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 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 On a predator-prey system interaction under fluctuating water level with nonselective harvesting

2020 ◽  
Vol 18 (1) ◽  
pp. 458-475
Author(s):  
Na Zhang ◽  
Yonggui Kao ◽  
Fengde Chen ◽  
Binfeng Xie ◽  
Shiyu Li
Keyword(s):  
Water Level ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Model Interaction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fluctuating Water Level

Abstract A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global stability of the positive equilibrium are obtained. The bionomic equilibrium and the optimal harvesting policy are also presented. Numerical simulations are carried out to show the feasibility of the main results.

scholarly journals Analisis Kestabilan Model Predator-Prey dengan Adanya Faktor Tempat Persembunyian Menggunakan Fungsi Respon Holling Tipe III

2021 ◽  
Vol 3 (2) ◽  
pp. 88
Author(s):  
Riris Nur Patria Putri ◽  
Windarto Windarto ◽  
Cicik Alfiniyah
Keyword(s):  
Numerical Simulation ◽  
Response Function ◽  
Equilibrium Points ◽  
Prey Population ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Asymptotic Stable ◽  
Simulation Results ◽  
Model Predator

Predation is interaction between predator and prey, where predator preys prey. So predators can grow, develop, and reproduce. In order for prey to avoid predators, then prey needs a refuge. In this thesis, a predator-prey model with refuge factor using Holling type III response function which has three populations, i.e. prey population in the refuge, prey population outside the refuge, and predator population. From the model, three equilibrium points were obtained, those are extinction of the three populations which is unstable, while extinction of predator population and coexistence are asymptotic stable under certain conditions. The numerical simulation results show that refuge have an impact the survival of the prey.

scholarly journals Mathematical analysis of harvested predator-prey system with prey refuge and intraspecific competition

2021 ◽  
Vol 47 (2) ◽  
pp. 728-737
Author(s):  
Alanus Mapunda ◽  
Thadei Sagamiko
Keyword(s):  
Mathematical Analysis ◽  
Intraspecific Competition ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behaviour ◽  
Predator Population ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System

In this paper, a predator-prey relationship in the presence of prey refuge was studied. The analysis of the dependence of locally stable equilibrium points on the parameters of the problem was carried out. Bifurcation and limit cycles for the model were analyzed to show the dynamical behaviour of the system. The results showed that the system is stable at a constant prey refuge m = 0.3 and prey harvesting rate H = 0.3. However, increasing m and decreasing H or vice versa, the predator-prey system remains stable. It was further observed that for a constant prey refuge m ≥ 0.78, the predator population undergoes extinction. Therefore, m was found to be a bifurcation parameter and m = 0.78 is a bifurcation value. Keywords: Prey refuge, bifurcation, harvesting, intraspecific competition, phase portrait

A mathematical study of a predator–prey model with disease circulating in the both populations

2015 ◽  
Vol 08 (02) ◽  
pp. 1550015 ◽  
Author(s):  
Krishna Pada Das ◽  
J. Chattopadhyay
Keyword(s):  
Equilibrium Point ◽  
Prey Population ◽  
Global Dynamics ◽  
Predator Population ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Force Of Infection ◽  
Predator Prey Model ◽  
High Infection ◽  
Infected Prey

Disease in ecological systems plays an important role. In the present investigation we propose and analyze a predator–prey mathematical model in which both species are affected by infectious disease. The parasite is transmitted directly (by contact) within the prey population and indirectly (by consumption of infected prey) within the predator population. We derive biologically feasible and insightful quantities in terms of ecological as well as epidemiological reproduction numbers that allow us to describe the dynamics of the proposed system. Our observations indicate that predator–prey system is stable without disease but high infection rate drive the predator population toward extinction. We also observe that predation of vulnerable infected prey makes the disease to eradicate into the community composition of the model system. Local stability analysis of the interior equilibrium point near the disease-free equilibrium point is worked out. To study the global dynamics of the system, numerical simulations are performed. Our simulation results show that for higher values of the force of infection in the prey population the predator population goes to extinction. Our numerical analysis reveals that predation rates specially on susceptible prey population and recovery of infective predator play crucial role for preventing the extinction of the susceptible predator and disease propagation.

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.

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 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 PENYELESAIAN MODIFIKASI MODEL PREDATOR PREY LESLIE-GOWER DENGAN SEBAGIAN PREY TERINFEKSI MENGGUNAKAN ADAMS BASHFORTH MOULTON ORDE EMPAT

2019 ◽  
Vol 19 (2) ◽  
pp. 53
Author(s):  
Liatri Arianti ◽  
Rusli Hidayat ◽  
Kosala Dwija Purnomo
Keyword(s):  
Fourth Order ◽  
Local Stability ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Spreading Model ◽  
Disease Spreading ◽  
Predator Prey Models ◽  
Infected Prey ◽  
Model Predator

Eco-epidemiology is a science that studies the spread of infectious diseases in a population in an ecosystem where two or more species interact like a predator prey. In this paper discusses about how to solve modification Leslie Gower of predator prey models (with Holling II response function) with some prey infected using fourth order Adams Bashforth Moulton method. This paper used a simple disease-spreading model that is Susceptible-Infected (SI). The model is divided into three populations: the sound prey (which is susceptible), the infected prey and predator population. Keywords: Adams Basforth Moulton, Eco-epidemiology Holling Tipe II, Local stability, Leslie-Gower, Predator-Prey model

A STAGE-STRUCTURED PREDATOR–PREY MODEL WITH DENSITY-DEPENDENT MATURATION DELAY

2011 ◽  
Vol 04 (03) ◽  
pp. 289-312 ◽  
Author(s):  
MANJU AGARWAL ◽  
SAPNA DEVI
Keyword(s):  
Predator Population ◽  
System Behavior ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Density Dependent ◽  
Maturation Delay ◽  
Predator Prey Model ◽  
Prey Model ◽  
Trivial Equilibrium ◽  
Quantitative Changes

In this paper, a stage-structured predator–prey model is proposed and analyzed with density-dependent maturation delay. We studied the dynamics of our model analytically and obtained conditions which influence the positivity and boundedness of all populations. Criteria for the existence of a non-trivial equilibrium and conditions for the uniqueness of this equilibrium are given. A linearized analysis on the equilibria, which is algebraically very complicated in the case of non-trivial equilibrium, is carried out. We proved that the system is globally asymptotically stable in the situation when non-trivial equilibrium does not exist. To accomplish our all analytical findings and to investigate the effect of density-dependent maturation delay on the system behavior, we presented a numerical simulation. It is concluded that variations in parameter, which we introduce in the system to observe the effect of density-dependent maturation delay, produces significant quantitative changes in system behavior and also qualitative changes in the behavior of immature predator population.

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