Finally, Flip bifurcation,chaos control strategy, and asymptotic stability of the only positive fixed point are verifiedwith the help of numerical simulations.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, a discrete-time prey-predator population model with immigration which is obtained by implementing forward Euler’s scheme has been considered.
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Key Findings & Data:The existence of fixed points of the presented model has been investigated.
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Conclusion & Impact:Finally, Flip bifurcation,chaos control strategy, and asymptotic stability of the only positive fixed point are verifiedwith the help of numerical simulations.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:prey-predator model · stability analysis · Immigration
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.
KILIÇ, H., TOPSAKAL, N., KANGALGİL, F. (2020). Stability Analysis of a Discrete Time Prey-Predator Population Model with Immigration. Cumhuriyet Science Journal, Cilt: 41 Sayı: 4, 884–900. https://ajindex.com/articles/960312
Lotka A. J., Elements of Mathematical Biology, Baltimore; Williams & Wilkins, 1925
Volterra V., Variazioni e fluttuazioni Del Numero D’individui in Specieanimali conviventi. Mem. R. Accad. Naz. Dei Lincei, 6 (2) (1926) 31-113
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
Sugie J., Saito Y., Uniqueness of limit cycles in a Rosenzweig-MacarthurModel with prey immigration, Siam. J. Appl. Math., 72 (1) (2012) 299-316
Stone L., Hart D., Effects of ·Immigration on Dynamics of Simple Population Models,Theoretical Population Biology, 55 (1999) 227-234
Gümüş Ak Ö., Kangalgil F., Dynamics of a host-parasite model connected with immigration, New Trends in Mathematical Sciences, 5 (3) (2017) 332-339
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
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)
McCallum H. I., Effects of immigration on chaotic population dynamics, J. Theor. Biol., 154 (1992) 277-284
Stone L., Hart D., Effects of immigration on the dynamics of simple population models, Theoretical Population Biology, 55 (1999) 227-234
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
Rohani P., Miramontes O., Immigration and the Persistence of chaos inpopulation models, J. Theor. Biol, 10, (1995)
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
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
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