# 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.  Google Scholar [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.  Google Scholar [3] H. Caswell, Matrix Populations Models,, Sinauer Associates Inc, (2001).   Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. M. Cushing, An Introduction to Structured Population Dynamics,, CBMS-NSF regional conference series in applied mathematics, (1998).  doi: 10.1137/1.9781611970005.  Google Scholar [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).   Google Scholar [8] K. D. Deimling, Nonlinear Functional Analysis,, Springer, (1985).  doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [10] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley, (2000).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. R. Gantmacher, The Theory of Matrices,, Vol. Two, (1989).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [21] Y. Lou and J. Wu, Global dynamics of a tick Ixodes scapularis model,, Can. Appl. Math. Quart., 19 (2011), 65.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, American Mathematical Society, (1995).   Google Scholar [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.  Google Scholar [33] H. L. Smith and P. Waltman, Theory of the Chemostat. Dynamics of Microbial Cometition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar [34] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Amer. Math. Soc., (2011).   Google Scholar [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.  Google Scholar [36] H. R. Thieme, Mathematical Population Biology,, Princeton University Press, (2003).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar [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.  Google Scholar [45] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003).  doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [3] H. Caswell, Matrix Populations Models,, Sinauer Associates Inc, (2001).   Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. M. Cushing, An Introduction to Structured Population Dynamics,, CBMS-NSF regional conference series in applied mathematics, (1998).  doi: 10.1137/1.9781611970005.  Google Scholar [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).   Google Scholar [8] K. D. Deimling, Nonlinear Functional Analysis,, Springer, (1985).  doi: 10.1007/978-3-662-00547-7.  Google Scholar [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.  Google Scholar [10] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley, (2000).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. R. Gantmacher, The Theory of Matrices,, Vol. Two, (1989).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [21] Y. Lou and J. Wu, Global dynamics of a tick Ixodes scapularis model,, Can. Appl. Math. Quart., 19 (2011), 65.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, American Mathematical Society, (1995).   Google Scholar [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.  Google Scholar [33] H. L. Smith and P. Waltman, Theory of the Chemostat. Dynamics of Microbial Cometition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar [34] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Amer. Math. Soc., (2011).   Google Scholar [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.  Google Scholar [36] H. R. Thieme, Mathematical Population Biology,, Princeton University Press, (2003).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar [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.  Google Scholar [45] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (2003).  doi: 10.1007/978-0-387-21761-1.  Google Scholar
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