Öz
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.
Anahtar Kelimeler
Neimark-Sacker bifurcation plant-herbivore model stability system of difference equation
Kaynakça
- 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.
- Caughley G, Lawton JH, 1981. Plant-herbivore systems, in Theoretical Ecolog. Sinauer Associates, Sunderland, 132-166.
- Cejas VO, Fort J, Mendez V, 2004. The role of the delay time in the modeling of biological range expansions. Ecology, 85: 258-264.
- 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.
- Danca M, Codreanu S, Bako B, 1997. Detailed analysis of a nonlinear prey-predator model. J Biol Phys, 23: 11-20.
- Das K, Sarkar AK, 2001. Stability and oscillation of an autotroph-herbivore model with time delay. Int J Sys Sci, 32: 585-590.
- 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.
- Kuznetsov YA, 1998. Elements of applied bifurcation theory. Springer-Verlag, Newyork.
- Li Y, 2011. Toxicity impact on a plant-herbivore model with disease in herbivores. Comput Math Appl, 62: 2671-2680.
- May RM, 2001. Stability and complexity in model ecosystems. Princeton University Press, 40, New Jercy.
- 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.
- Peng M, 2005. Multiple bifurcations and periodic “bubbling” in a delay population model. Chaos Soliton Fract, 25: 1123-1130.
- 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.
- 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.
- 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.
- Wen GL, 2005. Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys Rev E ,72: 026201-3.
Öz
Anahtar Kelimeler
Fark denklem sistemi kararlılık Neimark-Sacker çatallanma ot-otçul model
Kaynakça
- 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.
- Caughley G, Lawton JH, 1981. Plant-herbivore systems, in Theoretical Ecolog. Sinauer Associates, Sunderland, 132-166.
- Cejas VO, Fort J, Mendez V, 2004. The role of the delay time in the modeling of biological range expansions. Ecology, 85: 258-264.
- 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.
- Danca M, Codreanu S, Bako B, 1997. Detailed analysis of a nonlinear prey-predator model. J Biol Phys, 23: 11-20.
- Das K, Sarkar AK, 2001. Stability and oscillation of an autotroph-herbivore model with time delay. Int J Sys Sci, 32: 585-590.
- 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.
- Kuznetsov YA, 1998. Elements of applied bifurcation theory. Springer-Verlag, Newyork.
- Li Y, 2011. Toxicity impact on a plant-herbivore model with disease in herbivores. Comput Math Appl, 62: 2671-2680.
- May RM, 2001. Stability and complexity in model ecosystems. Princeton University Press, 40, New Jercy.
- 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.
- Peng M, 2005. Multiple bifurcations and periodic “bubbling” in a delay population model. Chaos Soliton Fract, 25: 1123-1130.
- 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.
- 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.
- 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.
- Wen GL, 2005. Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys Rev E ,72: 026201-3.
Ayrıntılar
| Birincil Dil | Türkçe |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik / Mathematics |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Mart 2018 |
| Gönderilme Tarihi | 25 Temmuz 2017 |
| Kabul Tarihi | 13 Kasım 2017 |
| Yayımlandığı Sayı | Yıl 2018 Cilt: 8 Sayı: 1 |