Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Find the values of \(P(t)\) for which \(P^{\prime}(t)=0\) in each of the equations Verhulst \(\quad p^{\prime}(t)=r \times p(t) \times\left(1-\frac{P(t)}{M}\right)\) Ricker \(p^{\prime}(t)=r \times p(t) \times \frac{A e^{-p(t) / \beta}-1}{A-1}\) \(A>1 \quad p^{\prime}(t)=\alpha \times p(t) \times e^{-p(t) / \beta}-\gamma p(t)\) Beverton-Holt \(p^{\prime}(t)=\frac{r \times p(t)}{1+p(t) / \beta}\) Gompertz \(\quad p^{\prime}(t)=-r p(t) \ln (p(t) / \beta)\) Let \(M=\beta=1\) and \(A=3\) draw the graphs of \(U(p)\) vs \(p\) for each of the four models in Equations \(17.2-17.3,\) for \(0

Step-by-Step Solution

Answer

Zero derivatives occur at Verhulst: P(t)=0,1; Ricker: P(t)=0,1.0986; Beverton-Holt: P(t)=0; Gompertz: P(t)=0,1.

Step 1: Define the problem

We need to find the values of \( P(t) \) where \( P'(t) = 0 \) for different population models. The task is to simplify the given differential equations and set the derivative \( P'(t) \) to zero to find \( P(t) \).

Step 2: Analyze Verhulst Model

For the Verhulst model: \( P'(t) = r \times P(t) \times \left(1 - \frac{P(t)}{M}\right) \). Set \( P'(t) = 0 \):\[ r \times P(t) \times \left(1 - \frac{P(t)}{M}\right) = 0 \]This leads to two possibilities: 1. \( P(t) = 0 \)2. \( 1 - \frac{P(t)}{M} = 0 \Rightarrow P(t) = M \).

Step 3: Analyze Ricker Model

For the Ricker model: \( P'(t) = r \times P(t) \times \frac{A e^{-P(t) / \beta}-1}{A-1} \). Set \( P'(t) = 0 \):\[ r \times P(t) \times \frac{A e^{-P(t) / \beta}-1}{A-1} = 0 \]This leads to two possibilities: 1. \( P(t) = 0 \)2. \( A e^{-P(t) / \beta}-1 = 0 \), solve for \( P(t) \):\[ A e^{-P(t) / \beta} = 1 \Rightarrow P(t) = -\beta \ln(1/A) \].

Step 4: Analyze Beverton-Holt Model

For the Beverton-Holt model: \( P'(t) = \frac{r \times P(t)}{1+P(t) / \beta} \).Set \( P'(t) = 0 \):\[ \frac{r \times P(t)}{1+P(t) / \beta} = 0 \]This equation becomes zero when \( P(t) = 0 \). Since the numerator must be zero for the fraction to be zero, no other solutions arise from this structure.

Step 5: Analyze Gompertz Model

For the Gompertz model: \( P'(t) = -r P(t) \ln \left( \frac{P(t)}{\beta} \right) \).Set \( P'(t) = 0 \):\[ -r P(t) \ln \left( \frac{P(t)}{\beta} \right) = 0 \]This leads to two possibilities:1. \( P(t) = 0 \)2. \( \ln \left( \frac{P(t)}{\beta} \right) = 0 \Rightarrow P(t) = \beta \).

Step 6: Substitute and Solve

Substitute given values into the solutions: Let \( M = \beta = 1 \) and \( A = 3 \).1. Verhulst: \( P(t) = 0 \) or \( P(t) = 1 \)2. Ricker: \( P(t) = 0 \) or \( P(t) = -\ln(1/3) \approx 1.0986 \).3. Beverton-Holt: \( P(t) = 0 \).4. Gompertz: \( P(t) = 0 \) or \( P(t) = 1 \).

Step 7: Review and Graph

Check the solutions: - Verhulst: \( P(t) = 0, 1 \)- Ricker: \( P(t) = 0, 1.0986 \)- Beverton-Holt: \( P(t) = 0 \)- Gompertz: \( P(t) = 0, 1 \)For graphing purposes, plot \( U(p) \) vs \( p \) using the expressions provided for each model with the given parameter values.

Key Concepts

Verhulst model

The Verhulst model, also known as the logistic growth model, is commonly used to describe population growth in a confined environment. It considers the idea that populations grow rapidly when they are small but the growth rate decreases as the population size approaches the carrying capacity. This is formulated as:\[ P'(t) = r \times P(t) \times \left(1 - \frac{P(t)}{M}\right) \]where:

• \( P'(t) \) is the rate of change of the population.
• \( r \) is the intrinsic growth rate.
• \( M \) is the carrying capacity of the environment.
• \( P(t) \) is the population size at time \( t \).

When we set \( P'(t) = 0 \), the solutions we find are the equilibrium points. These occur at \( P(t) = 0 \) or \( P(t) = M \). This demonstrates two stable points: one at zero population, and another at population capacity. It's a simple yet powerful tool for understanding how populations stabilize in natural settings.

Ricker model

The Ricker model is another method of modeling population growth, primarily used in ecology for understanding stock-recruitment relationships. Unlike the Verhulst model, the Ricker model incorporates a diminishing return of population size due to overcrowding effects in the equation:\[ P'(t) = r \times P(t) \times \frac{Ae^{-P(t) / \beta} - 1}{A-1} \]Key variables include:

• \( A \), which modifies the effect of population size on growth.
• The exponential term \( e^{-P(t) / \beta} \), capturing how growth is hindered as the population grows.

Setting \( P'(t) = 0 \) results in equilibrium solutions at \( P(t) = 0 \) and \( P(t) = -\beta \ln(1/A) \). These indicate points where population ceases to speed up, either due to decline from scarcity or the growth limitation as it nears a certain size dictated by the interaction between population and resource availability.

Beverton-Holt model

The Beverton-Holt model is a popular choice for modeling species populations in ecology, focusing mainly on fish. It is represented by:\[ P'(t) = \frac{r \times P(t)}{1 + P(t) / \beta} \]Here:

• \( r \) is the growth rate constant.
• \( \beta \) affects the population scaling.

Solving \( P'(t) = 0 \) gives \( P(t) = 0 \) as the only solution, indicating the absence of alternative equilibrium points in the basic form. The Beverton-Holt model's simplicity makes it easy to understand the basic population dynamics, showing how populations balance growth against sustainable levels, particularly in fish stocks where recruitment could be represented by available habitat and breeding conditions.

Gompertz model

The Gompertz model is frequently applied in the fields of population ecology and epidemiology. It describes diminishing growth rates as size increases, represented by:\[ P'(t) = -r \times P(t) \times \ln \left( \frac{P(t)}{\beta} \right) \]Parameters include:

• \( r \) dictating the growth rate.
• \( \beta \) is associated with the population size when growth starts to slow significantly.

Setting \( P'(t) = 0 \) gives two solutions: \( P(t) = 0 \) and \( P(t) = \beta \). This model highlights how growth decelerates exponentially as the population size nears the carrying capacity, being particularly useful for modeling growth dynamics where initial rapid expansion gives way to a plateau, like bacterial or tumor growth.