scholarly journals Dynamics in two competing predators-one prey system with two types of Holling and fear effect

2021 ◽  
Vol 2 (2) ◽  
pp. 58-67
Author(s):  
Adin Lazuardy Firdiansyah ◽  
Nurhidayati Nurhidayati
Keyword(s):  
Consumption Rate ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
The Other ◽  
Type I ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Local Stability Analysis ◽  
Prey System

In this article, it is formulated a predator-prey model of two predators consuming a single limited prey resource. On the other hand, two predators have to compete with each other for survival. The predation function for two predators is assumed to be different where one predator uses Holling type I while the other uses Holling type II. It is also assumed that the fear effect is considered in this model as indirect influence evoked by both predators. Non-negativity and boundedness is written to show the biological justification of the model. Here, it is found that the model has five equilibrium points existed under certain condition. We also perform the local stability analysis on the equilibrium points with three equilibrium points are stable under certain conditions and two equilibrium points are unstable. Hopf bifurcation is obtained by choosing the consumption rate of the second predator as the bifurcation parameter. In the last part, several numerical solutions are given to support the analysis results.

Predator–Prey System: Bachelor Herding of the Prey Imposes Ecological Constraints on Predation

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Rajat Kaushik ◽  
Sandip Banerjee
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Herd Behavior ◽  
Ecological Constraints ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Schooling Behavior ◽  
Prey System ◽  
Parent Groups

Bachelor herd behavior is very common among juvenile animals who have not become sexually matured but have left their parent groups. The complex grouping or schooling behavior provides vulnerable juveniles refuge from predation and opportunities for foraging, especially when their parents are not within the area to protect them. In spite of this, juvenile/immature prey may easily become victims because of their greenness while on the other hand, adult prey may be invulnerable to attack due to their tricky manoeuvring abilities to escape from the predators. In this study, we propose a stage-structured predator–prey model, in which predators attack only the bachelor herds of juvenile prey while adult prey save themselves due to small predator–prey size ratio and their fleeing capability, enabling them to avoid confrontation with the predators. Local and global stability analysis on the equilibrium points of the model are performed. Sufficient conditions for uniform permanence and the impermanence are derived. The model exhibits both transcritical as well as Hopf bifurcations and the corresponding numerical simulations are carried out to support the analytical results. Bachelor herding of juvenile prey as well as inaccessibility of adult prey restricts the uncontrolled predation so that prey abundance and predation remain balanced. This investigation on bachelor group defence brings out some unpredictable results, especially close to the zero steady state. Altogether, bachelor herding of the juvenile prey, which causes unconventional behavior near the origin, plays a significant role in establishing uniform permanence conditions, also increases richness of the dynamics in numerical simulations using the bifurcation theory and thereby, shapes ecosystem properties tremendously and may have a large influence on the ecosystem functioning.

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 Existence and Attractiveness of Order One Periodic Solution of a Holling I Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Huidong Cheng ◽  
Tongqian Zhang ◽  
Fang Wang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Impulsive Control ◽  
Management Strategies ◽  
Type I ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
State Dependent ◽  
Order One Periodic Solution

According to the integrated pest management strategies, a Holling type I functional response predator-prey system concerning state-dependent impulsive control is investigated. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution, and the attractivity of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results which show that our method used in this paper is more efficient than the existing ones for proving the existence and attractiveness of order one periodic solution.

On a delay ratio-dependent predator–prey system with feedback controls and shelter for the prey

2018 ◽  
Vol 11 (07) ◽  
pp. 1850095
Author(s):  
Changyou Wang ◽  
Linrui Li ◽  
Yuqian Zhou ◽  
Rui Li
Keyword(s):  
Numerical Solutions ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Feedback Controls ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Ratio Dependent ◽  
Predator Prey Models

In this paper, a class of three-species multi-delay Lotka–Volterra ratio-dependent predator–prey model with feedback controls and shelter for the prey is considered. A set of easily verifiable sufficient conditions which guarantees the permanence of the system and the global attractivity of positive solution for the predator–prey system are established by developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function. Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In addition, some numerical solutions of the equations describing the system are given to show that the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator–prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system. At the same time, the influence of the delays and shelters on the dynamics behavior of the system is also considered by solving numerically the predator–prey models.

scholarly journals Global Stability of Predator-Prey System with Alternative Prey

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Banshidhar Sahoo
Keyword(s):  
Global Stability ◽  
Bifurcation Analysis ◽  
Equilibrium Points ◽  
Handling Time ◽  
Stability Conditions ◽  
Alternative Prey ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey System

A predator-prey model in presence of alternative prey is proposed. Existence and local stability conditions for interior equilibrium points are derived. Global stability conditions for interior equilibrium points are also found. Bifurcation analysis is done with respect to predator’s searching rate and handling time. Bifurcation analysis confirms the existence of global stability in presence of alternative prey.

scholarly journals Analisis Kestabilan Model Predator-Prey dengan Infeksi Penyakit pada Prey dan Pemanenan Proporsional pada Predator

2020 ◽  
Vol 1 (1) ◽  
pp. 8-15
Author(s):  
Siti Maisaroh ◽  
Resmawan Resmawan ◽  
Emli Rahmi
Keyword(s):  
Equilibrium Points ◽  
The Other ◽  
Predator Prey ◽  
Population Extinction ◽  
Predator Prey Model ◽  
Runge Kutta Method ◽  
Extinction Point ◽  
Infected Prey ◽  
Disease In Prey ◽  
Model Predator

The dynamics of predator-prey model with infectious disease in prey and harvesting in predator is studied. Prey is divided into two compartments i.e the susceptible prey and the infected prey. This model has five equilibrium points namely the all population extinction point, the infected prey and predator extinction point, the infected prey extinction point, and the co-existence point. We show that all population extinction point is a saddle point and therefore this condition will never be attained, while the other equilibrium points are conditionally stable. In the end, to support analytical results, the numerical simulations are given by using the fourth-order Runge-Kutta method.

scholarly journals Asymmetrical Impact of Allee Effect on a Discrete-Time Predator-Prey System

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Wenting Wang ◽  
Yujuan Jiao ◽  
Xiuping Chen
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Stable State ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Stable Conditions ◽  
Periodic Dynamics ◽  
Asymmetrical Impact

A discrete-time predator-prey model is proposed with Leslie-type numerical response, and the asymmetrical influence of Allee effect on the proposed system is investigated. By mathematical analysis, locally stable conditions for the equilibrium points of the considered systems with or without Allee effect are obtained firstly. Furthermore, numerical simulation is used to verify the results and detect some new outcomes. The results show that Allee effect on predator leads the system to its stable state in much longer time. Conversely, the prey population with Allee effect makes it much faster. In particular, a large value of Allee effect on prey results in periodic dynamics of the system.

scholarly journals Dynamics of Infected Predator-Prey System with Nonlinear Incidence Rate and Prey in Refuge

2020 ◽  
Vol 6 (2) ◽  
pp. 123-134
Author(s):  
Adin Lazuardy Firdiansyah
Keyword(s):  
Incidence Rate ◽  
Numerical Solutions ◽  
Nonlinear Incidence Rate ◽  
Prey Refuge ◽  
Predator Prey ◽  
Nonlinear Incidence ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Future Work

A predator-prey system with nonlinear incidence rate and refuging in prey is proposed to describe behavior change of certain infected diseases on healthy prey when the number of infected prey is getting large, while predator can predate prey by accessing refuging in prey. Therefore, this paper discusses the dynamics behavior predator-prey model with the spread of infected disease that is denoted by nonlinear incidence rate and adding prey refuge. We find the existence of eight non-negative equilibrium in the model, which their local stability has been determined. Furthermore, we also observe the prey refuge properties in the model. We find that prey refuge can prevent extinction in prey populations. In the end, some numerical solutions are carried out to illustrate our analytic results. For future work, we can investigate the harvesting effect in both populations, which is disease control in the predator-prey model with the spread of infected disease.

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 The dynamics of a Leslie type predator–prey model with fear and Allee effect

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. Vinoth ◽  
R. Sivasamy ◽  
K. Sathiyanathan ◽  
Bundit Unyong ◽  
Grienggrai Rajchakit ◽  
...  
Keyword(s):  
Local Stability ◽  
Allee Effect ◽  
Jacobian Matrix ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Lyapunov Coefficient ◽  
Points Of View ◽  
Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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