Top

Journal of Applied Mathematics and Computing

Published in:

01-02-2016 | Original Research

Almost periodic solution analysis in a two-species competitive model of plankton alleopathy with impulses

Authors: Changjin Xu, Qiming Zhang, Peiluan Li

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we investigate a two-species competitive model of plankton alleopathy with impulses. By constructing a suitable Lyapunov function, we establish some sufficient conditions which guarantee the uniformly asymptotic stability of a unique positive almost periodic solution for the model. An example with its numerical simulations is given which is in a good agreement with our theoretical analysis. Our results are new and complement previously known results.

previous article Similarity-based minimization of fuzzy tree automata

next article Spectral band structure of periodic Schrödinger operators with two potentials on the degenerate zigzag nanotube

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Liu, M., Wang, K.: A note on a delay Lotka-Volterra competitive system with random perturbations. Appl. Math. Lett. 26(6), 589–594 (2013)MathSciNetCrossRefMATH

Du, N.H., Dang, N.H.: Dynamics of Kolmogorov systems of competitive type under the telegraph noise. J. Differ. Equ. 250(1), 386–409 (2011)MathSciNetCrossRefMATH

Tang, X.H., Cao, D.M., Zou, X.F.: Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay. J. Differ. Equ. 228(2), 580–610 (2006)MathSciNetCrossRefMATH

Huo, H.F.: Permanence and global attractivity of delay diffusive prey-predator systems with the michaelis-menten functional response. Comput. Math. Appl. 49(2–3), 407–416 (2005)MathSciNetCrossRefMATH

Zhao, J.D., Zhang, Z.C., Ju, J.: Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive Lotka-Volterra system. Appl. Math. Comput. 230, 587–596 (2014)MathSciNetCrossRef

Hou, Z.: On permanence of all subsystems of competitive Lotka-Volterra systems with delays. Nonlinear Anal. Real World Appl. 11(5), 4285–4301 (2010)MathSciNetCrossRefMATH

Balbus, J.: Attractivity and stability in the competitive systems of PDEs of Kolmogorov type. Appl. Math. Comput. 237, 69–76 (2014)MathSciNetCrossRef

Shi, C.L., Li, Z., Chen, F.D.: Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. Nonlinear Anal. Real World Appl. 13(5), 2214–2226 (2012)MathSciNetCrossRefMATH

Liu, M., Wang, K.: Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations. Math. Comput. Model. 57(3–4), 909–925 (2013)CrossRefMATH

10.

Li, Y.K., Zhang, T.W., Ye, Y.: On the existence and stability of a unique almost periodic sequence solution in discrete predator-prey models with time delays. Appl. Math. Model. 35(11), 5448–5459 (2011)MathSciNetCrossRefMATH

11.

Li, Y.K., Yang, L., Wu, W.Q.: Almost periodic solutions for a class of discrete systems with Allee-effect. Appl. Math. 59(2), 191–203 (2014)MathSciNetCrossRefMATH

12.

Zhang, C., Huang, N.J., O’Regan, D.: Almost periodic solutions for a Volterra model with mutual interference and Holling type III functional response. Appl. Math. Comput. 225, 503–511 (2013)MathSciNetCrossRef

13.

Li, Y.K., Ye, Y.: Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms. Commun. Nonlinear Sci. Numer. Simul. 18(11), 3190–3201 (2013)MathSciNetCrossRef

14.

Huang, X., Cao, J.D., Ho, D.W.C.: Existence and attractivity of almost periodic solution for recurrent neural networks with unbounded delays and variable coefficients. Nonlinear Dyn. 45(3–4), 337–351 (2006)MathSciNetCrossRefMATH

15.

Zhang, H., Shao, J.Y.: Almost periodic solutions for cellular neural networks with time-varying delays in leakage terms. Appl. Math. Comput. 219(24), 11471–11482 (2013)MathSciNetCrossRefMATH

16.

Li, Z., Han, M.A., Chen, F.D.: Almost periodic solutions of a discrete almost periodic logistic equation with delay. Appl. Math. Comput. 232, 743–751 (2014)MathSciNetCrossRef

17.

Zhang, H., Li, YiQ, Jing, B., Zhao, W.Z.: Global stability of almost periodic solution of multispecies mutualism system with time delays and impulsive effects. Appl. Math. Comput. 232, 1138–1150 (2014)MathSciNetCrossRef

18.

Yilmaz, E.: Almost periodic solutions of impulsive neural networks at non-prescribed moments of time. Neurocomputing 141, 148–152 (2014). in pressCrossRef

19.

Wang, C.: Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales. Commun. Nonlinear Sci. Numer. Simul. 19(8), 2828–2842 (2014)MathSciNetCrossRef

20.

Gao, J., Wang, Q.R., Zhang, L.W.: Existence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales. Appl. Math. Comput. 237, 639–649 (2014)MathSciNetCrossRef

21.

Liu, B.W.: New results on the positive almost periodic solutions for a model of hematopoiesis. Nonlinear Anal. Real World Appl. 17, 252–264 (2014)MathSciNetCrossRefMATH

22.

Ding, H.S., Nieto, J.J.: A new approach for positive almost periodic solutions to a class of Nicholson\(^{,}\)s blowflies model. J. Comput. Appl. Math. 253, 249–254 (2013)MathSciNetCrossRefMATH

23.

Lin, X.J., Du, Z.J., Lv, Y.S.: Global asymptotic stability of almost periodic solution for a multispecies competition-predator system with time delays. Appl. Math. Comput. 219(9), 4908–4923 (2013)MathSciNetCrossRef

24.

Tan, R.H., Liu, W.F., Wang, Q.L., Liu, Z.J.: Uniformly asymptotic stability of almost periodic solution for a competitive system with impulsive perturbations. Adv. Differ. Equ. 2014(1), 2 (2014). doi:10.1186/1687-1847-2014-2

25.

Zhang, L.J., Shi, B.: Existence and global exponential stability of almost periodic solution for CNNs with variable coefficients and delays. J. Appl. Math. Comput. 28(1–2), 461–472 (2008)MathSciNetCrossRefMATH

26.

Zhou, H., Jiang, W.: Existence and stability of positive almost periodic solution for stochastic Lasota-Wazewska model. J. Appl. Math. Comput. 47(1–2), 61–71 (2015)

27.

Wang, Q., Dai, B.X.: Almost periodic solution for n-species Lotka-Volterra competitive system with delay and feedback controls. Appl. Math. Comput. 200(1), 133–146 (2008)MathSciNetCrossRefMATH

28.

Wang, Q., Zhang, H.Y., Wang, Y.: Existence and stability of positive almost periodic solutions and periodic solutions for a logarithmic population model. Nonlinear Anal. Theory, Methods Appl. 72(12), 4384–4389 (2010)CrossRefMATH

29.

Wang, Q., Zhang, H.Y., Ding, M.M., Wang, Z.J.: Global attractivity of the almost periodic solution of a delay logistic population model with impulses. Nonlinear Anal. Theory, Methods Appl. 73(12), 3688–3697 (2010)MathSciNetCrossRefMATH

30.

Abbas, S., Sen, M., Banerjee, M.: Almost periodic solution of a non-autonomous model of phytoplankton allelopathy. Nonlinear Dynamics 67(1), 203–214 (2012)MathSciNetCrossRefMATH

31.

Wang, C., Agarwal, R.P.: Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive \(\nabla \)-dynamic equations on time scales. Adv. Differ. Equ. 2014, 153 (2014). doi:10.1186/1687-1847-2014-153 CrossRef

32.

Wang, C.: Existence and exponential stability of piecewise mean-square almost periodic solutions for impulsive stochastic Nicholson\(^{,}\)s blowflies model on time scales. Appl. Math. Comput. 248, 101–112 (2014)MathSciNetCrossRef

33.

Wang, C., Agarwal, R.P.: Exponential dichotomies of impulsive dynamic systems with applications on time scales. Math. Methods Appl. Sci. 1–22 (2014). doi:10.1002/mma.3325

34.

He, M.X., Chen, F.D., Li, Z.: Almost periodic solution of an impulsive differential equation model of plankton allelopathy. Nonlinear Anal. Real World Appl. 11(4), 2296–2301 (2010)MathSciNetCrossRefMATH

35.

Maynard, J.: Models in Ecology. Cambridge University Press, Cambridge (1974)MATH

36.

Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. Model. 84(1–3), 287–289 (1996)CrossRef

37.

Agarwal, R.P.: Difference Equations and Inequalities: Theory, Method and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, 2nd edn. Marcel Dekker, New York, NY, USA (2000)

38.

Li, Y.K., Lu, L.H.: Positive periodic solutions of discrete n-species food-chain systems. Appl. Math. Comput. 167(1), 324–344 (2005)MathSciNetCrossRefMATH

39.

Chen, X., Chen, F.D.: Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control. Appl. Math. Comput. 181(2), 1446–1454 (2006)MathSciNetCrossRefMATH

40.

Muroya, Y.: Persistence and global stability in Lotka-Volterra delay differential systems. Appl. Math. Lett. 17(7), 795–800 (2004)MathSciNetCrossRefMATH

41.

Muroya, Y.: Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems. Tokyo J. Math. 28(1), 189–200 (2005)MathSciNetCrossRefMATH

42.

Yang, X.T.: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 316(1), 161–177 (2006)MathSciNetCrossRefMATH

43.

Chen, F.D.: Permanence for the discrete mutualism model with time delays. Math. Comput. Model. 47(3–4), 431–435 (2008)CrossRefMATH

44.

Zhang, W.P., Zhu, D.M., Bi, P.: Multiple periodic positive solutions of a delayed discrete predator-prey system with type IV functional responses. Appl. Math. Lett. 20(10), 1031–1038 (2007)MathSciNetCrossRefMATH

45.

Chen, Y.M., Zhou, Z.: Stable periodic of a discrete periodic Lotka-Volterra competition system. J. Math. Anal. Appl. 277(1), 358–366 (2003)MathSciNetCrossRefMATH

46.

Li, X.P., Yang, W.S.: Permanence of a discrete model of mutualism with infinite deviating arguments. Discret. Dyn. Nat. Soc. 2010, 1–7 (2010). Article ID 931798

47.

Li, X.P., Yang, W.S.: Permanence of a discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls. Discret. Dyn. Nat. Soc. 2008, 1–8 (2008). Article ID 149267CrossRefMATH

48.

Chen, F.D.: Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems. Appl. Math. Comput. 182(1), 3–12 (2006)MathSciNetCrossRefMATH

49.

Wu, D.Y.: Bifurcation analysis of a two-species competitive discrete model of plankton allelopathy. Adv. Differ. Equ. 2014, 1–10 (2014). doi:10.1186/1687-1847-2014-70 CrossRef

50.

Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)MATH

51.

He, C.Y.: Almost Periodic Differential Equations. Higher Education Press, Beijing (1992). (Chinese version)

52.

Chen, F.D., Li, Z., Huang, Y.J.: Note on the permanence of a competitive system with infinite delay and feedback controls. Nonlinear Analysis: Real World Applications 8(2), 680–687 (2007)MathSciNetCrossRefMATH

53.

Samoilenko, A.M., Perestyuk, N.A.: Perestyuk, Impulsive Differential Equations. World Scientific, Singapore (1995)

Title: Almost periodic solution analysis in a two-species competitive model of plankton alleopathy with impulses
Authors: Changjin Xu
Qiming Zhang
Peiluan Li
Publication date: 01-02-2016
Publisher: Springer Berlin Heidelberg
Published in: Journal of Applied Mathematics and Computing / Issue 1-2/2016
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-015-0878-6

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1-2/2016

A new linearized implicit iteration method for nonsymmetric algebraic Riccati equations

On Hermitian positive definite solutions of the nonlinear matrix equation

Optimality conditions and duality results for non-differentiable interval optimization problems

Solvability of a coupled system of nonlinear fractional differential equations with fractional integral conditions

Bounds on quasi-cyclic codes over finite chain rings

Multiple periodic solutions of functional differential equations with impulses on time scales

Premium Partner