Analisis Interaksi Populasi Hare dan Lynx Menggunakan Model Lotka-Volterra dengan Menerapkan Pertumbuhan Logistik dan Respon Fungsional Holling Tipe I dan II
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Abstract
This study discusses the modeling of hare and lynx population interactions using the Lotka–Volterra model, developed from real population data for both species from 1861–1880 (source: Hudson Bay Company/GitHub). This study uses two models: Model I, which applies logistic growth to the prey population (hare) with a Holling type I functional response, and Model II, which applies logistic growth to the prey population (hare) with a Holling type II functional response. The use of two models in this study aims to determine which model better represents the interaction between prey and predator. Both models are analyzed for their existence and stability of solutions. Furthermore, the model parameters, namely 𝑎, the intrinsic growth rate of the prey population, 𝑏, the rate of decline of the prey population due to interactions with predators, 𝑐, the natural mortality rate of the predator population, d, the energy conversion efficiency from predation, and 𝐾, the environmental carrying capacity, are estimated using two methods, namely Least Square Estimation (LSE) and Maximum Likelihood Estimation (MLE). The accuracy of both methods is assessed by comparing their total MSE values. The estimated parameter values of both models with the smallest total MSE are used to determine numerical solutions for both models, which yield time series of hare and lynx populations. Comparison of the real data with numerical solutions from two models with different functional responses indicates that both models are relatively capable of representing the population dynamics of lynx and hares. However, the comparison results cannot conclude which model more realistically represents the dynamics of both populations simultaneously.
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References
[1] A. De Gaetano, “The Nature of Mathematical Models,” Mathematics, vol. 14, no. 11, hal. 1–35, 2026, doi: 10.3390/math14111882.
[2] P. Frejd dan C. Bergsten, “Professional modellers’ conceptions of the notion of mathematical modelling: Ideas for education,” ZDM - Math. Educ., vol. 50, no. 1–2, hal. 117–127, 2018, doi: 10.1007/s11858-018-0928-2.
[3] C. Waldock et al., “A quantitative review of abundance-based species distribution models,” Ecography (Cop.)., vol. 2022, no. 1, hal. 0–18, 2022, doi: 10.1111/ecog.05694.
[4] M. Anokye, L. Guerrini, M. Ferrara, A. L. Sackitey, dan A. A. Konadu, “Stability and Bifurcation in a Delayed Predator–Prey Model with Environmental Stress and Dynamic Carrying Capacity,” Differ. Equations Dyn. Syst., 2026, doi: 10.1007/s12591-026-00765-1.
[5] L. M. J. O’Connor et al., “The Untapped Potential of Food Webs in Systematic Conservation Planning,” Conserv. Lett., vol. 19, no. 1, 2026, doi: 10.1111/con4.70002.
[6] A. E. Scully, S. Fisher, D. A. W. Miller, dan D. H. Thornton, “Influence of biotic interactions on the distribution of Canada lynx (Lynx canadensis) at the southern edge of their range,” J. Mammal., vol. 99, no. 4, hal. 760–772, Agu 2018, doi: 10.1093/jmammal/gyy053.
[7] W. Bonnaffé, B. C. Sheldon, dan T. Coulson, “Neural ordinary differential equations for ecological and evolutionary time-series analysis,” Methods Ecol. Evol., vol. 12, no. 7, hal. 1301–1315, 2021, doi: 10.1111/2041-210X.13606.
[8] Boyce dan R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Wiley, 2008.
[9] D. Luengo, L. Martino, M. Bugallo, V. Elvira, dan S. Särkkä, “A survey of Monte Carlo methods for parameter estimation,” EURASIP J. Adv. Signal Process., vol. 2020, no. 1, 2020, doi: 10.1186/s13634-020-00675-6.
[10] W. Mu, Q. Wei, dan S. Xiong, “Some notes on concordance between optimization and statistics,” Math. Probl. Eng., vol. 2019, 2019, doi: 10.1155/2019/3485064.
[11] G. Casella dan R. Berger, Statistical Inference. Boca Raton: Chapman and Hall/CRC, 2024. doi: 10.1201/9781003456285.
[12] G. A. F. Seber dan A. J. Lee, Linear Regression Analysis. in Wiley Series in Probability and Statistics. Wiley, 2003. doi: 10.1002/9780471722199.
[13] K. Lakshmi, B. Mahaboob, D. Sateesh Kumar, G. Balagi Prakash, dan T. Nageswara Rao, “A new vision on ordinary least squares estimation of parameters of linear model,” AIP Conf. Proc., vol. 2375, no. 2, hal. 2015–2030, 2021, doi: 10.1063/5.0066922.
[14] E. J. Chapman dan C. J. Byron, “The flexible application of carrying capacity in ecology,” Glob. Ecol. Conserv., vol. 13, hal. e00365, Jan 2018, doi: 10.1016/j.gecco.2017.e00365.
[15] N. E. Papanikolaou, T. Kypraios, H. Moffat, A. Fantinou, D. P. Perdikis, dan C. Drovandi, “Predators’ Functional Response: Statistical Inference, Experimental Design, and Biological Interpretation of the Handling Time,” Front. Ecol. Evol., vol. 9, no. October, hal. 1–5, 2021, doi: 10.3389/fevo.2021.740848.