Tousheng Huang

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

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.

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

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

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.

Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect

A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

Diverse self-organized patterns and complex pattern transitions in a discrete ratio-dependent predator–prey system

Abstract The spatiotemporal complexity of a discrete ratio-dependent predator–prey system is investigated via development of a coupled map lattice model. Through stability analysis and bifurcation analysis, the critical conditions for stable homogeneous stationary and oscillatory states are determined. Meanwhile, pattern formation conditions are derived by Turing instability analysis. Based on the theoretical results, numerical simulations are performed, exhibiting rich patterns of spatiotemporal dynamics of the discrete system. On the route to chaos induced by Neimark–Sacker bifurcation, dynamic variation occurs from invariant cycles, experiencing periodic window and period-doubling process, to chaotic attractors. A variety of patterns are self-organized and demonstrate diverse types in configuration, including cold spots, labyrinth, cold stripes-spots, spirals, hot stripes, circles, arcs, disk, mosaics and fractals. Complex pattern transitions occur among the diverse patterns, suggesting sensitivity of pattern formation to parameter variations. Moreover, spatiotemporal chaos is found in pattern formation process, where tiny variations in initial conditions can result to the self-organization of different patterns. This approach reveals great diversity and complexity of pattern self-organization and pattern transition in predator–prey interactions, promoting comprehending on the spatiotemporal complexity of spatially extended predator–prey system.

Coupled Effects of Turing and Neimark-Sacker Bifurcations on Vegetation Pattern Self-Organization in a Discrete Vegetation-Sand Model

Wind-induced vegetation patterns were proposed a long time ago but only recently a dynamic vegetation-sand relationship has been established. In this research, we transformed the continuous vegetation-sand model into a discrete model. Fixed points and stability analyses were then studied. Bifurcation analyses are done around the fixed point, including Neimark-Sacker and Turing bifurcation. Then we simulated the parameter space for both bifurcations. Based on the bifurcation conditions, simulations are carried out around the bifurcation point. Simulation results showed that Neimark-Sacker bifurcation and Turing bifurcation can induce the self-organization of complex vegetation patterns, among which labyrinth and striped patterns are the key results that can be presented by the continuous model. Under the coupled effects of the two bifurcations, simulation results show that vegetation patterns can also be self-organized, but vegetation type changed. The type of the patterns can be Turing type, Neimark-Sacker type, or some other special type. The difference may depend on the relative intensity of each bifurcation. The calculation of entropy may help understand the variance of pattern types.

A Study of Dynamical Model on the Competitive Relationship Between Soil Erosion and Vegetation Growth in Humid Regions

In this study, a new dynamical model is established based on Thornes’ model. Then a detailed competitive and interactive relationship between soil erosion process and vegetation growth process is detected in humid regions. By employing the nonlinear dynamical analyses, a globally asymptotically stable equilibrium point is obtained under given parameters. This stable state indicates the vegetation growth process can coexist with soil erosion process. Furthermore, the transitions among different equilibrium states caused by the variation describe the competitive process in diverse environment. The results show that the soil erosion has effect on the formations of vegetation patterns in humid regions.Copyright

Fragility of Chaos in Multispecies Competition Influenced by Predation

In order to study the stability of chaotic behaviors, a nonlinear dynamical model of the competing multispecies with a predator is investigated. A series of numerical simulations is demonstrated via wave diagram and phase diagram. The results show that the chaos can change into either oscillation or ordinary equilibrium as the attacking rate of the predator increases. Moreover, chaos in the system becomes fragile and even vanishes when the attacking rate reaches 0.0186. This study also exhibits the transformation in phase diagram from a strange attractor to a stable equilibrium.Copyright

A Study on Nonlinear Population Dynamics Based on Energy Flow

In considering energy flow between organism and environment, population dynamics for single species is studied in the present research. When the total available energy of environment is infinite, a model obeying the Malthusian law is obtained. Under the condition of limited total available energy, a model approximate Logistic Model is also established. These two models give a better comprehension on the ecological effects of growth rate and carry capacity of environment.© 2010 ASME