A framework for long-lasting, slowly varying transient dynamics
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 lo...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2023540?viewType=HTML |
| _version_ | 1797809932403212288 |
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| author | Ankai Liu Felicia Maria G. Magpantay Kenzu Abdella |
| author_facet | Ankai Liu Felicia Maria G. Magpantay Kenzu Abdella |
| author_sort | Ankai Liu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-13T07:00:22Z |
| format | Article |
| id | doaj.art-a86e1d07a2d348f99ae9794a949655da |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-03-13T07:00:22Z |
| publishDate | 2023-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-a86e1d07a2d348f99ae9794a949655da2023-06-07T01:24:06ZengAIMS PressMathematical Biosciences and Engineering1551-00182023-05-01207121301215310.3934/mbe.2023540A framework for long-lasting, slowly varying transient dynamicsAnkai Liu0Felicia Maria G. Magpantay 1Kenzu Abdella21. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada2. Department of Mathematics and Statistics, Queen's University, Kingston, ON, K7L 3N6, Canada3. Department of Mathematics, Trent University, Peterborough, ON, K9L 0G2, CanadaMuch 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.https://www.aimspress.com/article/doi/10.3934/mbe.2023540?viewType=HTMLlong transiencenon-asymptotic dynamicsdifferential equationshoneymoon periods |
| spellingShingle | Ankai Liu Felicia Maria G. Magpantay Kenzu Abdella A framework for long-lasting, slowly varying transient dynamics Mathematical Biosciences and Engineering long transience non-asymptotic dynamics differential equations honeymoon periods |
| title | A framework for long-lasting, slowly varying transient dynamics |
| title_full | A framework for long-lasting, slowly varying transient dynamics |
| title_fullStr | A framework for long-lasting, slowly varying transient dynamics |
| title_full_unstemmed | A framework for long-lasting, slowly varying transient dynamics |
| title_short | A framework for long-lasting, slowly varying transient dynamics |
| title_sort | framework for long lasting slowly varying transient dynamics |
| topic | long transience non-asymptotic dynamics differential equations honeymoon periods |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2023540?viewType=HTML |
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