Mathematical Biosciences and Engineering, 2015, 12(4): 661-686. doi: 10.3934/mbe.2015.12.661.

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

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Stability and persistence in ODE modelsfor 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

A model of ordinary differential equations is formulated for populationswhich are structured by many stages. The model is motivated by tickswhich are vectors of infectious diseases, but is general enough to apply to many other species.Our analysis identifies a basic reproduction numberthat acts as a threshold between population extinction and persistence.We establish conditions for the existence and uniqueness of nonzeroequilibria and show that their local stability cannot be expected ingeneral. Boundedness of solutions remains an open problem though wegive some sufficient conditions.
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Keywords persistence; uniqueness; extinction; Basic reproduction number; boundedness; equilibria (existence; Lyapunov functions; and stability).

Citation: Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE modelsfor populations with many stages. Mathematical Biosciences and Engineering, 2015, 12(4): 661-686. doi: 10.3934/mbe.2015.12.661

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