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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

Yıl 2022, Cilt 43, Sayı 1, 88 - 97, 30.03.2022
https://doi.org/10.17776/csj.1026330

Öz

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.

Kaynakça

  • [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).

Yıl 2022, Cilt 43, Sayı 1, 88 - 97, 30.03.2022
https://doi.org/10.17776/csj.1026330

Öz

Kaynakça

  • [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).

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Natural Sciences
Yazarlar

Figen KANGALGİL
Dokuz Eylül University Bergama Vocational School
0000-0003-0116-8553
Türkiye


Feda İLHAN (Sorumlu Yazar)
BOLU ABANT IZZET BAYSAL UNIVERSITY
0000-0003-2867-6304
Türkiye

Yayımlanma Tarihi 30 Mart 2022
Başvuru Tarihi 20 Kasım 2021
Kabul Tarihi 22 Şubat 2022
Yayınlandığı Sayı Yıl 2022, Cilt 43, Sayı 1

Kaynak Göster

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 . DOI: 10.17776/csj.1026330