Niche Modeling: Predictions From Statistical Distributions - Chapter 4

Chapter 4 Topology The focus of topology here is the study of the subset structure of sets in the mathematical spaces. Topology can be used to describe and relate the different spaces used in niche modeling. A topology is a natural internal structure, precisely defining the entire group of subsets produced by standard operations of union and intersection. Of particular importance are those subsets, referred to as open sets, where every element has a neighborhood also in the set. More than one topology in X may be possible for a given set X. Examples of subsets in niche modeling that could form topologies are the geographic areas potentially occupied by a species, regions in environmental space, groups of species, and so on. Application of topological set theory helps to identify the basic assump-tions underlying niche modeling, and the relationships and constraints be-tween these assumptions. The chapter shows the standard definition of the niche as environmental envelopes around all ecologically relevant variables is equivalent to a box topology. A proof is offered that the Hutchinsonian en-vironmental envelope definition of a niche when extended to large or infinite dimensions of environmental variables loses desirable topological properties. This argues for the necessity of careful selection of a small set of environmental variables. 4.1 Formalism The three main entities in niche modeling are: S: the species, N: the niche of environment variables, and B: geographic space, where the environmental variables are defined. . 45 © 2007 by Taylor and Francis Group, LLC 46 Niche Modeling The relationships between these entities constitute whole fields of study in themselves. Most applications of niche modeling fall into one of the categories in Table 4.1. TABLE 4.1: Links between geographic, environmental and species spaces. S S interspecies relationships N habitat suitability B range predictions N − correlations geographic information B − − autocorrelation Niche modeling operates on the collection of sets within these spaces. That is, a set of individuals collectively termed a species, occupies a set of grid cells, collectively termed its range, of similar environmental conditions, termed its niche. Thus a niche model N is a triple: N = (S,N,B) The niche model is a general notion applicable to many phenomena. Here are three examples: • Biological species: e.g. the mountain lion Puma concolor, the environ-ment variables might be temperature and rainfall, and space longitude and latitude. • Consumer products: e.g. a model of digital camera, say the Nikon D50, environment variables for a D50 might be annual income and years of photographic experience, and space the identities of individual con-sumers. • Economic event: e.g. a phenomenon such as median home price in-creases greater than 20%, the variables relevant to home price increases wouldbeproximitytocoast, familyincome, andthespaceofthemetropoli-tan areas. A niche model can vary in dimension. Here are some examples of dimensions of the geographic space B: • zero dimensional such as a set, e.g. survey sites or individual people, • one dimensional such as time, e.g. change in temperature or populations, © 2007 by Taylor and Francis Group, LLC Topology 47 • two dimensional such as a spatial area, e.g. range of a species, • three dimensional such as change in range over time. While examples of contemporary niche modeling can be seen in each of these dimensions, many examples in this book are one dimensional, particularly in describing the factors that introduce uncertainty into models, because a simpler space is easier to visualize, analyze and comprehend. All results should extend to studies in higher dimensions. Dimensions of environmental space N, in Chapter 4, concern the implica-tions of extending finite dimensional niche concepts into infinite dimensions. Dimensions of species, one species for each dimension, relates to the field of community ecology through inter-specific relationships. Here we restrict examples to one species, and one S dimension. 4.2 Topology There are a number of other ways to describe niche modeling. There are a rich diversity of methods to predict species’ distribution and they could be listed and described. Alternatively, biological relationships between species and the environment could be emphasized, and approaches from population dynamics used as a starting point. While useful, these are not the approaches taken in this book, preferring to adhere to examination of fundamental prin-ciples behind niche modeling. Topology is concerned with the study of qualitative properties of geometric structures. One of the ways to address the question – What is niche modeling? – is to study its topological properties. 4.3 Hutchinsonian niche Historically, the quantitative basis of niche modeling lies in the Hutchinso-nian definition of a niche [Hut58]. Here that set of environmental characteris-tics where a species is capable of surviving was described as a ‘hypervolume’ of an n-dimensional shape in n environmental variables. This is a generalization of more easily visualizable lower dimensional volumes, i.e.: © 2007 by Taylor and Francis Group, LLC 48 Niche Modeling • one, an unbroken interval on the axis of an environmental variable, rep-resenting the environmental limits of survival of the species, • two, a rectangle, • three, a box, • n dimensions, hypervolumes. This formulation of the niche has been very influential, in part because in contrast to more informal definitions of the niche, it is easily operationalized by simply defining the limits of observations of the species along the axes of a chosen set of ecological factors. 4.3.1 Species space Hutchinson denotes a species as S1 so the set of species is therefore denoted S. In its simplest form the values of the species S1 are a two valued set, presence or absence: S1 = {0,1} Alternatively the presence of a species could be defined by probability: S1 = {p|p ∈ [0,1]} 4.3.2 Environmental space Using the notation of Hutchinson the niche is defined by the limiting values on independent environmental variables such as x1 and x2. The notation used for the limiting values are x1,x00 and x2,x00 for x1 and x2 respectively. The area defined by these values corresponds to a possible environmental state permitting the species to exist indefinitely. Extending this definition into more dimensions, the fundamental niche of species S1 is described as the volume defined by the n variables x1,x2,...,xn when n are all ecological factors relative to S1. This is called an n-dimensional hypervolume N1. © 2007 by Taylor and Francis Group, LLC Topology 49 4.3.3 Topological generalizations The notion Hutchinson had in mind is possibly the Cartesian product. If sets in environmental variables xi are defined as sets of spaces Xi, then N1 is a subset of the Cartesian product X of the set X1,...,Xn, denoted by X = X1 ×...×Xn, or X = Qi=1 Xi In a Cartesian product denoted by set X, a point in an environmental region is an n-tuple denoted (x1,....xn). The environmental region related to a species S1 is some subset of the entire Cartesian space of variables X. The collection of sets has the form Q i∈J i Setting a potentially infinite number i ∈ J to index the sets, rather than a finite i equals 1 to n is a slight generalization. The construct captures the idea that the space Xi could consist of an infinite number of intervals. This generalizes the n-dimensional hypervolume for a given species in S, so that the space may encompass a finite or infinite number of variables. Another generalization is to define each environmental variable xi as a topo-logical space. A topological space T provides simple mathematical properties on a collection of open subsets of the variable such that the empty set and the whole set are in T, and the union and the intersection of all subsets are in T. The set of open intervals: (xi,x00) where x0,x00 ∈ R is a topological space, called the standard topology on R. Where each of the spaces in Xi is a topology, this generates a topology called a box topology, describing the box-like shape created by the intervals. An element of the box topology is possibly what Hutchinson described as the the n-dimensional hypervolume N1 defining a niche. 4.3.4 Geographic space There are differences between the environmental space N and the geograph-ical space B. While the distribution of a species may be scattered over many © 2007 by Taylor and Francis Group, LLC ... - tailieumienphi.vn