# American Institute of Mathematical Sciences

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

## Stability and persistence in ODE models for populations with many stages

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

Received  February 2014 Revised  October 2014 Published  April 2015

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.
Citation: Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE models for populations with many stages. Mathematical Biosciences & Engineering, 2015, 12 (4) : 661-686. doi: 10.3934/mbe.2015.12.661
##### References:
 [1] A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population,, J. Differential Equations, 217 (2005), 431. doi: 10.1016/j.jde.2004.12.013. [2] S. M. Baer, B. W. Koii, Y. A. Kuznetsov and H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model,, SIAM J. Appl. Math., 66 (2006), 1339. doi: 10.1137/050627757. [3] H. Caswell, Matrix Populations Models,, Sinauer Associates Inc, (2001). [4] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth,, J. Differential Equations, 247 (2009), 956. doi: 10.1016/j.jde.2009.04.003. [5] J. Chu and P. Magal, Hopf bifurcation for a size-structured model with resting phase,, Discrete Contin. Dyn. Syst., 33 (2013), 4891. doi: 10.3934/dcds.2013.33.4891. [6] J. M. Cushing, An Introduction to Structured Population Dynamics,, CBMS-NSF regional conference series in applied mathematics, (1998). doi: 10.1137/1.9781611970005. [7] J. M. Cushing, R. F. Costantino, B. Dennis, R. A. Desharnais and S. M. Henson, Chaos in Ecology, Experimental Nonlinear Dynamics,, Academic Press, (2003). [8] K. D. Deimling, Nonlinear Functional Analysis,, Springer, (1985). doi: 10.1007/978-3-662-00547-7. [9] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example,, J. Math. Biol., 61 (2010), 277. doi: 10.1007/s00285-009-0299-y. [10] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley, (2000). [11] J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics. II. Chaos,, J. Math. Anal. Appl., 242 (2000), 255. doi: 10.1006/jmaa.1999.6657. [12] J. Dyson, R. Villella-Bressan and G. F. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes,, Math. Model. Nat. Phenom., 2 (2007), 69. doi: 10.1051/mmnp:2007004. [13] G. Fan, H. R. Thieme and H. Zhu, A differential delay model for ticks,, J. Math. Biol., (). doi: 10.1007/s00285-014-0845-0. [14] J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions,, Math. Biosci. Eng., 8 (2011), 503. doi: 10.3934/mbe.2011.8.503. [15] F. R. Gantmacher, The Theory of Matrices,, Vol. Two, (1989). [16] S. A. Gourley, R. Liu and J. Wu, Spatiotemporal patterns of disease spread: Interaction of physiological structure, spatial movements, disease progression and human intervention,, Structured population models in biology and epidemiology, 1936 (2008), 165. doi: 10.1007/978-3-540-78273-5_4. [17] W. S. C. Gurney and R. M. Nisbet, Fluctuation periodicity, generation separation, and the expression of larval competition,, Theor. Popul. Biol., 28 (1985), 150. doi: 10.1016/0040-5809(85)90026-7. [18] N. A. Hartemink, S. E. Randolph, S. A. Davis and J. A. P. Heesterbeek, The basic reproduction number for complex disease systems: Defining $R_0$ for tick-borne infections,, The American Nat., 171 (2008), 743. [19] W. H. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models for some parasitic infections: methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733. doi: 10.1002/cpa.3160380607. [20] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3. [21] Y. Lou and J. Wu, Global dynamics of a tick Ixodes scapularis model,, Can. Appl. Math. Quart., 19 (2011), 65. [22] P. Magal and S. Ruan (eds.), Structured Population Models in Biology and Epidemiology,, Springer Verlag, (2008). doi: 10.1007/978-3-540-78273-5. [23] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. [24] J. A. J. Metz and O. Diekmann, The Dynamics of Physilogically Structured Populations,, Springer Verlag, (1986). doi: 10.1007/978-3-662-13159-6. [25] H. G. Mwambi, J. Baumgartner and K. P. Hadeler, Ticks and tick-borne diseases: A vector-host interaction model for the brown ear tick,, Statistical Methods in Medical Research, 9 (2000), 279. [26] N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis,, Int. J. Parasit., 35 (2005), 375. doi: 10.1016/j.ijpara.2004.12.013. [27] R. Rosà, A. Pugliese, R. Normand and P. J. Hudson, Thresholds for disease persistence in models for tick-borne infections including non-viraemic transmission, extended feeding and tick aggregation,, Journal of Theoretical Biology, 224 (2003), 359. doi: 10.1016/S0022-5193(03)00173-5. [28] R. Rosà and A. Pugliese, Effects of tick population dynamics and host densities on the persistence of tick-borne infections,, Mathematical Biosciences, 208 (2007), 216. doi: 10.1016/j.mbs.2006.10.002. [29] K. Schumacher and H. R. Thieme, Some theoretical and numerical aspects of modeling dispersion in the development of ectotherms,, Computers and Mathematics with Applications, 15 (1988), 565. doi: 10.1016/0898-1221(88)90281-7. [30] C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541. doi: 10.1137/0152030. [31] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, American Mathematical Society, (1995). [32] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,, Springer New York, (2011). doi: 10.1007/978-1-4419-7646-8. [33] H. L. Smith and P. Waltman, Theory of the Chemostat. Dynamics of Microbial Cometition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. [34] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Amer. Math. Soc., (2011). [35] H. R. Thieme, Well-posedness of physiologically structured population models for Daphnia magna (How biological concepts can benefit by abstract mathematical analysis),, J. Math. Biology, 26 (1988), 299. doi: 10.1007/BF00277393. [36] H. R. Thieme, Mathematical Population Biology,, Princeton University Press, (2003). [37] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870. [38] S. L. Tucker and S. O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables,, SIAM J. Appl. Math., 48 (1988), 549. doi: 10.1137/0148032. [39] S. Tuljapurkar and H. Caswell (Eds.), Structured-population Models in Marine, Terrestrial, and Freshwater Systems,, Springer, (1997). doi: 10.1007/978-1-4615-5973-3. [40] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosc., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. [41] G. F. Webb, Population models structured by age, size, and spatial position,, Structured population models in biology and epidemiology, 1936 (2008), 1. doi: 10.1007/978-3-540-78273-5_1. [42] P. Willadsen and F. Jongejan, Immunology of the tick-host interaction and the control of ticks and tick-borne diseases,, Parasitology Today, 15 (1999), 258. doi: 10.1016/S0169-4758(99)01472-6. [43] X. Wu, V. R. S. K. Duvvuri and J. Wu, Modeling dynamical temperature influence on tick Ixodes scapularis population,, 2010 International Congress on Environmental Modelling and Software Modelling for Environment's Sake (D.A. Swayne, (2010). [44] X. Wu, V. R. Duvvuri, Y. Lou, N. H. Ogden, Y. Pelcat and J. Wu, Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada,, Journal of Theoretical Biology, 319 (2013), 50. doi: 10.1016/j.jtbi.2012.11.014. [45] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003). doi: 10.1007/978-0-387-21761-1.

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##### References:
 [1] A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population,, J. Differential Equations, 217 (2005), 431. doi: 10.1016/j.jde.2004.12.013. [2] S. M. Baer, B. W. Koii, Y. A. Kuznetsov and H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model,, SIAM J. Appl. Math., 66 (2006), 1339. doi: 10.1137/050627757. [3] H. Caswell, Matrix Populations Models,, Sinauer Associates Inc, (2001). [4] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth,, J. Differential Equations, 247 (2009), 956. doi: 10.1016/j.jde.2009.04.003. [5] J. Chu and P. Magal, Hopf bifurcation for a size-structured model with resting phase,, Discrete Contin. Dyn. Syst., 33 (2013), 4891. doi: 10.3934/dcds.2013.33.4891. [6] J. M. Cushing, An Introduction to Structured Population Dynamics,, CBMS-NSF regional conference series in applied mathematics, (1998). doi: 10.1137/1.9781611970005. [7] J. M. Cushing, R. F. Costantino, B. Dennis, R. A. Desharnais and S. M. Henson, Chaos in Ecology, Experimental Nonlinear Dynamics,, Academic Press, (2003). [8] K. D. Deimling, Nonlinear Functional Analysis,, Springer, (1985). doi: 10.1007/978-3-662-00547-7. [9] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example,, J. Math. Biol., 61 (2010), 277. doi: 10.1007/s00285-009-0299-y. [10] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley, (2000). [11] J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics. II. Chaos,, J. Math. Anal. Appl., 242 (2000), 255. doi: 10.1006/jmaa.1999.6657. [12] J. Dyson, R. Villella-Bressan and G. F. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes,, Math. Model. Nat. Phenom., 2 (2007), 69. doi: 10.1051/mmnp:2007004. [13] G. Fan, H. R. Thieme and H. Zhu, A differential delay model for ticks,, J. Math. Biol., (). doi: 10.1007/s00285-014-0845-0. [14] J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions,, Math. Biosci. Eng., 8 (2011), 503. doi: 10.3934/mbe.2011.8.503. [15] F. R. Gantmacher, The Theory of Matrices,, Vol. Two, (1989). [16] S. A. Gourley, R. Liu and J. Wu, Spatiotemporal patterns of disease spread: Interaction of physiological structure, spatial movements, disease progression and human intervention,, Structured population models in biology and epidemiology, 1936 (2008), 165. doi: 10.1007/978-3-540-78273-5_4. [17] W. S. C. Gurney and R. M. Nisbet, Fluctuation periodicity, generation separation, and the expression of larval competition,, Theor. Popul. Biol., 28 (1985), 150. doi: 10.1016/0040-5809(85)90026-7. [18] N. A. Hartemink, S. E. Randolph, S. A. Davis and J. A. P. Heesterbeek, The basic reproduction number for complex disease systems: Defining $R_0$ for tick-borne infections,, The American Nat., 171 (2008), 743. [19] W. H. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models for some parasitic infections: methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733. doi: 10.1002/cpa.3160380607. [20] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3. [21] Y. Lou and J. Wu, Global dynamics of a tick Ixodes scapularis model,, Can. Appl. Math. Quart., 19 (2011), 65. [22] P. Magal and S. Ruan (eds.), Structured Population Models in Biology and Epidemiology,, Springer Verlag, (2008). doi: 10.1007/978-3-540-78273-5. [23] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. [24] J. A. J. Metz and O. Diekmann, The Dynamics of Physilogically Structured Populations,, Springer Verlag, (1986). doi: 10.1007/978-3-662-13159-6. [25] H. G. Mwambi, J. Baumgartner and K. P. Hadeler, Ticks and tick-borne diseases: A vector-host interaction model for the brown ear tick,, Statistical Methods in Medical Research, 9 (2000), 279. [26] N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis,, Int. J. Parasit., 35 (2005), 375. doi: 10.1016/j.ijpara.2004.12.013. [27] R. Rosà, A. Pugliese, R. Normand and P. J. Hudson, Thresholds for disease persistence in models for tick-borne infections including non-viraemic transmission, extended feeding and tick aggregation,, Journal of Theoretical Biology, 224 (2003), 359. doi: 10.1016/S0022-5193(03)00173-5. [28] R. Rosà and A. Pugliese, Effects of tick population dynamics and host densities on the persistence of tick-borne infections,, Mathematical Biosciences, 208 (2007), 216. doi: 10.1016/j.mbs.2006.10.002. [29] K. Schumacher and H. R. Thieme, Some theoretical and numerical aspects of modeling dispersion in the development of ectotherms,, Computers and Mathematics with Applications, 15 (1988), 565. doi: 10.1016/0898-1221(88)90281-7. [30] C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541. doi: 10.1137/0152030. [31] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, American Mathematical Society, (1995). [32] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,, Springer New York, (2011). doi: 10.1007/978-1-4419-7646-8. [33] H. L. Smith and P. Waltman, Theory of the Chemostat. Dynamics of Microbial Cometition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. [34] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Amer. Math. Soc., (2011). [35] H. R. Thieme, Well-posedness of physiologically structured population models for Daphnia magna (How biological concepts can benefit by abstract mathematical analysis),, J. Math. Biology, 26 (1988), 299. doi: 10.1007/BF00277393. [36] H. R. Thieme, Mathematical Population Biology,, Princeton University Press, (2003). [37] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870. [38] S. L. Tucker and S. O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables,, SIAM J. Appl. Math., 48 (1988), 549. doi: 10.1137/0148032. [39] S. Tuljapurkar and H. Caswell (Eds.), Structured-population Models in Marine, Terrestrial, and Freshwater Systems,, Springer, (1997). doi: 10.1007/978-1-4615-5973-3. [40] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosc., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. [41] G. F. Webb, Population models structured by age, size, and spatial position,, Structured population models in biology and epidemiology, 1936 (2008), 1. doi: 10.1007/978-3-540-78273-5_1. [42] P. Willadsen and F. Jongejan, Immunology of the tick-host interaction and the control of ticks and tick-borne diseases,, Parasitology Today, 15 (1999), 258. doi: 10.1016/S0169-4758(99)01472-6. [43] X. Wu, V. R. S. K. Duvvuri and J. Wu, Modeling dynamical temperature influence on tick Ixodes scapularis population,, 2010 International Congress on Environmental Modelling and Software Modelling for Environment's Sake (D.A. Swayne, (2010). [44] X. Wu, V. R. Duvvuri, Y. Lou, N. H. Ogden, Y. Pelcat and J. Wu, Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes scapularis in Canada,, Journal of Theoretical Biology, 319 (2013), 50. doi: 10.1016/j.jtbi.2012.11.014. [45] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003). doi: 10.1007/978-0-387-21761-1.
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