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To solve this predator-prey model problem, let's go through each part systematically. Here’s an outline for addressing each item in the question:

### Given Model Equations
The predator-prey model is given by:
\[
\frac{du}{dt} = f_1(u, v) = r_1 u (1 - \alpha u - \beta v)
\]
\[
\frac{dv}{dt} = f_2(u, v) = -r_2 v (1 - \gamma u)
\]
where:
- \( u \): Represents the prey population.
- \( v \): Represents the predator population.
- \( r_1, r_2 \): Growth rates for prey and predator respectively.
- \( \alpha \): Carrying capacity-related parameter for the prey.
- \( \beta, \gamma \): Predation rate parameters.

### Part (i) - Numerical Solution and Plotting
#### Step-by-Step Approach:
1. **Parameter Setup**: Use the values provided:
   - \( r_1 = \frac{1}{5} \)
   - \( r_2 = \frac{5}{4} \)
   - \( \alpha = \frac{1}{10} \)
   - \( \beta = \frac{1}{4} \)
   - \( \gamma = \frac{1}{5} \)

2. **Initial Conditions**: Solve the model with different initial conditions for the first 100 days:
   - (A) \( u(0) = 7.5 \), \( v(0) = 2.5 \)
   - (B) \( u(0) = 1.5 \), \( v(0) = 1.5 \)
   - (C) \( u(0) = 3.5 \), \( v(0) = 8.5 \)

3. **Numerical Solution**: Use `NDSolve` in a computational tool (like Mathematica or Python’s SciPy library) to solve these differential equations for each initial condition. 

4. **Plotting**: Plot \( u(t) \) and \( v(t) \) for each case over 100 days.

#### Expected Observations
The plot will typically show oscillations characteristic of predator-prey dynamics, with periodic rises and falls in prey and predator populations over time.

### Part (ii) - Analysis of Solutions
- **Oscillatory Behavior**: Determine if the solution shows oscillations, as expected in many predator-prey models. 
- **Period Estimation**: Estimate the period of these oscillations by observing the intervals between peaks of \( u(t) \) or \( v(t) \) in the plot.

### Part (iii) - Equilibrium Solutions
The equilibrium solutions \((u_{eq}, v_{eq})\) are found by setting \(\frac{du}{dt} = 0\) and \(\frac{dv}{dt} = 0\):
1. Solve the equations:
   \[
   r_1 u (1 - \alpha u - \beta v) = 0
   \]
   \[
   -r_2 v (1 - \gamma u) = 0
   \]
2. From these, the non-trivial equilibrium (where both \( u_{eq} > 0 \) and \( v_{eq} > 0 \)) can be obtained.

3. **Interpretation**: Compare this equilibrium point to the oscillations observed in part (ii). Typically, the equilibrium point represents a center around which the populations oscillate in stable cycles (or near-stable cycles).

Would you like help solving these equations numerically in Python, or with further details on any of the steps above? 

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### Related Questions:
1. How does changing the initial conditions affect the oscillation period of predator-prey cycles?
2. What happens to the system behavior if \( \alpha \), \( \beta \), or \( \gamma \) values are slightly adjusted?
3. How would a higher value of \( r_1 \) (prey growth rate) affect the equilibrium solution?
4. Could there be a stable equilibrium without oscillations in this model? Under what parameter conditions?
5. How can this model be modified to include external factors affecting either the prey or predator population?

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### Tip
In predator-prey models, small changes in initial conditions often reveal how sensitive the system is, showing how closely populations interact dynamically over time.

The following predator-prey model aims to mimic this general observation:

du/dt = f1(u, v) = r1*u*(1 - α*u - β*v)
dv/dt = f2(u, v) = -r2*v*(1 - γ*u),

where the variable u represents the prey and v the predator. Time t is measured in days. The parameters r1 and r2 represent prey and predator growth rates, β and γ are corresponding predation rate parameters, and α represents a 'carrying capacity' related parameter of the prey population.

(i) Set r1 = 1/5, r2 = 5/4, α = 1/10, β = 1/4, and γ = 1/5. Use NDSolve[] to solve the model numerically and plot the solutions for each of the following initial conditions for the first 100 days:
A. u(0) = 7.5, v(0) = 2.5
B. u(0) = 1.5, v(0) = 1.5
C. u(0) = 3.5, v(0) = 8.5

(ii) Comment on what your solutions show. Do you observe oscillations? If yes, what’s the approximate period?

(iii) Find the equilibrium solution (u_eq, v_eq) of this model with u_eq > 0 and v_eq > 0. How does it relate to the above observed numerical solutions?

Solve a predator-prey model with specified growth and predation parameters using numerical methods. Learn how to analyze oscillations and calculate the equilibrium solution. This problem involves differential equations, numerical analysis, and equilibrium point analysis, suitable for undergraduate students.

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