Articles

ON A MULTI-DELAY LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH FEEDBACK CONTROLS AND PREY DIFFUSION

  • Changyou WANG ,
  • Nan LI ,
  • Yuqian ZHOU ,
  • Xingcheng PU ,
  • Rui LI
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  • 1. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China;
    2. Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China;
    3. College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received date: 2018-02-22

  Revised date: 2018-06-28

  Online published: 2019-05-06

Supported by

This work is supported by the Sichuan Science and Technology Program of China (2018JY0480), the Natural Science Foundation Project of CQ CSTC of China (cstc2015jcyjBX0135), the National Nature Science Fundation of China (61503053).

Abstract

This article is focusing on a class of multi-delay predator-prey model with feedback controls and prey diffusion. By developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the globally attractivity of positive solution for the predator-prey system. Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system.

Cite this article

Changyou WANG , Nan LI , Yuqian ZHOU , Xingcheng PU , Rui LI . ON A MULTI-DELAY LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH FEEDBACK CONTROLS AND PREY DIFFUSION[J]. Acta mathematica scientia, Series B, 2019 , 39(2) : 429 -448 . DOI: 10.1007/s10473-019-0209-3

References

[1] Song X, Chen L. Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay. Comput Math Appl, 1998, 35:33-40
[2] Cui J. The effect of dispersal on permanence in a Predator-Prey population growth model. Comput Math Appl, 2002, 44:1085-1097
[3] Chen F, Xie X. Permanence and extinction in nonlinear single and multiple species system with diffusion. Appl Math Comput, 2006, 177:410-426
[4] Song X, Chen L. Persistence and periodic orbits for two-species predator-prey system with diffusion. Canadian Appl Math Quart, 1998, 6:233-244
[5] Wei F, Lin Y, Que L, et al. Periodic solution and global stability for a nonautonomous competitive LotkaVolterra diffusion system. Appl Math Comput, 2010, 216:3097-3104
[6] Muhammadhaji A, Teng Z, Zhang L. Permanence in general nonautonomous predator-prey Lotka-Volterra systems with distributed delays and impulses. J Biol Syst, 2013, 21:1350012
[7] Muhammadhaji A, Teng Z, Rehim M. Dynamical behavior for a class of delayed Competitive-Mutulism systems. Differ Equ Dyn Syst, 2015, 23:281-301
[8] Xu R, Chaplain M, Davidson F A. Periodic solution of a Lotka-Volterra predator-prey model with dispersion and time delays. Appl Math Comput, 2004, 148:537-560
[9] Zhou X, Shi X, Song X. Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay. Appl Math Comput, 2008, 196:129-136
[10] Zhang Z, Wang Z. Periodic solutions of a two-species ratio-dependent predator-prey system with time delay in a two-patch environment. ANZIAM J, 2003, 45:233-244
[11] Liang R, Shen J. Positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Appl Math Comput, 2010, 217:661-676
[12] Muhammadhaji A, Mahemuti R, Teng Z. On a periodic predator-prey system with nonlinear diffusion and delays. Afrika Matematika, 2016, 27:1179-1197
[13] Gopalsamy K, Weng P. Global attractivity in a competition system with feedback controls. Comput Math Appl, 2003, 45:665-676
[14] Chen F. The permanence and global attractivity of Lotka-Volterra competition system with feedback controls. Nonli Anal:RWA, 2006, 7:133-143
[15] Nie L, Tenga Z, Hu L, et al. Permanence and stability in nonautonomous predator-prey Lotka-Volterra systems with feedback controls. Computers and Mathematics with Applications, 2009, 58:436-448
[16] Chen F, Gong X, Pu L, et al. Dynamic behaviors of a Lotka-Volterra predator-prey system with feedback controls. J Biomathematics, 2015, 30:328-332(in Chinese)
[17] Ding X, Wang F. Positive periodic solution for a semi-ratio-dependent predator-prey system with diffusion and time delays. Nonli Anal:RWA, 2008, 9:39-249
[18] Gopalsamy K, Weng P. Feedback regulation of logistic growth. Int J Math Math Sci, 1993, 16:177-192
[19] Xu J, Chen F. Permanence of a Lotka-Volterra cooperative system with time delays and feedback controls. Commun Math Biol Neurosci, 2015
[20] Xie W, Weng P. Existence of periodic solution for a predator-prey model with patch-diffusion and feedback control. Journal of South China Normal University (Natural Science Edition), 2012, 44:42-47(in Chinese)
[21] Spagnolo B, Fiasconaro A, Valenti D. Noise induced phenomena in Lotka-Volterra systems. Fluctuation and Noise Letters, 2003, 3:L177-L185
[22] Spagnolo B, Cirone M, La Barbera A, et al. Noise-induced effects in population dynamics. J Phys:Condens Matter, 2002, 14:2247-2255
[23] Spagnolo B, Valenti D, Fiasconaro A. Noise in ecosystems:A short review. Math Biosci Engi, 2004, 1:185-211
[24] Lande R, Engen S, Saether B E. Stochastic Population Dynamics in Ecology and Conservation. Oxford:Oxford University Press, 2003
[25] Ridolfi L, Dodorico P, Laio F. Noise-Induced Phenomena in the Environmental Sciences. Cambridge:Cambridge University Press, 2011
[26] Liu Y, Shan M, Lian X. Stochastic extinction and persistence of a parasite-host epidemiological model. Physica A, 2016, 462:586-602
[27] Fiasconaro A, Spagnolo B. Resonant activation in piecewise linear asymmetric potentials. Phys Rev E, 2011, 83:041122
[28] Pizzolato N, Fiasconaro A, Persano Adorno D, et al. Resonant activation in polymer translocation:new insights into the escape dynamics of molecules driven by an oscillating field. Phys Biol, 2010, 7:034001
[29] Fiasconaro A, Mazo J J, Spagnolo B. Noise-induced enhancement of stability in a metastable system with damping. Phys Rev E, 2010, 82:041120
[30] Ciuchi S, de Pasquale, Spagnolo B. Nonlinear relaxation in the presence of an absorbing barrier. Phys Rev E, 1993, 47:3915-3926
[31] Bashkirtseva I, Ryashko L. How environmental noise can contract and destroy a persistence zone in population models with Allee effect. Theoretical Population Biology, 2017, 115:61-68
[32] Dubkov A, Spagnolo B. Langevin approach to Lévy flights in fixed potentials:Exact results for stationary probability distributions. Acta Phys Pol B, 2007, 38:1745-1758
[33] La Barberaa A, Spagnolo B. Spatio-temporal patterns in population dynamics. Physica A, 2002, 314:120-124
[34] Valenti D, Fiasconaro A, Spagnolo B. Pattern formation and spatial correlation induced by the noise in two competing species. Acta Physica Pol B, 2004, 35:1481-1489
[35] Mantegna R N, Spagnolo B, Testa L. Stochastic resonance in magnetic systems described by Preisach hysteresis model. J Appl Phys, 2005, 97:10E519/2
[36] Bashkirtseva I, Ryashko L. Noise-induced shifts in the population model with a weak Allee effect. Phys A, 2018, 491:28-36
[37] Sun G Q, Jin Z, Li L, Liu Q X. The role of noise in a predator-prey model with Allee effect. J Biol Phys, 2009, 35:185-196
[38] Gao J B, Hwang S K, Liu J M. When can noise induce chaos? Phys Rev Lett, 1999, 82:1132-1135
[39] Lai Y C, Tel T. Transient Chaos:Complex Dynamics on Finite Time Scales. Berlin:Springer, 2011
[40] Augello G, Valenti D, Spagnolo B. Non-Gaussian noise effects in the dynamics of a short overdamped Josephson junction. Eurn Phys J B, 2010, 78:225-234
[41] Shi Q H, Peng C M. Wellposedness for semirelativistic Schrodinger equation with power-type nonlinearity. Nonl Anal:TMA, 2019, 178:133-144
[42] Shi Q H, Wang S. Nonrelativistic approximation in the energy space for KGS system. J Math Anal Appl, 2018, 462:1242-1253
[43] Wang C Y, Zhou Y Q, Li Y H, Li R. Well-posedness of a ratio-dependent Lotka-Volterra system with feedback control. Boundary Value Problems, 2018, 2018:ID 117
[44] Wang C Y, Li L R, Zhou Y Q, Li R. On a delay ratio-dependent predator-prey system with feedback controls and shelter for the prey. Int J Biomath, 2018, 11(7):ID 1850095
[45] Boukhatem Y, Benabderrahmane B. General decay for a viscoelastic equation of variable coefficients with a time-varying delay in the boundary feedback and acoustic boundary conditions. Acta Math Sci, 2017, 37B(5):1453-1471
[46] Jia X J, Jia R A. Improve efficiency of biogas feedback supply chain in rural China. Acta Math Sci, 2017, 37B(3):768-785
[47] Apalara T A. Uniform decay in weakly dissipative timoshenko system with internal distributed delay feedbacks. Acta Math Sci, 2016, 36B(3):815-830
[48] Wang C Y, Liu H, Pan S, Su X L, Li R. Globally Attractive of a ratio-dependent Lotka-Volterra predatorprey model with feedback control. Adv Biosci Bioeng, 2016, 4(5):59-66
[49] Li N, Wang C Y. New existence results of positive solution for a class of nonlinear fractional differential equations. Acta Math Sci, 2013, 33B(3):847-854
[50] Yang X S, Cao J D. Adaptive pinning synchronization of coupled neural networks with mixed delays and vector-form stochastic perturbations. Acta Math Sci, 2012, 32B(3):955-977
[51] Chen F. On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay. J Comput Appl Math, 2005, 180:33-49
[52] Nakata Y, Muroya Y. Permanence for nonautonomous Lotka-Volterra cooperative systems with delays. Nonli Anal:RWA, 2010, 11:528-534
[53] Khalil H K. Nonlinear Systems. 3rd ed. Englewood Cliffs:Prentice-Hall, 2002
[54] Basener W. Topology and its Applications. Hoboken:John Wiley and Sons, 2006
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