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All Journal Epsilon: Jurnal Matematika Murni dan Terapan JTAM (Jurnal Teori dan Aplikasi Matematika)
Puspitasari, Nurmaini
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Decision Sciences, Operations Research & Management Mathematics Transportation

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Dynamic Analysis of the Symbiotic Model of Commensalism and Parasitism with Harvesting in Commensal Populations Puspitasari, Nurmaini; Kusumawinahyu, Wuryansari Muharini; Trisilowati, Trisilowati
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 5, No 1 (2021): April
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v5i1.3893

Abstract

This article discussed about a dynamic analysis of the symbiotic model of commensalism and parasitism with harvesting in the commensal population. This model is obtained from a modification of the symbiosis commensalism model. This modification is by adding a new population, namely the parasite population. Furthermore, it will be investigated that the three populations can coexist. The analysis carried out includes the determination of all equilibrium points along with their existence and local stability along with their stability requirements. From this model, it is obtained eight equilibrium points, namely three population extinction points, two population extinction points, one population extinction point and three extinction points can coexist. Of the eight points, only two points are asymptotically stable if they meet certain conditions. Next, a numerical simulation will be performed to illustrate the model’s behavior. In this article, a numerical simulation was carried out using the RK-4 method. The simulation results obtained support the results of the dynamic analysis that has been done previously.
Dynamical Analysis of the Symbiotic Model of Commensalism in Four Populations with Michaelis-Menten type Harvesting in the First Commensal Population Puspitasari, Nurmaini; Kusumawinahyu, Wuryansari Muharini; Trisilowati, Trisilowati
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 5, No 2 (2021): October
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v5i2.4727

Abstract

This study discusses the dynamical analysis of the symbiosis commensalism and parasitism models in four populations with Michaelis-Menten type harvesting in the first commensal population. This model is formed from a construction of the symbiotic model of commensalism and parasitism by harvesting the commensal population. This construction is by adding a new population, namely the second commensal population. Furthermore, it will be investigated that the four populations can coexist. The first analysis is to identify the conditions of existence at all equilibrium points along with the conditions for their existence and local stability around the equilibrium point along with the stability requirements. From this model, it is obtained sixteen points of equilibrium, namely one point of extinction in the four populations, four points of extinction in all three populations, six points of extinction in both populations, four points of extinction in one population and one point where the four populations can coexist. Of the sixteen points, only four points can be asymptotically stable if they meet the stability conditions that have been determined. Finally, a numerical simulation is performed to describe the model behavior. In this study, the method used in numerical simulation is the RK-4 method. The numerical simulation results that have been obtained support the dynamical analysis results that have been carried out previously.
NON-STANDARD SCHEME DISCRETIZATION (NSFD) FOR COMMENSALISM SYMBIOSIS MODEL WITH HARVESTING IN COMMENSAL POPULATIONS Puspitasari, Nurmaini; Faisal, Faisal; Yulida, Yuni; Jannah, Nur Wahidiyatil; Balya, Muhammad Afief
EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN Vol 19, No 2 (2025)
Publisher : Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Lambung Mangkurat

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20527/epsilon.v19i2.16551

Abstract

Dynamic analysis on the model of commensalism symbiosis with the discretized Michaelis-Menten cropping by using different schemes to non-standard finite difference (NSFD) is the main focus in this article. The analysis is started by searching the equilibrium points with their existence terms and local stability with their stability terms. In this article, there are four equilibrium points. The points are the extinction point of both populations, the host extinction point, the commensal extinction point, and the point where both populations can coexist (the coexistence equilibrium point). The existence of a host extinction point and a point at which both populations can coexist depends on the conditions of existence that must be met. Among the four equilibrium points, the commensal extinction point and the coexistence equilibrium point are locally asymptotically stable provided that the specified stability conditions are met. In the final analysis, numerical simulations were performed using the 4th order Runge–Kutta scheme for the continuous model and the NSFD scheme for the discrete model. The results show that the NSFD scheme offers greater flexibility in choosing the integration time step to ensure convergence to a feasible solution, outperforming the 4th order Runge–Kutta scheme in this respect.
Co-Authors Balya, Muhammad Afief Faisal Faisal Jannah, Nur Wahidiyatil Trisilowati Trisilowati, Trisilowati Wuryasari Muharini Kusumawinahyu Yuni Yulida