_{t+1} = 3.5 \left(\frac{x_t}{50}\right) (1 - \frac{x_t}{50})\]. This is the specific form of the discrete logistic equation with the given parameters.

Key Concepts

Population Growth Modeling

The discrete logistic equation is a fundamental tool in understanding population growth dynamics. It describes how populations evolve over discrete time steps, factoring in limitations of resources such as food and space. The equation acknowledges that while populations can grow rapidly, they will eventually reach a stage where growth slows due to resource scarcity.

Key elements considered in this model include:

• The current population size (\(x_t\)
• The growth rate (\(R\)
• The maximum population size that the environment can support, called carrying capacity (\(K\)

These factors together provide a dynamic interplay of biological and environmental constraints affecting the population's trajectory. The beauty of this model lies in its ability to simulate realistic environmental pressures on population growth, allowing scientists to predict changes over time.

Canonical Form

The canonical form is a standardized version of the logistic equation that simplifies comparisons and calculations. It is expressed as:\(x_{t+1} = r x_t (1 - x_t)\), where:

• \(x_{t+1}\) is the population size at the next time step
• \(x_t\) is the current population size
• \(r\) represents the adjusted growth rate

This form erases specific parameters by scaling them, providing a clearer view of key dynamics without different scales interfering. In the context of the exercise, the canonical form is achieved by aligning the discrete logistic equation with \(x_{t+1} = r x_t (1 - x_t)\). While re-arranging, you compare the coefficients and adjust them so that the equation adopts a uniform structure applicable across different scenarios and models. The canonical form thus serves as a universal descriptor of logistic growth.

Growth Rate

Growth rate (\(R\)) is a critical factor in modeling populations. It reflects how fast the population increases per time step. Higher growth rates indicate rapidly expanding populations, whereas lower rates suggest slower growth.

In the logistic equation, the effective growth rate (\(r\)) is derived from \(R\) as \(r = 1 + R\). This conversion integrates constant growth adjustments to accommodate population scaling and carrying capacity effects.

• Increases in \(R\) raise \(r\) proportionately, expanding growth potential.
• A stable \(r\) accounts for initial exponential growth and eventual stabilization due to limited resources.

Understanding growth rate dynamics helps predict how fast a population approaches the carrying capacity and informs conservation and management strategies in real-world ecological systems.

Carrying Capacity

The notion of carrying capacity (\(K\)) is pivotal in understanding the logistics of population growth. It denotes the maximum number of individuals an environment can sustainably support.

As the population size approaches \(K\), growth rates decline due to resource limitations, leading to a stabilizing effect. In our equation, the population size is represented as a fraction of \(K\), converting the raw population figures into a dimensionless quantity:

• When \(x_t = K\), growth theoretically ceases, reflecting a balance with environmental carrying limits.
• When \(x_t < K\), growth continues, but at a reducing rate as reserves deplete.

Carrying capacity is crucial in the planning of resource management, as it provides insights into the limits of population expansion within a given ecological setting.