4. comparing and analyzing the solutions

3.2. Numerically solve equation (2.13) and its second-order approximation (2.14) using a mathematical software package. Compare solutions of the two equations that originate from the same initial point. Describe what you observe about the differences in the two solutions. Can you explain the differences you observed? Based on these numerical simulations, what conclusions can you draw regarding the accuracy of the second-order approximation of equations (2.13)? The equation in the image is:

$$\frac{dR}{dt} = \gamma\left( N_{0} - R - S_{0}e^{- \frac{R}{\rho}} \right)$$

STEP BY STEP ANSWER WITH EXPLANATION

Understand the Equations
Select Appropriate Parameter Values
Implement Numerical Solutions Using Python
Compare and Analyze the Solutions
Interpret the Results in Context of Theorem 2.1.1

Let's delve into each step in detail.

1. Understanding the Equations

Equation (2.13): Original Differential Equation

$$e^{- \frac{R}{\rho}} \approx 1 - \frac{R}{\rho} + \frac{R^{2}}{2\rho^{2}}$$

Substituting this into Equation (2.13):

2. Selecting Parameter Values

For numerical simulations, we need to assign numerical values to the parameters:

γ: Growth rate constant
N₀: Carrying capacity or maximum population/resource
S₀: Scaling parameter
ρ: Scaling factor in the exponent

Assumed Parameter Values:
Let's choose the following values for illustration:

γ = 0.1 (units: per day)
N₀ = 1000
S₀ = 500
ρ = 10

3. Implementing Numerical Solutions Using Python

Setup: Installing Required Libraries

pip install numpy scipy matplotlib

Python Code Implementation

Below is the Python script to solve both equations and compare their solutions:

Explanation of the Code

Parameter Definition: Define the constants γ, N₀, S₀, ρ with chosen values and set the initial condition R₀ = 50.
Time Span: Simulate over 100 days with 1000 evaluation points for smooth curves.
Defining the ODEs:
- eq_213: Represents Equation (2.13).
- eq_214: Represents Equation (2.14).
Numerical Solving: Use solve_ivp with the RK45 method (a common Runge-Kutta method) to solve both ODEs.
Plotting: Plot both solutions on the same graph for comparison.

4. Comparing and Analyzing the Solutions

Initial Agreement: Since both equations start from the same initial condition and the approximation is accurate near R = 0, the solutions should initially coincide or be very close.
Divergence Over Time: As R(t) increases, the higher-order terms neglected in the second-order approximation become significant, causing the approximation to deviate from the original equation.
Behavioral Differences:
- Equation (2.13): The presence of the exponential term $e^{- \frac{R}{\rho}}$ allows for more flexible growth dynamics, potentially leading to asymptotic approaches to a steady state.
- Equation (2.14): The quadratic term R² introduces a stronger nonlinearity, which may cause the approximation to predict faster or slower growth compared to the original ODE, depending on parameter values.

5. Interpretation of Results

Magnitude of Deviation: The second-order approximation may significantly deviate from the original solution as R(t) grows, especially if R(t) becomes comparable to or exceeds ρ.
Stability and Steady State: The steady-state solution of the original ODE is determined by solving $N_{0} - R - S_{0}e^{- \frac{R}{\rho}} = 0$, which is transcendental and typically requires numerical methods. The second-order approximation, on the other hand, provides a quadratic equation for the steady-state value.

6. Validity of Statement (2) of Theorem 2.1.1

Based on the numerical simulations, the second-order approximation is accurate near the initial condition (small R). However, as R grows, the approximation diverges from the original equation due to the increasing significance of higher-order terms.

Thus, the statement (2) of Theorem 2.1.1 may be valid within a limited range (small R) but is limited in its global applicability.