To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote the number of acres for growing peas as \(P\) and the number of acres for growing beans as \(B\). The objective is to maximize profit, which can be calculated based on the profit per acre of each crop.

The profit per acre from peas is $100, and from beans is $160. Thus, the total profit can be represented as \(100P + 160B\).

We have two main constraints:
1. **Bug-spray budget constraint**: The cost for bug-spray for peas is $30 per acre, and for beans is $50 per acre. The total available budget for bug-spray is $1300. This gives us the inequality \(30P + 50B \leq 1300\).
2. **Care-taking time constraint**: Each acre of peas requires 2 hours of care-taking, and each acre of beans requires 1.5 hours. With a total of 50 hours available for care-taking, we have \(2P + 1.5B \leq 50\).

Additionally, we know that the gardener has 30 acres in total to allocate between peas and beans, leading to the constraint \(P + B \leq 30\). Also, \(P \geq 0\) and \(B \geq 0\), as the number of acres cannot be negative.

Now, let's translate this problem into Gurobi code in Python:

```python
from gurobipy import *

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

# Define the decision variables
P = m.addVar(vtype=GRB.CONTINUOUS, name="Peas_Acres", lb=0)
B = m.addVar(vtype=GRB.CONTINUOUS, name="Beans_Acres", lb=0)

# Set the objective function: Maximize profit
m.setObjective(100*P + 160*B, GRB.MAXIMIZE)

# Add constraints
m.addConstr(30*P + 50*B <= 1300, "Bug_Spray_Budget")
m.addConstr(2*P + 1.5*B <= 50, "Care_Taking_Time")
m.addConstr(P + B <= 30, "Total_Acres")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Acres of Peas: {P.x}")
    print(f"Acres of Beans: {B.x}")
    print(f"Total Profit: ${100*P.x + 160*B.x:.2f}")
else:
    print("No optimal solution found")
```