Dynamical Analysis of the Symbiotic Model of Commensalism in Four Populations with Michaelis-Menten type Harvesting in the First Commensal Population

Nurmaini Puspitasari; Wuryansari Muharini Kusumawinahyu; Trisilowati Trisilowati

doi:10.31764/jtam.v5i2.4727

Dynamical Analysis of the Symbiotic Model of Commensalism in Four Populations with Michaelis-Menten type Harvesting in the First Commensal Population

Nurmaini Puspitasari, Wuryansari Muharini Kusumawinahyu, Trisilowati Trisilowati

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.

Keywords

Commensalism; parasitism; Michaelis-Menten; Local stabilty analysis.

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DOI: https://doi.org/10.31764/jtam.v5i2.4727