Four Positive Almost Periodic Solutions to an Impulsive Delayed Plankton Allelopathy System with Multiple Exploit (or Harvesting) Terms
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Four Positive Almost Periodic Solutions to an Impulsive Delayed Plankton Allelopathy System with Multiple Exploit (or Harvesting) Terms

Authors: Fengshuo Zhang, Zhouhong Li

Abstract:

In this paper, we obtain sufficient conditions for the existence of at least four positive almost periodic solutions to an impulsive delayed periodic plankton allelopathy system with multiple exploited (or harvesting) terms. This result is obtained through the use of Mawhins continuation theorem of coincidence degree theory along with some properties relating to inequalities.

Keywords: Almost periodic solutions, plankton allelopathy system, coincidence degree, impulse.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1130535

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[1] A. Mukhopadhyay, J. Chattopadhyay, P.K. Tapaswi, ”A delay differential equations model of plankton allelopathy”, Mathematical Biosciences,, Vol.149, pp. 167-189,1998.
[2] A.M. Samoilenko, N.A. Perestyuk, ”Impulsive Differential Equations”,World Scientific, Singapore, 1995.
[3] B.X. Yang, J.L. Li, An almost periodic solution for an impulsive two-species logarithmic population model with time -varying delay, Mathematical and Computer Modelling, Vol.55 n0o.7-8, pp. 1963-1968, 2012.
[4] C.Y. He, ”Almost Periodic Differential Equations”, Higher Education Publishing House, Beijing (in Chinese), 1992.
[5] D. Hu, Z. Zhang, ”Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms”, Nonlinear Anal. RWA., Vol.11, pp. 1560-1571, 2010.
[6] D.S. Wang, ”Four positive periodic solutions of a delayed plankton allelopathy system on time scales with multipoe exploited (or harvesting) terms”, IMA Journal of Applied mathematics, Vol.78, pp. 449-473, 2013.
[7] E. L. Rice, Alleopathy, second ed., Academic Press, New York, 1984.
[8] G.T. Stamov, I.M. Stamova, J.O. Alzaut, ”Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations”, Nonlinear Analysis: Hybrid Systems, Vol.6 no.2, pp. 818-823, 2012.
[9] J.B. Geng, Y.H. Xia, ”Almost periodic solutions of a nonlinear ecological model”, Commun Nonlinear Sci Numer Simulat, Vol.16, pp.2575-2597, 2011.
[10] J. Chattopadhyay, ”Effect of toxic substances on a two-species competitive system”, Ecol. Modelling, Vol.84, pp. 287-289, 1996.
[11] J. Dhar, K. S. Jatav, ”Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories”, Ecological Complexity, Vol.16, pp. 59-67, 2013.
[12] J.G. Jia, M.S. Wang, M.L. Li, ”Periodic solutions for impulsive delay differential equations in the control model of plankton allelopathy”, Chaos, Solitons and Fractals, Vol.32, pp. 962-968, 2007.
[13] J. Hou, Z.D. Teng, S.J. Gao, ”Permanence and global stability for nonautonomous Nspecies Lotka-Volterra competitive system with impulses”, Nonlinear Anal. RWA., Vol.11 no.3, pp. 1882-1896, 2010.
[14] J.M.Smith, Modles in Ecology, Cambridge University, Cambridge, 1974.
[15] J. ZHEN, Z.E. MA, ”Periodic Solutions for Delay Differential Equations Model of Plankton Allelopathy”, Computers and Mathematics with Applications , Vol.44, pp. 491-500, 2002.
[16] K.H. Zhao, Y.K. Li, ”Four positive periodic solutions to two species parasitical system with harvesting terms”, Comput. Math. with Appl., Vol.59 no.8, pp. 2703-2710, 2010.
[17] K.H. Zhao, Y. Ye, ”Four positive periodic solutions to a periodic Lotka-Volterra predatoryprey system with harvesting terms”, Nonlinear Anal. RWA., Vol.11, pp.2448-2455, 2010.
[18] L. Yang, S.M. Zhong, ”Dynamics of a delayed stage-structured model with impulsive harvesting and diffusion”, Ecological Complexity, Vol.19, pp. 111-123, 2014.
[19] M.X. He, F.D. Chen, Z. Li, ”Almost periodic solution of an impulsive differential equation model of plankton allelopathy”, Nonlinear Analysis: Real World Applications,, Vol.11, pp. 2296-2301, 2010.
[20] M. Zhao, X.T. Wang, H.G.Yu, J. Zhu, ”Dynamics of an ecological model with impulsive control strategy and distributed time delay”, Mathematics and Computers in Simulation, Vol.82 no.8, pp. 1432-1444, 2012.
[21] Q. Wang, Y.Y. Fang, D.C. Lu, ”Existence of four periodic solutions for a generalized delayed ratio-dependent predator-prey system”, Applied Mathematics and Computation, Vol.247, pp. 623-630 ,2014.
[22] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differetial Equitions, Springer Verlag, Berlin, 1977.
[23] S.Y. Tang, L.S. Chen, ”The periodic predator-prey Lotka-Volterra model with impulsive effect”, J. Mech. Med. Biol., Vol.2, pp. 1-30, 2002.
[24] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[25] X.H. Wang, J.W. Jia, ”Dynamic of a delayed predator-prey model with birth pulse and impulsive harvesting in a polluted environment”, Physica A: Statistical Mechanics and its Applications, Vol.422, pp. 1-15, 2015.
[26] X.Y. Song and L.S. Chen, ”Periodic solution of a delay differential equation of plankton allelopathy”, Acta Math. Sci. Ser. A, Vol.23, pp. 8-13, 2003.
[27] Y.K. Li, K.H. Zhao, ”2n positive periodic solutions to n species non-autonomous Lotka-Volterra unidirectional food chains with harvesting terms”, Math. Model. Anal., Vol.15, pp. 313-326, 2010.
[28] Y.K. Li, K.H. Zhao, ”Eight positive periodic solutions to three species non-autonomous Lotka-Volterra cooperative systems with harvesting terms”, Topol. Methods Nonlinear Anal., Vol.37, pp. 225-234, 2011.
[29] Y.K. Li, K.H. Zhao, ”Multiple positive periodic solutions to m-layer periodic Lotka-Volterra network-like multidirectional food-chain with harvesting terms”, Anal. Appl., Vol.9, pp. 71-96, 2011.
[30] Y.K. Li, K.H. Zhao, Y. Ye, ”Multiple positive periodic solutions of n species delay competition systems with harvesting terms”, Nonlinear Anal. RWA., Vol.12, pp. 1013-1022, 2011.
[31] Y.K. Li, ”Positive periodic solutions of a periodic neutral delay logistic equation with impulses”, Comput. Math. Appl., Vol.56 no.9, pp. 2189-2196, 2008.
[32] Y.K. Li, Y. Ye, ”Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms”, Commun. Nonlinear Sci. Numer. Simul., Vol.18 no.11, pp. 3190-3201, 2013.
[33] Y. Xie, X.G. Li, ”Almost periodic solutions of single population model with hereditary”, Appl. Math. Comput., Vol.203, pp. 690-697, 2008.
[34] Z.H. Li, K.H. Zhao, Y.K. Li, ”Multiple positive periodic solutions for a non-autonomous stage-structured predatory-prey system with harvesting terms”, Commun. Nonlinear Sci. Numer. Simul., Vol.15, pp. 2140-2148, 2010.
[35] Z.J. Du, M. Xu, ”Positive periodic solutions of n-species neutral delayed Lotka-Volterra competition system with impulsive perturbations”, Applied Mathematics and Computation, Vol.243, pp. 379-391, 2014.
[36] Z.J. Du, Y.S. Lv, ”Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time”, Applied Mathematical Modelling, Vol.37 no.3, pp. 1054-1068, 2013.
[37] Z.J. Liu, J.H. Wu, Y.P. Chen, M. Haque, ”Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy”, Nonlinear Analysis: Real World Applications, Vol.11, pp. 432-445, 2010.
[38] Z.J. Liu, L.S. Chen, ”Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays”, Journal of Computational and Applied Mathematics, Vol.197, pp. 446-456,2006.
[39] Z.L. He, L.F. Nie, Z.D. Teng, ”Dynamics analysis of a two-species competitive model with state-dependent impulsive effects”, Journal of the Franklin Institute, Vol.352 no.5, pp. 2090-2112, 2015.
[40] Z. Li, M.A. Han, F.D. Chen, ”Almost periodic solutions of a discrete almost periodic logistic equation with delay”, Applied Mathematics and Computation, Vol.232, pp. 743-751, 2014.
[41] Z.Q. Zhang, Z. Hou, ”Existence of four positive periodic solutions for a ratio-dependent predator-prey system with multiple exploited (or harvesting) terms”, Nonlinear Anal. RWA., Vol.11, pp. 1560-1571, 2010.
[42] Z. Zhang, T. Tian, ”Multiple positive periodic solutions for a generalized predator-prey system with exploited terms”, Nonlinear Anal. RWA., Vol.9, pp. 26-39, 2008.