Modelling, singular perturbation and bifurcation analyses of bitrophic food chains

Area of Science:

• Ecology
• Mathematical Biology
• Theoretical Ecology

Background:

• Predator-prey models are essential for understanding ecological dynamics.
• The Rosenzweig-MacArthur (RM) and Mass Balance (MB) chemostat models are widely used but exhibit different behaviors under slow-fast conditions.
• Singular perturbation problems arise when predator and prey dynamics have vastly different time scales.

Purpose of the Study:

• To compare the dynamics of the RM and MB predator-prey models, particularly in their fast-slow formulations.
• To investigate the occurrence of bifurcations and limit cycles in these models.
• To identify and analyze novel dynamical phenomena such as canard explosions.

Main Methods:

• Mathematical modeling using singular perturbation theory.
• Analysis of transcritical and Hopf bifurcations.
• Asymptotic expansion techniques to predict model dynamics.
• Comparison of model predictions under varying parameter values.

Main Results:

• Both RM and MB models exhibit transcritical and Hopf bifurcations, leading to limit cycles.
• The fast-slow RM model displays a "canard explosion," a previously unobserved dynamic in this model, altering limit cycle amplitude and shape.
• The MB model, while similar in many aspects, does not exhibit a canard explosion.
• Asymptotic expansion accurately predicts the canard explosion point in the RM model.

Conclusions:

• The fast-slow RM model exhibits complex dynamics, including canard explosions, not seen in the MB model.
• Differences in model formulation (explicit resource in MB) lead to distinct behaviors.
• These findings enhance our understanding of ecological oscillations and model limitations.