To solve this problem, we need to formulate a linear programming model that captures the constraints and objectives described. The goal is to maximize the number of chickens transported given the budget constraint and the relationship between train and truck trips.

Let's define the variables:
- \(T\) = Number of train trips
- \(K\) = Number of truck trips

The objective function aims to maximize the total number of chickens transported. Since each train trip can transport 500 chickens and each truck trip can transport 300 chickens, the objective function can be written as:
\[ \text{Maximize:} \quad 500T + 300K \]

The constraints are as follows:
1. **Budget Constraint**: The total cost of train and truck trips must not exceed $2000.
   - Cost per train trip = $100
   - Cost per truck trip = $80
   - Therefore, \(100T + 80K \leq 2000\)

2. **Relationship Between Train and Truck Trips**: The number of train trips cannot exceed the number of truck trips.
   - This can be represented as \(T \leq K\)

3. **Non-Negativity Constraints**: Both \(T\) and \(K\) must be non-negative since we cannot have a negative number of trips.

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

```python
from gurobipy import *

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

# Define variables
T = m.addVar(vtype=GRB.INTEGER, name="Train_Trips")
K = m.addVar(vtype=GRB.INTEGER, name="Truck_Trips")

# Objective function: Maximize the total number of chickens transported
m.setObjective(500*T + 300*K, GRB.MAXIMIZE)

# Constraints
m.addConstr(100*T + 80*K <= 2000, "Budget_Constraint")
m.addConstr(T <= K, "Train_vs_Truck_Constraint")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of Train Trips: {T.x}")
    print(f"Number of Truck Trips: {K.x}")
    print(f"Total Chickens Transported: {500*T.x + 300*K.x}")
else:
    print("No optimal solution found")
```