To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote the amount of pork in kilograms as $x_1$ and the amount of chicken in kilograms as $x_2$. The objective is to minimize the total cost of the mixture.

The symbolic representation of the variables is:
- $x_1$: Amount of pork in kilograms
- $x_2$: Amount of chicken in kilograms

Given that pork costs $40 per kg and chicken costs $50 per kg, the objective function (total cost) can be represented as:
\[ 40x_1 + 50x_2 \]

The constraints given are:
1. The mixture needs to contain at least 10 units of protein.
2. The mixture needs to contain at least 15 units of fat.

Per kilogram, pork contains 2 units of protein and 4 units of fat, and chicken contains 3 units of protein and 2 units of fat. Thus, the constraints can be represented as:
1. Protein constraint: \(2x_1 + 3x_2 \geq 10\)
2. Fat constraint: \(4x_1 + 2x_2 \geq 15\)

Additionally, since we cannot have negative amounts of pork or chicken, we also have the non-negativity constraints:
\[ x_1 \geq 0, x_2 \geq 0 \]

The symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'Amount of pork in kilograms'), ('x2', 'Amount of chicken in kilograms')],
    'objective_function': '40*x1 + 50*x2',
    'constraints': ['2*x1 + 3*x2 >= 10', '4*x1 + 2*x2 >= 15', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's implement this problem in Gurobi using Python:

```python
from gurobipy import *

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

# Add variables to the model
x1 = m.addVar(lb=0, name="pork")
x2 = m.addVar(lb=0, name="chicken")

# Set the objective function
m.setObjective(40*x1 + 50*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(2*x1 + 3*x2 >= 10, name="protein_constraint")
m.addConstr(4*x1 + 2*x2 >= 15, name="fat_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Pork: {x1.x} kg")
    print(f"Chicken: {x2.x} kg")
    print(f"Total Cost: ${m.objVal}")
else:
    print("No optimal solution found")

```