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

- $x_1$ as the number of deer shot,
- $x_2$ as the number of bears shot.

The objective function aims to maximize the total points earned from shooting deer and bears. Given that each deer is worth 4 points and each bear is worth 10 points, the objective function can be represented algebraically as:

\[ \text{Maximize: } 4x_1 + 10x_2 \]

The constraints are as follows:
1. Shoot at least 5 deer: $x_1 \geq 5$
2. Shoot at least 2 bears: $x_2 \geq 2$
3. Shoot at most 10 deer: $x_1 \leq 10$
4. Shoot at most 5 bears: $x_2 \leq 5$
5. Total animals shot do not exceed 12: $x_1 + x_2 \leq 12$

All variables are non-negative since you cannot shoot a negative number of animals.

Thus, the symbolic representation of the problem is:

```json
{
    'sym_variables': [('x1', 'deer'), ('x2', 'bears')],
    'objective_function': '4*x1 + 10*x2',
    'constraints': ['x1 >= 5', 'x2 >= 2', 'x1 <= 10', 'x2 <= 5', 'x1 + x2 <= 12']
}
```

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

```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(vtype=GRB.INTEGER, name="deer")
x2 = m.addVar(vtype=GRB.INTEGER, name="bears")

# Set the objective function
m.setObjective(4*x1 + 10*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 >= 5, "at_least_5_deer")
m.addConstr(x2 >= 2, "at_least_2_bears")
m.addConstr(x1 <= 10, "at_most_10_deer")
m.addConstr(x2 <= 5, "at_most_5_bears")
m.addConstr(x1 + x2 <= 12, "total_animals")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Deer: {x1.x}")
    print(f"Bears: {x2.x}")
    print(f"Total points: {4*x1.x + 10*x2.x}")
else:
    print("No optimal solution found")
```