To capture the problem description and provide a solution using Gurobi, we first need to understand and translate the given natural language description into mathematical terms that can be represented in code.

The objective function to minimize is: \(1 \times (\text{quantity of kiwis})^2 + 3 \times (\text{quantity of kiwis}) \times (\text{quantity of strawberries}) + 7 \times (\text{quantity of strawberries})^2 + 9 \times (\text{quantity of kiwis}) + 3 \times (\text{quantity of strawberries})\).

Let's denote:
- \(x_0\) as the quantity of kiwis,
- \(x_1\) as the quantity of strawberries.

Given constraints:
1. Tastiness rating of kiwis: \(1.97x_0\)
2. Tastiness rating of strawberries: \(4.63x_1\)
3. Minimum combined tastiness rating: \(1.97x_0 + 4.63x_1 \geq 71\)
4. Maximum combined tastiness rating: \(1.97x_0 + 4.63x_1 \leq 138\)
5. Additional constraint: \(8x_0^2 - 8x_1^2 \geq 0\)

Given that \(x_0\) can be non-integer but \(x_1\) must be an integer, we have a mixed-integer quadratic programming (MIQP) problem.

Here's how we can represent this problem in Gurobi:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="kiwis")
x1 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="strawberries")

# Objective function
m.setObjective(1*x0**2 + 3*x0*x1 + 7*x1**2 + 9*x0 + 3*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(1.97*x0 + 4.63*x1 >= 71, name="min_tastiness")
m.addConstr(1.97*x0 + 4.63*x1 <= 138, name="max_tastiness")
m.addConstr(8*x0**2 - 8*x1**2 >= 0, name="additional_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Kiwis: {x0.x}")
    print(f"Strawberries: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```