Research Article
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Year 2022, Volume: 43 Issue: 1, 88 - 97, 30.03.2022

Figen Kangalgil , Feda İlhan

https://doi.org/10.17776/csj.1026330
https://izlik.org/JA43DR98SM

Abstract

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).

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

Year 2022, Volume: 43 Issue: 1, 88 - 97, 30.03.2022

Figen Kangalgil , Feda İlhan

https://doi.org/10.17776/csj.1026330
https://izlik.org/JA43DR98SM

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).
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Figen Kangalgil 0000-0003-0116-8553

Feda İlhan 0000-0003-2867-6304

Submission Date November 20, 2021
Acceptance Date February 22, 2022
Publication Date March 30, 2022
DOI https://doi.org/10.17776/csj.1026330
IZ https://izlik.org/JA43DR98SM
Published in Issue Year 2022 Volume: 43 Issue: 1

Cite

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

Cited By

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

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