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In this paper, by using Mawhin-s continuation theorem of coincidence degree and a method based on delay differential inequality, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of cellular neural networks with distributed delays and impulses on time scales. The results of this paper generalized previously known results., {"references":["J. Cao, New results concerning exponential stability and periodic solutions\nof delayed cellular neural networks with delays, Phys Lett A 307 (2003)\n136-147.","Y. Li, S. Gao, Global Exponential Stability for Impulsive BAM Neural\nNetworks with Distributed Delays on Time Scales, Neural Processing\nLetters 31 (1) (2010) 65-91.","Z. Liu and A. Chen, Existence and global exponential stability of periodic\nsolutions of cellular neural networks with time-vary delays, J. Math Anal.\nAppl. 290 (2004) 247-262.","Y. Li, Existence and stability of periodic solutions for Cohen¨CGrossberg\nneural networks with multiple delays. Chaos, Solitons & Fractals 20\n(2004) 459-466.","Y. Li, P. Liu, Existence and stability of positive periodic solution for BAM\nneural networks with delays, Math Comput Model 40 (2004) 757-770.","Y. Li, C. Liu, L. Zhu, Global exponential stability of periodic solution for\nshunting inhibitory CNNs with delays. Phys Lett A 337 (2005) 46-54.","Y. Li, T. Zhang, Global exponential stability of fuzzy internal delayed\nneural networks with impulses on time scales, Int. J. Neural Syst.19 (6)\n(2009) 449-456.","Y. Chen, Global stability of neural networks with distributed delays,\nNeural Networks 15 (2002) 867-871.","C. Feng, R. Plamondon, On the stability analysis of delayed neural\nnetworks systems, Neural Networks 14 (2011) 1181-1188.\n[10] J. Zhang, Globally exponential stability of neural networks with variables\ndelays. IEEE Trans Circuit Syst-I 50 (2003) 288-291.\n[11] Y. Li, Global exponential stability of BAM neural networks with delays\nand impulses, Chaos, Solitons & Fractals 24 (2005) 279-285.\n[12] Y. Ren, Y. Li, Periodic solutions of recurrent neural networks with\ndistributed delays and impulses on time scales, Int. J. Comp. Math. Sci.\n5 (2011) 209-218.\n[13] Z. Guan, L. James, G. Chen, On impulsive auto-associative neural\nnetworks, Neural Networks 13 (2000) 63-69.\n[14] Y. Li, W. Xing, L. Lu, Existence and global exponential stability of\nperiodic solution of a class of neural networks with impulses, Chaos,\nSolitons & Fractals 27 (2006) 437-445.\n[15] Y. Li, W. Xing, Existence and global exponential stability of periodic\nsolution of CNNs with impulses, Chaos, Solitons & Fractals 33 (2007)\n1686-1693.\n[16] R. Gains, J. Mawhin, Coincidence degree and nonlinear differential\nequations, Springer-Verlag, Berlin, 1977.\n[17] H. Tokumarn, N. Adachi, T. Amemiya, Macroscopic stability of interconnected\nsystems, 6th IFAC Congress, paper ID44.4, 1975, 1-7."]}

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., {"references":["A. Mukhopadhyay, J. Chattopadhyay, P.K. Tapaswi, \"A delay differential\nequations model of plankton allelopathy\", Mathematical Biosciences,,\nVol.149, pp. 167-189,1998.","A.M. Samoilenko, N.A. Perestyuk, \"Impulsive Differential\nEquations\",World Scientific, Singapore, 1995.","B.X. Yang, J.L. Li, An almost periodic solution for an impulsive\ntwo-species logarithmic population model with time -varying delay,\nMathematical and Computer Modelling, Vol.55 n0o.7-8, pp. 1963-1968,\n2012.","C.Y. He, \"Almost Periodic Differential Equations\", Higher Education\nPublishing House, Beijing (in Chinese), 1992.","D. Hu, Z. Zhang, \"Four positive periodic solutions to a Lotka-Volterra\ncooperative system with harvesting terms\", Nonlinear Anal. RWA., Vol.11,\npp. 1560-1571, 2010.","D.S. Wang, \"Four positive periodic solutions of a delayed plankton\nallelopathy system on time scales with multipoe exploited (or harvesting)\nterms\", IMA Journal of Applied mathematics, Vol.78, pp. 449-473, 2013.","E. L. Rice, Alleopathy, second ed., Academic Press, New York, 1984.","G.T. Stamov, I.M. Stamova, J.O. Alzaut, \"Existence of almost periodic\nsolutions for strongly stable nonlinear impulsive differential-difference\nequations\", Nonlinear Analysis: Hybrid Systems, Vol.6 no.2, pp. 818-823,\n2012.","J.B. Geng, Y.H. Xia, \"Almost periodic solutions of a nonlinear ecological\nmodel\", Commun Nonlinear Sci Numer Simulat, Vol.16, pp.2575-2597,\n2011.\n[10] J. Chattopadhyay, \"Effect of toxic substances on a two-species\ncompetitive system\", Ecol. Modelling, Vol.84, pp. 287-289, 1996.\n[11] J. Dhar, K. S. Jatav, \"Mathematical analysis of a delayed stage-structured\npredator-prey model with impulsive diffusion between two predators\nterritories\", Ecological Complexity, Vol.16, pp. 59-67, 2013.\n[12] J.G. Jia, M.S. Wang, M.L. Li, \"Periodic solutions for impulsive delay\ndifferential equations in the control model of plankton allelopathy\",\nChaos, Solitons and Fractals, Vol.32, pp. 962-968, 2007.\n[13] J. Hou, Z.D. Teng, S.J. Gao, \"Permanence and global stability\nfor nonautonomous Nspecies Lotka-Volterra competitive system with\nimpulses\", Nonlinear Anal. RWA., Vol.11 no.3, pp. 1882-1896, 2010.\n[14] J.M.Smith, Modles in Ecology, Cambridge University, Cambridge, 1974. [15] J. ZHEN, Z.E. MA, \"Periodic Solutions for Delay Differential Equations\nModel of Plankton Allelopathy\", Computers and Mathematics with\nApplications , Vol.44, pp. 491-500, 2002.\n[16] K.H. Zhao, Y.K. Li, \"Four positive periodic solutions to two species\nparasitical system with harvesting terms\", Comput. Math. with Appl.,\nVol.59 no.8, pp. 2703-2710, 2010.\n[17] K.H. Zhao, Y. Ye, \"Four positive periodic solutions to a periodic\nLotka-Volterra predatoryprey system with harvesting terms\", Nonlinear\nAnal. RWA., Vol.11, pp.2448-2455, 2010.\n[18] L. Yang, S.M. Zhong, \"Dynamics of a delayed stage-structured model\nwith impulsive harvesting and diffusion\", Ecological Complexity, Vol.19,\npp. 111-123, 2014.\n[19] M.X. He, F.D. Chen, Z. Li, \"Almost periodic solution of an impulsive\ndifferential equation model of plankton allelopathy\", Nonlinear Analysis:\nReal World Applications,, Vol.11, pp. 2296-2301, 2010.\n[20] M. Zhao, X.T. Wang, H.G.Yu, J. Zhu, \"Dynamics of an ecological model\nwith impulsive control strategy and distributed time delay\", Mathematics\nand Computers in Simulation, Vol.82 no.8, pp. 1432-1444, 2012.\n[21] Q. Wang, Y.Y. Fang, D.C. Lu, \"Existence of four periodic solutions\nfor a generalized delayed ratio-dependent predator-prey system\", Applied\nMathematics and Computation, Vol.247, pp. 623-630 ,2014.\n[22] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differetial\nEquitions, Springer Verlag, Berlin, 1977.\n[23] S.Y. Tang, L.S. Chen, \"The periodic predator-prey Lotka-Volterra model\nwith impulsive effect\", J. Mech. Med. Biol., Vol.2, pp. 1-30, 2002.\n[24] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive\nDifferential Equations, World Scientific, Singapore, 1989.\n[25] X.H. Wang, J.W. Jia, \"Dynamic of a delayed predator-prey model with\nbirth pulse and impulsive harvesting in a polluted environment\", Physica\nA: Statistical Mechanics and its Applications, Vol.422, pp. 1-15, 2015.\n[26] X.Y. Song and L.S. Chen, \"Periodic solution of a delay differential\nequation of plankton allelopathy\", Acta Math. Sci. Ser. A, Vol.23, pp.\n8-13, 2003.\n[27] Y.K. Li, K.H. Zhao, \"2n positive periodic solutions to n species\nnon-autonomous Lotka-Volterra unidirectional food chains with\nharvesting terms\", Math. Model. Anal., Vol.15, pp. 313-326, 2010.\n[28] Y.K. Li, K.H. Zhao, \"Eight positive periodic solutions to three species\nnon-autonomous Lotka-Volterra cooperative systems with harvesting\nterms\", Topol. Methods Nonlinear Anal., Vol.37, pp. 225-234, 2011.\n[29] Y.K. Li, K.H. Zhao, \"Multiple positive periodic solutions to m-layer\nperiodic Lotka-Volterra network-like multidirectional food-chain with\nharvesting terms\", Anal. Appl., Vol.9, pp. 71-96, 2011.\n[30] Y.K. Li, K.H. Zhao, Y. Ye, \"Multiple positive periodic solutions of\nn species delay competition systems with harvesting terms\", Nonlinear\nAnal. RWA., Vol.12, pp. 1013-1022, 2011.\n[31] Y.K. Li, \"Positive periodic solutions of a periodic neutral delay\nlogistic equation with impulses\", Comput. Math. Appl., Vol.56 no.9, pp.\n2189-2196, 2008.\n[32] Y.K. Li, Y. Ye, \"Multiple positive almost periodic solutions to an\nimpulsive non-autonomous Lotka-Volterra predator-prey system with\nharvesting terms\", Commun. Nonlinear Sci. Numer. Simul., Vol.18 no.11,\npp. 3190-3201, 2013.\n[33] Y. Xie, X.G. Li, \"Almost periodic solutions of single population model\nwith hereditary\", Appl. Math. Comput., Vol.203, pp. 690-697, 2008.\n[34] Z.H. Li, K.H. Zhao, Y.K. Li, \"Multiple positive periodic solutions for a\nnon-autonomous stage-structured predatory-prey system with harvesting\nterms\", Commun. Nonlinear Sci. Numer. Simul., Vol.15, pp. 2140-2148,\n2010.\n[35] Z.J. Du, M. Xu, \"Positive periodic solutions of n-species neutral delayed\nLotka-Volterra competition system with impulsive perturbations\", Applied\nMathematics and Computation, Vol.243, pp. 379-391, 2014.\n[36] Z.J. Du, Y.S. Lv, \"Permanence and almost periodic solution of a\nLotka-Volterra model with mutual interference and time\", Applied\nMathematical Modelling, Vol.37 no.3, pp. 1054-1068, 2013.\n[37] Z.J. Liu, J.H. Wu, Y.P. Chen, M. Haque, \"Impulsive perturbations in\na periodic delay differential equation model of plankton allelopathy\",\nNonlinear Analysis: Real World Applications, Vol.11, pp. 432-445, 2010.\n[38] Z.J. Liu, L.S. Chen, \"Positive periodic solution of a general discrete\nnon-autonomous difference system of plankton allelopathy with delays\",\nJournal of Computational and Applied Mathematics, Vol.197, pp.\n446-456,2006.\n[39] Z.L. He, L.F. Nie, Z.D. Teng, \"Dynamics analysis of a two-species\ncompetitive model with state-dependent impulsive effects\", Journal of\nthe Franklin Institute, Vol.352 no.5, pp. 2090-2112, 2015.\n[40] Z. Li, M.A. Han, F.D. Chen, \"Almost periodic solutions of a discrete\nalmost periodic logistic equation with delay\", Applied Mathematics and\nComputation, Vol.232, pp. 743-751, 2014.\n[41] Z.Q. Zhang, Z. Hou, \"Existence of four positive periodic solutions\nfor a ratio-dependent predator-prey system with multiple exploited (or\nharvesting) terms\", Nonlinear Anal. RWA., Vol.11, pp. 1560-1571, 2010.\n[42] Z. Zhang, T. Tian, \"Multiple positive periodic solutions for a generalized\npredator-prey system with exploited terms\", Nonlinear Anal. RWA., Vol.9,\npp. 26-39, 2008."]}

In this paper, we consider quasilinear fractional differential equation with resonance boundary condition. After translating the quasilinear equation into the linear fractional differential system, by using coincidence degree theory, the existence result is established.

In this paper, the three species food chain model on time scales is established. The Monod–Haldane functional response and time delay are considered. With the help of coincidence degree theory, existence of periodic solutions is investigated, which unifies the continuous and discrete analogies., {"references":["","Jean-Pierre Francoise, Jaume Llibre, Analytical study of a triple Hopf\nbifurcation in a tritrophic food chain model, Applied Mathematics and\nComputation, 217(2011), 7146–7154.","B.Mukhopadhyay, R.Bhattacharyya, A stage–structured food chain model\nwith stage dependent predation: Existence of codimension one and codimension\ntwo bifurcations, Nonlinear Analysis: RWA, 12(2011), 3056–\n3072.","Changyong Xu, Meijuan Wang, Permanence for a delayed discrete three–\nlevel food–chain model with Beddington–DeAngelis functional response,\nApplied Mathematics and Computation, 187(2007), 1109–1119.","Rui Xu, Lansun Chen, Feilong Hao, Periodic solutions of a discrete time\nLotka–Volterra type food–chain model with delays, Applied Mathematics\nand Computation, 171(2005), 91–103.","Weiming Wang, Hailing Wang, Zhenqing Li, The dynamic complexity\nof a three–species Beddington–type food chain with impulsive control\nstrategy, Chaos, Solitons and Fractals, 32(2007), 1772–1785.","J.F. Andrews, A mathematical model for the continuous culture of\nmicroorganisms utilizing inhabitory substrates, Biotechnol Bioengrg.\n10(1986), 707–723.","Stefan Hilger, Analysis on measure chains–a unified approach to continuous\nand discrete calculus, Results Math., 18(1990), 18–56.","Martin Bohner, Meng Fan, Jiming Zhang, Periodicity of scalar dynamic\nequations and applications to population models, J. Math. Anal. Appl,\n330(2007), 1–9.","Yu Tong, Zhenjie Liu, Zhiying Gao, Yonghong Wang, Existence of\nperiodic solutions for a predator–prey system with sparse effect and\nfunctional response on time scales, Commun. Nonlinear Sci. Numer.\nSimulat., 17(2012), 3360–3366.\n[10] Kejun Zhuang, Periodic solutions for a three species predator–prey system\non time scales, Int. J. Dynamical Systems and Differential Equations,\n3(2011), 3-11.\n[11] Martin Bohner, Meng Fan, Jiming Zhang, Existence of periodic solutions\nin predator–prey and competition dynamic systems, Nonl. Anal: RWA,\n7(2006), 1193–1204.\n[12] Martin Bohner, Allan Peterson, Dynamic Equations on Time Scales: An\nIntroduction with Applications, Boston: Birkh¨auser, 2001.\n[13] Bingbing Zhang, Meng Fan, A remark on the application of coincidence\ndegree to periodicity of dynamic equtions on time scales, J. Northeast\nNormal University(Natural Science Edition), 39(2007), 1–3. (in Chinese)\n[14] R. E. Gaines, J. L. Mawhin, Coincidence Degree and Nonlinear Differential\nEquations, Lecture Notes in Mathematics, Berlin: Springer–Verlag,\n1977."]}

In this paper, by applying Mawhin-s continuation theorem of coincidence degree theory, we study the existence of almost periodic solutions for neural multi-delay logarithmic population model and obtain one sufficient condition for the existence of positive almost periodic solution for the above equation. An example is employed to illustrate our result., {"references":["S.Lu, W. Ge,Existence of positive periodic solutions for neutral logarithmic\npop- ulation model with multiple delays. J. Comput. Appl. Math. 166(2), 371-383 (2004).","Y. Luo, Z. Luo,Existence of positive periodic solutions for neutral multidelay\nlog- arithmic population model. Appl. Math. Comput. 216, 1310-1315 (2010)","K. Gopalsamy, Stability and Oscillations in Delay Differential Equations\nof Population Dynamics, Kluwer Acad. Publ., 1992.","M. Kot, Elements of Mathematical Ecology, Cambr. Univ. Press, 2001.","Y. Kuang, Delay Differential Equations With Applications in Population\nDynamics, Academic Press, Inc., 1993.","J. Alzabuta, G. Stamovb,E. Sermutlu, Positive almost periodic solutions\nfor a delay logarithmic population model.Math. and Comput. Model.,53(12), 161-167(2011).","Q. Wang,H. Zhang, Y. Wang, Existence and stability of positive almost\nperiodic solutions and periodic solutions for a logarithmicpopulation\nmodel. Nonlinear Anal: Theory, Methods and Appl.,72(12), 4384-\n4389(2010).","Gaines R, Mawhin J. Coincidence degree and nonlinear differential\nequations. Berlin: Springer Verlag; 1977.","Y. Xie, X. Li, Almost periodic solutions of single population model with\nhereditary effects, Appl. Math. Comp. 203,690-697(2008).\n[10] C. He, Almost Periodic Differential Equations, Higher Education Publishing\nHouse, Beijing, 1992 (in Chinese)."]}

This paper is devoted to a delayed periodic predatorprey system with non-monotonic numerical response on time scales. With the help of a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results improve and generalize some known ones., {"references":["Y. Chen, Multiple periodic solutions of delayed predator-prey systems\nwith type IV functional responses, Nonlinear Anal. Real World Appl. 5\n(2004) 45-53.","X. Ding, J. Jiang, Multiple periodic solutions in generalized Gausetype\npredator-prey systems with non-monotonic numerical responses,\nNonlinear Anal. Real World Appl. 10 (2009) 2819-2827.","X. Ding, J. Jiang, Positive periodic solutions for a generalized twospecies\nsemi-ratio-dependent predator-prey system in a two-patch environment,\nMath. Comput. Modelling 52 (2010) 361-369.","M. Fan, K. Wang, Periodic solutions of a discrete time non-autonomous\nratio-dependent predator-prey system, Math. Comput. Modelling 35\n(2002) 951-961.","X. Hu, G. Liu, J. Yan, Existence of multiple positive periodic solutions of\ndelayed predator-prey models with functional responses, Comput. Math.\nAppl. 52 (2006) 1453-1462.","X. Tang, X. Zou, On positive periodic solutions of Lotka-Volterra\ncompetition systems with deviating arguments, Proc. Amer. Math. Soc.\n134 (2006) 2967-2974.","W. Zhang, D. Zhu, P. Bi, Multiple positive periodic solutions of a\ndelayed discrete predator-prey system with type IV functional responses,\nAppl. Math. Lett. 20 (2007) 1031-1038.","M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An\nIntroduction with Applications, Birkh┬¿auser, Boston, 2001.","M. Bohner, A. Peterson, Advances in Dynamic Equations on Time\nScales, Birkh┬¿auser, Boston, 2003.\n[10] V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems\non Measure Chains, Kluwer Academic Publishers, Dordrecht,\n1996.\n[11] M. Bohner, M. Fan, J. Zhang, Existence of periodic solutions in predatorprey\nand competition dynamic systems, Nonlinear Anal. Real World\nAppl. 7 (2006) 1193-1204.\n[12] M. Fazly, M. Hesaaraki, Periodic solutions for predator-prey systems\nwith Beddington-DeAngelis functional response on time scales, Nonlinear\nAnal. Real World Appl. 9 (2008) 1224-1235.\n[13] Y. Li, H. Zhang, Existence of periodic solutions for a periodic mutualism\nmodel on time scales, J. Math. Anal. Appl. 343 (2008) 818-825.\n[14] J. Zhang, M. Fan, H. Zhu, Periodic solution of single population models\non time scales, Math. Comput. Modelling 52 (2010) 515-521.\n[15] L. Zhang, H. Li, X. Zhang, Periodic solutions of competition Lotka-\nVolterra dynamic system on time scales, Comput. Math. Appl. 57 (2009)\n1204-1211.\n[16] K. Zhuang, Periodicity for a semi-ratio-dependent predator-prey system\nwith delays on time scales, Int. J. Comput. Math. Sci. 4 (2010) 44-47.\n[17] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential\nEquations, Springer-Verlag, Berlin, 1977.\n[18] B. Zhang, M. Fan, A remark on the application of coincidence degree\nto periodicity of dynamic equations on time scales, J. Northeast Norm.\nUniv. Nat. Sci. 39(4) (2007) 1-3.\n[19] F.E. Smith, Population dynamics in daphnia magna and a new model\nfor population growth, Ecology 44 (1963) 651-663.\n[20] K. Gopalsamy, G. Ladas, On the oscillation and asymptotic behavior\nof İ\nN (t) = N(t)[a + bN(t − ┬┐) − cN\n2(t − ┬┐)], Qurt. Appl. Math. 3\n(1990) 433-440."]}

In this paper, using the Gaines and Mawhin,s continuation theorem of coincidence degree theory on time scales, the existence of periodic solutions for a two-prey one-predator system is studied. Some sufficient conditions for the existence of positive periodic solutions are obtained. The results provide unified existence theorems of periodic solution for the continuous differential equations and discrete difference equations., {"references":["M. Bohner and A Peterson, Dynamic Equations on Times Scales: An\nIntroduction with Applications. Boston: Birkh¨a user, 2001.","M. Bohner and A Peterson, Advances in Dynamic Equations on Time\nScales. Boston: Birkh¨a user, 2003.","S. Hilger, Analysis on measure chains-a unfified approach to continuous\nand discrete calculus. Results in Math. 18 (1990) 18-56.","E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral\nnonlinear dynamical equation on a time scale. J. Math. Anal. Appl. 319\n(1) (2006) 315-325.","Y. K. Li and H. T. Zhang, Existence of periodic solutions for a periodic\nmutualism model on time scales, J. Math. Anal. Appl. 343(2) (2008)\n818-825.","M. Fazly and M. Hesaaraki, Periodic solutions for predator-prey systems\nwith Beddington-DeAngelis functional response on time scales. Nonlinear\nAnal.: Real World Appl. 9(3) (2008) 1224-1235.","H. J. Li, A. P. Liu and Z. T. Hao, Existence for periodic solutions of a\nratio-dependent predator-prey system with time-varying delays on time\nscales. Anal. Appl. 8(3) (2010) 227-233.","W. P. Zhang, P. Bi and D. M. Zhu, Periodicity in a ratio-dependent\npredator-prey system with stage-structured predator on time scales.\nNonlinear Anal.: Real World Appl. 9(2) (2008) 344-353.","J. Liu, Y. K. Li and L. L. Zhao, On a periodic solution predator-prey\nsystem with time delays on time scales. Commun. Nonlinear Sci. Numer.\nSimulat. 14(8) (2009) 3432-3438.\n[10] H. Baek, Species extiction and permanence of an impulsively controlled\ntwo-prey one-predator system with seasonal effects. BioSystems 98(1)\n(2009) 7-18.\n[11] B. Aulbach and S. Hilger, Linear Dynamical Processes with Inhomogeneous\nTime Scales, Nonlinear Dynamics and Quantum Dynamical\nSystems. Berlin: Akademie Verlage, 1990.\n[12] V. Lakshmikantham, S. Sivasundaram and B. Kaymarkcalan, Dynamic\nSystems on Measure Chains. Boston: Kluwer Academic Publishers, 1996.\n[13] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear\nDifferential Equations. Berlin: Springer-verlag, 1997."]}

In this paper, a neutral impulsive competition system with distributed delays is studied by using Mawhin-s coincidence degree theory and the mean value theorem of differential calculus. Sufficient conditions on the existence of positive periodic solution of the system are obtained., {"references":["Y. Li, Positive periodic solution for neutral delay model, Acta Math.\nSinica 39 (1996) 789-795 (in Chinese).","Y. Li, Periodic solution of a periodic neutral delay equation, J. Math.\nAnal. Appl. 214 (1997) 11-21.","K. Gopalsamy, Stability and Oscillations in Delay Differential Equations\nof Population Dynamics, Kluwer Academic, Boston, 1992.","Y. Kuang, On neutral delay two species Lotka-Volterra competitive\nsystem, J. Austral. Math. Soc. Ser. B 32 (1991) 311-326.","Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional\ndifferential equation. Inter. J. Comp. Math. Sci., 5 (2011) 8-12.","Y. Li, On a periodic neutral delay Lotka-Volterra system, Nonlinear Anal.\n39 (2000) 767-778.","Z. Liu, Positive periodic solution for a neutral delay competitive system,\nJ. Math. Anal. Appl. 293 (2004) 181-189.","X. Ding, S. Cheng, Positive periodic solution for a nonautonomous\nlogistic model with linear feedback regulation, Appl. Math. J. Chinese\nUniv. Ser. B, 21 (2006) 302-312.","X. Ding, Periodicity in a delayded semi-ratio-dependent predator-prey\nsystem, Appl. Math. J. Chinese Univ. Ser. B, 20 (2005) 151-158.\n[10] Y. Li, W. Xing, L. Lu, Existence and global exponential stability of\nperiodic solution of a class of neural networks with impulses, Chaos,\nSolitions and Fractals (2006) 437-445.\n[11] H. Huo, Existence of positive periodic solution for a neutral delay\nLotka-Volterra system with impulses, Computers and Mathematics with\nApplications 48 (2004) 1833-1846.\n[12] H. Huo, W. Li, Existence of positive periodic solution of a neutral impulsive\ndelay predator-prey system, Applied Mathematics and Computation\n185 (2007) 499-507.\n[13] D. Wang, C. Feng, Three positive periodic solutions to a kind of\nImpulsive Differential Equations with delay, Journal of Jilin University,\nScience Edition, 3 (2011) 391-396.\n[14] Z. Liu, L. Chen, Periodic solution of a two-species competitive system\nwith toxicant and birth pulse, Chaos, Solitons and Fractals 32 (2007)\n1703-1712.\n[15] Y. Li, Z. Xing, Existence and global exponential stability of periodic\nsolutions of CNNs with impulses, Chaos, Solitons and Fractals 33 (2007)\n1686-1693.\n[16] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differential\nEquations, Springer-Verlag, Berlin, 1977."]}

The main purpose of this paper is to investigate a discrete time three–species food chain system with ratio dependence. By employing coincidence degree theory and analysis techniques, sufficient conditions for existence of periodic solutions are established., {"references":["Haifeng Huo, Wantong Li, J J Nieto. Periodic solutions of delayed\npredator-prey model with the Beddington-DeAngelis functional response.\nChaos, Solitons and Fractals, 33(2007), 505-512.","Changyong Xu, Meijuan Wang. Permanence for a delayed discrete three-\nlevel food-chain model with Beddington-DeAngelis functional response.\nApplied Mathematics and Computation, 187(2007), 1109-1119.","A. Maiti, A.K. Pal, G.P. Samanta. Effect of time-delay on a food chain\nmodel. Applied Mathematics and Computation, 200(2008), 189-203.","Chengjun Sun, Michel Loreau. Dynamics of a three-species food chain\nmodel with adaptive traits. Chaos, Solitons and Fractals, 41(2009), 2812-\n2819.","Zhijun Zeng. Dynamics of a non-autonomous ratio-dependent food chain\nmodel. Applied Mathematics and Computation, 215(2009), 1274-1287.","Raid Kamel Naji, Ranjit Kumar Upadhyay, Vikas Rai. Dynamical consequences\nof predator interference in a tri-trophic model food chain.\nNonlinear Analysis: Real World Applications, 11(2010), 809-818.","Fengyan Wang, Guoping Pang. Chaos and Hopf bifurcation of a hybrid\nratio-dependent three species food chain. Chaos, Solitons and Fractals,\n(36)2008, 1366-1376.","Pingzhou Liu, K. Gopalsamy. Global stability and chaos in a population\nmodel with piecewise constant arguments. Applied Mathematics and\nComputation, 101(1999), 63-88.","Bingbing Zhang, Meng Fan. A remark on the application of coincidence\ndegree to periodicity of dynamic equtions on time scales. J. Northeast\nNormal University(Natural Science Edition), 39(2007), 1-3.(in Chinese)\n[10] R E Gaines, J L Mawhin. Coincidence Degree and Nonlinear Differential\nEquations. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1977."]}

In this paper, the existence of periodic solutions of a delayed competitive system with the effect of toxic substances is investigated by using the Gaines and Mawhin,s continuation theorem of coincidence degree theory on time scales. New sufficient conditions are obtained for the existence of periodic solutions. The approach is unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations. Moreover, The approach has been widely applied to study existence of periodic solutions in differential equations and difference equations., {"references":["V. Voltera, Opere matematiche: mmemorie e note, vol. V, Acc. Naz. dei\nLincei, Roma, Cremon, 1962.","A.J. Lotka, Elements of Mathematical Biology, Dover, New York, 1962.","W. Ding, M. Han, Dynamic of a non-autonomous predator-prey system\nwith infinite delay and diffusion, Comput. Math. Appl., 56 (2008) 1335-\n1350.","Z.J. Liu, R.H. Tan, Y.P. Chen, L.S. Chen, On the stable periodic solutions\nof a delayed two-species model of facultative mutualism, Appl. Math.\nComput., 196 (2008) 105-117.","D. Summers, C. Justian, H. Brian, Chaos in periodically forced discretetime\necosystem models, Chaos, Solitons & Frctals, 11 (2000) 2331-2342.","M. Fan, K. Wang, Periodic solutions of a discrete time non-autonomous\nratio-dependent predator-prey system, Math. Comput. Model. 35 (2002)\n951-961.","W. P. Zhang, D. M. Zhu, P. Bi, Multiple positive solutions of a delayed\ndiscrete predator-prey system with type IV functional responses, Appl.\nMath. Lett. 20 (2007) 1031-1038.","X.Y. Song, L.S. Chen, Periodic solutions of a delay differential equation\nof plankton allelopathy. Acta. Math. Sci. Ser. A , 23 (2003) 8-13. (in\nChinese)","M. Bohner, A. Peterson, Dynamic Equations on Times Scales: An\nIntroduction with Applications, Birkh¨auser, Boston, 2001.\n[10] V.Lakshmikantham, S.Sivasundaram, B.Kaymarkcalan, Dyanmic System\non Measure Chains, Kluwer Academic Publishers, Boston, 1996.\n[11] B.Aulbach, S.Hilger, Linear Dynamical Processes with Inhomogeneous\nTime Scales, Nonlinear Dynamics and Quantum Dynamical Systems,\nAkademie Verlage, Berlin, 1990.\n[12] S.Hilger, Analysis on measure chains-a unfified approach to continuous\nand discrete calculus, Results Math. 18 (1990) 18-56.\n[13] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time\nScales, Birkh¨auser, Boston, 2003.\n[14] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential\nEquations, Berlin, Springer-verlag, 1997."]}

This paper deals with a delayed single population model on time scales. With the assistance of coincidence degree theory, sufficient conditions for existence of periodic solutions are obtained. Furthermore, the better estimations for bounds of periodic solutions are established., {"references":["V G Nazarenko. Influence of delay on auto-oscillation in cell populations.\nBiofisika, 21(1976), 352-356.","Yongli Song, Yahong Peng. Periodic solutions of a nonautonomous\nperiodic model of population with continuous and discrete time. J. Comp.\nAppl. Math., 188(2006), 256-264.","S H Saker. Periodic solutions, oscillation and attractivity of discrete\nnonlinear delay population model. Math. Comput. Model., 2008, 47: 278-\n297.","Martin Bohner, Allan Peterson. Dynamic Equations on Time Scales: An\nIntroduction with Applications. Boston: Birkh¨auser, 2001.","Stefan Hilger. Analysis on measure chains-a unified approach to continuous\nand discrete calculus. Results Math., 18(1990), 18-56.","Martin Bohner, Meng Fan, Jiming Zhang. Existence of periodic solutions\nin predator-prey and competition dynamic systems. Nonlinear Anal.\nRWA, 7(2006), 1193-1204.","Kejun Zhuang. Periodicity for a semi-ratio-dependent predator-prey\nsystem with delays on time scales. Int. J. Comput. Math. Sci., 4(2010),\n44-47.","Bingbing Zhang, Meng Fan. A remark on the application of coincidence\ndegree to periodicity of dynamic equtions on time scales. J. Northeast\nNormal University(Natural Science Edition), 39(2007), 1-3.(in Chinese)","R E Gaines, J L Mawhin. Coincidence Degree and Nonlinear Differential\nEquations. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1977."]}

In this paper, the semi–ratio–dependent predator-prey system with nonmonotonic functional response on time scales is investigated. By using the coincidence degree theory, sufficient conditions for existence of periodic solutions are obtained., {"references":["Xiaohua Ding, Chun Lu and Mingzhu Liu, Periodic solutions for a semi-\nratio-dependent predator-prey system with nonmonotonic functional response\nand time delay, Nonl. Anal: RWA, 9(2008), 762-775.","Haifeng Huo, Periodic solutions for a semi-ratio-dependent predator-\nprey system with functional responses, Applied Mathematics Letters,\n18(2005), 313-320.","Qian Wang, Meng Fan and Ke Wang, Dynamics of a class of nonautonomous\nsemi-ratio-dependent predator-prey systems with functional\nresponses, J. Math. Anal. Appl., 278(2003), 443-451.","Pingzhu Liu and K. Gopalsamy, Global stability and chaos in a population\nmodel with piecewise constant arguments, Applied Mathematics and\nComputation, 101(1999), 63-88.","Stefan Hilger, Analysis on measure chains-a unified approach to continuous\nand discrete calculus, Results Math., 18(1990), 18-56.","Martin Bohner, Meng Fan and Jiming Zhang, Existence of periodic\nsolutions in predator-prey and competition dynamic systems, Nonl. Anal:\nRWA, 7(2006), 1193-1204.","Martin Bohner, Meng Fan and Jiming Zhang, Periodicity of scalar\ndynamic equations and applications to population models, J. Math. Anal.\nAppl, 330(2007), 1-9.","Jie Liu, Yongkun Li and Lili Zhao, On a periodic predator-prey system\nwith time delays on time scales, Commu Nonlinear Sci Numer Simulat,\n14(2009), 3432-3438.","Martin Bohner and Allan Peterson, Dynamic Equations on Time Scales:\nAn Introduction with Applications, Boston: Birkh┬¿auser, 2001.\n[10] R. E. Gaines and J. L. Coincidence Degree and Nonlinear Differential\nEquations, Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1977.\n[11] Bingbing Zhang and Meng Fan, A remark on the application of\ncoincidence degree to periodicity of dynamic equtions on time scales,\nJ. Northeast Normal University(Natural Science Edition), 39(2007), 1-\n3.(in Chinese)"]}

In this paper, by using Mawhin-s continuation theorem of coincidence degree theory, we establish the existence of multiple positive periodic solutions of a competitor-competitor-mutualist Lotka-Volterra system with harvesting terms. Finally, an example is given to illustrate our results., {"references":["M. Gyllenberg, P. Yan, Y. Wang, Limit cycles for competitor-competitormutualist\nLotka-Volterra systems, Physica D 221 (2006) 135-145.","C.V. Pao, The global attractor of a competitor-competitor-mutualist\nreaction-diffusion system with time delays, Nonlinear Anal. 67 (2007)\n2623-2631.","W.Y. Chen, R. Peng, Stationary patterns created by cross-diffusion for the\ncompetitor- competitor-mutualist model, J. Math. Anal. Appl. 291 (2004)\n550-564.","S. Fu, Persistence in a periodic competitor-competitor-mutualist diffusion\nsystem, J. Math. Anal. Appl. 263, 234-245 (2001).","X. Lv, P. Yan, S. Lu, Existence and global attractivity of positive periodic\nsolutions of competitor-competitor-mutualist Lotka-Volterra systems with\ndeviating arguments, Math. Comput. Modelling 51 (2010) 823-832.","M. Li, Z. Lin, J.H. Liu, Coexistence in a competitor-competitor-mutualist\nmodel, Appl. Math. Modelling, in press, doi:10.1016/j.apm.2010.02.029","Z. Ma, Mathematical Modelling and Studing on Species Ecology, Education\nPress, Hefei, 1996, (in Chinese).","H.R. Thieme, Mathematics in population biology, in: Princeton Syries\nin Theoretial and Computational Biology, Princeton University Press,\nPrinceton, NJ, 2003.","R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differetial\nEquitions, Springer Verlag, Berlin, 1977.\n[10] Q. Wang, B. Dai, Y. Chen, Multiple periodic solutions of an impulsive\npredator-prey model with Holling-type IV functional response, Math.\nComput. Modelling 49 (2009) 1829-1836.\n[11] D. Hu, Z. Zhang, Four positive periodic solutions to a Lotka-Volterra\ncooperative system with harvesting terms, Nonlinear Anal. Real World\nAppl. 11 (2010) 1115-1121.\n[12] K.H. Zhao, Y. Ye, Four positive periodic solutions to a periodic Lotka-\nVolterra predatory-prey system with harvesting terms, Nonlinear Anal.\nReal World Appl. in press, doi:10.1016/j.nonrwa.2009.08.001.\n[13] K.H. Zhao, Y.K. Li, Four positive periodic solutions to two species\nparasitical system with harvesting terms, Comput. Math. Appl. 59 (2010)\n2703-2710."]}