Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics.

@article{Petrovskii2001WaveOC,
  title={Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics.},
  author={Sergei V. Petrovskii and Horst Malchow},
  journal={Theoretical population biology},
  year={2001},
  volume={59 2},
  pages={
          157-74
        }
}
The dynamics of a simple prey-predator system is described by a system of two reaction- diffusion equations with biologically reasonable non-linearities (logistic growth of the prey, Holling type II functional response of the predator). We show that, when the local kinetics of the system is oscillatory, for a wide class of initial conditions the evolution of the system leads to the formation of a non-stationary irregular pattern corresponding to spatio-temporal chaos. The chaotic pattern first… 
View on PubMed
math.le.ac.uk
209 Citations
Highly Influential Citations
5
Background Citations
68
Methods Citations
12
Results Citations
3

Figures from this paper

209 Citations

Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species

The spatio-temporal dynamics of three competitive species is considered. Mathematically, the community is described by a system of partial differential equations of Lotka-Volterra type. The…

Pattern Formation, Long-Term Transients, and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

This paper considers a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice, and obtains that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory.

Pattern Formation in a Semi-Ratio-Dependent Predator-Prey System with Diffusion

We investigate spatiotemporal dynamics of a semi-ratio-dependent predator-prey system with reaction-diffusion and zero-flux boundary. We obtain the conditions for Hopf, Turing, and wave bifurcations…

Modelling the spatiotemporal dynamics of diffusive prey-predator interactions: Pattern formation and ecological implications

In this paper, we investigate the formation of spatiotemporal patterns in ecological systems via numerical simulations. We study the diffusive prey-predator model and observe the rich dynamical…

Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study

A three-species predator–prey model is proposed and investigated to inspect the mutual interference between predators and a variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response.

Spatiotemporal complexity of a discrete space-time predator−prey system with self- and cross-diffusion

Wave of chaos in a spatial eco-epidemiological system: Generating realistic patterns of patchiness in rabbit-lynx dynamics.

Wave phenomena and edge of chaos in a diffusive predator-prey system under Allee effect

We investigate a spatial predator-prey system; predator under Allee effect. The predator-prey community dynamics is assumed to be of Holling-Tanner type. When community’s local dynamics displays…

Spatiotemporal complexity of a ratio-dependent predator-prey system.

The study shows that the spatially extended model has not only more complex dynamic patterns in the space, but also chaos and spiral waves, which may help to better understand the dynamics of an aquatic community in a real marine environment.

Transition to spatiotemporal chaos can resolve the paradox of enrichment

...

References

SHOWING 1-10 OF 91 REFERENCES

A minimal model of pattern formation in a prey-predator system

Chaotic wakes and other wave-induced behavior in a system of reaction–diffusion equations

The phenomenon of spatiotemporal chaos as it arises in systems of reaction–diffusion equations has been the subject of much study over recent years. Initially, chaotic responses in these systems were…

Wave-induced chemical chaos.

Spatiotemporal chaos in a two-variable, cubic autocatalator model with equal diffusivities of the species is described and Extremesensitivity to initial conditions in both space and time and a rapid falloff of the spatial correlation function are exhibited in the chaotic regime.

Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system

Motional instabilities in prey-predator systems.

  • H. Malchow
  • Environmental Science
    Journal of theoretical biology
  • 2000
Different scenarios of this general mechanism of spatio-temporal pattern formation in reaction-diffusion-advection systems will be demonstrated, using a simple two-species predator-prey system as an example model.

Non-equilibrium spatio-temporal patterns in models of non-linear plankton dynamics

The main aim of this paper is to give an extensive introduction to the subject, which focusses on minimal models of the temporally and spatially continuous dynamics of freshwater and marine plankton as well as other microorganisms, best described by non-linear reaction-diffusion and advection equations.

Traveling periodic waves in heterogeneous environments

Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits

We demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator–prey system. The analysis is in the three-dimensional phase space of the nonlinear…

Transition fronts and localized structures in bistable reaction-diffusion equations

Chemical instability induced by a differential flow.

A new kind of instability is predicted for a system involving activator and inhibitor kinetics in a reactive flow that is free from the restrictions of the Turing instability on the diffusion coefficients and can thus be expected to occur in a wide variety of natural and artificial systems.
...