### Mathematical Biosciences and Engineering

2015, Issue 4: 661-686. doi: 10.3934/mbe.2015.12.661

# Stability and persistence in ODE modelsfor populations with many stages

• Received: 01 February 2014 Accepted: 29 June 2018 Published: 01 April 2015
• MSC : Primary: 92D25; Secondary: 34D20, 34D23, 37B25, 93D30.

• 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.

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

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• 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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