To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints.

Let's define:
- $x_1$ as the number of boat trips.
- $x_2$ as the number of cargo plane trips.

The objective is to maximize the total number of boxes of litchis delivered, which can be represented by the objective function: $500x_1 + 200x_2$.

The constraints are:
1. Budget constraint: The total cost cannot exceed $200,000. This translates to $5000x_1 + 3000x_2 \leq 200,000$.
2. Trip constraint: The number of boat trips cannot exceed the number of cargo plane trips, which gives us $x_1 \leq x_2$.
3. Non-negativity constraints: Both $x_1$ and $x_2$ must be non-negative since they represent numbers of trips.

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of boat trips'), ('x2', 'number of cargo plane trips')],
  'objective_function': '500*x1 + 200*x2',
  'constraints': ['5000*x1 + 3000*x2 <= 200000', 'x1 <= x2', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we will use the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="boat_trips")
x2 = m.addVar(vtype=GRB.INTEGER, name="cargo_plane_trips")

# Set the objective function
m.setObjective(500*x1 + 200*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5000*x1 + 3000*x2 <= 200000, "budget")
m.addConstr(x1 <= x2, "trip_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of boat trips: {x1.x}")
    print(f"Number of cargo plane trips: {x2.x}")
    print(f"Total boxes delivered: {500*x1.x + 200*x2.x}")
else:
    print("No optimal solution found")
```