Ayrık Zamanlı Caputo-Fabrizio Kesirsel Mertebeden Ekli Allee Etkili Lojistik Denklemin Kompleks Dinamiği

Yıl 2023, Cilt: 39 Sayı: 1, 13 - 26, 02.05.2023

Hatice Karakaya , Senol Kartal , İlhan Öztürk

Öz

Bu çalışmada ekli Allee etkisine sahip Caputo-Fabrizio lojistik model ele alınmıştır. Adams-Bashfort nümerik yöntemiyle kesirsel mertebeden dinamik sistemden iki boyutlu bir fark denklem sistemi elde edilmiştir. Bu fark denklem sisteminin denge noktaları hesaplanmış ve her bir denge noktasının lokal asimptotik kararlılığı için gerekli olan cebirsel koşullar Schur-Chon kriterlerinin kullanılmasıyla elde edilmiştir. Ayrıca fark denklem sisteminin, pozitif denge noktası civarında Neimark-Sacker çatallanması sergilediği gösterilmiştir. Yine Allee fonksiyonunun sistemin dinamik yapısı üzerindeki etkisi araştırılmıştır. Allee fonksiyonunun kararlılık bölgesini genişlettiği ve daha geç çatallanmaların oluşmasına sebebiyet verdiği gözlemlenmiştir. Son olarak bütün teorik sonuçlar nümerik olarak test edilmiş ve biyolojik olarak yorumlanmıştır.

Anahtar Kelimeler

Fark denklem, Kararlılık, Neimark-Sacker çatallanma, Caputo-Fabrizio türev, Allee etkisi.

Kaynakça

• Djeddi, N., Hasan, S., Al-Smadi, M., Momani, S. 2020. Modified analytical approach for generalized quadratic and cubic logistic modela with Caputo-Fabrizio fractional derivative. Alex. Eng. Jour., 59(2020), 5111-5122.
• Vivek, D., Kanagarian, K., Harikrishnan, S. 2016 Numerical solutionof fractional-orderl logistic equations by fractional Euler's method. IJRASET, 4(2016), 775-780.
• Zhu, S., Cai, C., Spanos, P. D. 2015. A nonlinear fractional derivative viscoelatc model for rail pads in dynamic analysis of coupled vehicle-slap track system. Jour.of Sound. and Vibr., 335(2015), 304-320.
• Singh, J., Kumar, D., Hammouch, Z., Atangana, A. 2015. A fractional epidemiological model for computer viruses pertaininf to a new fractional derivative. Appl. Math. and Compt, 316 (2015), 504-515.
• Laj, J., Mao, S., Qui, J., Zhang, Q., Hu, Z., Chen, J. 2016. Investigation progress and Applications of Fractional Derivative Model in Geotechnical Engineering. Hindawi, (2016).
• Balcı, E., Özturk, İ., Kartal, S. 2019. Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative. Chaos, Solutions and fractals, 123 (2019), 43-51.
• Javidi, M., Nyamoradi, N. 2013. Dynamic analysis of order prey-predator interaction with harvesting. Appl. Math. Model, 37(2013), 8946-8956.
• Aydogan, S. M., Baleanu, D., Mousalou, A., Rezapour, S. 2017. On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations. Adv. Differ. Equ, (2017), 221.
• Caputo, M., Fabrizio M. 2015. A new Definition of Fractional Derivative without Singular Kernel. Prog. Fract. Differ. Appl, 2 (2015), 73-85.
• Bashiri, T., Vaezpour, S. M., Nieto, J. 2018. Approximating Solution of Fabrizio-Caputo Volterra's Model for Population Growth in a Closed System by Homotopy Analysis Method. Hindiwa, (2018).
• Dokuyucu, M. A., Celik, E., Bulut, H., Baskonus, H. M. 2018. Cancer treatment model with the Caputo-Fabrizio fractional derivative. Eur. Phys. J. Plus, 133:92 (2018).
• Baleanu, D., Mohammadi, H., Rezapour, S. 2020. A fractional differantial equation model for the Covid-19 transmission by using the Caputo-fabrizio derivative. Adv. Di_er. Equ, 299(2020).
• Khan, M. A., Hammouch, Z., Baleanu, D. 2019. Modeling the Dynamics of Hepatitis E via the Caputo-Fabrizo derivative. Math. Model. Nat. Phenom, 14 (2019).
• Harrouche, N., Momani, S., Hasan, S., Al-Smadi, M. 2021. Computational algorithm for solving drug pharmacokinetic model under uncertainty with nonsingular kernel type Caputo-Fabrizio fractional derivative. Alex. Eng. J, 60(2021), 4347-4362. Karakaya, H., Özturk, İ., Kartal, Ş., Gurcan, F., 2021. Dynamical Analysis of Discretized Logistic Model with Caputo-Fabrizio Fraction Derivative. Comp. Ecol.and Soft, 11(2021), 21-34.
• Yousef, A., Yousef, F.B. 2019. Bifurcation ans stability Analysis of a system of Fractional-Orde Differential Equations for a Plant-Herbivore Model with Allee Effect. mathematics, DOI: 10.3390/math7050454.
• Vinoth, S., Sivasamy, R., Sathiyanathan, K., Rajchakit, G., Hammachukiattikul, P., Vadivel, R., Gunasekaran, N. 2021. Dynamical analysis of a delayed food chain model with additve Allee effect. Adv. Dif.Equ, (2021), 2021:54.
• Dennis, B. 1989. Allee effects: Population growth, Critical Density and the Change of extinction. Nat. Resc. Mod, 3(1989).
• Cai, Y., Zhao, C., Wang, W., Wang, J. 2015. Dynamics of a Leslie-Gower predator-prey model with additive Allee effect. App. Math. Model, 39 (2015), 2092-2106.
• Lv, Y., Chen, L., Chen, F. 2020. Stability and bifurcation in a single species logistic model with additive Allee effect and feedback control. Adv. Differ. Equ, 129 (2020).
• Qin, L., Zhang, F., Wang, W., Song, W. 2017. Interaction between Allee effects caused by organism-environment feedback and by other ecological mechanisms. Plos. One, 12(2017).
• Koca, I. 2018. Effcient numerical approach for solving fractional partial di_erential equations with non-singular kernel derivatives. Choas Solutions Fractals 116 (2018), 278-286.
• Noupoue, Y., Y., Tandogan, Y., Awadalla, M. 2019. On numerical techniques for solving the fractional logistic differential equation,. Adv. Di_er. Equ, 108 (2019).