How Would The Introduction Of A Holling Type III Functional Response Term In A Rosenzweig-MacArthur Predator-prey Model Affect The Stability Of The System's Coexistence Equilibrium When The Predator's Hill Coefficient Is Greater Than 2, And The Prey's Intrinsic Growth Rate Is Density-dependent?

The introduction of a Holling type III functional response with a Hill coefficient greater than 2 in the Rosenzweig-MacArthur predator-prey model enhances the stability of the coexistence equilibrium. This is because the sigmoidal shape of the type III response leads to a stronger predator effect at higher prey densities, which can dampen oscillations and maintain the equilibrium. The Jacobian analysis shows that the key term affecting stability (A) becomes more negative, resulting in eigenvalues with negative real parts, indicating stable dynamics.

Answer:

The stability of the coexistence equilibrium is enhanced. The system becomes more stable due to the sigmoidal nature of the Holling type III functional response, which leads to more effective predator control at higher prey densities, damping oscillations and maintaining equilibrium.