v) Indices of faunal similarity
How are zoogeographic subdivisions such as regions and provinces delimited? An important way is to compile taxonomic lists of presence or absence of species, genera, families or other taxonomic groups and compute indices of similarity in pairwise comparisons between areas. These indices are then "clustered" into distinct units by any of a number of computer-based algorithms. Two examples of such indices are:
Jaccard's Index = C/[(N1 + N2) - C]
or Simpson's Index = C/N1
where, C = the number of species (or genera, families, etc) that occur in both areas, N1 = the total number of species (or genera, families, etc) in area 1, and N2 is the same quantity for area 2 and where N1 is the area with the largest number of the chosen taxonomic rank (e.g., species).
For instance, below is a diagonal matrix of faunal similarities (Simpson's index) among zoogeographic areas based on the numbers of mammal families. Larger numbers represent greater faunal similarity:
| Area | North America | West Indies | South America | Africa | Madagascar | Eurasia | SE Asian Islands | Philippines | New Guinea | Australia |
| North America | - | |||||||||
| West Indies | 67 | - | ||||||||
| South America | 81 | 73 | - | |||||||
| Africa | 31 | 27 | 25 | - | ||||||
| Madagascar | 38 | 27 | 35 | 65 | - | |||||
| Eurasia | 48 | 27 | 36 | 80 | 69 | - | ||||
| SE Asian Islands | 37 | 20 | 32 | 82 | 63 | 92 | - | |||
| Philippines | 40 | 20 | 32 | 88 | 50 | 96 | 100 | - | ||
| New Guinea | 36 | 21 | 36 | 64 | 50 | 64 | 79 | 64 | - | |
| Australia | 22 | 20 | 22 | 67 | 38 | 50 | 61 | 50 | 93 | - |
This matrix was then run through a clustering algorithm (Unweighted pair group method of averages, UPGMA), but I subtracted the value of Simpson's index from 100 to get a pairwise difference value instead (just a quirk of the program) to generate a "tree" that depicts clusters of areas based on how different (or similar) their faunas are. The method examines the table of differences and picks those two areas that are the least different (SE Asian Island sand Philippines, SI = 100 or differences = 0) and joins them into a single cluster. The method then looks for the next most similar pair of areas (including the new single SEA-Philippine cluster) and joins them (in this case it is Australia - New Guinea.. The method continues until all areas are joined in the tree. The shorter the horizontal lines are that join any two areas or clusters, the more similar they are. In this way, "natural" faunal groups emerge from the data and in most cases could be predicted from independent information. For instance, we know that New Guinea and Australia have a long period of association owing to their close proximity and the fact that they were connected to each other when sea levels where lower (via the Sahul Shelf). Similarly, Madagascar has had a long period of isolation from Africa and is, consequently, quite distinct faunsitically from Africa.

UPGMA tree of faunal distinctions mong 10 geographic areas (100 - Simpson's Similarity Index).
In this way, regions, provinces, and other zoogeographic subdivisions have been identified.
In another context, similarity indices have been used to document the decline of provincialism among areas. The figure below shows the change in Jacaard's similarity indices among 1128 pairwise comparisons of freshwater fish species among the 48 continental US states. Note that most comparisons have seen an increase in the percent similarity between states. This is the result of introductions of fishes by humans for (largely) sportfishing. States like Nevada, Utah, and Arizona now have over 50% of their total fish fauna now consisting of introduced species. Panel A = the total change in similarity, B = change in similarity owing to extinctions between states, C = change in similarity owing to introductions. Clearly, the pattern in A is most similar to that in C and cannot be attributed to biased extinctions (although that can also act to increase similarity).
This phenomenon is known as "homogenization" of faunas (see Rahel 2000). Taylor (2004) conducted a similar analysis for Canadian freshwater fish faunas and reported a mean value of homogenization of 1.2% across Canada, but 3.8% differentiation between ecoregions within BC.

(iv) Disjunction
Disjunctions occur when closely related taxa are distributed in widely separated areas. Disjunctions can occur at different spatial scales. For instance, the order Lepidosireniformes (lungfishes) are disjunct across South America, Africa, and Australia.

Disjunct distribution of the Lepidosireniformes (lungfishes, shaded areas) in South America, Africa, and Australia.
On the other hand, two subspecies of cutthroat trout are disjunct in western North America.

Two subspecies of cutthroat trout (Oncorhynchus clarkii clarkii and O. c. lewisi) have disjunct distributions in western North America.
Also note endemism and disjunction often depend on the taxonomic level of consideration. Lungfishes as an order are disjunct across the Southern Hemisphere, but the Australian lungfish (Ceratodontidae) is endemic to eastern Australia.

The Australian lungfish, Neoceratodus forsteri.
Disjunctions may have their cause in one of three processes:
1. A continuous geographic range that is "split-up" by a major geological process (e.g., a vicariant event) such as continental drift, mountain building, or flooding.
2. Biased extinctions of populations in the middle of a once continuous range (perhaps by a process in 1.)
3. Long distance dispersal of one of the lineages from its place of origin to another, distant locality.
The relict distribution of lungfishes is a good example of range splitting. Recall the widespread distribution of fossil lungfish across the globe. The distribution and geological age of these fossils are consistent with the idea that the relict distribution of lungfishes is due to the breakup of Pangea, the drifting of continents and the present distribution of lungfishes in tropical freshwaters.
Biased extinctions have occurred in many instances. The disjunct distrubtion of O. clarki shown above may have resulted from extinction of populations in the central highlands area of the current Okanagan Basin in central BC and Washington. At one time, a more continuous mesic forest existed from the west coast to the foot of the Rocky Mountains. Uplift of the central Okanagan Highlands resulted in climate change in that region to the more xeric conditions that exist today. It has been hypothesized that this change in climate produced disjunctions in a at one time, mesic-loving, continuous population of organisms like the cutthroat trout. Disjunct populations of the tailed frog (Ascaphus truei), that are also associated with mesic forests, across the same range are another possible example.
The galaxid fish, Galaxias maculatus, is found in coastal areas of South America, southern Africa, New Zealand/Australia (see below, heavy lines). This disjunction may have been due to the breakup of Gondwanaland as well, but these fish mature in freshwater, spawn in estuaries, and the larvae are salt-water tolerant and capable of dispersal in marine waters. The distribution of these larvae across the southern oceans could well have been driven by dispersal in the circum-subAntarctic "West Wind Drift" current (see below).
Disjunction in the distribution of the galaxid smelt (Galaxias maculatus) in South America, Africa, and Australia/New Zealand.
Finally, disjunctions rely on the idea that the taxa are closely-related and are, therefore, critically dependent on good systematics. For instance, the map below shows the distribution of freshwater crayfish. Note the distribution both in the Southern Hemisphere and Northern Hemisphere. This distribution was thought to have resulted from two centres of origin (one in each hemisphere) of distantly related crayfish such that "freshwater crayfish" had evolved twice in two distinct areas (from local marine crustaceans) and, therefore, they were not disjunct as a result of range splitting of a once continuously-distributed freshwater crayfish.

Distribution of freshwater crayfish.
Crandall et al. (2000), however, tested this idea by generating a molecular phylogeny of crayfish using DNA sequence data. The phylogeny (shown below) clearly showed that crayfish in both hemispheres were derived from a common ancestor (the two groups stem from a common point on the tree and are monophyletic) and thus have evolved only once (not twice as implied by the two centres of origin hypothesis). This means that, in fact, the freshwater crayfish are closely related and likely once occupied a continuous range that has become fragmented to produce the current disjunction. Fossils dating to the Triassic (265 million years ago) and dating of molecular divergences suggests that the timing of the range splitting is consistent with the timing of Pangean breakup. Here is a case, therefore, where a proper understanding of the systematic relationships within a group (i.e., the monophyly of freshwater crayfish) is fundamental to characterizing their global distribution as disjunct. A good example, as well, of the multidisciplinary nature of robust zoogeographic analyses.
Phylogeny of freshwater crayfish (families Cambaridae, Astacidae, and Parastacidae) demonstrates that they are a monophyletic group.
References
Abell, R. and 27 co-authors. 2008. Freshwater ecoregions of the world: a new map of biogeographic units for freshwater biodiversity conservation. Bioscience 58: 403-414.
Barber, P.H. et al. 2000. A marine Wallace's Line? Nature 406: 692-693.
Brown, J. and M. Lomolino. 1998. Biogeography. 2nd Edition. Chapter 10.
Crandall, K.A., D.J. Harris, and J.W. Fetzner Jr. 2000. The monophyletic origin of freshwater crayfish estimated from nuclear and mitochondrial DNA sequences. Proc. Roy. Soc. Lond. B. 267: 1679-1686.
Rahel, F.J. 2000. Homogenization of fish faunas across the United States. Science 288: 854-855.
Taylor, E.B. 2004. An analysis of homogenization and differentiation of Canadian freshwater fish faunas with an emphasis on British Columbia. Can. J. Fish. Aquat. Sci. 61: 68-79.