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

2019 ◽  
Vol 29 (07) ◽  
pp. 1950091 ◽  
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.

scholarly journals Global Dynamics of an Exploited Prey-Predator Model with Constant Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Uttam Das ◽  
T. K. Kar ◽  
U. K. Pahari
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Global Dynamics ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Predator Model ◽  
Theoretical Results ◽  
Type Ii Functional Response

This paper describes a prey-predator model with Holling type II functional response incorporating constant prey refuge and harvesting to both prey and predator species. We have analyzed the boundedness of the system and existence of all possible feasible equilibria and discussed local as well as global stabilities at interior equilibrium of the system. The occurrence of Hopf bifurcation of the system is examined, and it was observed that the bifurcation is either supercritical or subcritical. Influences of prey refuge and harvesting efforts are also discussed. Some numerical simulations are carried out for the validity of theoretical results.

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

2019 ◽  
Vol 20 (1) ◽  
pp. 89-104 ◽  
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.

scholarly journals Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Dahlia Khaled Bahlool ◽  
Huda Abdul Satar ◽  
Hiba Abdullah Ibrahim
Keyword(s):  
Equilibrium Points ◽  
Global Dynamics ◽  
Persistence Conditions ◽  
Local Bifurcation Analysis ◽  
Uniqueness Of The Solution ◽  
Harvesting Process ◽  
Predator Model ◽  
The Stability ◽  
Predator System ◽  
Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

Fear Induced Stabilization in an Intraguild Predation Model

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.

scholarly journals Bifurcation Analysis of a Delayed Predator-Prey Model with Holling Type III Functional Response and Predator Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Uttam Das ◽  
T. K. Kar
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Normal Form Theory ◽  
Predator Species ◽  
Center Manifold Theorem ◽  
Type Iii ◽  
Predator Prey Model ◽  
Form Theory ◽  
Predator Model ◽  
The Stability

This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results.

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

2021 ◽  
Vol 3 (1) ◽  
pp. 16-25
Author(s):  
Adin Lazuardy Firdiansyah
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Interesting Phenomenon ◽  
Type Iii ◽  
Interior Equilibrium ◽  
The Stability

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.

scholarly journals Stochastic analysis of a prey–predator model with herd behaviour of prey

2016 ◽  
Vol 21 (3) ◽  
pp. 345-361 ◽  
Author(s):  
Shyam Pada Bera ◽  
Alakes Maiti ◽  
Guruprasad Samanta
Keyword(s):  
Birth Rate ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Social Activity ◽  
Statistical Linearization ◽  
Predator Species ◽  
The Social ◽  
Predator Model ◽  
The Stability ◽  
White Noises

In nature, a number of populations live in groups. As a result when predators attack such a population the interaction occur only at the outer surface of the herd. Again, every model in biology, being concerned with a subsystem of the real world, should include the effect of random fluctuating environment. In this paper, we study a prey–predator model in deterministic and stochastic environment. The social activity of the prey population has been incorporated by using the square root of prey density in the functional response. A brief analysis of the deterministic model including the stability of equilibrium points is presented. In random environment, the birth rate of prey species and death rate of predator species are perturbed by Gaussian white noises. We have used the method of statistical linearization to study the stability and non-equilibrium fluctuation of the populations in stochastic model. Numerical computations carried out to illustrate the analytical findings. The biological implications of analytical and numerical findings are discussed critically.

scholarly journals Stability of a fractional order harvested prey–predator model in the existence of infection in prey

2021 ◽  
Vol 1 (1) ◽  
Author(s):  
Subhashis Das ◽  
◽  
Sanat Mahato ◽  
Prasenjit Mahato
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Type I ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Different Types ◽  
Predator Model ◽  
Predictor Corrector ◽  
Uniqueness Criteria ◽  
Type Ii Functional Response

The growing relationship between prey and their predator is one of the important aspects in the field of ecology and mathematical biology. On the other hand, the utility of fractional calculus in different types of mathematical modelling have been applied extensively. In this paper, a fractional order prey–predator model is developed with the consideration of Holling type-I and Holling type-II functional response of the predator. As infection spreads through prey, the prey population is divided into two parts. In addition, we exploit the effect of harvesting to control the excessive spread of the infection. The existence and uniqueness criteria, the boundedness of the solution of the proposed model are investigated. A number of five possible equilibrium points of the proposed model are determined along with the feasibility conditions for each equilibrium points. The local stability at these equilibrium points and global stability at interior equilibrium point are investigated. Numerical simulation is presented with the help of modified Predictor-corrector method in MATLAB software to understand the dynamics of the proposed model.

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

2013 ◽  
Vol 21 (02) ◽  
pp. 1350013 ◽  
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.

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

2021 ◽  
Vol 1 (1) ◽  
pp. 16-25
Author(s):  
Adin Lazuardy Firdiansyah
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Interesting Phenomenon ◽  
Type Iii ◽  
Interior Equilibrium ◽  
The Stability

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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