July  2019, 24(7): 3089-3114. doi: 10.3934/dcdsb.2018302

Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls

1. 

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

2. 

Departamento de Matemática and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

* Corresponding author: Teresa Faria

Received  April 2018 Revised  June 2018 Published  October 2018

Fund Project: Professor Yoshiaki Muroya passed away in October 2015, while the research for this paper was being conducted. The second author wishes to dedicate this paper to his memory.

In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original system is reduced to a lower dimensional one which maintains its global dynamics features.

Citation: Yoshiaki Muroya, Teresa Faria. Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3089-3114. doi: 10.3934/dcdsb.2018302
References:
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S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Anal., 34 (1998), 191-228.  doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar

[2]

S. Ahmad and A. C. Lazer, Average growth and extinction in a competitive Lotka-Volterra system, Nonlinear Anal., 62 (2005), 545-557.  doi: 10.1016/j.na.2005.03.069.  Google Scholar

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I. Al-DarabsahX. Tang and Y. Yuan, A prey-predator model with migrations and delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 737-761.  doi: 10.3934/dcdsb.2016.21.737.  Google Scholar

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H. Bereketoglu and I. Györi, Global asymptotic stability in a nonautonomous Lotka-Volterra type systems with infinite delay, J. Math. Anal. Appl., 210 (1997), 279-291.  doi: 10.1006/jmaa.1997.5403.  Google Scholar

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A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.  Google Scholar

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F. ChenZ. Li and Y. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Anal. RWA, 8 (2007), 680-687.  doi: 10.1016/j.nonrwa.2006.02.006.  Google Scholar

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F. ChenX. Xie and H. Wang, Global stability in a competition model of plankton allelopathy with infinite delay, J. Syst. Sci. Complex., 28 (2015), 1070-1079.  doi: 10.1007/s11424-015-3125-1.  Google Scholar

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T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay, J. Dynam. Differential Equations, 22 (2010), 299-324.  doi: 10.1007/s10884-010-9166-1.  Google Scholar

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T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure, Appl. Math. Comput., 245 (2014), 575-590.  doi: 10.1016/j.amc.2014.08.009.  Google Scholar

[10]

T. Faria, Persistence and permanence for a class of functional differential equations with infinite delay, J. Dynam. Differential Equations, 28 (2016), 1163-1186.  doi: 10.1007/s10884-015-9462-x.  Google Scholar

[11]

T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 301-330.  doi: 10.1017/S0308210513001194.  Google Scholar

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K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publ., Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

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K. Gopalsamy and P. Weng, Global attractivity in a competition system with feedback controls, Comput. Math. Appl., 45 (2003), 665-676.  doi: 10.1016/S0898-1221(03)00026-9.  Google Scholar

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J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

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J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New-York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

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Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

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J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, London Mathematical Society, Cambridge University Press, Cambridge, 1988.  Google Scholar

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Z. Hou, Permanence and extinction in competitive Lotka-Volterra systems with delays, Nonlinear Anal. RWA, 12 (2011), 2130-2141.  doi: 10.1016/j.nonrwa.2010.12.027.  Google Scholar

[19]

H. HuZ. Teng and S. Gao, Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls, Nonlinear Anal. RWA, 10 (2009), 2508-2520.  doi: 10.1016/j.nonrwa.2008.05.011.  Google Scholar

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Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. Differential Equations, 119 (1995), 503-532.  doi: 10.1006/jdeq.1995.1100.  Google Scholar

[21]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[22]

Z. LiM. Han and F. Chen, Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays, Nonlinear Anal. RWA, 14 (2013), 402-413.  doi: 10.1016/j.nonrwa.2012.07.004.  Google Scholar

[23]

F. Montes de Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay, Nonlinear Anal., 75 (2012), 758-768.  doi: 10.1016/j.na.2011.09.009.  Google Scholar

[24]

F. Montes de Oca and L. Pérez, Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2663-2690.  doi: 10.3934/dcdsb.2015.20.2663.  Google Scholar

[25]

F. Montes de Oca and M. Vivas, Extinction in a two dimensional Lotka-Volterra system with infinite delay, Nonlinear Anal. RWA, 7 (2006), 1042-1047.  doi: 10.1016/j.nonrwa.2005.09.005.  Google Scholar

[26]

Y. Muroya, Partial survival and extinction of species in nonutonomous Lotka-Volterra systems with delays, Dynam. Systems Appl., 12 (2003), 295-306.   Google Scholar

[27]

Y. Muroya, Global stability of a delayed nonlinear Lotka-Volterra system with feedback controls and patch structure, Appl. Math. Comput., 239 (2014), 60-73.  doi: 10.1016/j.amc.2014.04.036.  Google Scholar

[28]

Y. Muroya, A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model), Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 999-1008.  doi: 10.3934/dcdss.2015.8.999.  Google Scholar

[29]

Y. MuroyaT. Kuniya and J. Wang, Stability analysis of a delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure, J. Math. Anal. Appl., 425 (2015), 415-439.  doi: 10.1016/j.jmaa.2014.12.019.  Google Scholar

[30]

L. NieJ. Peng and Z. Teng, Permanence and stability in multi-species non-autonomous Lotka-Volterra competitive systems with delays and feedback controls, Math. Comput. Modelling, 49 (2009), 295-306.  doi: 10.1016/j.mcm.2008.05.004.  Google Scholar

[31]

C. ShiZ. Li and F. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls, Nonlinear Anal. RWA, 13 (2012), 2214-2226.  doi: 10.1016/j.nonrwa.2012.01.016.  Google Scholar

[32]

T. Su and X. Yang, Finite-time synchronization of competitive neural networks with mixed delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3655-3667.  doi: 10.3934/dcdsb.2016115.  Google Scholar

[33]

X.H. Tang and X. Zou, Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedback, J. Differential Equations, 192 (2003), 502-535.  doi: 10.1016/S0022-0396(03)00042-1.  Google Scholar

[34]

Z. Teng and L. Chen, Global stability of periodic Lotka-Volterra systems with delays, Nonlinear Anal., 45 (2001), 1081-1095.  doi: 10.1016/S0362-546X(99)00441-1.  Google Scholar

[35]

Z. Teng and M. Rehim, Persistence in nonautonomous predator-prey systems with infinite delay, J. Comput. Appl. Math., 197 (2006), 302-321.  doi: 10.1016/j.cam.2005.11.006.  Google Scholar

[36]

R. R. Vance and E. A. Coddington, A nonautonomous model of population growth, J. Math. Biol., 27 (1989), 491-506.  doi: 10.1007/BF00288430.  Google Scholar

[37]

K. WangZ. Teng and H. Jiang, On the permanence for n-species non-autonomous Lotka-Volterra competitive system with infinite delay and feedback controls, Int. J. Biomath., 1 (2008), 29-43.  doi: 10.1142/S1793524508000060.  Google Scholar

[38]

Z. Zhang, Existence and global attractivity of a positive periodic solution for a generalized delayed population model with stocking and feedback control, Math. Comput. Modelling, 48 (2008), 749-760.  doi: 10.1016/j.mcm.2007.10.015.  Google Scholar

show all references

References:
[1]

S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Anal., 34 (1998), 191-228.  doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar

[2]

S. Ahmad and A. C. Lazer, Average growth and extinction in a competitive Lotka-Volterra system, Nonlinear Anal., 62 (2005), 545-557.  doi: 10.1016/j.na.2005.03.069.  Google Scholar

[3]

I. Al-DarabsahX. Tang and Y. Yuan, A prey-predator model with migrations and delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 737-761.  doi: 10.3934/dcdsb.2016.21.737.  Google Scholar

[4]

H. Bereketoglu and I. Györi, Global asymptotic stability in a nonautonomous Lotka-Volterra type systems with infinite delay, J. Math. Anal. Appl., 210 (1997), 279-291.  doi: 10.1006/jmaa.1997.5403.  Google Scholar

[5]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.  Google Scholar

[6]

F. ChenZ. Li and Y. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Anal. RWA, 8 (2007), 680-687.  doi: 10.1016/j.nonrwa.2006.02.006.  Google Scholar

[7]

F. ChenX. Xie and H. Wang, Global stability in a competition model of plankton allelopathy with infinite delay, J. Syst. Sci. Complex., 28 (2015), 1070-1079.  doi: 10.1007/s11424-015-3125-1.  Google Scholar

[8]

T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay, J. Dynam. Differential Equations, 22 (2010), 299-324.  doi: 10.1007/s10884-010-9166-1.  Google Scholar

[9]

T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure, Appl. Math. Comput., 245 (2014), 575-590.  doi: 10.1016/j.amc.2014.08.009.  Google Scholar

[10]

T. Faria, Persistence and permanence for a class of functional differential equations with infinite delay, J. Dynam. Differential Equations, 28 (2016), 1163-1186.  doi: 10.1007/s10884-015-9462-x.  Google Scholar

[11]

T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 301-330.  doi: 10.1017/S0308210513001194.  Google Scholar

[12]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publ., Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[13]

K. Gopalsamy and P. Weng, Global attractivity in a competition system with feedback controls, Comput. Math. Appl., 45 (2003), 665-676.  doi: 10.1016/S0898-1221(03)00026-9.  Google Scholar

[14]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[15]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New-York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[16]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[17]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, London Mathematical Society, Cambridge University Press, Cambridge, 1988.  Google Scholar

[18]

Z. Hou, Permanence and extinction in competitive Lotka-Volterra systems with delays, Nonlinear Anal. RWA, 12 (2011), 2130-2141.  doi: 10.1016/j.nonrwa.2010.12.027.  Google Scholar

[19]

H. HuZ. Teng and S. Gao, Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls, Nonlinear Anal. RWA, 10 (2009), 2508-2520.  doi: 10.1016/j.nonrwa.2008.05.011.  Google Scholar

[20]

Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. Differential Equations, 119 (1995), 503-532.  doi: 10.1006/jdeq.1995.1100.  Google Scholar

[21]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103 (1993), 221-246.  doi: 10.1006/jdeq.1993.1048.  Google Scholar

[22]

Z. LiM. Han and F. Chen, Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays, Nonlinear Anal. RWA, 14 (2013), 402-413.  doi: 10.1016/j.nonrwa.2012.07.004.  Google Scholar

[23]

F. Montes de Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay, Nonlinear Anal., 75 (2012), 758-768.  doi: 10.1016/j.na.2011.09.009.  Google Scholar

[24]

F. Montes de Oca and L. Pérez, Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2663-2690.  doi: 10.3934/dcdsb.2015.20.2663.  Google Scholar

[25]

F. Montes de Oca and M. Vivas, Extinction in a two dimensional Lotka-Volterra system with infinite delay, Nonlinear Anal. RWA, 7 (2006), 1042-1047.  doi: 10.1016/j.nonrwa.2005.09.005.  Google Scholar

[26]

Y. Muroya, Partial survival and extinction of species in nonutonomous Lotka-Volterra systems with delays, Dynam. Systems Appl., 12 (2003), 295-306.   Google Scholar

[27]

Y. Muroya, Global stability of a delayed nonlinear Lotka-Volterra system with feedback controls and patch structure, Appl. Math. Comput., 239 (2014), 60-73.  doi: 10.1016/j.amc.2014.04.036.  Google Scholar

[28]

Y. Muroya, A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model), Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 999-1008.  doi: 10.3934/dcdss.2015.8.999.  Google Scholar

[29]

Y. MuroyaT. Kuniya and J. Wang, Stability analysis of a delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure, J. Math. Anal. Appl., 425 (2015), 415-439.  doi: 10.1016/j.jmaa.2014.12.019.  Google Scholar

[30]

L. NieJ. Peng and Z. Teng, Permanence and stability in multi-species non-autonomous Lotka-Volterra competitive systems with delays and feedback controls, Math. Comput. Modelling, 49 (2009), 295-306.  doi: 10.1016/j.mcm.2008.05.004.  Google Scholar

[31]

C. ShiZ. Li and F. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls, Nonlinear Anal. RWA, 13 (2012), 2214-2226.  doi: 10.1016/j.nonrwa.2012.01.016.  Google Scholar

[32]

T. Su and X. Yang, Finite-time synchronization of competitive neural networks with mixed delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3655-3667.  doi: 10.3934/dcdsb.2016115.  Google Scholar

[33]

X.H. Tang and X. Zou, Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedback, J. Differential Equations, 192 (2003), 502-535.  doi: 10.1016/S0022-0396(03)00042-1.  Google Scholar

[34]

Z. Teng and L. Chen, Global stability of periodic Lotka-Volterra systems with delays, Nonlinear Anal., 45 (2001), 1081-1095.  doi: 10.1016/S0362-546X(99)00441-1.  Google Scholar

[35]

Z. Teng and M. Rehim, Persistence in nonautonomous predator-prey systems with infinite delay, J. Comput. Appl. Math., 197 (2006), 302-321.  doi: 10.1016/j.cam.2005.11.006.  Google Scholar

[36]

R. R. Vance and E. A. Coddington, A nonautonomous model of population growth, J. Math. Biol., 27 (1989), 491-506.  doi: 10.1007/BF00288430.  Google Scholar

[37]

K. WangZ. Teng and H. Jiang, On the permanence for n-species non-autonomous Lotka-Volterra competitive system with infinite delay and feedback controls, Int. J. Biomath., 1 (2008), 29-43.  doi: 10.1142/S1793524508000060.  Google Scholar

[38]

Z. Zhang, Existence and global attractivity of a positive periodic solution for a generalized delayed population model with stocking and feedback control, Math. Comput. Modelling, 48 (2008), 749-760.  doi: 10.1016/j.mcm.2007.10.015.  Google Scholar

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