Primary: 58F15, 58F17; Secondary: 53C35.

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents

1. Mathematics Department, University of Louisiana at Lafayette, Lafayette, LA 70504

## Abstract    Related pages

This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence in a class of dissipative discrete-time dynamical systems on the positive orthant of $\mathbb{R}^m$, generated by maps. Here a unified approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of $\mathbb{R}^m_+$ to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.
Figure/Table
Supplementary
Article Metrics

Citation: Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences and Engineering, 2011, 8(3): 807-825. doi: 10.3934/mbe.2011.8.807

• 1. Gregory Roth, Paul L. Salceanu, Sebastian J. Schreiber, Robust Permanence for Ecological Maps, SIAM Journal on Mathematical Analysis, 2017, 49, 5, 3527, 10.1137/16M1066440
• 2. Jude D. Kong, Paul Salceanu, Hao Wang, A stoichiometric organic matter decomposition model in a chemostat culture, Journal of Mathematical Biology, 2018, 76, 3, 609, 10.1007/s00285-017-1152-3
• 3. Muhammad Dur-e-Ahmad, Mudassar Imran, Adnan Khan, Analysis of a Mathematical Model of Emerging Infectious Disease Leading to Amphibian Decline, Abstract and Applied Analysis, 2014, 2014, 1, 10.1155/2014/145398
• 4. Paul L. Salceanu, Robust uniform persistence in discrete and continuous nonautonomous systems, Journal of Mathematical Analysis and Applications, 2013, 398, 2, 487, 10.1016/j.jmaa.2012.09.005
• 5. Azmy S. Ackleh, Robert J. Sacker, Paul Salceanu, On a discrete selection–mutation model, Journal of Difference Equations and Applications, 2014, 20, 10, 1383, 10.1080/10236198.2014.933819
• 6. Mudassar Imran, Muhammad Usman, Muhammad Dur-e-Ahmad, Adnan Khan, Transmission Dynamics of Zika Fever: A SEIR Based Model, Differential Equations and Dynamical Systems, 2017, 10.1007/s12591-017-0374-6
• 7. Azmy S. Ackleh, John Cleveland, Horst R. Thieme, Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces, Journal of Differential Equations, 2016, 261, 2, 1472, 10.1016/j.jde.2016.04.008
• 8. AZMY S. ACKLEH, J. M. CUSHING, PAUL L. SALCEANU, ON THE DYNAMICS OF EVOLUTIONARY COMPETITION MODELS, Natural Resource Modeling, 2015, 28, 4, 380, 10.1111/nrm.12074
• 9. Azmy S. Ackleh, Paul L. Salceanu, Competitive exclusion and coexistence in ann-species Ricker model, Journal of Biological Dynamics, 2015, 9, sup1, 321, 10.1080/17513758.2015.1020576
• 10. Mudassar Imran, Muhammad Usman, Tufail Malik, Ali R. Ansari, Mathematical analysis of the role of hospitalization/isolation in controlling the spread of Zika fever, Virus Research, 2018, 10.1016/j.virusres.2018.07.002