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.

scholarly journals Dynamical Behaviors of a Fractional-Order Three Dimensional Prey-Predator Model

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Noor S. Sh. Barhoom ◽  
Sadiq Al-Nassir
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Predator Species ◽  
Dynamical Behaviors ◽  
Proposed Model ◽  
Constant Rate ◽  
Predator Model ◽  
Theoretical Results

In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynamics of our proposed model.

scholarly journals Effects of Refuge Prey on Stability of the Prey-Predator Model Subject to Immigrants: A Mathematical Modelling Approach

2021 ◽  
Vol 47 (4) ◽  
pp. 1376-1391
Author(s):  
Mussa Amos Stephano ◽  
Il Hyo Jung
Keyword(s):  
Mathematical Model ◽  
Mathematical Modelling ◽  
Functional Response ◽  
Volterra Model ◽  
Predator Population ◽  
Type I ◽  
Proposed Model ◽  
Predator Model ◽  
Model Subject ◽  
Type Ii Functional Response

Prey-predator system is enormously complex and nonlinear interaction between species. Such complexity regularly requires development of new approaches which involves more factors in analysis of its population dynamics. In this paper, we formulate a modified Lotka-Volterra model that incorporates factors such as refuge prey and immigrants. We investigate the effects of refuge prey and immigrants by varying the refuge factor, with and without immigrants. The results show that with Holling’s type I functional response, the proposed model is asymptotically convergent when a refuge prey factor is introduced. Moreover, with Holling’s type II functional response, the proposed mathematical model is unstable and does not converge. However, with Holling’s type III functional response in a system, the proposed mathematical model is asymptotically stable. These results point out the following remarks: The effects of refuge prey on stability of the dynamical system vary depending on the type of functional response, and when the predator population increases, the likelihood of prey extinction declines when the proportion of preys in refuge population increases. Hence, the factor of refuge prey is crucial for controlling the population of the predator and obtaining balances between prey and predator in the ecosystem. Keywords: Refuge prey, stability, prey-predator, immigrants, Mathematical modelling

scholarly journals A Fractional-Order Predator-Prey Model with Ratio-Dependent Functional Response and Linear Harvesting

2019 ◽  
Author(s):  
Agus Suryanto ◽  
Isnani Darti ◽  
Hasan S. Panigoro ◽  
Adem Kilicman
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Stability Criteria ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Proposed Model ◽  
Prey Interaction ◽  
Ratio Dependent ◽  
Predictor Corrector

We consider a model of predator-prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity as well as the boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon's theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurance of Hopf bifurcation around the interior point is also shown analytically. At the end, we implement the Predictor-Corrector scheme to perform some numerical simulations.

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 A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1100 ◽  
Author(s):  
Agus Suryanto ◽  
Isnani Darti ◽  
Hasan S. Panigoro ◽  
Adem Kilicman
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Stability Criteria ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Proposed Model ◽  
Prey Interaction ◽  
Ratio Dependent ◽  
Predictor Corrector

We consider a model of predator–prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity and boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon’s theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurrence of Hopf bifurcation around the interior point is also shown analytically. At the end, we implemented the Predictor–Corrector scheme to perform some numerical simulations.

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.

scholarly journals DETERMINISTIC CHAOS VERSUS STOCHASTIC OSCILLATION IN A PREY-PREDATOR-TOP PREDATOR MODEL

2011 ◽  
Vol 16 (3) ◽  
pp. 343-364 ◽  
Author(s):  
Ranjit Kumar Upadhyay ◽  
Malay Banerjee ◽  
Rana Parshad ◽  
Sharada Nandan Raw
Keyword(s):  
Deterministic Model ◽  
Driving Forces ◽  
Equilibrium Points ◽  
Chaotic Oscillation ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Model System ◽  
Dynamical Analysis ◽  
Interior Equilibrium ◽  
Predator Model

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings.

scholarly journals Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Haiping Ye ◽  
Yongsheng Ding
Keyword(s):  
Fractional Order ◽  
Equation Model ◽  
Viral Diversity ◽  
Human Immune System ◽  
Order Model ◽  
Hiv Model ◽  
Interior Equilibrium ◽  
Infected Cells ◽  
The One ◽  
Predictor Corrector

We introduce fractional order into an HIV model. We consider the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T-lymphocytes (CTLs) and rate of elimination of infected cells by CTLs, based on a fractional-order differential equation model. For the one-virus model, our analysis shows that the interior equilibrium which is unstable in the classical integer-order model can become asymptotically stable in our fractional-order model and numerical simulations confirm this. We also present simulation results of the chaotic behaviors produced from the fractional-order HIV model with viral diversity by using an Adams-type predictor-corrector method.

DYNAMICAL BEHAVIOR IN A HARVESTED DIFFERENTIAL-ALGEBRAIC PREY-PREDATOR MODEL

2009 ◽  
Vol 02 (04) ◽  
pp. 463-482 ◽  
Author(s):  
CHAO LIU ◽  
QINGLING ZHANG ◽  
XUE ZHANG ◽  
XIAODONG DUAN
Keyword(s):  
Bifurcation Theory ◽  
State Feedback ◽  
Dynamical Behavior ◽  
Algebraic Model ◽  
Economic Interest ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Differential Algebraic System ◽  
Predator Model ◽  
Predator System

A differential-algebraic model which considers a prey-predator system with harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, local stability of the proposed model around the interior equilibrium is investigated. Furthermore, the instability mechanism of the proposed model due to the variation of economic interest of harvesting is studied. With the purpose of stabilizing the proposed model around the interior equilibrium and maintaining the economic interest of harvesting at an ideal level, a state feedback controller is designed. Finally, numerical simulations are carried out to show the consistency with theoretical analysis.

scholarly journals Effects of Additional Foods to Predators on Nutrient-Consumer-Predator Food Chain Model

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Banshidhar Sahoo
Keyword(s):  
Death Rate ◽  
Local Stability ◽  
Equilibrium Points ◽  
Chain Model ◽  
Additional Food ◽  
Boundedness Property ◽  
Interior Equilibrium ◽  
Predator Model ◽  
Food Chain Model ◽  
Unstable Dynamics

We have proposed a nutrient-consumer-predator model with additional food to predator, at variable nutrient enrichment levels. The boundedness property and the conditions for local stability of boundary and interior equilibrium points of the system are derived. Bifurcation analysis is done with respect to quality and quantity of additional food and consumer’s death rate for the model. The system has stable as well as unstable dynamics depending on supply of additional food to predator. This model shows that supply of additional food plays an important role in the biological controllability of the system.

Sign in / Sign up

Export Citation Format

Share Document