Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response

Liu, Xia; Liu, Yanwei; Li, Qiaoping

doi:10.1155/2015/245421

Cited by 4 publications

(6 citation statements)

References 41 publications

(60 reference statements)

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“…, the model (17) becomes Then the map (20) can undergo the Hopf bifurcation when the following discriminatory quantity is not zero: From the preceding analysis and theorem in [20] we have the following result.…”

Section: Flip Bifurcation and Hopf Bifurcationmentioning

confidence: 98%

“…However, for a mathematical biology model, if the size of population is rarely small, or the population has no overlapping generation, or people study population changes within certain intervals of time, the discrete-time model would indeed be more suitable and realistic than the continuous-time model [4][5][6][7][8][9][10][11][12]. On the other hand, numerical solutions or approximate solutions of discrete-time models can be obtained more easily, and much work has shown that discrete-time prey-predator models can produce a much richer set of patterns than those observed in continuous-time models [13][14][15][16][17][18]. Thus, it is necessary to consider the corresponding discrete model of system (2).…”

Section: Introductionmentioning

confidence: 99%

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Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response

Zhao

Yan

2018

Adv Differ Equ

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In this paper, we study a discrete predator-prey system with modified Holling-Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.

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Section: Flip Bifurcation and Hopf Bifurcationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response

Zhao

Yan

2018

Adv Differ Equ

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“…• Holling III interaction functional a(x(t)) 2 y(t) 1+at h (x(t)) 2 [30]. • Generalized Holling III interaction functional a(x(t)) 2 y(t) 1+bx(t)+c(x(t)) 2 [32]. • Beddington-DeAngelis interaction functional ax(t)y (t) 1+bx(t)+cy(t) [28].…”

Section: Introductionmentioning

confidence: 99%

Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

Djilali

Ghanbari

2021

Adv Differ Equ

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The behavior of any complex dynamic system is a natural result of the interaction between the components of that system. Important examples of these systems are biological models that describe the characteristics of complex interactions between certain organisms in a biological environment. The study of these systems requires the use of precise and advanced computational methods in mathematics. In this paper, we discuss a prey–predator interaction model that includes two competitive predators and one prey with a generalized interaction functional. The primary presumption in the model construction is the competition between two predators on the only prey, which gives a strong implication of the real-world situation. We successfully establish the existence and stability of the equilibria. Further, we investigate the impact of the memory measured by fractional time derivative on the temporal behavior. We test the obtained mathematical results numerically by a proper numerical scheme built using the Caputo fractional-derivative operator and the trapezoidal product-integration rule.

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“…In recent years, discrete-time population systems also come to the fore due to the following reasons [9][10][11][12][13][14][15][16]: Firstly, discrete-time systems are more suitable than continuoustime systems to describe populations with nonoverlapping generations. Secondly, they can produce more complex and rich dynamical behaviors than continuous-time systems.…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Baek

2018

Mathematical Problems in Engineering

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The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R 2 + are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, the maximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors.

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Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response

Cited by 4 publications

References 41 publications

Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response

Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response

Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

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