Mathematical Biosciences and Engineering (MBE)

Stability and persistence in ODE models for populations with many stages

Pages: 661 - 686, Volume 12, Issue 4, August 2015      doi:10.3934/mbe.2015.12.661

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Guihong Fan - Department of Mathematics and Philosophy, Columbus State University, Columbus, Georgia 31907, United States (email)
Yijun Lou - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China (email)
Horst R. Thieme - Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, United States (email)
Jianhong Wu - Mathematics and Statistics, York University, and Centre for Disease Modelling, York Institute of Health Research, Toronto, Ontario, Canada (email)

Abstract: A model of ordinary differential equations is formulated for populations which are structured by many stages. The model is motivated by ticks which are vectors of infectious diseases, but is general enough to apply to many other species. Our analysis identifies a basic reproduction number that acts as a threshold between population extinction and persistence. We establish conditions for the existence and uniqueness of nonzero equilibria and show that their local stability cannot be expected in general. Boundedness of solutions remains an open problem though we give some sufficient conditions.

Keywords:  Basic reproduction number, persistence, extinction, Lyapunov functions, boundedness, equilibria (existence, uniqueness, and stability).
Mathematics Subject Classification:  Primary: 92D25; Secondary: 34D20, 34D23, 37B25, 93D30.

Received: February 2014;      Accepted: October 2014;      Available Online: April 2015.