The Stability of Solutions for a Fractional Predator-Prey System

Guo, Yingjia

doi:10.1155/2014/124145

Cited by 9 publications

(9 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For comparison, Fig. 2(b) shows the dynamic behavior for α = 0.85 with (x 0 , y 0 ) = (40,35), (40,37), (38,36), (42,36). The simulations presented in the two figures above indicate that the stability of the equilibrium points does not depend on the order α ∈ (0, 1).…”

Section: The Effect Of Different Parameter Valuesmentioning

confidence: 95%

“…1. The dynamic behaviors of interacting prey (x) and predator (y) for the order α = 0.7 with the initial values (x 0 , y 0 ) = (50,40), (50,55), (75,55), (80,45) are shown in Fig. 1(a).…”

Section: The Effect Of Different Parameter Valuesmentioning

confidence: 99%

“…In further simulations, we use the same set of parameters as in Fig. 4, but we change the initial values to (x 0 , y 0 ) = (28,40) and (100, 120) and the derivative orders to α = 0.9 and 0.99. The simulations reveal that there exists an attracting limit cycle of model (3) as shown in Fig.…”

Section: The Effect Of Varying the Order αmentioning

confidence: 99%

“…The predator-prey model has been applied to describe the relationship between two species in biological systems in which one a predator feeds on the other its prey [39]. Fractional order predator-prey models have been used by many researchers, who also discussed the stability and numerical solutions of the models [32,33,40,41]. Moreover, some studies showed the conditions for existence of a Hopf bifurcation [42][43][44] and the appearance of a limit cycle [45].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Baisad

Moonchai

2018

Adv Differ Equ

View full text Add to dashboard Cite

The Kolmogorov model has been applied to many biological and environmental problems. We are particularly interested in one of its variants, that is, a Gauss-type predator-prey model that includes the Allee effect and Holling type-III functional response. Instead of using classic first order differential equations to formulate the model, fractional order differential equations are utilized. The existence and uniqueness of a nonnegative solution as well as the conditions for the existence of equilibrium points are provided. We then investigate the local stability of the three types of equilibrium points by using the linearization method. The conditions for the existence of a Hopf bifurcation at the positive equilibrium are also presented. To further affirm the theoretical results, numerical simulations for the coexistence equilibrium point are carried out.

show abstract

Section: The Effect Of Different Parameter Valuesmentioning

confidence: 95%

“…1. The dynamic behaviors of interacting prey (x) and predator (y) for the order α = 0.7 with the initial values (x 0 , y 0 ) = (50,40), (50,55), (75,55), (80,45) are shown in Fig. 1(a).…”

Section: The Effect Of Different Parameter Valuesmentioning

confidence: 99%

Section: The Effect Of Varying the Order αmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Baisad

Moonchai

2018

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…The interaction between the predator and prey is one of basic relationships among biological species, which becomes one of the hot issues in ecology and biomathematics. The predator-prey model is widely used in renewable resources management [1][2][3][4][5], marine resource conservation [6][7][8], biological control [9][10][11][12], the research about animal infectious diseases [13][14][15], and so on. Freedman and Wolkowicz [16] firstly put forward a where φ(x) denotes the predator response function, which reflects the capture ability of the predator to prey.…”

Section: Introductionmentioning

confidence: 99%

Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

Liu

Huang

2019

J Inequal Appl

View full text Add to dashboard Cite

A new way to study the harvested predator-prey system is presented by analyzing the dynamics of two-prey and one-predator model, in which two teams of prey are interacting with one team of predators and the harvesting functions for two prey species takes different forms. Firstly, we make a brief analysis of the dynamics of the two subsystems which include one predator and one prey, respectively. The positivity and boundedness of the solutions are verified. The existence and stability of seven equilibrium points of the three-species model are further studied. Specifically, the global stability analysis of the coexistence equilibrium point is investigated. The problem of maximum sustainable yield and dynamic optimal yield in finite time is studied. Numerical simulations are performed using MATLAB from four aspects: the role of the carrying capacity of prey, the simulation about the model equations around three biologically significant steady states, simulation for the yield problem of model system, and the comparison between the two forms of harvesting functions. We obtain that the new form of harvesting function is more realistic than the traditional form in the given model, which may be a better reflection of the role of human-made disturbance in the development of the biological system. MSC: 34D20; 49M30; 92B05

show abstract

Unraveling the combined actions of a Holling type III predator–prey model incorporating Allee response and memory effects

Ali

Raut²,

Sarkar

et al. 2020

Comp and Math Methods

View full text Add to dashboard Cite

In this article, we have studied a memory dependent prey–predator model with logistic type prey growth subject to the Allee effect and Holling type‐III functional response. To include the memory effect, we have used here a fractional order system. The model contains at most four equilibrium points among which one is trivial, two are axial, and the last one is interior. Here we have studied behavior of these four equilibrium points using eigen analysis method. Our investigation shows that the trivial equilibrium point is stable for strong Allee effect, and unstable for weak Allee effect. Between the two axial equilibrium points, one is stable under certain condition but the other is always unstable. This unstable equilibrium point exists only for strong Allee case. The interior equilibrium point arises when the axial equilibrium point is unstable. Stability condition of this interior equilibrium point depends on the parametric relationship. The schematic diagram divides the a‐p plane into five sub‐regions in both the cases of memory dependent and memory less system. The number and nature of the equilibrium points in different sub‐regions are different. Finally, we have drawn the bifurcation diagrams considering all the equilibrium points in different sub‐regions and the results are concluded at the end.

show abstract

The Stability of Solutions for a Fractional Predator-Prey System

Cited by 9 publications

References 13 publications

Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

Unraveling the combined actions of a Holling type III predator–prey model incorporating Allee response and memory effects

Contact Info

Product

Resources

About