scholarly journals The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Salih Djilali ◽  
Behzad Ghanbari
Keyword(s):  
Infectious Disease ◽  
Fractional Order ◽  
Graphical Representation ◽  
Transmission Rate ◽  
Equilibrium Points ◽  
Hunting Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Interaction ◽  
The Stability

AbstractIn this research, we discuss the influence of an infectious disease in the evolution of ecological species. A computational predator-prey model of fractional order is considered. Also, we assume that there is a non-fatal infectious disease developed in the prey population. Indeed, it is considered that the predators have a cooperative hunting. This situation occurs when a pair or group of animals coordinate their activities as part of their hunting behavior in order to improve their chances of making a kill and feeding. In this model, we then shift the role of standard derivatives to fractional-order derivatives to take advantage of the valuable benefits of this class of derivatives. Moreover, the stability of equilibrium points is studied. The influence of this infection measured by the transmission rate on the evolution of predator-prey interaction is determined. Many scenarios are obtained, which implies the richness of the suggested model and the importance of this study. The graphical representation of the mathematical results is provided through a precise numerical scheme. This technique enables us to approximate other related models including fractional-derivative operators with high accuracy and efficiency.

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.

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.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

scholarly journals Further Results on Bifurcation for a Fractional-Order Predator-Prey System concerning Mixed Time Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Zhenjiang Yao ◽  
Bingnan Tang
Keyword(s):  
Fractional Order ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Predator Prey ◽  
Mixed Time Delays ◽  
Predator Prey Model ◽  
Prey System ◽  
Change Of Variable ◽  
Set Up ◽  
The Stability

In the present work, we mainly focus on a new established fractional-order predator-prey system concerning both types of time delays. Exploiting an advisable change of variable, we set up an isovalent fractional-order predator-prey model concerning a single delay. Taking advantage of the stability criterion and bifurcation theory of fractional-order dynamical system and regarding time delay as bifurcation parameter, we establish a new delay-independent stability and bifurcation criterion for the involved fractional-order predator-prey system. The numerical simulation figures and bifurcation plots successfully support the correctness of the established key conclusions.

Mutual interference and its effects on searching efficiency between predator–prey interaction with bifurcation analysis and chaos control

2021 ◽  
pp. 107754632110564
Author(s):  
Waqas Ishaque ◽  
Qamar Din ◽  
Muhammad Taj
Keyword(s):  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Mutual Interference ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Existence And Stability ◽  
Prey Interaction

In this paper, we study the dynamic of the predator–prey model based on mutual interference and its effects on searching efficiency. The parametric conditions, existence, and stability for trivial and boundary equilibrium points are studied. Also, it has shown that by applying the center manifold theorem and bifurcation theory, system undergoes Neimark–Sacker bifurcation across the neighborhood of a positive fixed point. Moreover, due to the bifurcation and chaos which objectively exist in a system, three chaos control strategies are designed and used. Moreover, to validate our theoretical and analytical discussions, numerical simulations are applied to show complex and chaotic behavior. Finally, theoretical discussions are validated with experimental field data.

Time-delayed predator–prey interaction with the benefit of antipredation response in presence of refuge

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Sudeshna Mondal ◽  
Guruprasad Samanta
Keyword(s):  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Prey Population ◽  
Predator Prey ◽  
Terrestrial Vertebrates ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Refuge Area ◽  
Prey Interaction ◽  
Transcritical Bifurcations

AbstractA field experiment on terrestrial vertebrates observes that direct predation on predator–prey interaction can not only affect the population dynamics but the indirect effect of predator’s fear (felt by prey) through chemical and/or vocal cues may also reduce the reproduction of prey and change their life history. In this work, we have described a predator–prey model with Holling type II functional response incorporating prey refuge. Irrespective of being considering either a constant number of prey being refuged or a proportion of the prey population being refuged, a different growth rate and different carrying capacity for the prey population in the refuge area are considered. The total prey population is divided into two subclasses: (i) prey x in the refuge area and (ii) prey y in the predatory area. We have taken the migration of the prey population from refuge area to predatory area. Also, we have considered a benefit from the antipredation response of the prey population y in presence of cost of fear. Feasible equilibrium points of the proposed system are derived, and the dynamical behavior of the system around equilibria is investigated. Birth rate of prey in predatory region has been regarded as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the interior equilibrium point. Moreover, the conditions for occurrence of transcritical bifurcations have been determined. Further, we have incorporated discrete-type gestational delay on the system to make it more realistic. The dynamical behavior of the delayed system is analyzed. Finally, some numerical simulations are given to verify the analytical results.

scholarly journals Dynamics of Predator-Prey Model Interaction with Harvesting Effort

2020 ◽  
Vol 6 (2) ◽  
pp. 93-103
Author(s):  
Muhammad Ikbal ◽  
Riskawati
Keyword(s):  
Response Function ◽  
Equilibrium Points ◽  
Economic Value ◽  
Type I ◽  
Model Interaction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Predator Model ◽  
The Stability ◽  
Different Response

In this research, we study and construct a dynamic prey-predator model. We include an element of intraspecific competition in both predators. We formulated the Holling type I response function for each predator. We consider all populations to be of economic value so that they can be harvested. We analyze the positive solution, the existence of the equilibrium points, and the stability of the balance points. We obtained the local stability condition by using the Routh-Hurwitz criterion approach. We also simulate the model. This research can be developed with different response function formulations and harvest optimization.

scholarly journals Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

CAUCHY ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 260-269
Author(s):  
Ismail Djakaria ◽  
Muhammad Bachtiar Gaib ◽  
Resmawan Resmawan
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence And Stability ◽  
The Stability ◽  
Population Equilibrium ◽  
Predator Behavior

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

scholarly journals Limit cycles in a Kolmogorov-type model

1990 ◽  
Vol 13 (3) ◽  
pp. 555-566 ◽  
Author(s):  
Xun-Cheng Huang
Keyword(s):  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Equilibrium Points ◽  
Type Model ◽  
Kolmogorov Type ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stability Of Equilibrium ◽  
Uniqueness Of Limit Cycles ◽  
The Stability

In this paper, a Kolmogorov-type model, which includes the Gause-type model (Kuang and Freedman, 1988), the general predator-prey model (Huang 1988, Huang and Merrill 1989), and many other specialized models, is studied. The stability of equilibrium points, the existence and uniqueness of limit cycles in the model are proved.

scholarly journals The hunting cooperation of a predator under two prey's competition and fear-effect in the prey-predator fractional-order model

2022 ◽  
Vol 7 (4) ◽  
pp. 5463-5479
Author(s):  
Ali Yousef ◽  
◽  
Ashraf Adnan Thirthar ◽  
Abdesslem Larmani Alaoui ◽  
Prabir Panja ◽  
...  
Keyword(s):  
Fractional Order ◽  
Equation System ◽  
Order System ◽  
Differential Equation System ◽  
Order Model ◽  
Predator Prey ◽  
Fear Effect ◽  
Fractional Order Model ◽  
Prey Interaction ◽  
The Stability

<abstract><p>This paper investigates a fractional-order mathematical model of predator-prey interaction in the ecology considering the fear of the prey, which is generated in addition by competition of two prey species, to the predator that is in cooperation with its species to hunt the preys. At first, we show that the system has non-negative solutions. The existence and uniqueness of the established fractional-order differential equation system were proven using the Lipschitz Criteria. In applying the theory of Routh-Hurwitz Criteria, we determine the stability of the equilibria based on specific conditions. The discretization of the fractional-order system provides us information to show that the system undergoes Neimark-Sacker Bifurcation. In the end, a series of numerical simulations are conducted to verify the theoretical part of the study and authenticate the effect of fear and fractional order on our model's behavior.</p></abstract>

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