## Symbolic Representation

To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation.

Let's define the symbolic variables as follows:
- $x_1$ represents the number of long-term cruise tickets sold.
- $x_2$ represents the number of week-long cruise tickets sold.

The objective is to maximize profit. Given that a profit of $500 is made on each long-term cruise ticket and a profit of $150 is made on each week-long cruise ticket, the objective function can be represented as:

Maximize: $500x_1 + 150x_2$

The constraints based on the problem description are:
1. The cruise ship can take at most 1500 people: $x_1 + x_2 \leq 1500$
2. There are a minimum of 35 long-term cruise tickets available: $x_1 \geq 35$
3. At least 4 times as many people prefer to buy week-long cruise tickets than long-term cruise tickets: $x_2 \geq 4x_1$
4. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

However, since $x_1$ and $x_2$ represent the number of tickets, the non-negativity constraints are implicitly satisfied by the other constraints.

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'long-term cruise tickets'), ('x2', 'week-long cruise tickets')],
    'objective_function': '500*x1 + 150*x2',
    'constraints': [
        'x1 + x2 <= 1500',
        'x1 >= 35',
        'x2 >= 4*x1'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

def solve_cruise_ship_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name='long_term_tickets', lb=0, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='week_long_tickets', lb=0, vtype=gurobi.GRB.INTEGER)

    # Objective function: Maximize 500*x1 + 150*x2
    model.setObjective(500*x1 + 150*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x1 + x2 <= 1500, name='capacity_constraint')
    model.addConstr(x1 >= 35, name='min_long_term_constraint')
    model.addConstr(x2 >= 4*x1, name='week_long_vs_long_term_constraint')
    model.addConstr(x1 >= 35, name='min_long_term_constraint_explicit')

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution found. Long-term tickets: {x1.varValue}, Week-long tickets: {x2.varValue}")
    else:
        print("No optimal solution found.")

solve_cruise_ship_problem()
```