Main Article Content
Aims: In this paper, a discrete-time prey-predator model with predator immigration has been considered.
Methodology: The model was mathemati-cally analysed and simulated in Mapple.
Results: The complex dynamical behavior of the presented model has been analyzed. Stability and existence of coexistence positive fixed point have been investigated. Moreover, using bifurcation theory, it has been shown that the model undergoes Flip bifurcation. Also, direction of Flip bifurcation has been given. Some numerical simulations including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the model have been given to support of the
analytical finding. The computation of the maximum Lyapunov exponents has confirmed the presence of chaotic behavior in the system.
Conclusion: The model was successfully built, analysed and simulated showing results that matched the theory.
Volterra V, Variazioni e fluttuazioni del numero di’ individui in specie animali conviventi. Mem.
R. Accad. Naz. Dei Lincei, Ser. VI. 1926;2:31-113.
Zhu G, Wei J. Global stability and Bifurcation Analysis of a delayed predator-Prey System with prey immigration, Eloctronic Journal of Qualitative Theory of Differential Equations.
, 13 1-20.
Sugie J, SaitoY. Uniqueness of limit cycles in a Rosenzweig-Macarthur Model with prey immigration, Siam. J. Appl. Math. 2012;72(1):299-316.
Stone L, Hart D. Effects of ˙Immigration on dynamics of simple population models. Theoretical Population Biology. 1999;55: 227-234.
G¨um¨u¸s Ak, ¨ O., Kangalgil F. Dynamics of a host-parasite model connected with immigration.
New Trends in Mathematical Sciences. 2017;5(3):332-339.
Misra JC, Mitra A. Instabilities in single-species and host-parasite systems: Period-doubling bifurcations and chao. Computers and Mathematics with Applications. 2006;52:525-538 .
Holt RD. Immigration and the dynamics of peripheral populations. in Advances in Herpetology and Evolutionary Biology (Rhodin and Miyata, Eds.), Museum of Comparative Zoology,Harvard University, Cambridge, MA.; 1983.
Mc Callum HI. Effects of immigration on chaotic population dynamics. J. Theor. Biol.
Stone L, Hart D. Effects of immigration on the dynamics of simple population models.
Theoretical Population Biology 1999;55:227-234.
Ruxton GD. Low levels of immigration between chaotic populations can reduce system extinctions by inducing asynchronous regular cycles. Proc. R. Soc. London B. 1994;256:189-193.
Rohani P, Miramontes O. Immigration and the Persistence of chaos in population models. J.
Theor. Biol (in press); 1995.
Tahara T, Gavina M, Kawano T, Tubay J, Rabajante JF, Ito H, Morito S, Ichinose G, Okabe T, Togashi T, Tainaka K, Shimizu A, Nagatani T, Yoshimura J. Asymptotic stability of a modified Lotka-Volterra Model with small immigrations. Scientific Reports. 2018;8:7029.
Isık S. A study of stability and bifurcation analysis in discrete-time predator–prey system involving the Allee effect. Int. Journal of Biomathematics. 2019;12(1):1-15.
Kangalgil F. Neimark-Sacker bifurcation and stability analysis of a discrete-time prey-predator model with Allee effect in prey. Adv. Difference Equations. 2019;92:1-12.
Kangalgil F. The local stability analysis of a nonlinear discrete-time population model with delay and Allee effect. Cumhuriyet Sciences Journal. 2017;38(3):480-487.
C¸ elik C, Merdan H, Duman O, Akın ¨ O. Allee effects on pop¨ulation dynmics in countinuous (overlapping) case. Chaos, Solitons & Fractals Chaos. 2008;37:65-74.
Merdan H, Duman O. On the stability analysis of a general discrete-time population model involving predation and Allee effects. Chaos, Solitons & Fractals Chaos. 2009;40:1169-1175.
Qifa Lin. “Allee effect increasing the final density of the species subject to Allee effect in a Lotka-Volterra commensal symbiosis model”. Advance in Difference Equations. 2018:196.
Kangalgil F. Flip bifurcation and stability in a discrete-time prey-predator model with allee effect. Cumhuriyet Sciences Journal. 2019;40(1):141-149.
Kangalgil F, Kartal S. Stability and bifurcation analysis in a host parasitoid model with Hassell growth function. Adv. Difference Equations. 2018:240-255.
Holling CS. The functional response of predators to prey density and its role in mimicry and population. Mem. Entomol. Soc. Can. 1965;45:1-60.
Hu Z, Teng Z, Zhang L. Stability and Bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. Nonlinear Analysis: Real World Applications.
Jiangang Zhang, Tian Deng, Yandong Chu, Shuang Qin, Wenju Du, Hongwei Luo. Stability and bifurcation analysis of a discrete predator-prey model with Holling type III functional response. Journal of nonlinear Science and Applications. 2016;9:6228-6243.
Kartal S. Mathematical modeling and analysis of tumor-immune system interastion by using Lotka-Volterra Predator-prey like model with piecewise constant arguments. Periodicals of Engineering and Natural Sciences. 2014;2(1):7-12.
Kartal S, Gurcan F. Global behaviour of a predator–prey like model with piecewise constant arguments. Journal of Biological Dynamics. 2015;9(1):159-171. doi:10.1186/s13662-018-1692-x Atabaigi A. Multiple bifurcations and dynamics of a discrete-time predator-prey system with group defense and non-monotonic functional response. Differ Equ Dyn Syst; 2016.
Rana SM. Bifurcation and complex dynamics of a discrete-time predator-prey system.
Computational Ecology and Software. 2015;5(2);187-200.
Rana SM, Kulsum U. Bifurcation analysis and chaos control in a discrete-time predator-prey system of leslie type with simplified holling type IV functional response. Hindawi; 2017. Article
Khoshsiar Ghaziani R, Govaerts W, Sonck C. Resonance and bifurcation in a discrete- time predator-prey system with Holling functional response. Nonlinear Analysis:Real World Applications. 2012;13:1451-1465.
Kartal S. Flip and Neimark-Sacker bifurcation in a differential equation with piecewise constant arguments model. J. Difference Equ. Appl. 2017;23:763-778.