Bifurcation Analysis of a Plant-Herbivore Model Constructed with System of Difference Equations

Abstract

In this study, a plant-herbivore mathematical model constructed with the system of difference

equation is considered. Using Center Manifold theorem, the eigenvalue assignment, transversality and nonresonance

conditions that required for the existence of Neimark Sacker bifurcation in the system are analyzed and it has been

theoretically shown that these conditions are satisfied. In order to show the accuracy of these theoretical conditions,

some parameter values have been determined and the bifurcation diagram and phase diagrams of the system have

been obtained for these parameter values.

Keywords

• Neimark-Sacker bifurcation,plant-herbivore model,stability,system of difference equation

Fark Denklem Sistemleriyle Oluşturulmuş Ot-Otçul Modelinin Çatallanma Analizi

Abstract

Bu çalışmada, fark denklem sistemiyle oluşturulmuş bir ot-otçul matematiksel modeli göz önüne alınmıştır.

Center Manifold teoremi kullanılarak sistemde Neimark Sacker çatallanmasının oluşması için gerekli olan özdeğer

eşliği, transversality ve nonresonance koşulları analiz edilmiş ve teorik olarak bu koşulların sağlandığı gösterilmiştir.

Elde edilen bu teorik koşulların doğruluğunu göstermek için bazı parametre değerleri belirlenmiş ve bu parametre

değerleri için sistemin çatallanma diagramı, faz diyagramları elde edilmiştir.

Keywords

• Fark denklem sistemi,kararlılık,Neimark-Sacker çatallanma,ot-otçul model

References

1. Agiza HN, Elabbasy EM, Metwally HE., Elsadany AA, 2009. Chaotic dynamics of a discrete prey-predator model with Holling type II. Nonlinear Anal Real, 10: 116-119.
2. Caughley G, Lawton JH, 1981. Plant-herbivore systems, in Theoretical Ecolog. Sinauer Associates, Sunderland, 132-166.
3. Cejas VO, Fort J, Mendez V, 2004. The role of the delay time in the modeling of biological range expansions. Ecology, 85: 258-264.
4. Chattopadhayay J, Sarkar R, Hoballah MEF, Turlings TCJ, Bersier LF, 2001. Parasitoids may determine plant fitness-a mathematical model based on experimental data. J Theor Biol, 212: 295-302.
5. Danca M, Codreanu S, Bako B, 1997. Detailed analysis of a nonlinear prey-predator model. J Biol Phys, 23: 11-20.
6. Das K, Sarkar AK, 2001. Stability and oscillation of an autotroph-herbivore model with time delay. Int J Sys Sci, 32: 585-590.
7. He Z, Li B, 2014. Complex dynamic behavior of a discrete time predator-prey system of Holling-III type. Adv Differ Equ, 180. Kartal S, 2016. Dynamics of a plant-herbivore model with differential-difference equations. Cogent Mathematics, 3: Article Number: 1136198.
8. Kuznetsov YA, 1998. Elements of applied bifurcation theory. Springer-Verlag, Newyork.

9. Li Y, 2011. Toxicity impact on a plant-herbivore model with disease in herbivores. Comput Math Appl, 62: 2671-2680.
10. May RM, 2001. Stability and complexity in model ecosystems. Princeton University Press, 40, New Jercy.
11. Mukherjee D, Das P, Kesh D, 2011. Dynamics of a plant-herbivore model with holling type II functional response. Computational and Mathematical Biology, 2: 1-9.
12. Peng M, 2005. Multiple bifurcations and periodic “bubbling” in a delay population model. Chaos Soliton Fract, 25: 1123-1130.
13. Sohel Rana SM, 2015. Bifurcation and complex dynamics of a discrete-time predator-prey system. Comput Ecol Softw, 5: 187-200. Sui G, Fan M, Loladze I, Kuang Y, 2007. The Dynamics of a stoichiometric plant-herbivore model and its discrete analog. Math Biosci Eng, 4: 29-46.
14. Sun GQ, Chakraborty A, Liu QX, Jin Z, Anderson K.E., 2014. Influence of time delay and nonlinear diffusion on herbivore outbreak. Commun Nonlinear Sci Numer Simulat, 19: 1507-1518.
15. Xin B, Ma J, Gao Q, 2009. The complexity of an investment competition dynamical model with imperfect information in a security market. Chaos Soliton Fract, 42: 2425-2438.
16. Wen GL, 2005. Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys Rev E ,72: 026201-3.