To solve the given optimization problem, we first need to define the symbolic representation of the variables and the objective function, followed by the constraints.

Let's denote:
- $x_1$ as the number of days fishing is done in the Indian ocean,
- $x_2$ as the number of days fishing is done in the Arctic ocean.

The objective function aims to minimize the total cost. Given that it costs $6000 per day to operate in the Indian ocean and $9000 per day to operate in the Arctic ocean, the objective function can be represented algebraically as:
\[ \text{Minimize} \quad 6000x_1 + 9000x_2 \]

The constraints are based on the demand for fish, crab, and shrimp, as well as the operational capacity of each ocean. Specifically:
- For fish: $3.4x_1 + 6x_2 \geq 20$ (to meet the demand of 20 tons),
- For crab: $2.2x_1 + 5x_2 \geq 12$ (to meet the demand of 12 tons),
- For shrimp: $1.5x_1 + 3x_2 \geq 10$ (to meet the demand of 10 tons).

Additionally, since the number of days cannot be negative, we have:
- $x_1 \geq 0$,
- $x_2 \geq 0$.

Symbolic representation in JSON format:
```json
{
    'sym_variables': [('x1', 'days fishing in Indian ocean'), ('x2', 'days fishing in Arctic ocean')],
    'objective_function': '6000*x1 + 9000*x2',
    'constraints': ['3.4*x1 + 6*x2 >= 20', '2.2*x1 + 5*x2 >= 12', '1.5*x1 + 3*x2 >= 10', 'x1 >= 0', 'x2 >= 0']
}
```

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

```python
from gurobipy import *

# Create a model
m = Model("Omega_Seafood")

# Define the variables
x1 = m.addVar(lb=0, name="days_fishing_Indian")
x2 = m.addVar(lb=0, name="days_fishing_Arctic")

# Set the objective function
m.setObjective(6000*x1 + 9000*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(3.4*x1 + 6*x2 >= 20, "fish_demand")
m.addConstr(2.2*x1 + 5*x2 >= 12, "crab_demand")
m.addConstr(1.5*x1 + 3*x2 >= 10, "shrimp_demand")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Days fishing in Indian ocean: {x1.x}")
    print(f"Days fishing in Arctic ocean: {x2.x}")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")
```