Patrick Martinez[b] and
Judith Vancostenoble [c]
A new age-dependent population model with diffusion and gestation processes
Received: 15 January 2016
Accepted: 27 June 2016
Mathematics Subject Classification (2010): 35Q92, 35B40, 35B51.
Keywords: Population dynamics, coupled equations, qualitative properties, asymptotic behaviour.
[a]: Università degli Studi di Bari, Dipartimento di Matematica , Via Orabona 4, 70125 Bari, Italy
[b], [c]: University of Toulouse Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse cedex 9, France
Abstract: In this paper, we introduce a new age-structured population model with diffusion and gestation processes and make a complete study of the qualitative properties of its solutions. The model is in the spirit of a model introduced in [13, 15] and studied in . We aim here to correct some weakness of the model that was pointed out in .
 B. Ainseba, Exact and approximate controllability of the age and space population dynamics structured model, J. Math. Anal. Appl. 275 (2002), no. 2, 562–574. MR
 B. Ainseba and S. Anita, Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal. 6 (2001), no. 6, 357–368. MR
 B. Ainseba and M. Iannelli, Exact controllability of a nonlinear population-dynamics problem, Differential Integral Equations 16 (2003), no. 11, 1369–1384. MR
 B. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl. 248 (2000), no. 2, 455–474. MR
 S. Anita, Analysis and control of age-dependent population dynamics, in "Mathematical Modelling: Theory and Applications", 11, Kluwer Academic Publishers, Dordrecht 2000. MR
 S. Anita, Optimal impulse-control of population dynamics with diffusion, in "Differential equations and control theory", V. Barbu, ed., Pitman Res. Notes Math. Ser., 250, Longman, Harlow 1991, 1–6. MR
 S. Anita, Optimal control of a nonlinear population dynamics with diffusion, J. Math. Anal. Appl. 152 (1990), 176–208. MR
 K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, Springer-Verlag, New York 2000. MR
 G. Fragnelli, Delay equations with nonautonomous past, PhD. Thesis, Mathematischen Fakultät der Eberhard-Karls-Universität Tübingen, 2002.
 G. Fragnelli, P. Martinez and J. Vancostenoble, Qualitative properties of a population dynamics system describing pregnancy, Math. Models and Methods in Appl. Sci. 15 (2005), no. 4, 507–554. MR
 G. Fragnelli and G. Nickel, Partial functional differential equations with nonautonomous past in \(L^p\)-phase spaces, Differential Integral Equations 16 (2003), 327–348. MR
 G. Fragnelli, A spectral mapping theorem for semigroups solving PDEs with nonautonomous past, Abstr. Appl. Anal. 2003, no. 16, 933–951. MR
 G. Fragnelli and L. Tonetto, A population equation with diffusion, J. Math. Anal. Appl. 289 (2004), 90–99. MR
 P. Martinez, J.-P. Raymond and J. Vancostenoble, Regional null controllability of a linearized Crocco-type equation, SIAM J. Control Optim. 42 (2003), no. 2, 709–728. MR
 G. Nickel and A. Rhandi, Positivity and stability of delay equations with nonautonomous past, Math. Nachr. 278 (2005), no. 7-8, 864–876. MR
 S. Piazzera, An age-dependent population equation with delayed birth process, Math. Methods Appl. Sci. 27 (2004), no. 4, 427–439. MR
 M. A. Pozio, Some conditions for global asymptotic stability of equilibria of integro-differential equations, J. Math. Anal. Appl. 95 (1983), no. 2, 501–527. MR
 J. Wu, Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York 1996. MR