Hierarchical spatial scale

The hierarchical patch dynamic paradigm (HPDP) meets the above requirements (Wu and Loucks 1995, Wu and David 2002). It is a model for describing at a fundamental level the interactions and dynamics of ecological systems at landscape and regional scales. The HPDP inherently incorporates and predicts a variety of temporal and spatial scales, heterogeneity, and a wide range of dynamics. The basic tenets of HPDP are listed in Table 2.1. This framework is an alternative to models of ecological systems that incorporate a balance of nature, inherent stability, or multiple equilibriums. 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

Figu re 5.9 (a) Brightfield image of a HeLa cell, attached to a Cap2 window, in buffer. Scale bar= 10 im (b) Raman spectral image, obtained via hierarchical cluster analysis (HCA) from a hyperspectral data cube collected atSOOnm spatial resolution (488 nm... 

They observed cylindrical microdomains upon microphase separation of the BCR However, addition of the homopolymer modified both the diameter and spacing of the moieties whereas the spatial order and orientation remains. Selective removal of the PMMA either from the BCR or both from BCR and homopolymer leads to pores with two distinct pore sizes. Other BCP/homopolymer blends have also been reported to produce hierarchically structured surfaces with two-fold nanometer and micrometer length scales in porous polymer films [42],... 

Shein, E. V., Milanovsky, E. Yu. Spatial heterogeneity of properties on different hierarchic levels as a basis of soils structure and functions. Scale effects in the study of soils. Moscow Publishing house of Moscow state University, 2001, P. 47-61. (in Russian). 

The alternative to the hierarchical modeling is known as concurrent modeling and consists of combining different numerical models that simultaneously describe different sub-domains. Each numerical model runs in its sub-domain and exchanges information with its parent/child sub-domain at defined boundaries. The critical issue here is to define the criteria and protocols in order to automate the application of more detailed numerical models in a simulation domain in time and realizing the so-called temporal multi-scale calculations or/ and space referred as spatial multi-scale ones. 

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