Noise-induced pattern formation and unpredictability
of competition in a modified Volterra-Lotka model
Andrey A. Polezhaev and Andrey B. Goryachev
I.E.Tamm Theoretical Department, P.N.Lebedev Physical Institute
Leninsky prospect 53, 117924 Moscow, Russia
The problem of biological pattern formation and in particular of spatial patterns formed by living populations for a long time has been a matter of scientific interest. Various mechanisms have been proposed to account for emergence and stability of typical natural ecosystems mosaic patterns when they cannot be attributed just to heterogeneity of environment.
The purpose of the present research is to study by means of mathematical modelling whether competition between species can be responsible for formation of spatial patterns in ecosystems. A model is proposed in which the population specific growth rate depends on the population density, i.e. cooperative behavior resulting from sexual breeding is taken into account, contrary to the classical Volterra-Lotka type model. Due to this modification the degeneracy inherent in the classical model is eliminated and qualitatively novel regimes are observed. Analysis of the model demonstrates that in this case there appear domains in the parametric space in which the model has either two stable stationary states, one of them being "mixed" (corresponding to coexistence of species), or three stable stationary states, when, depending on the initial conditions, either the "mixed" or any of the "pure" states is realized. In order to describe spatial interaction of species, random motion in the form of a diffusion-like term is introduced into the model. It is shown that in the case when the corresponding parameters of competing species do not differ significantly the model can be reduced to a single Ginzburg-Landau type equation.
The model is examined both in absence and in presence of noise mimiking inherent fluctuations of birth and death. The following results are obtained.
1. The deterministic model (without noise) has only uniform stable stationary solutions. While evolving to these spatially uniform states the system forms domains of different phases (corresponding to different species), which in a 2-d medium have a characteristic parquet-like mosaic appearance. Though unstable, these transient structures can be rather long-living formations and should not be disregarded as biologically insignificant.
2. A Poisson white noise applied to the system changes qualitatively its behavior. First, being multiplicative, it shifts deterministic bifurcation maps, tending to reduce hysteresis loops. Second, it can induce transition of the system from homogeneous to spatially inhomogeneous multiphase state. Unlike the deterministic case such a structure is not merely a transient state to spatial homogeneity. The structure is nonstationary: domains mainly populated by one of the species emerge, grow, and then split into smaller ones or ``melt'' back into the mixed phase to make room for growing domains of the concurrent species. Thus noise not only induces the formation of spatial patterns, but also switches on endless turbulent-like rearangement of the system.
3. If an initially unpopulated habitat is occupied by competing species, the process appears to be extremely sensitive to initial conditions. It is shown that if the latter are taken in the form of a spatial white noise with vanishing intensity, the final state of the system becomes completely unpredictable and sensitive to any fluctuations.