To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. Let's denote:

- $x_1$ as the number of bear plush toys made,
- $x_2$ as the number of dog plush toys made.

The objective function aims to maximize profit. Given that each bear brings in $4 and each dog brings in $5, the objective function can be represented as:

\[ \text{Maximize:} \quad 4x_1 + 5x_2 \]

The constraints are:
1. Time constraint: Each bear takes 15 minutes to make, and each dog takes 12 minutes to make. The total time available is 1000 minutes.
\[ 15x_1 + 12x_2 \leq 1000 \]
2. Production ratio constraint: The company must make at least two times as many bears as dogs.
\[ x_1 \geq 2x_2 \]
3. Non-negativity constraints: The number of plush toys cannot be negative.
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

This problem can be represented symbolically as:
```json
{
    'sym_variables': [('x1', 'number of bear plush toys'), ('x2', 'number of dog plush toys')],
    'objective_function': '4*x1 + 5*x2',
    'constraints': ['15*x1 + 12*x2 <= 1000', 'x1 >= 2*x2', 'x1 >= 0', 'x2 >= 0']
}
```

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

# Create a model
m = Model("plush_toys")

# Define variables
x1 = m.addVar(name="bear_plush_toys", vtype=GRB.INTEGER, lb=0)
x2 = m.addVar(name="dog_plush_toys", vtype=GRB.INTEGER, lb=0)

# Set objective function
m.setObjective(4*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*x1 + 12*x2 <= 1000, name="time_constraint")
m.addConstr(x1 >= 2*x2, name="production_ratio")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of bear plush toys: {x1.x}")
    print(f"Number of dog plush toys: {x2.x}")
    print(f"Maximum profit: ${4*x1.x + 5*x2.x}")
else:
    print("No optimal solution found")
```