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

Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki

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.

2017 ◽  
Vol 10 (08) ◽  
pp. 1750119 ◽  
Author(s):  
Wensheng Yang

The dynamical behaviors of a diffusive predator–prey model with Beddington–DeAngelis functional response and disease in the prey is considered in this work. By applying the comparison principle, linearized method, Lyapunov function and iterative method, we are able to achieve sufficient conditions of the permanence, the local stability and global stability of the boundary equilibria and the positive equilibrium, respectively. Our result complements and supplements some known ones.


2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Guohong Zhang ◽  
Xiaoli Wang

We study a general Gause-type predator-prey model with monotonic functional response under Dirichlet boundary condition. Necessary and sufficient conditions for the existence and nonexistence of positive solutions for this system are obtained by means of the fixed point index theory. In addition, the local and global bifurcations from a semitrivial state are also investigated on the basis of bifurcation theory. The results indicate diffusion, and functional response does help to create stationary pattern.


2020 ◽  
Vol 30 (06) ◽  
pp. 2050082
Author(s):  
Zhihui Ma

A delay-induced nonautonomous predator–prey system with variable habitat complexity is proposed based on mathematical and ecological issues, and this system is more realistic than the published models. Firstly, the permanence of the nonautonomous predation system is studied and some sufficient conditions are obtained. Secondly, the dynamical behaviors of the corresponding autonomous predation system are investigated, including the positivity and boundedness, and local and global stabilities. Thirdly, the properties of Hopf bifurcation of the autonomous predation system without/with delay are investigated and the explicit formulas which determine the stability and the direction of periodic solutions are obtained. Finally, a numerical example is given to test our theoretical results.


2020 ◽  
Vol 30 (14) ◽  
pp. 2050205
Author(s):  
Zuchong Shang ◽  
Yuanhua Qiao ◽  
Lijuan Duan ◽  
Jun Miao

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.


2019 ◽  
Vol 29 (10) ◽  
pp. 1950136 ◽  
Author(s):  
Jun Zhou

This paper deals with a diffusive predator–prey model with Bazykin functional response. The parameter regions for the stability and instability of the unique constant steady state are derived. The Turing (diffusion-driven) instability which induces spatial inhomogeneous patterns, the existence of time-periodic orbits which produce temporal inhomogeneous patterns, the existence and nonexistence of nonconstant steady state positive solutions are proved. Numerical simulations are presented to verify and illustrate the theoretical results.


2021 ◽  
Vol 18 (1) ◽  
pp. 12-21
Author(s):  
Nur Suci Ramadhani ◽  
Toaha Toaha ◽  
Kasbawati Kasbawati

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.


2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
Jiangang Zhang ◽  
Juan Nan ◽  
Wenju Du ◽  
Yandong Chu ◽  
Hongwei Luo

We introduce a discretization process to discretize a modified fractional-order optically injected semiconductor lasers model and investigate its dynamical behaviors. More precisely, a sufficient condition for the existence and uniqueness of the solution is obtained, and the necessary and sufficient conditions of stability of the discrete system are investigated. The results show that the system’s fractional parameter has an effect on the stability of the discrete system, and the system has rich dynamic characteristics such as Hopf bifurcation, attractor crisis, and chaotic attractors.


2014 ◽  
Vol 24 (07) ◽  
pp. 1450093 ◽  
Author(s):  
Yongli Song ◽  
Yahong Peng ◽  
Xingfu Zou

In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.


Author(s):  
Yuqing Liu ◽  
Xianyi Li

In this paper, we use a semidiscretization method to derive a discrete predator–prey model with Holling type II, whose continuous version is stated in [F. Wu and Y. J. Jiao, Stability and Hopf bifurcation of a predator-prey model, Bound. Value Probl. 129(2019) 1–11]. First, the existence and local stability of fixed points of the system are investigated by employing a key lemma. Then we obtain the sufficient conditions for the occurrence of the transcritical bifurcation and Neimark–Sacker bifurcation and the stability of the closed orbits bifurcated by using the Center Manifold theorem and bifurcation theory. Finally, we present numerical simulations to verify corresponding theoretical results and reveal some new dynamics.


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