Bifurcation and chaos in a discrete-time predator–prey system of Holling and Leslie type

Hu, Dongpo; Cao, Hongjun

doi:10.1016/j.cnsns.2014.09.010

Cited by 63 publications

(32 citation statements)

References 13 publications

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“…A similar model with delay is studied in [31,32], and a three dimensional similar model with the same functional responses is studied in [33][34][35]. In [36], several Turing and Hopf bifurcation patterns were obtained by the continuous model (3). We can see that this model is well recognized.…”

Section: Introductionmentioning

confidence: 99%

“…The dynamic behaviors of the predator-prey system have captured the interest of both biologists and ecologists [1][2][3][4][5]. There are a substantial number of predator-prey system models.…”

Section: Introductionmentioning

confidence: 99%

“…The local dynamics of this continuous model (3) has been studied in [28,29], and the global stability of a similar model has been investigated in [30]. A similar model with delay is studied in [31,32], and a three dimensional similar model with the same functional responses is studied in [33][34][35].…”

Section: Introductionmentioning

confidence: 99%

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Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response

Zhang

et al. 2017

Entropy

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Abstract:The formation of self-organized patterns in predator-prey models has been a very hot topic recently. The dynamics of these models, bifurcations and pattern formations are so complex that studies are urgently needed. In this research, we transformed a continuous predator-prey model with Lesie-Gower functional response into a discrete model. Fixed points and stability analyses were studied. Around the stable fixed point, bifurcation analyses including: flip, Neimark-Sacker and Turing bifurcation were done and bifurcation conditions were obtained. Based on these bifurcation conditions, parameters values were selected to carry out numerical simulations on pattern formation. The simulation results showed that Neimark-Sacker bifurcation induced spots, spirals and transitional patterns from spots to spirals. Turing bifurcation induced labyrinth patterns and spirals coupled with mosaic patterns, while flip bifurcation induced many irregular complex patterns. Compared with former studies on continuous predator-prey model with Lesie-Gower functional response, our research on the discrete model demonstrated more complex dynamics and varieties of self-organized patterns.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response

Zhang

et al. 2017

Entropy

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“…In recent times, a few number of articles in literature discussed the dynamics of discrete-time predator-prey systems of Leslie type [11,12]. For example, a discretetime predator-prey system of Holling and Leslie type with constant-yield prey harvesting was investigated in [11], and a discrete predator-prey model with Holling-Tanner functional response was studied in [12].…”

Section: Introductionmentioning

confidence: 99%

“…For example, a discretetime predator-prey system of Holling and Leslie type with constant-yield prey harvesting was investigated in [11], and a discrete predator-prey model with Holling-Tanner functional response was studied in [12]. In their studies, the authors paid their attention to drive the existence of flip and NeimarkSacker bifurcations by using center manifold theory.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

Rana

Kulsum

2017

Discrete Dynamics in Nature and Society

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The dynamic behavior of a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response is examined. We algebraically show that the system undergoes a bifurcation (flip or Neimark-Sacker) in the interior of R 2 + . Numerical simulations are presented not only to validate analytical results but also to show chaotic behaviors which include bifurcations, phase portraits, period 2, 4, 6, 8, 10, and 20 orbits, invariant closed cycle, and attracting chaotic sets. Furthermore, we compute numerically maximum Lyapunov exponents and fractal dimension to justify the chaotic behaviors of the system. Finally, a strategy of feedback control is applied to stabilize chaos existing in the system.

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Torus doubling route to chaos and chaos eradication in delayed discrete‐time predator–prey models

Ghosh

Sarda

Sahu

2022

Math Methods in App Sciences

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This paper analyses a class of discrete‐time delayed predator–prey models where time delay is incorporated into the prey's density‐dependent growth term. We explore how time delay, carrying capacity, and harvesting can generate or suppress chaotic motion. Analytical conditions for local stability of the fixed point are derived by using the Routh–Hurwitz criterion. Increasing the time delay has the great potential to change the coexisting stable fixed point to unstable via a Neimark–Sacker bifurcation. Thus, the systems may experience quasi‐periodic and chaotic motion when the coexisting fixed point is unstable. The existence of a quasi‐periodic loop, which generates a 2‐quasi‐periodic loop, and then 4, 8, and 16 quasi‐periodic loops are successively detected as the carrying capacity increases. Further increase of carrying capacity induces chaos. Thus, a torus doubling route to chaos is explored in our article, which has been relatively less reported in the existing literature. Nevertheless, we show that harvesting either prey or predator is more likely to suppress chaos and stabilize the coexisting steady state. We draw all the bifurcation diagrams when delay, carrying capacity, and harvesting strength are varied. The qualitative nature of the phase portraits is determined by computing all the Lyapunov exponents. The analysis of these delayed models in a discrete‐time framework might attract a wide range of researchers to work in this direction. In the conclusion section, we also disclose some remaining questions and future perspectives.

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Bifurcation and chaos in a discrete-time predator–prey system of Holling and Leslie type

Cited by 63 publications

References 13 publications

Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response

Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response

Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

Torus doubling route to chaos and chaos eradication in delayed discrete‐time predator–prey models

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