# 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…

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## References

SHOWING 1-10 OF 91 REFERENCES

### Diffusion-induced chaos in a spatial predator–prey system

- Environmental ScienceProceedings of the Royal Society of London. Series B: Biological Sciences
- 1993

A continuous predator-prey model in which both species diffuse along a spatial gradient is shown to exhibit temporal chaos at a fixed point in space. The model incorporates a nonlinear functional…

### Wave-induced chemical chaos.

- PhysicsPhysical review letters
- 1996

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

- Environmental Science
- 1996

### Motional instabilities in prey-predator systems.

- Environmental ScienceJournal 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

- Environmental Science
- 2000

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

- Mathematics
- 1986

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…

### A model of patchiness for prey—predator plankton populations

- Environmental Science, Mathematics
- 1975