Quiescence, excitability, and heterogeneity in ecological models

Area of Science:

• Dynamical Systems and Mathematical Ecology
• Theoretical Ecology
• Mathematical Biology

Background:

• Quiescent phases in dynamical systems typically stabilize equilibria and reduce oscillations.
• These stabilizing effects are observed when all species share uniform rates for entering and exiting quiescence.
• Divergent rates can lead to destabilization, a phenomenon explored in this study.

Purpose of the Study:

• To investigate the impact of quiescent phases with differential rates on the stability of equilibria in dynamical systems.
• To explore the relationship between differential quiescence rates and Turing instability.
• To extend findings to ecological, epidemic, delay, and reaction-diffusion models.

Main Methods:

• Analysis of dynamical systems and ecological models with quiescent phases.
• Investigation of bifurcation phenomena related to differing quiescence rates.
• Application of stability analysis to stationary points and periodic orbits.
• Utilizing geometric arguments to explain effects on periodic orbits.

Main Results:

• Differential quiescence rates can destabilize equilibria, unlike uniform rates.
• This destabilization is linked to Turing instability, particularly in two-species systems.
• Quiescent phases can stabilize systems against oscillations, consistent with spatial heterogeneity principles.
• Periodic orbits tend to shrink with the introduction of quiescent phases.

Conclusions:

• Differential quiescence rates introduce complex dynamics, potentially leading to instability.
• The study connects quiescence dynamics to established concepts like Turing instability.
• Findings have implications for understanding oscillations and stability in various mathematical models.