Global asymptotic stability for Gurtin-MacCamy’s population dynamics model

Ma, Zhaohai; Magal, Pierre

doi:10.1090/proc/15629

Global asymptotic stability for Gurtin-MacCamy’s population dynamics model
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by Zhaohai Ma and Pierre Magal;

Proc. Amer. Math. Soc. 152 (2024), 765-780

DOI: https://doi.org/10.1090/proc/15629

Published electronically: November 7, 2023

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Abstract:

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker’s type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.

References

M. E. Fisher and B. S. Goh, Stability results for delayed-recruitment models in population dynamics, J. Math. Biol. 19 (1984), no. 1, 147–156. MR 737172, DOI 10.1007/BF00275937
Morton E. Gurtin and Richard C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal. 54 (1974), 281–300. MR 354068, DOI 10.1007/BF00250793
M. Iannelli, Mathematical theory of age-structured population dynamics, Appl. Math. Monogr. C.N.R., vol. 7, Giadini Editori E Stampatori, Pisa, 1994.
Zhihua Liu, Pierre Magal, and Shigui Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys. 62 (2011), no. 2, 191–222. MR 2786149, DOI 10.1007/s00033-010-0088-x
A. J. Lotka, Relation between birth rates and death rates, Science 26 (1907), 21–22.
Pierre Magal, Perturbation of a globally stable steady state and uniform persistence, J. Dynam. Differential Equations 21 (2009), no. 1, 1–20. MR 2482006, DOI 10.1007/s10884-008-9127-0
P. Magal, C. C. McCluskey, and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal. 89 (2010), no. 7, 1109–1140. MR 2674945, DOI 10.1080/00036810903208122
Pierre Magal and Shigui Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc. 202 (2009), no. 951, vi+71. MR 2559965, DOI 10.1090/S0065-9266-09-00568-7
Pierre Magal and Shigui Ruan, Theory and applications of abstract semilinear Cauchy problems, Applied Mathematical Sciences, vol. 201, Springer, Cham, 2018. With a foreword by Glenn Webb. MR 3887640, DOI 10.1007/978-3-030-01506-0
Pierre Magal, Ousmane Seydi, and Feng-Bin Wang, Monotone abstract non-densely defined Cauchy problems applied to age structured population dynamic models, J. Math. Anal. Appl. 479 (2019), no. 1, 450–481. MR 3987042, DOI 10.1016/j.jmaa.2019.06.034
Pierre Magal and Xiao-Qiang Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal. 37 (2005), no. 1, 251–275. MR 2172756, DOI 10.1137/S0036141003439173
F. R. Sharpe and A. J. Lotka, A problem in age-distribution, Lond. Edinb. Dublin Philos. Mag. J. Sci. 21(124) (1911), 435–438.
Hal L. Smith and Horst R. Thieme, Dynamical systems and population persistence, Graduate Studies in Mathematics, vol. 118, American Mathematical Society, Providence, RI, 2011. MR 2731633, DOI 10.1090/gsm/118
Horst R. Thieme, Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. MR 1993355, DOI 10.1515/9780691187655
G. F. Webb, Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1985. MR 772205
Xiao-Qiang Zhao, Dynamical systems in population biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 16, Springer-Verlag, New York, 2003. MR 1980821, DOI 10.1007/978-0-387-21761-1