scholarly journals Analisis dinamik model predator-prey tipe Gause dengan wabah penyakit pada prey

2021 ◽  
Vol 2 (1) ◽  
pp. 20-28
Author(s):  
Rusdianto Ibrahim ◽  
Lailany Yahya ◽  
Emli Rahmi ◽  
Resmawan Resmawan
Keyword(s):  
Deterministic Model ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Population Extinction ◽  
Predator Prey Model ◽  
Extinction Point ◽  
Infected Prey ◽  
Biological Equilibrium

This article studies the dynamics of a Gause-type predator-prey model with infectious disease in the prey. The constructed model is a deterministic model which assumes the prey is divided into two compartments i.e. susceptible prey and infected prey, and both of them are hunted by predator bilinearly. It is investigated that there exist five biological equilibrium points such as all population extinction point, infected prey and predator extinction point, infected prey extinction point, predator extinction point, and co-existence point. We find that all population extinction point always unstable while others are conditionally locally asymptotically stable. Numerical simulations, as well as the phase portraits, are given to support the analytical results.

scholarly journals Analisis Kestabilan Model Predator-Prey dengan Infeksi Penyakit pada Prey dan Pemanenan Proporsional pada Predator

2020 ◽  
Vol 1 (1) ◽  
pp. 8-15
Author(s):  
Siti Maisaroh ◽  
Resmawan Resmawan ◽  
Emli Rahmi
Keyword(s):  
Equilibrium Points ◽  
The Other ◽  
Predator Prey ◽  
Population Extinction ◽  
Predator Prey Model ◽  
Runge Kutta Method ◽  
Extinction Point ◽  
Infected Prey ◽  
Disease In Prey ◽  
Model Predator

The dynamics of predator-prey model with infectious disease in prey and harvesting in predator is studied. Prey is divided into two compartments i.e the susceptible prey and the infected prey. This model has five equilibrium points namely the all population extinction point, the infected prey and predator extinction point, the infected prey extinction point, and the co-existence point. We show that all population extinction point is a saddle point and therefore this condition will never be attained, while the other equilibrium points are conditionally stable. In the end, to support analytical results, the numerical simulations are given by using the fourth-order Runge-Kutta method.

scholarly journals Global stability of a fractional-order logistic growth model with infectious disease

2020 ◽  
Vol 1 (2) ◽  
pp. 49-56
Author(s):  
Hasan S. Panigoro ◽  
Emli Rahmi
Keyword(s):  
Infectious Disease ◽  
Fractional Order ◽  
Growth Model ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Population Extinction ◽  
Logistic Growth Model ◽  
Extinction Point

Infectious disease has an influence on the density of a population. In this paper, a fractional-order logistic growth model with infectious disease is formulated. The population grows logistically and divided into two compartments i.e. susceptible and infected populations. We start by investigating the existence, uniqueness, non-negativity, and boundedness of solutions. Furthermore, we show that the model has three equilibrium points namely the population extinction point, the disease-free point, and the endemic point. The population extinction point is always a saddle point while others are conditionally asymptotically stable. For the non-trivial equilibrium points, we successfully show that the local and global asymptotic stability have the similar properties. Especially, when the endemic point exists, it is always globally asymptotically stable. We also show the existence of forward bifurcation in our model. We portray some numerical simulations consist of the phase portraits, time series, and a bifurcation diagram to validate the analytical findings.

Application of Mathematical Models Two Predators and Infected Prey by Pesticide Control in Nilaparvata Lugens Spreading in Bantul Regency

2020 ◽  
Vol 2 (1) ◽  
pp. 41-50
Author(s):  
Irham Taufiq ◽  
Denik Agustito
Keyword(s):  
Equilibrium Points ◽  
Nilaparvata Lugens ◽  
Logistic Function ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Using Data ◽  
Infected Prey ◽  
Model Predator

AbstractIn this paper, we develop a mathematical model to analyze interactions between planthopper pests as prey and menochilus sexmaculatus and mirid ladybug as two predators where prey is controlled by pesticides. The interaction between predator and prey is modeled using the Holling type II response function. The predator and prey growth are modeled using a logistic function. From this model, we obtain eight equilibrium points. The three of these equilibrium points are analyzed using linearization and locally asymptotically stable. We simulate this model using data to predict the dynamics of planthopper population and its predators. Simulation result shows that all of these populations will survive because they are influenced by pesticide control and predation rates.Keywords: control of pest; predator-prey model; the Holling type II; the logistic function.                                                                                     AbstrakPada penelitian ini, kami membangun model matematika untuk menganalisis interaksi antara hama wereng sebagai mangsa (prey) dan menochilus sexmaculatus dan mirid ladybug sebagai dua pemangsa (predator) dimana mangsa dikontrol oleh pestisida. Interaksi antara predator dan prey dimodelkan menggunakan fungsi respon Holling tipe II sedangkan pertumbuhan predator dan prey dimodelkan menggunakan fungsi logistik. Dari model tersebut diperoleh delapan titik ekuilibrium. Tiga titik ekuilibrium dari titik-titik equilibrium tersebut dianalisis menggunakan metode linierisasi dan bersifat stabil asimtotik lokal. Kemudian model ini diaplikasikan pada data.  Untuk memudahkan interpretasi antara mangsa dan dua pemangsa dilakukan simulasi numerik untuk memprediksikan dinamika populasi wereng dan predatornya. Hasil simulasi menunjukkan bahwa semua populasi tersebut akan bertahan hidup karena dipengaruhi oleh kontrol pestisida dan tingkat pemangsaan.Kata Kunci: kontrol pestisida; model predator-prey; Holling tipe II; fungsi logistik.

scholarly journals Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 785
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Equilibrium Point ◽  
Power Law ◽  
Equilibrium Points ◽  
Essential Difference ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

OCCURRENCE OF CHAOS AND ITS POSSIBLE CONTROL IN A PREDATOR-PREY MODEL WITH DENSITY DEPENDENT DISEASE-INDUCED MORTALITY ON PREDATOR POPULATION

2010 ◽  
Vol 18 (02) ◽  
pp. 399-435 ◽  
Author(s):  
KRISHNA PADA DAS ◽  
SAMRAT CHATTERJEE ◽  
J. CHATTOPADHYAY
Keyword(s):  
Equilibrium Points ◽  
Dynamical Behaviour ◽  
Predator Population ◽  
Epidemiological Models ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Chaotic Nature ◽  
Mortality Function ◽  
Infected Prey

Eco-epidemiological models are now receiving much attention to the researchers. In the present article we re-visit the model of Holling-Tanner which is recently modified by Haque and Venturino1 with the introduction of disease in prey population. Density dependent disease-induced predator mortality function is an important consideration of such systems. We extend the model of Haque and Venturino1 with density dependent disease-induced predator mortality function. The existence and local stability of the equilibrium points and the conditions for the permanence and impermanence of the system are worked out. The system shows different dynamical behaviour including chaos for different values of the rate of infection. The model considered by Haque and Venturino1 also exhibits chaotic nature but they did not shed any light in this direction. Our analysis reveals that by controlling disease-induced mortality of predator due to ingested infected prey may prevent the occurrence of chaos.

scholarly journals Examination of Stability of the Mathematical Predator-Prey Model by Observing the Distance between Predator and Prey

2019 ◽  
Vol 8 (6S3) ◽  
pp. 65-70
Keyword(s):  
Growth Model ◽  
Equilibrium Points ◽  
Predation Rate ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Defense Behavior ◽  
Long Distance ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

Maintaining distance is one of the strategies that can be applied by prey to defend themselves or to avoid predatory attacks. This defense behavior can affect predation rates. The distance or difference in the number of prey and predator populations will affect the level of balanced ecosystem. The distance is also affecting predation rate, when there’s a long distance between prey and predator thus the predation rate decreases and vice versa. The purpose of this thesis is to analyze the stability of the mathematical equilibrium on predator-prey model by observing the distance. There are two types of model being observed, type one uses exponential growth model and type two is using a logistic growth model. The analytics results obtain three equilibrium points, namely the unstable extinction equilibrium point, and the asymptotically stable predator extinction with certain conditions and asymptotically stable coexistence with certain conditions. Then numerical simulation is conducted to support the analytical results.

scholarly journals Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator

2020 ◽  
Vol 1 (1) ◽  
pp. 16-24
Author(s):  
Hasan S. Panigoro ◽  
Dian Savitri
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Biological Condition ◽  
Existence Of Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Caputo Fractional Order Derivative ◽  
Theoretical Results ◽  
Extinction Point

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results.

scholarly journals Predator-Prey Model with Refuge, Fear and Z-Control

2021 ◽  
Vol 26 (1) ◽  
pp. 40-57
Author(s):  
Ibrahim M. Elmojtaba ◽  
Kawkab Al-Amri ◽  
Qamar J.A. Khan
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Predator Population ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Long Run ◽  
Constant State

In this paper, we consider a predator-prey model incorporating fear and refuge.  Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium.  We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity.  Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium.  However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.

scholarly journals Dynamic Analysis of the Symbiotic Model of Commensalism and Parasitism with Harvesting in Commensal Populations

2021 ◽  
Vol 5 (1) ◽  
pp. 193
Author(s):  
Nurmaini Puspitasari ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Trisilowati Trisilowati
Keyword(s):  
Numerical Simulation ◽  
Dynamic Analysis ◽  
Local Stability ◽  
Equilibrium Points ◽  
Parasite Population ◽  
Asymptotically Stable ◽  
Population Extinction ◽  
Simulation Results ◽  
Extinction Point

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.This article discussed about a dynamic analysis of the symbiotic model of The dynamics of the symbiotic model of commensalism and parasitism with harvesting in the commensal population. is the main focus of this study. 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 begins by identifying the conditions for the existence 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.[VM1]  [VM1]To add a mathematical effect to the article. There can be added mathematical models produced in the study at the end of this section.

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