Persistence versus extinction for a class of discrete-time structured population models

详细信息    查看全文

• 作者：Wen Jin ; Hal L. Smith ; Horst R. Thieme
• 关键词：Basic reproduction number ; Net reproductive number ; Basic turnover number ; Krein ; Rutman theorem ; Plant population ; Seed bank ; Persistence threshold ; Eigenfunctional ; Stability ; 37L15 ; 37N25 ; 92C80 ; 92D25
• 刊名：Journal of Mathematical Biology
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：72
• 期：4
• 页码：821-850
• 全文大小：648 KB
• 参考文献：Bacaër N (2009) Periodic matrix population models: growth rate, basic reproduction number, and entropy. Bull Math Biol 71:1781–1792CrossRef MathSciNet MATH
  Bacaër N, Ait EH (2012) Dads, On the biological interpretation of a definition for the parameter R0 in periodic population models. J Math Biol 65:601–621CrossRef MathSciNet MATH
  Bonsall FF (1958) Linear operators in complete positive cones. Proc Lond Math Soc 8:53–75CrossRef MathSciNet
  Cushing JM (2011) On the relationship between \(r\) and \(R_0\) and its role in the bifurcation of stable equilibria of Darwinian matrix models. J Biol Dyn 5:277–297CrossRef MathSciNet
  Cushing JM, Ackleh AS (2012) A net reproductive number for periodic matrix models. J Biol Dyn 6:166–188CrossRef MathSciNet
  Cushing JM, Zhou Y (1994) The net reproductive value and stability in matrix population models. Nat Res Mod 8:297–333
  Deimling KD (1985) Nonlinear functional analysis. Springer, Berlin HeidelbergCrossRef MATH
  Diekmann O, Getto P, Gyllenberg M (2007) Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J Math Anal 39:1023–1069CrossRef MathSciNet MATH
  Diekmann O, Gyllenberg M, Huang H, Kirkilionis M, Metz JAJ, Thieme HR (2001) On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory. J Math Biol 43:157–189CrossRef MathSciNet MATH
  Diekmann O, Gyllenberg M, Metz JAJ (2003) Steady-state analysis of structured population models. Theor Pop Biol 63:309–338CrossRef MATH
  Diekmann O, Gyllenberg M, Metz JAJ, Thieme HR (1998) On the formulation and analysis of general deterministic structured population models. I. Linear theory. J Math Biol 36:349–388CrossRef MathSciNet MATH
  Diekmann O, Gyllenberg M, Metz JAJ, Nakaoka S, de Roos AM (2010) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. J Math Biol 61:277–318CrossRef MathSciNet MATH
  Eager EA, Rebarber R, Tenhumberg B (2014) Global asymptotical stability of plant-seed bank models. J Math Biol 69:1–13CrossRef MathSciNet MATH
  Gyllenberg M, Lant T, Thieme HR (2006) Perturbing evolutionary systems on dual spaces by cumulative outputs. Diff Integr Eqn 19:401–436MathSciNet MATH
  Gyllenberg M, Metz JAJ (2001) On fitness in structured metapopulations. J Math Biol 43:545–560CrossRef MathSciNet MATH
  Jin W (2014) Persistence of discrete dynamical systems in infinite dimensional state spaces. Dissertation, Arizona State University
  Jin W, Thieme HR (2015) An extinction/persistence threshold for sexually reproducing populations: the cone spectral radius (under review)
  Kato T (1976) Perturbation theory for linear operators. Springer, Berlin HeidelbergCrossRef MATH
  Krasnosel’skij MA, Lifshits JA, Sobolev AV (1989) Positive linear systems: the method of positive operators. Heldermann Verlag, BerlinMATH
  Krause U (2015) Positive dynamical systems in discrete time. Theory, models and applications. De Gruyter, BerlinCrossRef MATH
  Krein MG (1939) Sur les opérations linéaires transformant un certain ensemble conique en lui-même. CR (Doklady) Acad Sci URSS (NS) 23:749–752MathSciNet
  Krein MG, Rutman MA (1948) Linear operators leaving invariant a cone in a Banach space (Russian). Uspehi Mat Nauk (NS) 3:3–95, Am Math Soc Transl (1950)
  Lemmens B, Nussbaum RD (2012) Nonlinear Perron-Frobenius theory. Cambridge University Press, CambridgeCrossRef MATH
  Lemmens B, Nussbaum RD (2013) Continuity of the cone spectral radius. Proc Am Math Soc 141:2741–2754CrossRef MathSciNet MATH
  Mallet-Paret J, Nussbaum RD (2010) Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index. J Fixed Point Theory Appl 7:103–143CrossRef MathSciNet MATH
  Nussbaum RD (1981) Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem. In: Fadell E, Fournier G (eds) Fixed Point Theory. Springer, New York, pp 309–331CrossRef
  Nussbaum RD (1998) Eigenvectors of order-preserving linear operators. J Lond Math Soc 2:480–496CrossRef MathSciNet
  Rebarber R, Tenhumberg B, Townley B (2012) Global asymptotic stability of density dependent integral population projection models. Theor Popul Biol 81:81–87CrossRef MATH
  Schaefer HH (1966) Topological vector spaces. Macmillan, New YorkMATH
  Smith HL, Thieme HR (2011) Dynamical systems and population persistence. AMS, ProvidenceMATH
  Smith HL, Thieme HR (2013) Persistence and global stability for a class of discrete time structured population models. Disc Cont Dyn Syst 33:4627–4646CrossRef MathSciNet MATH
  Smoller J (1983) Shock waves and reaction-diffusion equations. Springer, New YorkCrossRef MATH
  Thieme HR (2009) Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity. SIAM J Appl Math 70:188–211CrossRef MathSciNet MATH
  Thieme HR (2013) Eigenvectors and eigenfunctionals of homogeneous order-preserving maps. arXiv:​1302.​3905v1 [math.FA]
  Thieme HR (2014) Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations (under review)
  
• 作者单位：Wen Jin (1)
  Hal L. Smith (1)
  Horst R. Thieme (1)
  
  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Biology
  Applications of Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1416

文摘

We provide sharp conditions distinguishing persistence and extinction for a class of discrete-time dynamical systems on the positive cone of an ordered Banach space generated by a map which is the sum of a positive linear contraction A and a nonlinear perturbation G that is compact and differentiable at zero in the direction of the cone. Such maps arise as year-to-year projections of population age, stage, or size-structure distributions in population biology where typically A has to do with survival and individual development and G captures the effects of reproduction. The threshold distinguishing persistence and extinction is the principal eigenvalue of \(({\mathbb {I}}-A)^{-1}G'(0)\) provided by the Krein-Rutman Theorem, and persistence is described in terms of associated eigenfunctionals. Our results significantly extend earlier persistence results of the last two authors which required more restrictive conditions on G. They are illustrated by application of the results to a plant model with a seed bank. Keywords Basic reproduction number Net reproductive number Basic turnover number Krein-Rutman theorem Plant population Seed bank Persistence threshold Eigenfunctional Stability