Skip to main content
Log in

A structured population model and a related functional differential equation: Global attractors and uniform persistence

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript


A structured population model of a single population having two distinct life stages is considered. The model equations, consisting of a hyperbolic partial differential equation coupled to an ordinary differential equation, can be reduced to a single, scalar functional differential equation. This allows us to use the well-developed dynamical systems theory for functional differential equations in order to study the dynamical system generated by the more complicated coupled system. A precise relation is established between the dynamical systems generated by each system of equations and a correspondence between their respective global attractors is made. The two systems are topologically equivalent on their respective attractors. These relationships are used to determine sharp sufficient conditions for the uniform persistence of the population.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Blythe, S. P., Nisbet, R. M., and Gurney, W. S. C. (1984). The dynamics of population models with distribution maturation periods.Theor. Pop. Biol. 25, 289–311.

    Google Scholar 

  • Calsina, à. (1991). A nonlinear model for size-dependent population dynamics.International Conference on Differential Equations, Barcelona, eds. C. Perelló, C. Simó, J. Solà-Morales, World Scientific, 1993, Singapore.

    Google Scholar 

  • Hale, J. K. (1977).Theory of Functional Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin.

    Google Scholar 

  • Hale, J. K. (1980).Ordinary Differential Equations, R. E. Krieger, Malabar, FL.

    Google Scholar 

  • Hale, J. K. (1988). Asymptotic behavior of dissipative systems.AMS Math. Surv. Monogr., No. 25.

  • Kirk, J., Orr, J. S., and Forrest, J. (1970). The role of chalone in the control of the bone marrow stem cell population.Math. Biosci. 6, 129–143.

    Google Scholar 

  • Metz, J. A. J., and Diekmann, O. (1986). The Dynamics of Physiologically Structured Populations,Lecture Notes in Biomathematics 68, Springer-Verlag, New York.

    Google Scholar 

  • Nisbet, R. M., and Gurney, W. S. C. (1983). The systematic formulation of population models for insects with dynamically varying instar duration.Theor. Popul. Biol. 23, 114–135.

    Google Scholar 

  • Saperstone, S. H. (1981). Semidynamical Systems in Infinite Dimensional Spaces,Appl. Math. Sci. 37, Springer-Verlag, New York.

    Google Scholar 

  • Smith, H. L. (1987). Monotone semiflows generated by functional differential equations.J. Diff. Eq. 66, 420–442.

    Google Scholar 

  • Smith, H. L. (to appear 1994). Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, to appear.Rocky Mt. Math. J. (in press).

  • Smith, H. L. (1993b). Reduction of structured population models to threshold-type delay equations and functional differential equations. A case study.Math. Biosc. 113, 1–24.

    Google Scholar 

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

    Google Scholar 

  • Thieme, H. R. (1993). Persistence under relaxed point dissipativity.SIAM J. Math. Anal. 24, 407–435.

    Google Scholar 

  • Webb, G. F. (1985).Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

About this article

Cite this article

Smith, H.L. A structured population model and a related functional differential equation: Global attractors and uniform persistence. J Dyn Diff Equat 6, 71–99 (1994).

Download citation

  • Received:

  • Issue Date:

  • DOI:

Key words