This article describes the dynamics of local stability equilibrium point models of interaction between prey populations and their predators. The model involves response functions in the form of ...Holling type III and anti-predator behavior. The existence and stability of the equilibrium point of the model can be obtained by reviewing several cases. One of the factors that affect the existence and local stability of the model equilibrium point is the carrying capacity (k) parameter. If x3∗, y3∗ > 0 is a constant solution of the model and ∈ (0,x3∗), then there is a unique boundary equilibrium point Ek (k , 0). Whereas, if k ∈ (x4∗, y4∗, then Ek (k, 0) is unstable and E3 (x3∗, y3∗) is stable. Furthermore, if k ∈ ( x4∗, ∞), then Ek ( k, 0) remains stable and E4 (x4∗, y4∗) is unstable, but the stability of the equilibrium point E3 (x3∗, y3∗) is branching. The equilibrium point E3 (x3∗, y3∗) can be stable or unstable depending on all parameters involved in the model. Variations of k parameter values are given in numerical simulation to verify the results of the analysis. Numerical simulation indicates that if k = 0,92 then nontrivial equilibrium point Ek (0,92 ; 0) stable. If k = 0,93 then Ek (0,93 ; 0) unstable and E3∗(0,929; 0,00003) stable. If k = 23,94, then Ek (23,94 ; 0) and E3∗(0,929; 0,143) stable, but E4∗(23,93 ; 0,0005) unstable. If k = 38 then Ek(38,0) stable, but E3∗(0,929; 0,145) and E4∗(23,93 ; 0,739) unstable.Keywords: anti-predator behavior, carrying capacity, and holling type III.
In this article, we present an SVIR epidemic model with deadly deseases and non constant population. We only discuss the local stability analysis of the model. Initially the basic formulation of the ...model is presented. Two equilibrium point exists for the system; disease free and endemic equilibrium point. The local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. If the value of R0 less than one then the desease free equilibrium point is locally asymptotically stable, and if its exceeds, the endemic equilibrium point is locally asymptotically stable. The numerical results are presented for illustration.
In the world of education, a lot of knowledge is learned by students. One of them is Mathematics which has been taught starting from basic education to higher education. Fractions are a subject in ...the field of mathematics that is considered the most difficult to learn, both understanding the basic concepts and understanding the concept of their use, namely understanding operations in fractions. To overcome the above difficulties, in studying or teaching fractions to students about the concepts and operations of counting, it is necessary to have a method, so that learning mathematics becomes an exciting learning. One of them is the Interactive Media based on Macromedia Flash. The short-term objective of this service activity is to provide a new atmosphere for students in learning, namely introducing the use of Macromedia Flash-based Interactive Media in teaching fractions. Meanwhile, the long-term goal is for the teachers of SD Pemb.V Yapis Waena to obtain a new method for comparison in learning fractions. This activity was held on 18-19 September 2020 which was attended by 20 grade V students of SD Pemb.V Yapis Waena. The methods used are lectures, discussions, question and answer methods, demonstrations and practice, as well as observation and evaluation of the results of the multimedia teaching aids. The conclusion is that by learning the fraction concept using interactive media based on Macromedia Flash, the teacher will be more concrete in explaining the concepts and operations of fractions to students.Keywords: SD Yapis, Fractions, Method, Macromedia Flash
Abstrak
Artikel ini termasuk dalam ruang lingkup matematika epidemiologi. Tujuan ditulisnya artikel ini untuk mendeskripsikan dinamika lokal penyebaran suatu penyakit dengan beberapa asumsi yang ...diberikan. Dalam pembahasan, dianalisis titik ekuilibrium model epidemi SVIR dengan adanya imigrasi pada kompartemen vaksinasi. Dengan langkah pertama, model SVIR diformulasikan, kemudian titik ekuilibriumnya ditentukan, selanjutnya, bilangan reproduksi dasar ditentukan. Pada akhirnya, kestabilan titik ekuilibirum yang bergantung pada bilangan reproduksi dasar ditentukan secara eksplisit. Hasilnya adalah jika bilangan reproduksi dasar kurang dari satu maka terdapat satu titik ekuilbirum dan titik ekuilbrium tersebut stabil asimtotik lokal. Hal ini berarti bahwa dalam kondisi tersebut penyakit akan cenderung menghilang dalam populasi. Sebaliknya, jika bilangan reproduksi dasar lebih dari satu, maka terdapat dua titik ekuilibrium. Dalam kondisi ini, titik ekuilibrium endemik stabil asimtotik lokal dan titik ekuilibrium bebas penyakit tidak stabil. Hal ini berarti bahwa dalam kondisi tersebut penyakit akan tetap ada dalam populasi.
Kata Kunci : Model SVIR, Stabil Asimtotik Lokal
Abstract
This article is included in the scope of mathematical epidemiology. The purpose of this article is to describe the dynamics of the spread of disease with some assumptions given. In this paper, we present an epidemic SVIR model with the presence of immigration in the vaccine compartment. First, we formulate the SVIR model, then the equilibrium point is determined, furthermore, the basic reproduction number is determined. In the end, the stability of the equilibrium point is determined depending on the number of basic reproduction. The result is that if the basic reproduction number is less than one then there is a unique equilibrium point and the equilibrium point is locally asymptotically stable. This means that in those conditions the disease will tend to disappear in the population. Conversely, if the basic reproduction number is more than one, then there are two equilibrium points. The endemic equilibrium point is locally asymptotically stable and the disease-free equilibrium point is unstable. This means that in those conditions the disease will remain in the population.
Keywords: SVIR Model, Locally Asymptotically stable.
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