Stability and persistence in ODE modelsfor populations with many stages

Guihong Fan; Yijun Lou; Horst R. Thieme; Jianhong Wu

doi:10.3934/mbe.2015.12.661

Article Contents

2015, Volume 12, Issue 4: 661-686. Doi: 10.3934/mbe.2015.12.661 open access

This issue Previous Article The evolutionary dynamics of a population model with a strong Allee effect Next Article Mathematical probit and logistic mortality models of the Khapra beetle fumigated with plant essential oils

Stability and persistence in ODE models for populations with many stages

1.
Department of Mathematics and Philosophy, Columbus State University, Columbus, Georgia 31907
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
3.
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804
4.
Mathematics and Statistics, York University, and Centre for Disease Modelling, York Institute of Health Research, Toronto, Ontario

Received: January 31, 2014

Revised: September 30, 2014

Published: March 2015

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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.

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Mathematics Subject Classification: Primary: 92D25; Secondary: 34D20, 34D23, 37B25, 93D30.

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