BIFURCATION ANALYSIS OF A TWO-DIMENSIONAL PREDATOR–PREY MODEL WITH HOLLING TYPE IV FUNCTIONAL RESPONSE AND NONLINEAR PREDATOR HARVESTING

2020 ◽  
Vol 28 (04) ◽  
pp. 839-864
Author(s):  
UTTAM GHOSH ◽  
PRAHLAD MAJUMDAR ◽  
JAYANTA KUMAR GHOSH
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Conversion Efficiency ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
Trivial Equilibrium

The aim of this paper is to investigate the dynamical behavior of a two-species predator–prey model with Holling type IV functional response and nonlinear predator harvesting. The positivity and boundedness of the solutions of the model have been established. The considered system contains three kinds of equilibrium points. Those are the trivial equilibrium point, axial equilibrium point and the interior equilibrium points. The trivial equilibrium point is always saddle and stability of the axial equilibrium point depends on critical value of the conversion efficiency. The interior equilibrium point changes its stability through various parametric conditions. The considered system experiences different types of bifurcations such as Saddle-node bifurcation, Hopf bifurcation, Transcritical bifurcation and Bogdanov–Taken bifurcation. It is clear from the numerical analysis that the predator harvesting rate and the conversion efficiency play an important role in stability of the system.

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 A type IV functional response with different shapes in a predator–prey model

2020 ◽  
Vol 505 ◽  
pp. 110419 ◽  
Author(s):  
Merlin C. Köhnke ◽  
Ivo Siekmann ◽  
Hiromi Seno ◽  
Horst Malchow
Keyword(s):  
Functional Response ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
Different Shapes

Stability and Bifurcation Analysis in a Nonlinear Harvested Predator–Prey Model with Simplified Holling Type IV Functional Response

2020 ◽  
Vol 30 (14) ◽  
pp. 2050205
Author(s):  
Zuchong Shang ◽  
Yuanhua Qiao ◽  
Lijuan Duan ◽  
Jun Miao
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Bifurcation Analysis ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
The Stability ◽  
Parameter Values

In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

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.

Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response

2014 ◽  
Vol 07 (05) ◽  
pp. 1450047 ◽  
Author(s):  
Lakshmi Narayan Guin ◽  
Prashanta Kumar Mandal
Keyword(s):  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Turing Instability ◽  
Spatial Pattern Analysis ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper, spatial patterns of a diffusive predator–prey model with sigmoid (Holling type III) ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The (local and global) asymptotic stability behavior of the corresponding non-spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pattern analysis through numerical simulations are depicted and analyzed. Furthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.

scholarly journals Dynamics Analysis of Modified Leslie-Gower Model with Simplified Holling Type IV Functional Response

2021 ◽  
Vol 18 (1) ◽  
pp. 12-21
Author(s):  
Nur Suci Ramadhani ◽  
Toaha Toaha ◽  
Kasbawati Kasbawati
Keyword(s):  
Population Density ◽  
Equilibrium Point ◽  
Functional Response ◽  
Prey Population ◽  
Dynamics Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Simulation Results ◽  
The Stability

In this paper, the modified Leslie-Gower predator-prey model with simplified Holling type IV functional response is discussed. It is assumed that the prey population is a dangerous population. The equilibrium point of the model and the stability of the coexistence equilibrium point are analyzed. The simulation results show that both prey and predator populations will not become extinct as time increases. When the prey population density increases, there is a decrease in the predatory population density because the dangerous prey population has a better ability to defend itself from predators when the number is large enough.

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