Bifurcation Dynamics and Pattern Formation of a Three-Species Food Chain System with Discrete Spatiotemporal Variables

2019 ◽  
Vol 74 (11) ◽  
pp. 945-959
Author(s):  
Huayong Zhang ◽  
Ge Pan ◽  
Tousheng Huang ◽  
Tianxiang Meng ◽  
Jieru Wang ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Food Chain ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Chain System ◽  
Self Organized ◽  
New Perspective ◽  
Centre Manifold Theorem

AbstractThe bifurcation dynamics and pattern formation of a discrete-time three-species food chain system with Beddington–DeAngelis functional response are investigated. Via applying the centre manifold theorem and bifurcation theorems, the occurrence conditions for flip bifurcation and Neimark–Sacker bifurcation as well as Turing instability are determined. Numerical simulations verify the theoretical results and reveal many interesting dynamic behaviours. The flip bifurcation and the Neimark–Sacker bifurcation both induce routes to chaos, on which we find period-doubling cascades, invariant curves, chaotic attractors, sub–Neimark–Sacker bifurcation, sub–flip bifurcation, chaotic interior crisis, sub–period-doubling cascade, periodic windows, sub–periodic windows, and various periodic behaviours. Moreover, the food chain system exhibits various self-organized patterns, including regular and irregular patterns of stripes, labyrinth, and spiral waves, suggesting the populations can coexist in space as many spatiotemporal structures. These analysis and results provide a new perspective into the complex dynamics of discrete food chain systems.

scholarly journals Chaos and Bifurcations of a Leslie-Gower Food Chain with Strong Allee Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Chunshan Xue ◽  
Xia Liu
Keyword(s):  
Food Chain ◽  
Allee Effect ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Biological Phenomenon ◽  
Chain System ◽  
Saddle Node ◽  
Strong Allee Effect

Allee effect, as an important biological phenomenon, has been considered in many ecosystems, whereas, its influence on interactions of three or more species is little investigated. In this paper we modify a three-species Leslie-Gower type food chain system by incorporating the strong Allee effect into the source. Our results show that the existence of Allee effect contributes to the occurrence of more complex dynamics of the system, including Hopf, saddle-node, transcritical, saddle-node-Hopf, period-doubling, and period-halving bifurcations and chaos.

Bifurcation, Chaos and Turing Instability Analysis for a Space-Time Discrete Toxic Phytoplankton-Zooplankton Model with Self-Diffusion

2019 ◽  
Vol 29 (13) ◽  
pp. 1950184
Author(s):  
Shihong Zhong ◽  
Jinliang Wang ◽  
You Li ◽  
Nan Jiang
Keyword(s):  
Numerical Simulations ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Turing Instability ◽  
Space Time ◽  
Discrete Version ◽  
Flip Bifurcation ◽  
Center Manifold Theorem ◽  
Toxic Phytoplankton ◽  
Stable Conditions

The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system.

scholarly journals Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Huayong Zhang ◽  
Shengnan Ma ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Zichun Gao ◽  
...  
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Routes To Chaos ◽  
Prey System ◽  
The One

We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

scholarly journals Complex dynamics of a three species food-chain model with Holling type IV functional response

2011 ◽  
Vol 16 (3) ◽  
pp. 553-374
Author(s):  
Ranjit Kumar Upadhyay ◽  
Sharada Nandan Raw
Keyword(s):  
Food Chain ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chain Model ◽  
Narrow Region ◽  
Type Iv ◽  
Numerical Bifurcation ◽  
Parameter Values ◽  
Food Chain Model

In this paper, dynamical complexities of a three species food chain model with Holling type IV predator response is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence criterion of the food chain model is obtained. Numerical bifurcation analysis reveals the chaotic behavior in a narrow region of the bifurcation parameter space for biologically realistic parameter values of the model system. Transition to chaotic behavior is established via period-doubling bifurcation and some sequences of distinctive period-halving bifurcation leading to limit cycles are observed.

scholarly journals Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Shengnan Ma ◽  
Ge Pan ◽  
Zhaodeng Wang ◽  
...  
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Period 2 ◽  
New Perspective ◽  
Matrix Center

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

scholarly journals Pattern Formation and Bistability in a Generalist Predator-Prey Model

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 20
Author(s):  
Vagner Weide Rodrigues ◽  
Diomar Cristina Mistro ◽  
Luiz Alberto Díaz Rodrigues
Keyword(s):  
Pattern Formation ◽  
Temporal Dynamics ◽  
Complex Dynamics ◽  
Species Conservation ◽  
Turing Instability ◽  
Food Sources ◽  
Generalist Predators ◽  
Generalist Predator ◽  
Predator Prey ◽  
Predator Prey Model

Generalist predators have several food sources and do not depend on one prey species to survive. There has been considerable attention paid by modellers to generalist predator-prey interactions in recent years. Erbach and collaborators in 2013 found a complex dynamics with bistability, limit-cycles and bifurcations in a generalist predator-prey system. In this paper we explore the spatio-temporal dynamics of a reaction-diffusion PDE model for the generalist predator-prey dynamics analyzed by Erbach and colleagues. In particular, we study the Turing and Turing-Hopf pattern formation with special attention to the regime of bistability exhibited by the local model. We derive the conditions for Turing instability and find the region of parameters for which Turing and/or Turing-Hopf instability are possible. By means of numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. In the Turing-Hopf range of the parameters, we observed either stable patterns or homogeneous periodic distributions. Our findings reveal that movement can break the effect of hysteresis observed in the local dynamics, what can have important implication in pest management and species conservation.

scholarly journals Predator–prey pattern formation driven by population diffusion based on Moore neighborhood structure

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Zhengran Hu ◽  
Ge Pan ◽  
Shengnan Ma ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Turing Instability ◽  
Global Dynamics ◽  
Neighborhood Structure ◽  
Von Neumann ◽  
Predator Prey ◽  
System A ◽  
Self Organized ◽  
Moore Neighborhood ◽  
The Way

Abstract Diffusion-driven instability is a basic nonlinear mechanism for pattern formation. Therefore, the way of population diffusion may play a determinative role in the spatiotemporal dynamics of biological systems. In this research, we launch an investigation on the pattern formation of a discrete predator–prey system where the population diffusion is based on the Moore neighborhood structure instead of the von Neumann neighborhood structure widely applied previously. Under pattern formation conditions which are determined by Turing instability analysis, numerical simulations are performed to reveal the spatiotemporal complexity of the system. A pure Turing instability can induce the self-organization of many basic types of patterns as described in the literature, as well as new spiral-spot and labyrinth patterns which show the temporally oscillating and chaotic property. Neimark–Sacker–Turing and flip–Turing instability can lead to the formation of circle, spiral and much more complex patterns, which are self-organized via spatial symmetry breaking on the states that are homogeneous in space and non-periodic in time. Especially, the emergence of spiral pattern suggests that spatial order can generate from temporal disorder, implying that even when the predator–prey dynamics in one site is chaotic, the spatially global dynamics may still be predictable. The results obtained in this research suggest that when the way of population diffusion changes, the pattern formation in the predator–prey systems demonstrates great differences. This may provide realistic significance to explain more general predator–prey coexistence.

scholarly journals Sicily Before the Greeks. The Interaction with Aegean and the Levant in the Pre-colonial Era

2020 ◽  
Vol 6 (1) ◽  
pp. 172-205
Author(s):  
Davide Tanasi
Keyword(s):  
Material Culture ◽  
Complex Dynamics ◽  
Eastern Mediterranean ◽  
Colonial Era ◽  
Local Material ◽  
Gradual Process ◽  
Multiple Variables ◽  
New Perspective ◽  
First Time ◽  
The Relationship

AbstractThe relationship between Sicily and the eastern Mediterranean – namely Aegean, Cyprus and the Levant – represents one of the most intriguing facets of the prehistory of the island. The frequent and periodical contact with foreign cultures were a trigger for a gradual process of socio-political evolution of the indigenous community. Such relationship, already in inception during the Neolithic and the Copper Age, grew into a cultural phenomenon ruled by complex dynamics and multiple variables that ranged from the Mid-3rd to the end of the 2nd millennium BCE. In over 1,500 years, a very large quantity of Aegean and Levantine type materials have been identified in Sicily alongside with example of unusual local material culture traditionally interpreted as resulting from external influence. To summarize all the evidence during such long period and critically address it in order to attempt historical reconstructions is a Herculean labor.Twenty years after Sebastiano Tusa embraced this challenge for the first time, this paper takes stock on two decades of new discoveries and research reassessing a vast amount of literature, mostly published in Italian and in regional journals, while also address the outcomes of new archaeometric studies. The in-depth survey offers a new perspective of general trends in this East-West relationship which conditioned the subsequent events of the Greek and Phoenician colonization of Sicily.

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