Four Positive Almost Periodic Solutions to an Impulsive Delayed Plankton Allelopathy System with Multiple Exploit (or Harvesting) Terms
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:75587", author = "Fengshuo Zhang and Zhouhong Li", title = "Four Positive Almost Periodic Solutions to an Impulsive Delayed Plankton Allelopathy System with Multiple Exploit (or Harvesting) Terms", 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.", volume = "10", number = "11", pages = "597-8", }
