Last modified: 2014-03-28

#### Abstract

The habitats of the real biological populations and community generally are fragmented. They can be presented as the system of local populations coupled by migration. Such system of subpopulations is called metapopulation. Here the population dynamics is more complicated than a single local population or biological community. For study of a spatiotemporal dynamics of metapopulation it can be used coupled map lattices. Multistability, synchronization, intermittency and clustering are observed in these models. Furthermore, process of clustering or forming synchronous element groups looks like a core-satellite distributions in metapopulation.

The paper investigates the phenomenon of clustering in such model. A local population dynamic is described by Riker’s model. The coupling between subpopulations apparently cannot be global and represents a seasonal movement or migration of individuals into neighboring patches.

To construct attraction basins for some phases of clustering we propose a method based on estimation of statistic distances between initial state and the states on stable attraction. It is used the coefficient of determination as a distance measure. This method too enables one to identify parameter regions of stable clustering phase. This technique has been successfully tested on an example of formation two-cluster states.

As result it is shown the forming of clusters is complex depends on initial distribution of individuals on flat area. In the case of forming two equal clusters, comparabled with ordered phase, the attraction basins coincide only partially with basins of antiphase modes in system of two coupled Riker’s map. The bifurcation scenario is described for this phase. It is shown the first bifurcation is always the period-doubling. It is arising of synchronous 2-cycle with only coherent phase. With increasing reproductive rate there is antiphase 2-cycle coexistsing with synchronous cycles of different lengths in the system. The existing of nonsynchronous modes makes possible emergence of stable order, partially order and turbulent phases. However, the bifurcation boundaries of coherent and not coherent phases are different.

With growth of reproductive rate the coherent phase basin is transformed from simply connected and two-dimensional domains to univariate regions. The univariate region is the part of the first quadrant bisector of the phase space. Here it is formed the dynamic mode closed to chaotic synchronization. This happens when the order phase gets the multistable nature and 2-cycle coexists with 4-cycle. The basin of 4-cycle has a heavily trimmed or an even fractal view. In area of Arnold’s tongues the coherent phase basins again takes the form of the two-dimensional areas. A decreasing the rate of individual movement leads to the emergence of two closed limiting invariant curves around of 2-cycle elements. Both clusters have quasiperiodic antiphase dynamics. The destruction of invariant curves produces 4-cycle.

In the case of form not equal clusters, the complexity of dynamic is realized by only Feigenbaum’s scenario. Also for such clusters we shown the basins of attraction are qualitatively similar to basins nonsynchronous modes of nonidentical asymmetrically coupled Riker’s map in asymptotic behavior.

*The study was supported by RFBR, research project No. 14-01-31443 mol_a.*