To solve the optimization problem described, we will use the Gurobi Python API to create a linear programming model. The objective function and constraints are defined based on the provided information.

The objective is to minimize the total cost, which is calculated as 7.64 times the number of bean stalks plus 4.45 times the quantity of orange trees plus 6.11 times the number of pansies plus 7.77 times the quantity of apple trees.

There are several constraints related to the resilience index and the quantities of each type of plant or tree. These include lower bounds on combinations of resilience indices from different plants and trees, as well as integer requirements for the quantities of each item.

Here's how we can represent this problem in Gurobi:

```python
from gurobipy import *

# Create a model
m = Model("Optimization Problem")

# Define variables (integer)
bean_stalks = m.addVar(lb=0, vtype=GRB.INTEGER, name="bean_stalks")
orange_trees = m.addVar(lb=0, vtype=GRB.INTEGER, name="orange_trees")
pansies = m.addVar(lb=0, vtype=GRB.INTEGER, name="pansies")
apple_trees = m.addVar(lb=0, vtype=GRB.INTEGER, name="apple_trees")

# Objective function: Minimize total cost
m.setObjective(7.64 * bean_stalks + 4.45 * orange_trees + 6.11 * pansies + 7.77 * apple_trees, GRB.MINIMIZE)

# Constraints based on resilience index and quantities
m.addConstr(14.59 * bean_stalks + 17.7 * pansies >= 21)
m.addConstr(14.59 * bean_stalks + 8.06 * apple_trees >= 51)
m.addConstr(14.59 * bean_stalks + 19.86 * orange_trees >= 43)
m.addConstr(19.86 * orange_trees + 8.06 * apple_trees >= 49)
m.addConstr(14.59 * bean_stalks + 19.86 * orange_trees + 17.7 * pansies >= 50)
m.addConstr(19.86 * orange_trees + 17.7 * pansies + 8.06 * apple_trees >= 50)
m.addConstr(14.59 * bean_stalks + 19.86 * orange_trees + 17.7 * pansies >= 25)
m.addConstr(19.86 * orange_trees + 17.7 * pansies + 8.06 * apple_trees >= 25)
m.addConstr(14.59 * bean_stalks + 19.86 * orange_trees + 17.7 * pansies + 8.06 * apple_trees >= 25)
m.addConstr(-3 * bean_stalks + 1 * pansies >= 0)
m.addConstr(-7 * orange_trees + 2 * pansies >= 0)

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Bean Stalks: {bean_stalks.x}")
    print(f"Orange Trees: {orange_trees.x}")
    print(f"Pansies: {pansies.x}")
    print(f"Apple Trees: {apple_trees.x}")
else:
    print("No optimal solution found")
```