Araştırma Makalesi
BibTex RIS Kaynak Göster

Güçlü Allee Etkili Av-Avcı Modelinin Kararlılığı, Neimark-Sacker Çatallanma Analizi ve Kaos Kontrol

Yıl 2022, , 775 - 787, 30.12.2022
https://doi.org/10.18185/erzifbed.1207680

Öz

Bu çalışmada, çoklu güçlü Allee etkisi içeren bir av-avcı modelinin dinamik davranışları araştırılmıştır. Modelin sabit noktalarının varlığı ve topolojik sınıflandırması incelenmiştir. $\beta$ çatallanma parametresi olarak seçildiğinde, modelin benzersiz pozitif sabit noktada bir Neimark-Sacker çatallanması olacağı gösterilmiştir. Çatallanma teorisi, Neimark-Sacker çatallanma varoluş koşullarını ve çatallanmanın yönünü sunmak için kullanılır. Ek olarak, bazı sayısal simülasyonlar, analitik sonucu desteklemek için sunulmuştur. Bunu takiben, modelin çatallanma diyagramı ve üçgen şeklindeki kararlılık bölgesi verilmiştir.

Kaynakça

  • [1] Arancibia-Ibarra, C., (2019), The basins of attraction in a modified May–Holling–Tanner predator– prey model with Allee affect, Nonlinear Analysis, 185, 15-28.
  • [2] Kundu, S., Maitra, S., (2019), Asymptotic behaviors of a two prey one predator model with cooperation among the prey species in a stochastic environment, Journal of Applied Mathematics and Computing, 61(1), 505-531.
  • [3] Martinez-Jeraldo, N., Aguirre, P., (2019), Allee effect acting on the prey species in a Leslie–Gower predation model, Nonlinear Analysis: Real World Applications, 45, 895-917.
  • [4] Elaydi, S., (1996), An introduction to difference equations, Springer-Verlag, New York, 10, 978-1.
  • [5] Kuznetsov, Y. A., Kuznetsov, I. A., Kuznetsov, Y, (1998), Elements of applied bifurcation theory (Vol. 112, pp. xx+-591), New York: Springer.
  • [6] Wiggins, S., Wiggins, S., Golubitsky, M., (2003), Introduction to applied nonlinear dynamical systems and chaos (Vol. 2, No. 3), New York: Springer.
  • [7] Zhou, S. R., Liu, Y. F., Wang, G., (2005), The stability of predator–prey systems subject to the Allee effects, Theoretical Population Biology, 67(1), 23-31.
  • [8] Wang, S., Yu, H., (2021), Complexity Analysis of a Modified Predator-Prey System with Beddington– DeAngelis Functional Response and Allee-Like Effect on Predator, Discrete Dynamics in Nature and Society.
  • [9] Allee, W. C., (1931), Animal Aggregations: A study in General Sociology, University of Chicago Press, USA.
  • [10] Courchamp, F., Berec, L., Gascoigne, J., (2008), Allee effects in ecology and conservation, OUP Oxford.
  • [11] Amarasekare, P., (1998), Interactions between local dynamics and dispersal: insights from single species models, Theoretical Population Biology, 53(1), 44-59.
  • [12] Drake, J. M., (2004), Allee effects and the risk of biological invasion, Risk Analysis: An International Journal, 24(4), 795-802.
  • [13] Shi, J., Shivaji, R., (2006), Persistence in reaction diffusion models with weak Allee effect, Journal of Mathematical Biology, 52(6), 807-829.
  • [14] Taylor, C. M., Hastings, A., (2005), Allee effects in biological invasions, Ecology Letters, 8(8), 895- 908.
  • [15] Celik, C., Duman, O., (2009), Allee effect in a discrete-time predator–prey system, Chaos, Solitons Fractals, 40(4), 1956-1962.
  • [16] Wang, W. X., Zhang, Y. B., Liu, C. Z., (2011), Analysis of a discrete-time predator–prey system with Allee effect, Ecological Complexity, 8(1), 81-85.
  • [17] Pal, S., Sasmal, S. K., Pal, N., (2018), Chaos control in a discrete-time predator–prey model with weak Allee effect, International Journal of Biomathematics, 11(07), 1850089.
  • [18] Kangalgil, F., ˙Ilhan, F., (2022), Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey, Cumhuriyet Science Journal, 43(1), 88-97.
  • [19] Kangalgil, F., Topsakal, N., ¨ Ozt¨urk, N., (2022), Analyzing bifurcation, stability, and chaos control for a discrete-time prey-predator model with Allee effect, Turkish Journal of Mathematics, 46(6), 2047-2068.
  • [20] Kangalgil, F., Is, ık, S., (2022), Effect of immigration in a predator-prey system: Stability, bifurcation and chaos, AIMS Mathematics, 7(8), 14354-14375.
  • [21] Is, ık, S., Kangalgil, F., (2022), On the analysis of stability, bifurcation, and chaos control of discretetime predator-prey model with Allee effect on predator, Hacettepe Journal of Mathematics and Statistics, 1-21.
  • [22] Kangalgil, F., Is, ık, S., (2020), Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system, Hacettepe Journal of Mathematics and Statistics, 49(5), 1761-1776.
  • [23] Kangalgil, F., (2019), Neimark–Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey, Advances in Difference Equations, 2019(1), 1-12.
  • [24] Khan, A. Q., (2016), Neimark-Sacker bifurcation of a two-dimensional discrete-time predator-prey model, SpringerPlus, 5(1), 1-10.
  • [25] Guckenheimer, J., Holmes, P., (2013), Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Vol. 42), Springer Science Business Media.

Stability, Neimark-Sacker Bifurcation Analysis of a Prey-Predator Model with Strong Allee Effect and Chaos Control

Yıl 2022, , 775 - 787, 30.12.2022
https://doi.org/10.18185/erzifbed.1207680

Öz

In this study, the dynamical behaviors of a prey–predator model with multiple strong Allee effect are investigated. The fixed points of the model are examined for existence and topological classification. By selecting as the bifurcation parameter $\beta$, it is demonstrated that the model can experience a Neimark-Sacker bifurcation at the unique positive fixed point. Bifurcation theory is used to present the Neimark-Sacker bifurcation conditions of existence and the direction of the bifurcation. Additionally, some numerical simulations are provided to back up the analytical result. Following that, the model's bifurcation diagram and the triangle-shaped stability zone are provided.

Kaynakça

  • [1] Arancibia-Ibarra, C., (2019), The basins of attraction in a modified May–Holling–Tanner predator– prey model with Allee affect, Nonlinear Analysis, 185, 15-28.
  • [2] Kundu, S., Maitra, S., (2019), Asymptotic behaviors of a two prey one predator model with cooperation among the prey species in a stochastic environment, Journal of Applied Mathematics and Computing, 61(1), 505-531.
  • [3] Martinez-Jeraldo, N., Aguirre, P., (2019), Allee effect acting on the prey species in a Leslie–Gower predation model, Nonlinear Analysis: Real World Applications, 45, 895-917.
  • [4] Elaydi, S., (1996), An introduction to difference equations, Springer-Verlag, New York, 10, 978-1.
  • [5] Kuznetsov, Y. A., Kuznetsov, I. A., Kuznetsov, Y, (1998), Elements of applied bifurcation theory (Vol. 112, pp. xx+-591), New York: Springer.
  • [6] Wiggins, S., Wiggins, S., Golubitsky, M., (2003), Introduction to applied nonlinear dynamical systems and chaos (Vol. 2, No. 3), New York: Springer.
  • [7] Zhou, S. R., Liu, Y. F., Wang, G., (2005), The stability of predator–prey systems subject to the Allee effects, Theoretical Population Biology, 67(1), 23-31.
  • [8] Wang, S., Yu, H., (2021), Complexity Analysis of a Modified Predator-Prey System with Beddington– DeAngelis Functional Response and Allee-Like Effect on Predator, Discrete Dynamics in Nature and Society.
  • [9] Allee, W. C., (1931), Animal Aggregations: A study in General Sociology, University of Chicago Press, USA.
  • [10] Courchamp, F., Berec, L., Gascoigne, J., (2008), Allee effects in ecology and conservation, OUP Oxford.
  • [11] Amarasekare, P., (1998), Interactions between local dynamics and dispersal: insights from single species models, Theoretical Population Biology, 53(1), 44-59.
  • [12] Drake, J. M., (2004), Allee effects and the risk of biological invasion, Risk Analysis: An International Journal, 24(4), 795-802.
  • [13] Shi, J., Shivaji, R., (2006), Persistence in reaction diffusion models with weak Allee effect, Journal of Mathematical Biology, 52(6), 807-829.
  • [14] Taylor, C. M., Hastings, A., (2005), Allee effects in biological invasions, Ecology Letters, 8(8), 895- 908.
  • [15] Celik, C., Duman, O., (2009), Allee effect in a discrete-time predator–prey system, Chaos, Solitons Fractals, 40(4), 1956-1962.
  • [16] Wang, W. X., Zhang, Y. B., Liu, C. Z., (2011), Analysis of a discrete-time predator–prey system with Allee effect, Ecological Complexity, 8(1), 81-85.
  • [17] Pal, S., Sasmal, S. K., Pal, N., (2018), Chaos control in a discrete-time predator–prey model with weak Allee effect, International Journal of Biomathematics, 11(07), 1850089.
  • [18] Kangalgil, F., ˙Ilhan, F., (2022), Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey, Cumhuriyet Science Journal, 43(1), 88-97.
  • [19] Kangalgil, F., Topsakal, N., ¨ Ozt¨urk, N., (2022), Analyzing bifurcation, stability, and chaos control for a discrete-time prey-predator model with Allee effect, Turkish Journal of Mathematics, 46(6), 2047-2068.
  • [20] Kangalgil, F., Is, ık, S., (2022), Effect of immigration in a predator-prey system: Stability, bifurcation and chaos, AIMS Mathematics, 7(8), 14354-14375.
  • [21] Is, ık, S., Kangalgil, F., (2022), On the analysis of stability, bifurcation, and chaos control of discretetime predator-prey model with Allee effect on predator, Hacettepe Journal of Mathematics and Statistics, 1-21.
  • [22] Kangalgil, F., Is, ık, S., (2020), Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system, Hacettepe Journal of Mathematics and Statistics, 49(5), 1761-1776.
  • [23] Kangalgil, F., (2019), Neimark–Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey, Advances in Difference Equations, 2019(1), 1-12.
  • [24] Khan, A. Q., (2016), Neimark-Sacker bifurcation of a two-dimensional discrete-time predator-prey model, SpringerPlus, 5(1), 1-10.
  • [25] Guckenheimer, J., Holmes, P., (2013), Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Vol. 42), Springer Science Business Media.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Deniz Elmacı 0000-0002-9234-6361

Figen Kangalgil 0000-0003-0116-8553

Yayımlanma Tarihi 30 Aralık 2022
Yayımlandığı Sayı Yıl 2022

Kaynak Göster

APA Elmacı, D., & Kangalgil, F. (2022). Stability, Neimark-Sacker Bifurcation Analysis of a Prey-Predator Model with Strong Allee Effect and Chaos Control. Erzincan University Journal of Science and Technology, 15(3), 775-787. https://doi.org/10.18185/erzifbed.1207680