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

Let's define:
- $x_1$ as the number of acres for growing peas,
- $x_2$ as the number of acres for growing beans.

The objective is to maximize profit. The profit per acre from peas is $100, and from beans is $160. Thus, the objective function can be represented as:
\[ \text{Maximize:} \quad 100x_1 + 160x_2 \]

The constraints are based on the available resources:
1. Bug-spray budget constraint: Each acre of peas requires $30 of bug-spray, and each acre of beans requires $50 of bug-spray. The total budget for bug-spray is $1300.
\[ 30x_1 + 50x_2 \leq 1300 \]

2. Care-taking time constraint: Each acre of peas requires 2 hours of care-taking, and each acre of beans requires 1.5 hours of care-taking. The total available time for care-taking is 50 hours.
\[ 2x_1 + 1.5x_2 \leq 50 \]

3. Non-negativity constraints: The number of acres for peas and beans cannot be negative.
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

4. Total land constraint: The gardener has 30 acres in total to allocate between peas and beans.
\[ x_1 + x_2 \leq 30 \]

Symbolic representation:
```json
{
    'sym_variables': [('x1', 'acres of peas'), ('x2', 'acres of beans')],
    'objective_function': '100*x1 + 160*x2',
    'constraints': [
        '30*x1 + 50*x2 <= 1300',
        '2*x1 + 1.5*x2 <= 50',
        'x1 >= 0',
        'x2 >= 0',
        'x1 + x2 <= 30'
    ]
}
```

Now, let's write the Gurobi code to solve this linear programming problem:

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(100*x1 + 160*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(30*x1 + 50*x2 <= 1300, name="bug_spray_budget")
m.addConstr(2*x1 + 1.5*x2 <= 50, name="care_taking_time")
m.addConstr(x1 + x2 <= 30, name="total_land")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found.")
    print(f"Acreage of peas: {x1.x}")
    print(f"Acreage of beans: {x2.x}")
    print(f"Maximum profit: ${100*x1.x + 160*x2.x:.2f}")
else:
    print("No optimal solution found.")
```