American Institute of Mathematical Sciences

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October  2013, 33(10): 4627-4646. doi: 10.3934/dcds.2013.33.4627

Persistence and global stability for a class of discrete time structured population models

Hal L. Smith 1, and  Horst R. Thieme 2,

1. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804
2. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287

Received  September 2012 Revised  January 2013 Published  April 2013

We obtain sharp conditions distinguishing extinction from persistence and provide sufficient conditions for global stability of a positive fixed point for a class of discrete time dynamical systems on the positive cone of an ordered Banach space generated by a map which is, roughly speaking, a nonlinear, rank one perturbation of a linear contraction. Such maps were considered by Rebarber, Tenhumberg, and Towney (Theor. Pop. Biol. 81, 2012) as abstractions of a restricted class of density dependent integral population projection models modeling plant population dynamics. Significant improvements of their results are provided.
Keywords: structured population model, stability., persistence attractor, Discrete time, persistence.
Mathematics Subject Classification: Primary: 37N25, 92D25; Secondary: 37B2.
Citation: Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
References:
[1]

H. Caswell, "Matrix Population Models, Construction, Analysis, and Interpretation," 2nd ed., Sinauer Assoc. Inc., Sunderland MA, 2001.

[2]

J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conf. Series in Applied Math. 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.

[3]

J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.

[4]

N. Davydova, O. Diekmann and S. van Gils, On circulant populations. I. The algebra of semelparity, Lin. Alg. Appl., 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020.

[5]

W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, Differential Equations in Banach Spaces (A. Favini, E. Obrecht, eds.), 61-73, Lecture Notes in Mathematics 1223, Springer, Berlin Heidelberg, (1986). doi: 10.1007/BFb0099183.

[6]

O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle, J. Difference Equ. Appl., 11 (2005), 327-335. doi: 10.1080/10236190412331335409.

[7]

O. Diekmann and Ph. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Diff. Equations, 215 (2005), 268-319. doi: 10.1016/j.jde.2004.10.025.

[8]

O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars,, SIAM J. Math. Anal., 39 (): 1023.  doi: 10.1137/060659211.

[9]

O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189. doi: 10.1007/s002850170002.

[10]

O. Diekmann, M. Gyllenberg and J. A. J. Metz, Steady-state analysis of structured population models, Theor. Pop. Biol., 63 (2003), 309-338. doi: 10.1016/S0040-5809(02)00058-8.

[11]

O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory, J. Math. Biol., 36 (1998), 349-388. doi: 10.1007/s002850050104.

[12]

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), 277-318. doi: 10.1007/s00285-009-0299-y.

[13]

O. Diekmann, M. Gyllenberg and H. R. Thieme, Lack of uniqueness in transport equations with a nonlocal nonlinearity, Math. Models Methods Appl. Sci., 10 (2000), 581-591. doi: 10.1142/S0218202500000318.

[14]

O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM J. Appl. Dyn. Sys., 8 (2009), 1160-1189. doi: 10.1137/080722734.

[15]

O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, "Delay Equations: Functional-, Complex-, and Nonlinear Analysis," Springer, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[16]

O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence 1988.

[18]

H. J. A. M. Heijmans, Some results from spectral theory, in "One-Parameter Semigroups" (eds. Ph. Clément, et al.), North-Holland, Amsterdam, (1987), 282-291.

[19]

M. A. Krasnosel'skij, "Positive Solutions of Operator Equations," Noordhoff, Groningen, 1964.

[20]

M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems: The Method of Positive Operators," Heldermann Verlag, Berlin, 1989.

[21]

C.-K. Li and H. Schneider, Applications of Perron-Frobenious theory to population dynamics, J. Math. Biol., 44 (2002), 450-462. doi: 10.1007/s002850100132.

[22]

R. Rebarber, B. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models, Theoretical Population Biology, 81 (2012), 81-87. doi: 10.1016/j.tpb.2011.11.002.

[23]

H. H. Schaefer, "Topological Vector Spaces," Springer-Verlag, New York, 1971.

[24]

H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biology, 53 (2006), 747-758. doi: 10.1007/s00285-006-0004-3.

[25]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, 118, American Mathematical Society, Providence R.I. 2011.

[26]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.

[27]

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), 299-317. doi: 10.1007/BF00277393.

[28]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton 2003.

[29]

H. R. Thieme, Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.

[30]

K. Yosida, "Functional Analysis," Springer, Heidelberg, New York, 1968.

[31]

E. Zeidler., "Nonlinear Functional Analysis and Its Applications," I Springer-Verlag, New York, 1993.

show all references

References:
[1]

H. Caswell, "Matrix Population Models, Construction, Analysis, and Interpretation," 2nd ed., Sinauer Assoc. Inc., Sunderland MA, 2001.

[2]

J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conf. Series in Applied Math. 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.

[3]

J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.

[4]

N. Davydova, O. Diekmann and S. van Gils, On circulant populations. I. The algebra of semelparity, Lin. Alg. Appl., 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020.

[5]

W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, Differential Equations in Banach Spaces (A. Favini, E. Obrecht, eds.), 61-73, Lecture Notes in Mathematics 1223, Springer, Berlin Heidelberg, (1986). doi: 10.1007/BFb0099183.

[6]

O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle, J. Difference Equ. Appl., 11 (2005), 327-335. doi: 10.1080/10236190412331335409.

[7]

O. Diekmann and Ph. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Diff. Equations, 215 (2005), 268-319. doi: 10.1016/j.jde.2004.10.025.

[8]

O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars,, SIAM J. Math. Anal., 39 (): 1023.  doi: 10.1137/060659211.

[9]

O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189. doi: 10.1007/s002850170002.

[10]

O. Diekmann, M. Gyllenberg and J. A. J. Metz, Steady-state analysis of structured population models, Theor. Pop. Biol., 63 (2003), 309-338. doi: 10.1016/S0040-5809(02)00058-8.

[11]

O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory, J. Math. Biol., 36 (1998), 349-388. doi: 10.1007/s002850050104.

[12]

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), 277-318. doi: 10.1007/s00285-009-0299-y.

[13]

O. Diekmann, M. Gyllenberg and H. R. Thieme, Lack of uniqueness in transport equations with a nonlocal nonlinearity, Math. Models Methods Appl. Sci., 10 (2000), 581-591. doi: 10.1142/S0218202500000318.

[14]

O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM J. Appl. Dyn. Sys., 8 (2009), 1160-1189. doi: 10.1137/080722734.

[15]

O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, "Delay Equations: Functional-, Complex-, and Nonlinear Analysis," Springer, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[16]

O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.

[17]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence 1988.

[18]

H. J. A. M. Heijmans, Some results from spectral theory, in "One-Parameter Semigroups" (eds. Ph. Clément, et al.), North-Holland, Amsterdam, (1987), 282-291.

[19]

M. A. Krasnosel'skij, "Positive Solutions of Operator Equations," Noordhoff, Groningen, 1964.

[20]

M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems: The Method of Positive Operators," Heldermann Verlag, Berlin, 1989.

[21]

C.-K. Li and H. Schneider, Applications of Perron-Frobenious theory to population dynamics, J. Math. Biol., 44 (2002), 450-462. doi: 10.1007/s002850100132.

[22]

R. Rebarber, B. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models, Theoretical Population Biology, 81 (2012), 81-87. doi: 10.1016/j.tpb.2011.11.002.

[23]

H. H. Schaefer, "Topological Vector Spaces," Springer-Verlag, New York, 1971.

[24]

H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biology, 53 (2006), 747-758. doi: 10.1007/s00285-006-0004-3.

[25]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, 118, American Mathematical Society, Providence R.I. 2011.

[26]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.

[27]

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), 299-317. doi: 10.1007/BF00277393.

[28]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton 2003.

[29]

H. R. Thieme, Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.

[30]

K. Yosida, "Functional Analysis," Springer, Heidelberg, New York, 1968.

[31]

E. Zeidler., "Nonlinear Functional Analysis and Its Applications," I Springer-Verlag, New York, 1993.

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