A framework for long-lasting, slowly varying transient dynamics

Ankai Liu; Felicia Maria G. Magpantay; Kenzu Abdella; Ankai Liu; Felicia Maria G. Magpantay; Kenzu Abdella

doi:10.3934/mbe.2023540

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 12130-12153. doi: 10.3934/mbe.2023540

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A framework for long-lasting, slowly varying transient dynamics

1.
Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada
2.
Department of Mathematics and Statistics, Queen's University, Kingston, ON, K7L 3N6, Canada
3.
Department of Mathematics, Trent University, Peterborough, ON, K9L 0G2, Canada

Academic Editor: Xiang-Sheng Wang

Received: 27 February 2023 Revised: 02 May 2023 Accepted: 08 May 2023 Published: 16 May 2023

Much of the focus of applied dynamical systems is on asymptotic dynamics such as equilibria and periodic solutions. However, in many systems there are transient phenomena, such as temporary population collapses and the honeymoon period after the start of mass vaccination, that can last for a very long time and play an important role in ecological and epidemiological applications. In previous work we defined transient centers which are points in state space that give rise to arbitrarily long and arbitrarily slow transient dynamics. Here we present the mathematical properties of transient centers and provide further insight into these special points. We show that under certain conditions, the entire forward and backward trajectory of a transient center, as well as all its limit points must also be transient centers. We also derive conditions that can be used to verify which points are transient centers and whether those are reachable transient centers. Finally we present examples to demonstrate the utility of the theory, including applications to predatory-prey systems and disease transmission models, and show that the long transience noted in these models are generated by transient centers.
- long transience,
- non-asymptotic dynamics,
- differential equations,
- honeymoon periods
Citation: Ankai Liu, Felicia Maria G. Magpantay, Kenzu Abdella. A framework for long-lasting, slowly varying transient dynamics[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 12130-12153. doi: 10.3934/mbe.2023540

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Abstract

Much of the focus of applied dynamical systems is on asymptotic dynamics such as equilibria and periodic solutions. However, in many systems there are transient phenomena, such as temporary population collapses and the honeymoon period after the start of mass vaccination, that can last for a very long time and play an important role in ecological and epidemiological applications. In previous work we defined transient centers which are points in state space that give rise to arbitrarily long and arbitrarily slow transient dynamics. Here we present the mathematical properties of transient centers and provide further insight into these special points. We show that under certain conditions, the entire forward and backward trajectory of a transient center, as well as all its limit points must also be transient centers. We also derive conditions that can be used to verify which points are transient centers and whether those are reachable transient centers. Finally we present examples to demonstrate the utility of the theory, including applications to predatory-prey systems and disease transmission models, and show that the long transience noted in these models are generated by transient centers.

References

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