  We propose a simple mathematical model by applying Michaelis-Menton equations
of enzyme kinetics to study the mutualistic interaction between the leaf cutter
ant and its fungus garden at the early stage of colony expansion. We derive the
sufficient conditions on the extinction and coexistence of these two species.
In addition, we give a region of initial condition that leads to the extinction
of two species when the model has an interior attractor. Our global analysis
indicates that the division of labor by workers ants and initial conditions are
two important {factors} that determine whether leaf cutter ants colonies and
their fungus garden survive and grow can exist or not. We validate the model by
doing the comparing between model simulations and data on fungal and ant colony
growth rates under laboratory conditions. We perform sensitive analysis and
parameter estimation of the model based on the experimental data to gain more
biological insights on the ecological interactions between leaf cutter ants and
their fungus garden. Finally, we give conclusions and {discuss} potential
future {work}.
