To solve James' problem, we need to maximize his profit by determining the optimal area for planting aster flowers and stonecrops within the given budget and field size constraints.

### Symbolic Representation of the Problem

Let's denote:
- \(x_1\) as the area in square feet for planting aster flowers.
- \(x_2\) as the area in square feet for planting stonecrops.

The objective function, which represents James' total profit, can be written as:
\[60x_1 + 80x_2\]

This is because each square foot of aster flowers yields a profit of $60, and each square foot of stonecrops yields a profit of $80.

The constraints are:
1. The total area used for both types of plants cannot exceed 120 square feet:
\[x_1 + x_2 \leq 120\]
2. The total cost of seeds for both aster flowers and stonecrops must not exceed James' budget of $5000:
\[20x_1 + 45x_2 \leq 5000\]
3. The areas cannot be negative (since we can't plant a negative area):
\[x_1 \geq 0, x_2 \geq 0\]

Thus, the symbolic representation of James' optimization problem is:
```json
{
  'sym_variables': [('x1', 'area for aster flowers'), ('x2', 'area for stonecrops')],
  'objective_function': '60*x1 + 80*x2',
  'constraints': ['x1 + x2 <= 120', '20*x1 + 45*x2 <= 5000', 'x1 >= 0', 'x2 >= 0']
}
```

### Gurobi Code for the Optimization Problem

To solve this problem using Gurobi in Python, we'll use the following code:
```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, name="aster_flowers_area")
x2 = m.addVar(lb=0, name="stonecrops_area")

# Set the objective function
m.setObjective(60*x1 + 80*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 120, "total_area_constraint")
m.addConstr(20*x1 + 45*x2 <= 5000, "budget_constraint")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Aster flowers area: {x1.x} square feet")
    print(f"Stonecrops area: {x2.x} square feet")
    print(f"Maximum profit: ${m.objVal}")
else:
    print("No optimal solution found.")
```