Research Article
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Stability Analysis of a Discrete Time Prey-Predator Population Model with Immigration

Year 2020, Volume: 41 Issue: 4, 884 - 900, 29.12.2020

Hatice Kılıç , Nilüfer Topsakal , Figen Kangalgil

https://doi.org/10.17776/csj.779203
Cited By: 2

Abstract

In this paper, a discrete-time prey-predator population model with immigration which is obtained by implementing forward Euler’s scheme has been considered. The existence of fixed points of the presented model has been investigated. Moreover, the stability analysis of the fixed points of the population model has been examined and the topological classification of the fixed points of the model has been made. Moreover, the OGY feedback control method is to implement to controlchaos caused by the Flip bifurcation. Finally, Flip bifurcation,chaos control strategy, and asymptotic stability of the only positive fixed point are verifiedwith the help of numerical simulations.

Keywords

prey-predator model stability analysis Immigration Fixed point Chaotic behavior

Supporting Institution

CÜBAP

Project Number

F-621

References

  • [1] Lotka A. J., Elements of Mathematical Biology, Baltimore; Williams & Wilkins, 1925.
  • [2] Volterra V., Variazioni e fluttuazioni Del Numero D’individui in Specieanimali conviventi. Mem. R. Accad. Naz. Dei Lincei, 6 (2) (1926) 31-113.
  • [3] Zhu G., Wei J., Global stability and Bifurcation Analysis of a delayed Predator-Prey System with prey immigration, Electronic Journal of Qualitative Theory of Differential Equations, 13 (2016) 1-20.
  • [4] Sugie J., Saito Y., Uniqueness of limit cycles in a Rosenzweig-MacarthurModel with prey immigration, Siam. J. Appl. Math., 72 (1) (2012) 299-316.
  • [5] Stone L., Hart D., Effects of ·Immigration on Dynamics of Simple Population Models,Theoretical Population Biology, 55 (1999) 227-234 .
  • [6] Gümüş Ak Ö., Kangalgil F., Dynamics of a host-parasite model connected with immigration, New Trends in Mathematical Sciences, 5 (3) (2017) 332-339.
  • [7] Misra J. C., Mitra A. Instabilities in Single-Species and Host-ParasiteSystems, Period-Doubling Bifurcations and Chaos, Computers and Mathematics with Applications, 52 (2006) 525-538.
  • [8] Holt R. D., Immigration and the dynamics of peripheral populations inAdvances in Herpetology and Evolutionary Biology.(Rhodin and Miyata,Eds.), Museum of Comparative Zoology, Harvard University, Cambridge, MA., (1983).
  • [9] McCallum H. I., Effects of immigration on chaotic population dynamics, J. Theor. Biol., 154 (1992) 277-284.
  • [10] Stone L., Hart D., Effects of immigration on the dynamics of simple population models, Theoretical Population Biology, 55 (1999) 227-234.
  • [11] Ruxton G. D., Low levels of immigration between chaotic populations canreduce system extinctions by inducing asynchronous regular cycles, Proc.R. Soc., London B., 256 (1994) 189-193.
  • [12] Rohani P., Miramontes O., Immigration and the Persistence of chaos inpopulation models, J. Theor. Biol, 10, (1995).
  • [13] Tahara T., Gavina M., Kawano T., Tubay J., Rabajante J.F., Ito H., Morito S., Ichinose G., Okabe T., Togashi T., Tainaka K., Shimizu A., Nagatani T. And Yoshimura J., Asymptotic stability of a modifed Lotka-Volterra Model with small immigrations, Scientic Reports, 8 (2018) 7029.
  • [14] Işık S., A study of stability and bifurcation analysis in discrete-time predator-prey system involving the Allee effect, Int. Journal of Biomathematics, 12 (2019) 1.
  • [15] Kangalgil F., Neimark-Sacker bifurcation and stability analysis of a discrete-time prey predator model with Allee effect in prey, Adv. Difference Equations, (2019) 1-12.
  • [16] Kangalgil F., The Local Stability Analysis of a Nonlinear Discrete-TimePopulation Model with Delay and Allee Effect, Cumhuriyet Sciences Journal, 38 (3) (2017) 480-487.
  • [17] Çelik C., Merdan H., Duman O. And Akın Ö., Allee effects on population dynmics in countinuous (overlapping) case, Chaos, Solitons&Fractals Chaos, 37 (2008) 65-74.
  • [18] Merdan H., Duman O., On the stability analysis of a general discrete-time population model involving predation and Allee effects, Chaos, Solitons &Fractals Chaos, 40 (2009) 1169-1175.
  • [19] Lin Q., Allee Effect increasing the final density of the species subject to Allee Effect in a Lotka-Volterra commensal symbiosis model, Advance in Difference Equations, (2018) 196.
  • [20] Ott E., Grebogi C., Yorke J. A., Controlling chaos, Physical Review Letters, 4 (11) (1990) 1196–1199.
  • [21] Kartal S., Flip and Neimark-Sacker Bifurcation in a Differential Equation with piecewise constant Arguments Model, Journal of Difference Equations and Applications, 2 (4) (2017) 763-778.
  • [22] Kangalgil F., Flip Bifurcation and Stability in a Discrete-Time Prey-Predator Model with Allee Effect, Cumhuriyet Sciences Journal, 40 (1) (2019)141-149.
  • [23] Kangalgil F., Topsakal N., Stability Analysis and Flip Bifurcation of a Discrete-Time Prey-Predator Model with Predator Immigration, Asian Journal of Math. and Comp. Research, 27 (3) (2020) 1-10.
  • [24] Kangalgil F., Isık S., Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system, Hacet. J. Math. Stat, 49 (5) (2020) 1761-1776.
  • [25] Kılıç H., Göçe Bağlı Ayrık Zamanlı Av-Avcı Popülasyon Modelinin Kararlılığı Sivas Cumhuriyet Üniversitesi Fen Bilimler Enstitüsü, Yüksek Lisans Tezi, Sivas, (2020).

Year 2020, Volume: 41 Issue: 4, 884 - 900, 29.12.2020

Hatice Kılıç , Nilüfer Topsakal , Figen Kangalgil

https://doi.org/10.17776/csj.779203
Cited By: 2

Abstract

Project Number

F-621

References

  • [1] Lotka A. J., Elements of Mathematical Biology, Baltimore; Williams & Wilkins, 1925.
  • [2] Volterra V., Variazioni e fluttuazioni Del Numero D’individui in Specieanimali conviventi. Mem. R. Accad. Naz. Dei Lincei, 6 (2) (1926) 31-113.
  • [3] Zhu G., Wei J., Global stability and Bifurcation Analysis of a delayed Predator-Prey System with prey immigration, Electronic Journal of Qualitative Theory of Differential Equations, 13 (2016) 1-20.
  • [4] Sugie J., Saito Y., Uniqueness of limit cycles in a Rosenzweig-MacarthurModel with prey immigration, Siam. J. Appl. Math., 72 (1) (2012) 299-316.
  • [5] Stone L., Hart D., Effects of ·Immigration on Dynamics of Simple Population Models,Theoretical Population Biology, 55 (1999) 227-234 .
  • [6] Gümüş Ak Ö., Kangalgil F., Dynamics of a host-parasite model connected with immigration, New Trends in Mathematical Sciences, 5 (3) (2017) 332-339.
  • [7] Misra J. C., Mitra A. Instabilities in Single-Species and Host-ParasiteSystems, Period-Doubling Bifurcations and Chaos, Computers and Mathematics with Applications, 52 (2006) 525-538.
  • [8] Holt R. D., Immigration and the dynamics of peripheral populations inAdvances in Herpetology and Evolutionary Biology.(Rhodin and Miyata,Eds.), Museum of Comparative Zoology, Harvard University, Cambridge, MA., (1983).
  • [9] McCallum H. I., Effects of immigration on chaotic population dynamics, J. Theor. Biol., 154 (1992) 277-284.
  • [10] Stone L., Hart D., Effects of immigration on the dynamics of simple population models, Theoretical Population Biology, 55 (1999) 227-234.
  • [11] Ruxton G. D., Low levels of immigration between chaotic populations canreduce system extinctions by inducing asynchronous regular cycles, Proc.R. Soc., London B., 256 (1994) 189-193.
  • [12] Rohani P., Miramontes O., Immigration and the Persistence of chaos inpopulation models, J. Theor. Biol, 10, (1995).
  • [13] Tahara T., Gavina M., Kawano T., Tubay J., Rabajante J.F., Ito H., Morito S., Ichinose G., Okabe T., Togashi T., Tainaka K., Shimizu A., Nagatani T. And Yoshimura J., Asymptotic stability of a modifed Lotka-Volterra Model with small immigrations, Scientic Reports, 8 (2018) 7029.
  • [14] Işık S., A study of stability and bifurcation analysis in discrete-time predator-prey system involving the Allee effect, Int. Journal of Biomathematics, 12 (2019) 1.
  • [15] Kangalgil F., Neimark-Sacker bifurcation and stability analysis of a discrete-time prey predator model with Allee effect in prey, Adv. Difference Equations, (2019) 1-12.
  • [16] Kangalgil F., The Local Stability Analysis of a Nonlinear Discrete-TimePopulation Model with Delay and Allee Effect, Cumhuriyet Sciences Journal, 38 (3) (2017) 480-487.
  • [17] Çelik C., Merdan H., Duman O. And Akın Ö., Allee effects on population dynmics in countinuous (overlapping) case, Chaos, Solitons&Fractals Chaos, 37 (2008) 65-74.
  • [18] Merdan H., Duman O., On the stability analysis of a general discrete-time population model involving predation and Allee effects, Chaos, Solitons &Fractals Chaos, 40 (2009) 1169-1175.
  • [19] Lin Q., Allee Effect increasing the final density of the species subject to Allee Effect in a Lotka-Volterra commensal symbiosis model, Advance in Difference Equations, (2018) 196.
  • [20] Ott E., Grebogi C., Yorke J. A., Controlling chaos, Physical Review Letters, 4 (11) (1990) 1196–1199.
  • [21] Kartal S., Flip and Neimark-Sacker Bifurcation in a Differential Equation with piecewise constant Arguments Model, Journal of Difference Equations and Applications, 2 (4) (2017) 763-778.
  • [22] Kangalgil F., Flip Bifurcation and Stability in a Discrete-Time Prey-Predator Model with Allee Effect, Cumhuriyet Sciences Journal, 40 (1) (2019)141-149.
  • [23] Kangalgil F., Topsakal N., Stability Analysis and Flip Bifurcation of a Discrete-Time Prey-Predator Model with Predator Immigration, Asian Journal of Math. and Comp. Research, 27 (3) (2020) 1-10.
  • [24] Kangalgil F., Isık S., Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system, Hacet. J. Math. Stat, 49 (5) (2020) 1761-1776.
  • [25] Kılıç H., Göçe Bağlı Ayrık Zamanlı Av-Avcı Popülasyon Modelinin Kararlılığı Sivas Cumhuriyet Üniversitesi Fen Bilimler Enstitüsü, Yüksek Lisans Tezi, Sivas, (2020).
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Hatice Kılıç 0000-0003-1566-1103

Nilüfer Topsakal 0000-0001-7216-044X

Figen Kangalgil 0000-0003-0116-8553

Project Number F-621
Publication Date December 29, 2020
Submission Date August 11, 2020
Acceptance Date December 17, 2020
Published in Issue Year 2020Volume: 41 Issue: 4

Cite

APA Kılıç, H., Topsakal, N., & Kangalgil, F. (2020). Stability Analysis of a Discrete Time Prey-Predator Population Model with Immigration. Cumhuriyet Science Journal, 41(4), 884-900. https://doi.org/10.17776/csj.779203

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Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey
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https://doi.org/10.17776/csj.1026330