Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Dynamical Behavior of a Delayed Predator-prey Model in Periodically Fluctuating Environments

Discontinuity, Nonlinearity, and Complexity 8(3) (2019) 325--340 | DOI:10.5890/DNC.2019.09.008

A. Moussaoui$^{1}$, M. A. Aziz Alaoui$^{2}$

$^{1}$ Department of Mathematics, Faculty of Sciences, University of Tlemcen, Algeria

$^{2}$ Normandie Univ, France; ULH, LMAH, F-76600 Le Havre, FR CNRS 3335, ISCN, 25 rue Philippe Lebon 76600 Le Havre, France

Download Full Text PDF



In this paper we develop a non-autonomous predator-prey system with time delay to study the influence of water level fluctuations on the interactions between fish species living in an artificial lake. We derive persistence and extinction conditions of the species. Using coincidence degree theory, we determine conditions for which the system has at least one periodic solution. Numerical simulations are presented to illustrate theoretical results.


  1. [1]  Coops, H. and Hosper, S.H. ( 2002), Water-level management as a tool for the restoration of shallow lakes in the Netherlands. Lake Reservoir Management, 18, 293-298.
  2. [2]  Coops, H., Beklioglu, M., and Crisman, T.L. (2003), The role of water-level fluctuations in shallow lake ecosystems workshop conclusions. Hydrobiologia, 506, 23-27.
  3. [3]  Kahl, U., Hlsmann, S., Radke, R.J., and Benndorf, J. (2008), The impact of water level fluctuations on the year class strength of roach: Implications for fish stock management. Limnologica, 38, 258-268.
  4. [4]  Wantzen, K.M., Rothhaupt, K,O., Mörtl, M., Cantonati, M., Toth, L.G., and Fischer, P. (2008), Ecological effects of water-level fluctuations in lakes: an urgent issue, Hydrobiologia, 613, 1-4.
  5. [5]  Wlosinski, J.H. and Koljord, E.R. (1996), Effects of water levels on ecosystems, an annotated bibliography, long term resource monitoring program. Technical report 96-T007.
  6. [6]  Chiboub Fellah, N., Bouguima, S.M., and Moussaoui, A. (2012), The effect of water level in a prey-predator interaction: A nonlinear analysis study, Chaos, Solitons & Fractals 45, 205-212.
  7. [7]  Moussaoui, A., and Bouguima, S.M. (2015), A prey-predator interaction under fluctuating level water., Mathematical Methods in the Applied Sciences, 38, 123-137.
  8. [8]  Moussaoui, A. (2015), A reaction-diffusion equations modelling the effect of fluctuating water levels on prey-predator interactions. Applied Mathematics and Computation, 268, 1110-1121.
  9. [9]  Moussaoui, A., Bassaid, S, and Ait Dads, E.H. (2015), The impact of water level fluctuations on a delayed prey-predator model, Nonlinear Anal. Real World Appl. 21, 170-184.
  10. [10]  Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston.
  11. [11]  Liu, G. and Yan, J. (2011), Positive periodic solutions of neutral predator-prey model with Beddington-DeAngelis functional response, Computers and Mathematics with Applications, 61, 2317-2322.
  12. [12]  Hale, J. (1977), Theory of Functional Differential Equations, Springer-Verlag, Heidelberg.
  13. [13]  MacDonald, N. (2008), Biological Delay Systems: Linear Stability Theory. Cambridge University Press.
  14. [14]  Gaines, R. and Mawhin, J. (1977), Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin.
  15. [15]  Chen, F.D., Li,Z., Chen, X., and Jitka, L. (2007),Dynamic behaviours of a delay differential equationmodel of plankton allelopathy. J. Comput. Appl. Math., 206, 733-754.