Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

a. Show that the equilibrium numbers of the iteration $$ p_{t+1}=F\left(p_{t}\right)=p_{t}+R \times p_{t} \times\left(1-p_{t}\right)-K $$ are $$ p_{* 1}=\frac{R-\sqrt{R^{2}-4 K R}}{2 R} \quad \text { and } \quad p_{* 2}=\frac{R+\sqrt{R^{2}-4 K R}}{2 R} $$ b. Show that for there to be any positive equilibrium, the harvest \(K\) must be less than or equal to one-fourth the low density growth rate \(R\).

Step-by-Step Solution

Answer

The equilibria are found using the quadratic formula, and a positive equilibrium exists if the harvest rate \( K \leq \frac{R}{4} \).

Step 1: Define Equilibrium Condition

To find the equilibrium, we set the expression for the population at the next time step equal to the population at the current step: \( p_{t+1} = p_t \). This implies that \( F(p_t) = p_t \). So, we set \[ p_t + R \times p_t \times (1 - p_t) - K = p_t. \] Simplifying, we get \[ R \times p_t \times (1 - p_t) - K = 0. \]

Step 2: Rearrange the Equation

Rearrange the equation from Step 1 into a quadratic form: \[ R \times p_t \times (1 - p_t) = K. \] This simplifies to \[ R \times p_t - R \times p_t^2 = K. \] Hence, the equation becomes \[ R \times p_t^2 - R \times p_t + K = 0. \]

Step 3: Solve the Quadratic Equation

The equation from Step 2 is a standard quadratic form \( Ap^2 + Bp + C = 0 \) with \( A = R \), \( B = -R \), and \( C = K \). Using the quadratic formula \( p = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), we find the equilibrium points. Substitute \( A \), \( B \), and \( C \) into the formula to get: \[ p = \frac{R \pm \sqrt{R^2 - 4RK}}{2R}. \] Thus, the equilibrium points are \[ p_{*1} = \frac{R - \sqrt{R^2 - 4KR}}{2R} \] and \[ p_{*2} = \frac{R + \sqrt{R^2 - 4KR}}{2R}. \]

Step 4: Determine Condition for Positive Equilibrium

For the equilibrium \( p_{*1} \) and \( p_{*2} \) to be real and positive, the discriminant \( R^2 - 4KR \) must be non-negative, i.e., \( R^2 \geq 4KR \). Simplifying, we get \( K \leq \frac{R}{4} \). This condition ensures that the harvest rate does not exceed one-fourth of the low-density growth rate, allowing for the existence of positive equilibria.

Key Concepts

Equilibrium Analysis

In mathematical modeling, equilibrium analysis is essential for understanding systems that change over time. It helps identify steady states, where the system remains constant if undisturbed by external influences. For a population model, an equilibrium occurs when the population size no longer changes. In our equation, \( p_{t+1} = F(p_t) = p_t + R \times p_t \times (1-p_t) - K \), we set \( p_{t+1} = p_t \) to find equilibrium points. This means that the population size in the next time step is the same as the current time step.

Once the equilibrium condition is defined, rearranging the equation helps us see equilibrium as a solution to a given problem. The reformulated equation \( R \times p_t \times (1 - p_t) = K \) represents a balance between growth and external factors like harvesting. If \( K = 0 \), the population continues to grow until it stabilizes naturally, while positive \( K \) represents harvest impacts.

Understanding equilibrium in population dynamics can highlight critical factors impacting sustainability. It also informs management decisions by making effects of changes in system parameters clearer.

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, often used to find equilibrium. They have the general form \( Ax^2 + Bx + C = 0 \) and solutions derived using the quadratic formula: \[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

In our specific exercise, the population equilibrium under harvesting results in a quadratic equation: \( R \times p_t^2 - R \times p_t + K = 0 \). Here, \( A = R \), \( B = -R \), and \( C = K \). By solving this quadratic, we identify equilibrium points through which the stability of a population can be better understood:

• \( p_{*1} = \frac{R - \sqrt{R^2 - 4KR}}{2R} \)
• \( p_{*2} = \frac{R + \sqrt{R^2 - 4KR}}{2R} \)

For these solutions to be meaningful in a biological or environmental context, it's necessary for them to be real (non-imaginary) and positive. Ensuring this often requires analyzing the discriminant \( B^2 - 4AC \) to guarantee it is non-negative.

Population Dynamics

Population dynamics explores how and why populations change over time, often through births, deaths, and external factors such as harvesting. Analyzing these changes with mathematical models helps predict long-term trends and manage resources efficiently. The model we analyzed reflects how growth rate and harvesting affect population stability.

In the equation \( p_{t+1} = p_{t} + R \times p_{t} \times (1-p_{t}) - K \), each term reveals an aspect of population changes:

• The term \( R \times p_{t} \times (1-p_{t}) \) captures natural growth, assuming a logistic growth model where growth decreases as population size approaches carrying capacity.
• The subtraction of \( K \) accounts for harvesting or loss, impacting whether the population stabilizes, grows, or declines.

The model also explains conditions under which populations maintain positive equilibrium. For example, the constraint \( K \leq \frac{R}{4} \) ensures a sustainable harvest, preventing overuse.

By understanding these dynamics, everyone from ecologists to policymakers can make informed decisions on managing and conserving populations amid changing environments.