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State-dependent neutral delay equations from population dynamics

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Abstract

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE—shift system) is a limiting case of a system of two standard delay equations.

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Notes

  1. In (Pielou 1977, p. 38 ) the author speaks of a ‘concave’ curve, but the Fig. 2.7 in the same paper shows clearly a convex curve.

  2. Throughout this paper, a prime refers to differentiation with respect to \(U\), and a dot to differentiation with respect to the time variable \(t\).

  3. Think of a government that changes the voting age at a given moment (which is a very rapid change). If the voting age is decreased, a class of juveniles become instantaneously adults, which seems acceptable. But if the voting age in increased, then a certain class of adults lose their voting right, i.e., become instantaneously juveniles.

References

  • Aiello WG, Freedman HI (1990) A time-delay model of single-species growth with stage structure. Math Biosci 101(2):139–153. doi:10.1016/0025-5564(90)90019-U

    Article  MathSciNet  MATH  Google Scholar 

  • Aiello WG, Freedman HI, Wu J (1992) Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J Appl Math 52(3):855–869

    Article  MathSciNet  MATH  Google Scholar 

  • Arino J, Wang L, Wolkowicz GSK (2006) An alternative formulation for a delayed logistic equation. J Theor Biol 241(1):109–119

  • Arino O, Hadeler KP, Hbid ML (1998) Existence of periodic solutions for delay differential equations with state-dependent delay. J Differ Equ 144(2):263–301

    Article  MathSciNet  MATH  Google Scholar 

  • Barbarossa MV (2013) On a class of neutral equations with state-dependent delay in population dynamics. Dissertation, Technical University of Munich

  • Barbarossa MV, Walther HO (2014) Linearized stability for a new class of neutral equations with state-dependent delay. Differ Equ Dyn Syst. doi:10.1007/s12591-014-0204-z

  • Bélair J (1991) Population models with state-dependent delays. In: Arino O, Axelrod DE, Kimmel M (eds) Mathematical population dynamics. Marcel Dekker, New York, pp 165–176

    Google Scholar 

  • Bellen A, Zennaro M (2003) Numerical methods for delay differential equations. Clarendon Press, Oxford

    Book  MATH  Google Scholar 

  • Blythe SP, Nisbet RM, Gurney WSC (1984) The dynamics of population models with distributed maturation periods. Theor Popul Biol 25(3):289–311

    Article  MathSciNet  MATH  Google Scholar 

  • Bocharov GA, Hadeler KP (2000) Structured population models, conservation laws, and delay equations. J Differ Equ 168(1):212–237

    Article  MathSciNet  MATH  Google Scholar 

  • Brokate M, Colonius F (1990) Linearizing equations with state-dependent delays. Appl Math Optim 21(1):45–52

    Article  MathSciNet  MATH  Google Scholar 

  • Crouse DT, Crowder LB, Caswell H (1987) A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology 68(5):1412–1423

    Article  Google Scholar 

  • Cruz MA, Hale JK (1970) Existence, uniqueness and continuous dependence for hereditary systems. Ann Mat Pur Appl 85:63–82

    Article  MathSciNet  MATH  Google Scholar 

  • Gopalsamy K, Zhang BG (1988) On a neutral delay logistic equation. Dyn Syst 2(3):183–195

    Article  MathSciNet  Google Scholar 

  • Gourley SA, Kuang Y (2009) Dynamics of a neutral delay equation for an insect population with long larval and short adult phases. J Differ Equ 246:4653–4669

    Article  MathSciNet  MATH  Google Scholar 

  • Gurtin ME, MacCamy RC (1974) Non-linear age-dependent population dynamics. Arch Ration Mech Anal 54(3):281–300

    Article  MathSciNet  MATH  Google Scholar 

  • Hadeler KP (2008) Neutral delay equations from and for population dynamics. Electron J Qual Theo Diff Equ 11:1–18

    Google Scholar 

  • Hadeler KP, Bocharov GA (2003) Where to put delays in population models, in particular in the neutral case. Can Appl Math Quart 11:159–173

    MathSciNet  MATH  Google Scholar 

  • Hadeler KP, Tomiuk J (1977) Periodic solutions of difference-differential equations. Arch Ration Mech Anal 65:87–95

    Article  MathSciNet  MATH  Google Scholar 

  • Hale JK (1969) Ordinary differential equations. Wiley, New York

    MATH  Google Scholar 

  • Hale JK, Meyer KR (1967) A class of functional equations of neutral type. Mem Am Math Soc 76:1–65

    MathSciNet  Google Scholar 

  • Hartung F, Krisztin T, Walther HO, Wu J (2006) Functional differential equations with state-dependent delays: theory and applications. In: Cañada A, Drábek P, Fonda A (eds) Handbook of differential equations: ordinary differential equations, vol 3, chap 5. Elsevier, Amsterdam, pp 435–545

  • Hu Q, Zhao XQ (2013) Global dynamics of a state dependent delay model with unimodal feedback. J Math Anal Appl 399:133–146

    Article  MathSciNet  MATH  Google Scholar 

  • Hutchinson GE (1948) Circular causal systems in ecology. Ann N Y Acad Sci 50(4):221–246

    Article  Google Scholar 

  • Hutchinson GE (1978) An introduction to population ecology. Yale University Press, New Haven

    MATH  Google Scholar 

  • Krisztin T, Wu J (1994) Monotone semiflows generated by neutral equations with different delays in neutral and retarded parts. Acta Math Univ Comen New Ser 63:207–220

    MathSciNet  MATH  Google Scholar 

  • Kuang Y (1993) Delay differential equations with applications in population dynamics. Academic Press, Boston

    MATH  Google Scholar 

  • MacDonald N (1978) Time lags in biological models, Lecture Notes in Biomathematics, vol 27. Springer, Berlin

    Book  Google Scholar 

  • MacDonald N (1989) Biological delay systems: linear stability theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Mackey MC, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197(4300):287–289

    Article  Google Scholar 

  • Mallet-Paret J, Nussbaum RD (1986) Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation. Ann Mat Pur Appl 145(1):33–128

    Article  MathSciNet  MATH  Google Scholar 

  • Mallet-Paret J, Nussbaum RD (1992) Boundary layer phenomena for differential-delay equations with state-dependent time lags. I. Arch Rat Mech Anal 120(2):99–146

    Article  MathSciNet  MATH  Google Scholar 

  • May RM (1973) Stability and complexity in model ecosystems. Princeton University Press, Princeton

    Google Scholar 

  • Metz JAJ, Diekmann O (1991) Exact finite-dimensional representations for physiologically structured populations, the abstract formulation of linear chain trickery. In: Differential Equations with Applications to Physics and Engineering, Lecture Notes in Pure and Applied Mathematics, vol 139, Tayer & Francis, pp 269–289

  • Nisbet RM, Gurney WSC (1983) The systematic formulation of population models for insects with dynamically varying instar duration. Theor Popul Biol 23(1):114–135

    Article  MathSciNet  MATH  Google Scholar 

  • Nussbaum RD (1973) Periodic solutions of some nonlinear autonomous functional differential equations II. J Differ Equ 14:360–394

    Article  MathSciNet  MATH  Google Scholar 

  • Nussbaum RD (1974) Periodic solutions of some nonlinear autonomous functional differential equations. Ann Mat Pur Appl 101(1):263–306

    Article  MathSciNet  MATH  Google Scholar 

  • Perez FJ, Malta CP, Coutinho FA (1978) Qualitative analysis of oscillations in isolated populations of flies. J Theor Biol 71(4):505–514

    Article  MathSciNet  Google Scholar 

  • Pielou EC (1977) Mathematical ecology, 2nd edn. Wiley, New York

    Google Scholar 

  • Smith FE (1963) Population dynamics in Daphnia magna and a new model for population growth. Ecology 44(4):651–663

    Article  Google Scholar 

  • Smith HL (1993) Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. Math Biosci 113:1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Sulsky D, Vance RR, Newman WI (1989) Time delays in age-structured populations. J Theor Biol 141:403–422

    Article  MathSciNet  Google Scholar 

  • Trippel EA (1995) Age at maturity as a stress indicator in fisheries. Bioscience 45(11):759–771

    Article  Google Scholar 

  • Walther HO (2003a) Differentiable semiflows for differential equations with state-dependent delays. Univ Iagell Acta Math 41:57–66

    MathSciNet  Google Scholar 

  • Walther HO (2003b) The solution manifold and \(C^1\)-smoothness for differential equations with state-dependent delay. J Differ Equ 195(1):46–65

    Article  MathSciNet  MATH  Google Scholar 

  • Walther HO (2004) Smoothness properties of semiflows for differential equations with state dependent delay. J Math Sci 124:5193–5207

    Article  MathSciNet  Google Scholar 

  • Walther HO (2010) Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays. J Dyn Differ Equ 22(3):439–462

    Article  MathSciNet  MATH  Google Scholar 

  • Wright EM (1955) A non-linear difference-differential equation. J für Die Reine Und Angew Math (Crelles J) 194:66–87

    MATH  Google Scholar 

  • Wu J (1996) Theory and applications of partial functional differential equations, applied mathematics series, vol 119. Springer, Berlin

    Book  Google Scholar 

Download references

Acknowledgments

MVB was supported by the ERC Starting Grant No. 259559 as well as by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP-4.2.4. A/2-11-1-2012-0001 National Excellence Program.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    M. V. Barbarossa

  2. Department of Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    K. P. Hadeler

  3. Institute of Mathematics, Chair for Mathematical Modelling, Technische Universität München, Boltzmannstraße 3, 85748, Garching b. München, Germany

    C. Kuttler

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  1. M. V. Barbarossa
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Correspondence to M. V. Barbarossa.

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Barbarossa, M.V., Hadeler, K.P. & Kuttler, C. State-dependent neutral delay equations from population dynamics. J. Math. Biol. 69, 1027–1056 (2014). https://doi.org/10.1007/s00285-014-0821-8

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  • DOI: https://doi.org/10.1007/s00285-014-0821-8

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