A global stability criterion for a family of delayed population models
Authors:
Eduardo Liz, Manuel Pinto, Victor Tkachenko and Sergei Trofimchuk
Journal:
Quart. Appl. Math. 63 (2005), 56-70
MSC (2000):
Primary 34K20; Secondary 92D25
DOI:
https://doi.org/10.1090/S0033-569X-05-00951-3
Published electronically:
January 19, 2005
MathSciNet review:
2126569
Full-text PDF Free Access
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Abstract: For a family of single-species delayed population models, a new global stability condition is found. This condition is sharp and can be applied in both monotone and nonmonotone cases. Moreover, the consideration of variable or distributed delays is allowed. We illustrate our approach on the Mackey-Glass equations and the Lasota-Wazewska model.
- K. Cooke, P. van den Driessche, and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999), no. 4, 332–352. MR 1727839, DOI https://doi.org/10.1007/s002850050194
- K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1163190
- K. Gopalsamy, S. I. Trofimchuk, and N. R. Bantsur, A note on global attractivity in models of hematopoiesis, Ukraïn. Mat. Zh. 50 (1998), no. 1, 5–12 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 50 (1998), no. 1, 3–12. MR 1669243, DOI https://doi.org/10.1007/BF02514684
- K. Gopalsamy and Pei Xuan Weng, Global attractivity and level crossings in a model of haematopoiesis, Bull. Inst. Math. Acad. Sinica 22 (1994), no. 4, 341–360. MR 1311545
- I. Győri and G. Ladas, Oscillation theory of delay differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. With applications; Oxford Science Publications. MR 1168471
- István Győri and Sergei I. Trofimchuk, Global attractivity in $x’(t)=-\delta x(t)+pf(x(t-\tau ))$, Dynam. Systems Appl. 8 (1999), no. 2, 197–210. MR 1695779
- Jack K. Hale and Sjoerd M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR 1243878
- Anatoli Ivanov, Eduardo Liz, and Sergei Trofimchuk, Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Math. J. (2) 54 (2002), no. 2, 277–295. MR 1904953
- Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
- M. R. S. Kulenović, G. Ladas, and Y. G. Sficas, Global attractivity in population dynamics, Comput. Math. Appl. 18 (1989), no. 10-11, 925–928. MR 1021318, DOI https://doi.org/10.1016/0898-1221%2889%2990010-2
- Jing-Wen Li and Sui Sun Cheng, Global attractivity in an RBC survival model of Wazewska and Lasota, Quart. Appl. Math. 60 (2002), no. 3, 477–483. MR 1914437, DOI https://doi.org/10.1090/qam/1914437
- Eduardo Liz, Clotilde Martínez, and Sergei Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential Integral Equations 15 (2002), no. 7, 875–896. MR 1895571
- Eduardo Liz, Victor Tkachenko, and Sergei Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal. 35 (2003), no. 3, 596–622. MR 2048402, DOI https://doi.org/10.1137/S0036141001399222
- Eduardo Liz, Manuel Pinto, Gonzalo Robledo, Sergei Trofimchuk, and Victor Tkachenko, Wright type delay differential equations with negative Schwarzian, Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 309–321. MR 1952376, DOI https://doi.org/10.3934/dcds.2003.9.309
mag M.C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977), 287–289.
- X. H. Tang and Xingfu Zou, Stability of scalar delay differential equations with dominant delayed terms, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 4, 951–968. MR 2006211, DOI https://doi.org/10.1017/S0308210500002766
cooke K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999), 332-352.
gopalb K. Gopalsamy, “Stability and oscillations in delay differential equations of population dynamics," Mathematics and its Applications, 74, Kluwer, Dordrecht, 1992.
gtb K. Gopalsamy, S.I. Trofimchuk and N.R. Bantsur, A note on global attractivity in modems of hematopoiesis, Ukrain. Math. J. 50 (1998), 5-12.
gw K. Gopalsamy and P.-X. Weng, Global attractivity and level crossings in a model of haematopoiesis, Bull. Inst. Math. Acad. Sinica 22 (1994), 341-360.
G I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Univ. Press, London, 1991.
GT I. Győri and S. Trofimchuk, Global attractivity in $x’(t)=-\delta x(t)+pf(x(t-\tau ))$. Dynam. Systems Appl. 8, (1999), 197-210.
hl J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New-York, 1993.
ilt A. Ivanov, E. Liz and S. Trofimchuk, Halanay inequality, Yorke $3/2$ stability criterion, and differential equations with maxima, Tohoku Math. J. 54 (2002), 277–295.
kuang Y. Kuang, “Delay differential equations with applications in population dynamics," Academic Press, 1993.
KLS M.R.S. Kulenovic, G. Ladas and Y.G. Sficas, Global attractivity in population dynamics, Comput. Math. Appl. 18 (1989), 925-928.
li J.W. Li and S.S. Cheng, Global attractivity in an RBC survival model of Wazewska and Lasota, Quart. Appl. Math. 60 (2002), 477-483.
lmt E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential and Integral Equations, 15 (2002), 875-896.
ltt E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596–622.
dcds E. Liz, M. Pinto, G. Robledo, V. Tkachenko and S. Trofimchuk, Wright type delay differential equations with negative Schwarzian, Discrete Contin. Dynam. Systems, 9 (2003), 309–321.
mag M.C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977), 287–289.
tz X. H. Tang and X. Zou, Stability of scalar delay differential equations with dominant delayed terms, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 951-968.
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Additional Information
Eduardo Liz
Affiliation:
Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, Campus Marcosende, 36280 Vigo, Spain
Email:
eliz@dma.uvigo.es
Manuel Pinto
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
Email:
pintoj@abello.dic.uchile.cl
Victor Tkachenko
Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka str. 3, Kiev, Ukraine
Email:
vitk@imath.kiev.ua
Sergei Trofimchuk
Affiliation:
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
MR Author ID:
211398
Email:
trofimch@inst-mat.utalca.cl
Keywords:
Global stability,
delay differential equations,
Schwarz derivative,
Mackey-Glass equations,
Lasota–Wazewska model
Received by editor(s):
January 15, 2004
Published electronically:
January 19, 2005
Article copyright:
© Copyright 2005
Brown University
