Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey

Figen Kangalgil; Feda İlhan

doi:10.17776/csj.1026330

EN

Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey

Abstract

This article is about the dynamics of a discrete-time prey-predator system with Allee effect and immigration parameter on prey population. Particularly, we study existence and local asymptotic stability of the unique positive fixed point. Furthermore, the conditions for the existence of bifurcation in the system are derived. In addition, it is shown that the system goes through period-doubling bifurcation by using bifurcation theory and center manifold theorem. Eventually, numerical examples are given to illustrate theoretical results.

Keywords

Prey-predator model, Stability analysis, Fixed point, Allee effect, Period-doubling bifurcation.

References

[1] Lotka A.J., Elements of physical biology, 1st ed. Baltimore: Williams and Wilkins Co., (1925).
[2] Volterra V., Variazioni e Fluttuazioni del Numero Dindividui in Spece Animali Conviventi, Mem R Accad Naz dei Lincei, 2 (6) (1926).
[3] Murray J.D., Mathematical biology. New York: Springer-Verlag, (1993).
[4] Walde S.J., Murdoch W.W., Spatial Density Dependence in Parasitoids, Annu. Rev. of Entomol., 33 (1988) 441-466.
[5] Kangalgil F., Flip Bifurcation and Stability in a Discrete-Time Prey-Predator Model with Allee Effect, Cumhuriyet Sci. J., 40 (2019) 141-149.
[6] Beddington J.R., Free C.A., Lawton J.H., Dynamic Complexity in Predator-Prey Models Framed in Difference Equations, Nature, 255 (1975) 58-60.
[7] Blackmore D., Chen J., Perez J., Savescu M., Dynamical Properties of Discrete Lotka-Volterra Equations, Chaos Solution. Fract., 12 (2001) 2553-2568.
[8] Danca M., Codreanu S., Bako B., Detailed Analysis of a Nonlinear Prey-Predator Model, J. Biol. Phys., 23 (1997) 11-20.
[9] Hadeler K.P., Gerstmann I., The Discrete Rosenzweig Model, Math. Biosci., 98 (1) (1990) 49-72.
[10] Işık S., A Study of Stability and Bifurcation Analysis in Discrete-Time Predator-Prey System Involving the Allee Effect, Int. J. Biomath., 12 (01) (2019).

[11] Işık S., Kangalgil F., On the Analysis of Stability, Bifurcation, and Chaos Control of Discrete-Time Predator-Prey Model with Allee Effect on Predator, Hacet. J. Math. Stat., 51 (2) (2022) 404-420.
[12] Selvam A.G.M., Jacintha M., Dhineshbabu R., Bifurcation Analysis and Chaotic Behaviour in Discrete-Time Predator Prey System, Int. J. Comput. Eng. Res., 9 (4) (2019).
[13] Zhu G., Wei J., Global Stability and Bifurcation Analysis of a Delayed Predator-Prey System with Prey Immigration, Electron. J. Qual. Theory Differ. Equ., 13 (2016) 1-20.
[14] Sugie J., Saito Y., Uniqueness of Limit Cycles in a Rosenzweig-Macarthur Model with Prey Immigration, SIAM J. Appl. Math., 72 (1) (2012) 299-316.
[15] Stone L., Hart D., Effects of Immigration on Dynamics of Simple Population Models, Theor. Popul. Biol., 55 (3) (1999) 227-234.
[16] Ak Gümüş Ö., Kangalgil F., Dynamics of a Host-Parasite Model Connected with Immigration, New Trend. Math. Sci., 5 (3) (2017) 332-339.
[17] Misra J.C., Mitra A., Instabilities in Single-Species and Host-Parasite Systems: Period-Doubling Bifurcations and Chaos, Comput. Math. with Appl., 52 (3) (2006) 525-538.
[18] Holt R.D., Immigration and the Dynamics of Peripheral Populations, Advances in Herpetology and Evolutionary Biology (Rhodin and Miyata, Eds.), Museum of Comparative Zoology, Harvard University, Cambridge: M.A., (1983).
[19] McCallum H.I., Effects of Immigration on Chaotic Population Dynamics, J. Theor. Biol., 154 (1992) 277-284.
[20] Stone L., Hart D., Effects of Immigration on the Dynamics of Simple Population Models, Theor. Popul. Biol., 55 (3) (1999) 227-234.
[21] Ruxton G.D., Low Levels of Immigration between Chaotic Populations can Reduce System Extinctions by Inducing Asynchronous Regular Cycles, Proc. Royal Soc. B, 256 (1994) 189-193.
[22] Rohani P., Miramontes O., Immigration and the Persistence of Chaos in Population Models, J. Theor. Biol., 175 (2) (1995) 203-206.
[23] Zhou S., Liu Y., Wang G., The Stability of Predator-Prey ssstems Subject to the Allee Effects, Theor. Popul. Biol., 67 (1) (2005) 23-31.
[24] Sen M., Banarjee M., Morozou A., Bifurcation Analysis of a Ratio-Dependent Prey-Predator Model with the Allee Effect, Ecol. Complex., 11 (2012) 12-27.
[25] Cheng L., Cao H., Bifurcation Analysis of a Discrete-Time Ratio-Dependent Prey-Predator Model with the Allee Effect, Commun, Nonlinear Sci. Numer. Simul., 38 (2016) 288-302.
[26] Kangalgil F., Ak Gümüş Ö., Allee Effect in a New Population Model and Stability Analysis, Gen. Math. Notes, 35 (1) (2016) 54-65.
[27] Lin Q., Allee Effect Increasing the Final Density of the Species Subject to Allee Effect in a Lotka-Volterra Commensal Symbiosis, Model, Adv. Differ. Equ., 196 (2018).
[28] Kangalgil F., Işık S., Controlling Chaos and Neimark-Sacker Bifurcation in a Discrete-Time Predator-Prey System, Hacettepe J. Mathematics and Statistics, 49 (5) (2020) 1761-1776.
[29] Din Q., Global Stability of Beddington Model, Qual. Theory Dyn. Syst., 16 (2017) 391-415.
[30] Din Q., Global Stability and Neimark-Sacker Bifurcation of a Host-Parasitoid Model, Int. J. Syst. Sci., 48 (6) (2016) 1194-1202.
[31] Din Q., Neimark-Sacker Bifurcation and Chaos Control in Hassell-Varley Model, J. Differ. Equ. Appl., 23 (4) (2017) 741-762.
[32] Din Q., Ak Gümüş Ö., Khalil H., Neimark-Sacker Bifurcation and Chaotic Behaviour of a Modified Host-Parasitoid Model, Z. Naturforsch., 72 (1) (2016) 25-37.
[33] Din Q., Complexity and chaos control in a discrete-time prey-predator model, Commun. Nonlinear Sci. Numer. Simul., 49 (2017) 113-134.
[34] Elabbasy E.M., Elsadany A.A., Zhang Y., Bifurcation Analysis and Chaos in a Discrete Reduced Lorenz System, Appl. Math. Comput., 228 (2014) 184-194.
[35] Din Q., Donchev T., Kolev D., Stability, Bifurcation Analysis and Chaos Control in Chlorine Dioxide-Iodine-Malonic Acid Reaction, MATCH Commun. Math. Comput. Chem., 79 (2018) 577-606.
[36] Allen L.J.S., An introduction to mathematical biology, Texas Tech. University, (2007).
[37] Kılıç H., Topsakal N., Kangalgil F., Stability Analysis of a Discrete-Time Prey-Predator Population Model with Immigration, Cumhuriyet Sci. J., 41 (4) (2020) 884-900.
[38] Sucu G., Bir Ayrık Av-Avcı Modelinin Kararlılık ve Çatallanma Analizi, Yüksek Lisans Tezi, TOBB Ekonomi ve Teknoloji Üniversitesi, Fen Bilimleri Enstitüsü, 2016.
[39] He Z., Lai X., Bifurcation and Chaotic Behavior of a Discrete-Time Prey-Predator System, Nonlinear Anal. Real World Appl., 12 (2011) 403-417.
[40] Kuznetsov Y.A., Elements of applied bifurcation theory, 2nd ed. New York: Springer-Verlag, (1998).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Figen Kangalgil
0000-0003-0116-8553
Türkiye

Feda İlhan ^*
0000-0003-2867-6304
Türkiye

Publication Date

March 30, 2022

Submission Date

November 20, 2021

Acceptance Date

February 22, 2022

Published in Issue

Year 2022 Volume: 43 Number: 1

DOI

https://doi.org/10.17776/csj.1026330

IZ

https://izlik.org/JA43DR98SM

APA

Kangalgil, F., & İlhan, 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. https://doi.org/10.17776/csj.1026330

AMA

1.Kangalgil F, İlhan F. Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey. CSJ. 2022;43(1):88-97. doi:10.17776/csj.1026330

Chicago

Kangalgil, Figen, and Feda İlhan. 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. https://doi.org/10.17776/csj.1026330.

EndNote

Kangalgil F, İlhan F (March 1, 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.

IEEE

[1]F. Kangalgil and F. İlhan, “Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey”, CSJ, vol. 43, no. 1, pp. 88–97, Mar. 2022, doi: 10.17776/csj.1026330.

ISNAD

Kangalgil, Figen - İlhan, Feda. “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 (March 1, 2022): 88-97. https://doi.org/10.17776/csj.1026330.

JAMA

1.Kangalgil F, İlhan F. Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey. CSJ. 2022;43:88–97.

MLA

Kangalgil, Figen, and Feda İlhan. “Period-Doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-Predator Model With Allee Effect and Immigration Parameter on Prey”. Cumhuriyet Science Journal, vol. 43, no. 1, Mar. 2022, pp. 88-97, doi:10.17776/csj.1026330.

Vancouver

1.Figen Kangalgil, Feda İlhan. Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey. CSJ. 2022 Mar. 1;43(1):88-97. doi:10.17776/csj.1026330

Cited By

Effect of immigration in a predator-prey system: Stability, bifurcation and chaos

AIMS Mathematics

https://doi.org/10.3934/math.2022791

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

Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

https://doi.org/10.18185/erzifbed.1207680

Fixed points stability, bifurcation analysis, and chaos control of a Lotka–Volterra model with two predators and their prey

International Journal of Biomathematics

https://doi.org/10.1142/S1793524523500328

Dynamics of a Discrete Predator–Prey Model with an Allee Effect in Prey: Stability, Bifurcation Analysis and Chaos Control

International Journal of Bifurcation and Chaos

https://doi.org/10.1142/S021812742550097X

As of 2026, Cumhuriyet Science Journal will be published in six issues per year, released in February, April, June, August, October, and December