Model Modifikasi Holling-Tanner dengan Interaksi Kanibalisme dan Sumber Makanan Alternatif pada Predator
DOI:
https://doi.org/10.24014/jsms.v8i2.17356Abstract
Penelitian ini mengkaji model pertumbuhan populasi model predator-prey Holling-Tonner yang dimodifikasi. Pertumbuhan model predator-prey yang dianalisis mengadopsi pertumbuhan logistik. Sementara pada fungsi pemangsaan mengikuti model Holling Type II. Variabel yang tidak kalah penting dalam penelitian ini adalah adanya sumber alternatif bagi predator. Konsep seperti ini sangat realistis untuk dipertimbangkan, mengingat banyak spesies predator yang berganti sumber makanan, dikarenakan sumber makanan yang terbatas. Analisis kestabilan dilakukan pada model untuk melihat keberlangsungan populasi dalam jangka waktu yang lama dengan tingkat interaksi yang sesuai. Dari empat titik equilibrium non-negatif yang dihasilkan pada model, pengujian kestabilan hanya dilakukan pada satu titik equilibrium. Secara matematis dilakukan uji Routh-Hurwitz, dengan matriks Jacobian yang bersesuaian. Simulasi numerik menjadi asumsi realistis awal bagi keberlangsungan hidup masing-masing spesies dan melihat karakter dari masing-masing spesies tersebut. Titik equilibrium yang dipilih menjadi pegujian pada kesetabilan asimtotik lokal dan kurva pertumbuhan. Kemungkinan adanya perpindahan sumber makanan sangat mungkin terjadi, sementara interaksi kanibalisme pada model yang dikembangkan tidak memberikan pengaruh yang signifikan pada masing-masing spesies.
Kata Kunci: Kanibalisme, makanan alternatif dan Predator-Prey.
References
M. Soleh and P. R. Mandasari, “Model Matematika Pengaruh Program Rehabilitasi dan Penerapan Hukuman terhadap Jumlah Pemakai Narkoba,” J. Sains Mat. dan Stat, vol. 4, no. 2, pp. 9–17, 2018.
R. A. Pratama, M. Fransina, V. Ruslau, and U. Musamus, “Application of Beddington DeAngelis Response Function in Ecological Mathematical System : Study Fish Endemic Oliv Predator Species in Merauke,” JTAM (Jurnal Teor. dan Apl. Mat., vol. 6, no. 1, pp. 51–60, 2022.
Y. D. Jeong, S. Kim, I. H. Jung, and G. Cho, “Optimal Harvesting Strategy for Hairtail, Trichiurus Lepturus, in Korea Sea using Discrete-time Age-structured Model,” Appl. Math. Comput., vol. 392, p. 125743, 2021, doi: 10.1016/j.amc.2020.125743.
Y. Jia, Y. Li, and J. Wu, “Effect of Predator Cannibalism and Prey Growth on The Dynamic Behavior for a Predator-stage Structured Population Model with Diffusion,” J. Math. Anal. Appl., vol. 449, no. 2, pp. 1479–1501, 2017, doi: 10.1016/j.jmaa.2016.12.036.
J. F. Zhang, “Spatial Patterns of a Fractional Type Cross-diffusion Holling–Tanner Model,” Comput. Math. with Appl., vol. 76, no. 4, pp. 957–965, 2018, doi: 10.1016/j.camwa.2018.05.033.
L. Zhang and S. Fu, “Global Bifurcation for a Holling–Tanner Predator–prey Model with Prey-taxis,” Nonlinear Anal. Real World Appl., vol. 47, pp. 460–472, 2019, doi: 10.1016/j.nonrwa.2018.12.002.
C. Arancibia-Ibarra, M. Bode, J. Flores, G. Pettet, and P. van Heijster, “Turing Patterns in a Diffusive Holling–Tanner Predator-prey Model with an Alternative Food Source for The Predator,” Commun. Nonlinear Sci. Numer. Simul., vol. 99, p. 105802, 2021, doi: 10.1016/j.cnsns.2021.105802.
S. Ai, Y. Du, and R. Peng, “Traveling Waves for a Generalized Holling–Tanner Predator–prey Model,” J. Differ. Equ., vol. 263, no. 11, pp. 7782–7814, 2017, doi: 10.1016/j.jde.2017.08.021.
C. Arancibia–Ibarra and J. Flores, “Dynamics of a Leslie–Gower Predator–prey Model with Holling Type II Functional Response, Allee Effect and a Generalist Predator,” Math. Comput. Simul., vol. 188, pp. 1–22, 2021, doi: 10.1016/j.matcom.2021.03.035.
M. Cappelletti Montano and B. Lisena, “Diffusive Holling–Tanner Predator–prey Models in Periodic Environments,” Appl. Math. Lett., vol. 87, pp. 42–49, 2019, doi: 10.1016/j.aml.2018.07.024.
S. Li, C. Wang, and K. Wu, “Relaxation Oscillations of a Slow–fast Predator–prey Model with a Piecewise Smooth Functional Response,” Appl. Math. Lett., vol. 113, p. 106852, 2021, doi: 10.1016/j.aml.2020.106852.
K. P. Wijaya et al., “Food Sharing and Time Budgeting in Predator-prey Interaction,” Commun. Nonlinear Sci. Numer. Simul., vol. 97, p. 105757, 2021, doi: 10.1016/j.cnsns.2021.105757.
H. Cheng and R. Yuan, “Traveling Waves of Some Holling–Tanner Predator–prey System with Nonlocal Diffusion,” Appl. Math. Comput., vol. 338, pp. 12–24, 2018, doi: 10.1016/j.amc.2018.04.049.
J. Fu, D. Jiang, N. Shi, T. Hayat, and A. Alsaedi, “Qualitative Analysis of a Stochastic Ratio-dependent Holling-Tanner System,” Acta Math. Sci., vol. 38, no. 2, pp. 429–440, 2018, doi: 10.1016/S0252-9602(18)30758-6.
B. Roy, S. K. Roy, and D. B. Gurung, “Holling–Tanner model with Beddington–DeAngelis Functional Response and Time Delay Introducing Harvesting,” Math. Comput. Simul., vol. 142, pp. 1–14, 2017, doi: 10.1016/j.matcom.2017.03.010.
V. S. Ananth and D. K. K. Vamsi, “Influence of Quantity of Additional Food in Achieving Biological Conservation and Pest Management in Minimum-time for Prey-predator Systems Involving Holling type III response,” Heliyon, vol. 7, no. 8, p. e07699, 2021, doi: 10.1016/j.heliyon.2021.e07699.
J. Li, X. Zhu, X. Lin, and J. Li, “Impact of Cannibalism on Dynamics of a Structured Predator–prey System,” Appl. Math. Model., vol. 78, pp. 1–19, 2020, doi: 10.1016/j.apm.2019.09.022.
F. Zhang, Y. Chen, and J. Li, “Dynamical Analysis of a Stage-structured Predator-prey Model with Cannibalism,” Math. Biosci., vol. 307, no. August 2018, pp. 33–41, 2019, doi: 10.1016/j.mbs.2018.11.004.
P. Mishra, S. N. Raw, and B. Tiwari, “On a Cannibalistic Predator–prey Model with Prey Defense and Diffusio,” Appl. Math. Model., vol. 90, pp. 165–190, 2021, doi: 10.1016/j.apm.2020.08.060.
K. Takatsu, “Predator Cannibalism can Shift Prey Community Composition Toward Dominance by Small Prey Species,” Ecol. Evol., vol. 12, no. 5, pp. 1–11, 2022, doi: 10.1002/ece3.8894.
M. Rayungsari, A. Suryanto, W. M. Kusumawinahyu, and I. Darti, “Dynamical Analysis of a Predator-Prey Model Incorporating Predator Cannibalism and Refuge,” Axioms, vol. 11, no. 3, 2022, doi: 10.3390/axioms11030116.
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