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Analisis Kestabilan Model Mangsa Pemangsa dengan Pemanenan Ambang Batas pada Populasi Pemangsa Yusrianto Yusrianto; Syamsuddin Toaha; Kasbawati Kasbawati
Jurnal Matematika, Statistika dan Komputasi Vol. 16 No. 1 (2019): JMSK, July, 2019
Publisher : Department of Mathematics, Hasanuddin University

Abstrak Penelitian ini mengkaji model satu mangsa dan satu pemangsa yang saling berkompetisi. Fungsi predasi dari pemangsa diasumsikan menggunakan fungs1 respon Holling tipe II. Dengan asumsi bahwa adanya kompetisi intraspesifik pada popuasi pemangsa serta dilakukan pemanenan ambang batas pada popuasi pemangsa. Pada model tersebut dilakukan analisis tentang syarat kewujudan dan kestabilan titik keseimbangan interior. Analisis kestabilan titik keseimbangan interior dilakukan dengan metode linearisasi dan dengan memperhatikan nilai eigen dari matriks Jacobi yang diperoleh. Terdapat sepuluh titik kesetimbangan yang diperoleh pada model, satu diantaranya dapat dinterpretasikan. Titik tersebut dinyatakan stabil asimtotik. Berdasarkan hasil anasis menggunakan beberapa parameter, diketahui bahwa ada suatu waktu pemanenan ambang batas harus dihentikan karna sudah tidak memenuhi syarat kriteria ambang batas yang telah ditentukan.Kata kunci : Model mangsa pemangsa, Pemanenan ambang batas, Titik kesetimbanganAbstract This study examines the model of one prey and one predator who mutates each other. The predation function of predators is assumed to use the Holling type II response function. Assuming that the existence of intraspecific competition in predatory population and theshold harvesting for predatory population is carried out. In this model, an analysis of the actual conditions and stability of the interior balance point is carried out. Analysis of the interior stability balance points was carried out by linearization method and by taking into account the eigenvalues of the Jacobian matrix obtained. There are ten equilibrium points of engagement obtained on the model, one of which can be interpreted. This point is stated as asymptotically stable. Based on the results of analysis using several parameters, it is known that there is a time when harvesting the threshold must be stopped because it has not fulfill the specified criteria for threshold.Keyword : Prey-predator model threshold harvesting, equibrium point

ANALISIS KESTABILAN DAN KONTROL OPTIMAL MODEL LESLIE-GOWER FUNGSI RESPON HOLLING III DENGAN PEMANENAN PADA POPULASI PREDATOR DAN PREY Himmatul Ulya Febriyanti; Syamsuddin Toaha; Kasbawati Kasbawati
Jurnal Matematika, Statistika dan Komputasi Vol. 16 No. 1 (2019): JMSK, July, 2019
Publisher : Department of Mathematics, Hasanuddin University

This article modified the leslie-gower model on harvesting with predator and prey population. This study aims at construct a modification of leslie-gower model with holing III response function. In addition, there is an effort harvesting in predator and prey population, analyzing an equilibrium point, finding bionomic equilibrium and the condition where the present value is maximum from net income by controlling harvesting in both populations. In the modified leslie-gower model there is an equilibrium point which is asymptotically stable and when there have harvesting, the equilibrium point is also asymptotically stable. Bionomic equilibrium from harvesting on the modified leslie-gower model is maximizing the profit function π of harvesting on a model with the maximum pontryagin principle resulting an optimal equilibrium) affected by instantaneous rate of discount δ.

Stability Analysis of Mathematical Models of the Dynamics of Spread of Meningitis with the Effects of Vaccination, Campaigns and Treatment Sulma Sulma; Syamsuddin Toaha; Kasbawati Kasbawati
Jurnal Matematika, Statistika dan Komputasi Vol. 17 No. 1 (2020): JMSK, SEPTEMBER, 2020
Publisher : Department of Mathematics, Hasanuddin University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20956/jmsk.v17i1.10031

Meningitis is an infectious disease caused by bacteria, viruses, and protosoa and has the potential to cause an outbreak. Vaccination and campaign are carried out as an effort to prevent the spread of meningitis, treatment reduces the number of deaths caused by the disease and the number of infected people. This study aims to analyze and determine the stability of equilibrium point of the mathematical model of the spread of meningitis using five compartments namely susceptibles, carriers, infected without symptoms, infected with symptoms, and recovered with the effect of vaccination, campaign, and treatment. The results obtained from the analysis of the model that there are two equilibrium points, namely non endemic and endemic equilibrium points. If a certain condition is met then the non endemic equilibrium point will be asymptotically stable. Numerical simulations show that the spread of disease decreases with the influence of vaccination, campaign, and treatment.

Kestabilan Model Mangsa Pemangsa dengan fungsi respon Holling tipe IV dan penyakit pada pemangsa A. Muh. Amil Siddik; Syamsuddin Toaha; Andi Muhammad Anwar
Jurnal Matematika, Statistika dan Komputasi Vol. 17 No. 2 (2021): JANUARY 2021
Publisher : Department of Mathematics, Hasanuddin University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20956/jmsk.v17i2.11716

Stability of equilibrium points of the prey-predator model with diseases that spreads in predators where the predation function follows the simplified Holling type IV functional response are investigated. To find out the local stability of the equilibrium point of the model, the system is then linearized around the equilibrium point using the Jacobian matrix method, and stability of the equilibrium point is determined via the eigenvalues method. There exists three non-negative equilibrium points, except , that may exist and stable. Simulation results show that with the variation of several parameter values infection rate of disease , the diseases in the system may become endemic, or may become free from endemic.

Stability Analysis of Divorce Dynamics Models Syamsir Muaraf; Syamsuddin Toaha; Kasbawati Kasbawati
Jurnal Matematika, Statistika dan Komputasi Vol. 17 No. 2 (2021): JANUARY 2021
Publisher : Department of Mathematics, Hasanuddin University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20956/jmsk.v17i2.11984

This article examines the mathematical model of divorce. This model consists of four population classes, namely the Married class (M), the population class who experiences separation of separated beds (S), the population class who is divorced by Divorce (D), and the population class who experiences depression or stress due to divorce Hardship (H). This study focuses on the stability analysis of divorce-free and endemic equilibrium points. Local stability was analyzed using linearization and eigenvalues methods. In addition, the basic reproduction number is provided via the next generation matrix method. The existence and stability of the equilibrium point are determined from . The results showed that the rate of interaction between population M and populations other than H is very influential on efforts to minimize divorce. Divorce can be minimized when the transmission rate is reduced to . Reducing the transmission rate and increasing the rate of transfer from split bed class to married class can turn divorce endemic cases into non-endemic cases. A numerical simulation is given to confirm the analysis results.

Kontrol Optimal Model Matematika Merokok dengan Perokok Berhenti Sementara dan Perokok Berhenti Permanen Andi Utari Samsir; Syamsuddin Toaha; Kasbawati Kasbawati
Jurnal Matematika, Statistika dan Komputasi Vol. 18 No. 1 (2021): September 2021
Publisher : Department of Mathematics, Hasanuddin University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20956/j.v18i1.13974

Abstract This article discusses the optimal control of a mathematical model on smoking. This model consists of six population classes, namely potential to become smoker snuffing class irregular smokers regular smokers temporary quitters and permanent quitters The completion of this research uses the Pontryagin minimum principle and numerically using the forward-backward Sweep method. Numerical simulations of the optimal problem show that with the implementation of education campaigns and anti-nicotine medicine, the smokers can be decreased more quickly and the smoking population who quit permanently can be increased. The implementation of both through large amounts needs to be done from the beginning. The use of control in the form of education campaigns is of great value until the end of the research period means that it needs to be done continuously to reduce the number of smokers in the population.

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