Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases. In the end, several numerical solutions are presented to check the analytical results.

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Dynamical of Ratio-Dependent Eco-epidemical Model with Prey Refuge

CAUCHY ◽

10.18860/ca.v6i4.10827 ◽

2021 ◽

Vol 6 (4) ◽

pp. 227-237

Author(s):

Adin Lazuardy Firdiansyah

Keyword(s):

Hopf Bifurcation ◽

Equilibrium Points ◽

Uniform Persistence ◽

Asymptotically Stable ◽

Bifurcation Parameter ◽

Prey Refuge ◽

Globally Asymptotically Stable ◽

Interior Equilibrium ◽

Ratio Dependent ◽

Infected Prey

This paper discusses the dynamic analysis of three species in the eco-epidemiology model by considering the ratio-dependent function and prey refuge. The prey refuge is applied under the fact that infected prey has protection instincts that allow it to reduce predation risk. Here, we get the boundedness and three equilibrium points where are existence under certain conditions. In the model, three equilibrium points are locally asymptotically stable and one of the equilibrium points is globally asymptotically stable. We find that the system undergoes Hopf bifurcation around the interior equilibrium point by choosing as a bifurcation parameter. We also find a condition for uniform persistence. Finally, several simulations of numerical are performed not only to illustrate the analytical results but also to illustrate the effect of the prey refuge.

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Fear Induced Stabilization in an Intraguild Predation Model

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500534 ◽

2020 ◽

Vol 30 (04) ◽

pp. 2050053

Author(s):

Mainul Hossain ◽

Nikhil Pal ◽

Sudip Samanta ◽

Joydev Chattopadhyay

Keyword(s):

Growth Rate ◽

Hopf Bifurcation ◽

Intraguild Predation ◽

Equilibrium Points ◽

Stable Limit ◽

Interior Equilibrium ◽

The Rich ◽

The Stability ◽

The Cost ◽

The Impact

In the present paper, we investigate the impact of fear in an intraguild predation model. We consider that the growth rate of intraguild prey (IG prey) is reduced due to the cost of fear of intraguild predator (IG predator), and the growth rate of basal prey is suppressed due to the cost of fear of both the IG prey and the IG predator. The basic mathematical results such as positively invariant space, boundedness of the solutions, persistence of the system have been investigated. We further analyze the existence and local stability of the biologically feasible equilibrium points, and also study the Hopf-bifurcation analysis of the system with respect to the fear parameter. The direction of Hopf-bifurcation and the stability properties of the periodic solutions have also been investigated. We observe that in the absence of fear, omnivory produces chaos in a three-species food chain system. However, fear can stabilize the chaos thus obtained. We also observe that the system shows bistability behavior between IG prey free equilibrium and IG predator free equilibrium, and bistability between IG prey free equilibrium and interior equilibrium. Furthermore, we observe that for a suitable set of parameter values, the system may exhibit multiple stable limit cycles. We perform extensive numerical simulations to explore the rich dynamics of a simple intraguild predation model with fear effect.

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Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

Mathematical Problems in Engineering ◽

10.1155/2016/7068479 ◽

2016 ◽

Vol 2016 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Wei Tan ◽

Jianguo Gao ◽

Wenjun Fan

Keyword(s):

Hopf Bifurcation ◽

Chaotic System ◽

Center Manifold ◽

Distributed Delay ◽

Effective Control ◽

Asymptotically Stable ◽

Bifurcation Parameter ◽

Normal Form Theorem ◽

The Stability ◽

Variation Tendency

We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.

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Analisis Kestabilan Model SEIR Penyebaran Penyakit Campak dengan Pengaruh Imunisasi dan Vaksin MR

Jurnal Matematika Statistika dan Komputasi ◽

10.20956/jmsk.v16i1.6594 ◽

2019 ◽

Vol 16 (1) ◽

pp. 107

Author(s):

Willyam Daniel Sihotang ◽

Ceria Clara Simbolon ◽

July Hartiny ◽

Desrinawati Tindaon ◽

Lasker Pangarapan Sinaga

Keyword(s):

Infectious Disease ◽

Equilibrium Point ◽

Numerical Solutions ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Seir Model ◽

Rate Of Spread ◽

The Stability ◽

Disease Free

Measles is a contagious infectious disease caused by a virus and has the potential to cause an outbreak. Immunization and vaccination are carried out as an effort to prevent the spread of measles. This study aims to analyze and determine the stability of the SEIR model on the spread of measles with the influence of immunization and MR vaccines. The results obtained from model analysis, namely there are two disease free and endemic equilibrium points. If the conditions are met, the measles-free equilibrium point will be asymptotically stable and the measles endemic equilibrium point will be stable. Numerical solutions show a decrease in the rate of spread of measles due to the effect of immunization and the addition of MR vaccines.

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DYNAMIC CONSEQUENCES OF PREY REFUGIA IN A TWO-PREDATOR–ONE-PREY SYSTEM

Journal of Biological System ◽

10.1142/s0218339013500137 ◽

2013 ◽

Vol 21 (02) ◽

pp. 1350013 ◽

Cited By ~ 1

Author(s):

T. K. KAR ◽

ABHIJIT GHORAI ◽

SOOVOOJEET JANA

Keyword(s):

Hopf Bifurcation ◽

Center Manifold ◽

Sufficient Conditions ◽

Prey Refuge ◽

Center Manifold Theorem ◽

Interior Equilibrium ◽

Prey System ◽

The Stability ◽

Normal Form Methods ◽

Type Ii Functional Response

We consider a two predator and one prey model with Holling type II functional response incorporating a constant prey refuge. Depending upon the constant prey refuge m, which provides a criterion for protecting m of prey from predation, sufficient conditions for stability and global stability of equilibria are obtained. We find the critical value of this refuge parameter m for which the dynamical system undergoes a Hopf bifurcation and then makes use of center manifold theorem and normal form methods to find the direction of the Hopf bifurcation as well as the stability of the resulting limit cycle. The influence of the prey refuge parameter is also investigated at the interior equilibrium. Numerical simulations were carried out to illustrate and support the analytical results.

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Dynamics analysis of a delayed virus model with two different transmission methods and treatments

Advances in Difference Equations ◽

10.1186/s13662-019-2438-0 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 66

Author(s):

Tongqian Zhang ◽

Junling Wang ◽

Yuqing Li ◽

Zhichao Jiang ◽

Xiaofeng Han

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Reproduction Number ◽

Dynamics Analysis ◽

Asymptotically Stable ◽

Bifurcation Parameter ◽

Local Hopf Bifurcation ◽

Virus Model ◽

The Stability ◽

The Basic Reproduction Number

AbstractIn this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

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Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2017-0224 ◽

2019 ◽

Vol 20 (1) ◽

pp. 89-104 ◽

Cited By ~ 1

Author(s):

Hafizul Molla ◽

Md. Sabiar Rahman ◽

Sahabuddin Sarwardi

Keyword(s):

Hopf Bifurcation ◽

Uniform Boundedness ◽

Type Ii ◽

Prey Refuge ◽

Predator Species ◽

Interior Equilibrium ◽

Predator Prey Model ◽

Predator Model ◽

The Stability ◽

Manifold Theory

AbstractWe propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.

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Hopf Bifurcation Analysis in a Modified Optically Injected Semiconductor Lasers Model

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.631-632.254 ◽

2014 ◽

Vol 631-632 ◽

pp. 254-260

Author(s):

Jiang Ang Zhang ◽

Wen Ju Du ◽

Kutorzi Edwin Yao

Keyword(s):

Nonlinear Dynamics ◽

Hopf Bifurcation ◽

Semiconductor Lasers ◽

Bifurcation Analysis ◽

Autonomous System ◽

Dynamic Properties ◽

Equilibrium Points ◽

Bifurcation Parameter ◽

Numerical Example ◽

The Stability

In this paper, a modified optically injected semiconductor lasers model is studied in detail. More precisely, we study the stability of the equilibrium points and basic dynamic properties of the autonomous system by means of nonlinear dynamics theory. The existence of Hopf bifurcation is investigated by choosing the appropriate bifurcation parameter. Furthermore, formulas for determining the stability and the conditions for generating Hopf bifurcation of the equilibria are derived. Then, a numerical example is given.

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Stability and Hopf Bifurcation of a Delayed Prey–Predator Model with Disease in the Predator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419500913 ◽

2019 ◽

Vol 29 (07) ◽

pp. 1950091 ◽

Cited By ~ 65

Author(s):

Chuangxia Huang ◽

Hua Zhang ◽

Jinde Cao ◽

Haijun Hu

Keyword(s):

Hopf Bifurcation ◽

Equilibrium Points ◽

Mass Action ◽

Functional Responses ◽

Predator Species ◽

Interior Equilibrium ◽

Positivity Of Solutions ◽

Predator Model ◽

The Stability ◽

Type Ii Functional Response

Dealing with the epidemiological prey–predator is very important for us to understand the dynamical characteristics of population models. The existing literature has shown that disease introduction into the predator group can destabilize the established prey–predator communities. In this paper, we establish a new delayed SIS epidemiological prey–predator model with the assumptions that the disease is transmitted among the predator species only and different type of predators have different functional responses, viz. the infected predator consumes the prey according to Holling type-II functional response and the susceptible predator consumes the prey following the law of mass action. The positivity of solutions, the existence of various equilibrium points, the stability and bifurcation at those equilibrium points are investigated at length. Using the incubation period as bifurcation parameter, it is observed that a Hopf bifurcation may occur around the equilibrium points when the parameter passes through some critical values. We also discuss the direction and stability of the Hopf bifurcation around the interior equilibrium point. Simulations are arranged to show the correctness and effectiveness of these theoretical results.

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