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Avoiding Mistakes in Population Modeling:
Spatial Structure and Multiple Populations

List of common mistakes in population modeling

Ignoring spatial structure

Spatial structure refers to the spatial distribution of the population, so that instead of one mixed (panmictic) population, there are several subpopulations that are relatively isolated from each other (i.e., the dispersal among the subpopulations is more restricted or limited than within a subpopulation). There are several ways models ignore spatial structure.

First, if a population is composed of multiple subpopulations, then modeling it as a single population often gives the wrong results in terms of risk of decline and extinction (Burgman et al.1993; Akçakaya 2000b). In addition, certain types of threats or impacts (such as habitat fragmentation and dispersal barriers) and certain types of conservation measures (reserve design, translocations, habitat corridors, etc.) can only be evaluated by taking account the interactions between neighboring populations. Even measures that target individual populations may need to be evaluated in a metapopulation context, because the presence of other populations may change the relative effectiveness of alternative options (e.g., see Regan & Auld 2004). Determining harvest levels (for example, fishing quotas) without taking spatial structure into account may lead to overharvest (Smedbol & Stephenson 2004).

Second, a model can include multiple populations, but ignore their locations, which determine the rate of dispersal among the populations, and the correlation or similarity of the environmental conditions they experience. Both of these factors have important effects on population viability, so it is important to correctly estimate these factors (which often requires taking the locations of the populations into account).

Third, a model can include multiple populations, their locations, and interactions among the populations, but ignore the spatial variation in within-population characteristics such as survival rate, fecundity, carrying capacity, maximum growth rate, etc. Such variation may be a function of local habitat factors (such as habitat quality, habitat area, edge:core ratio, etc.) or other local factors (such as harvest rate, predation rate, etc.). Differences in these types of within-population characteristics may have important effects on metapopulation viability, e.g., through source-sink dynamics. Such differences may also result from human impacts (such as pollution or harvest), in which case ignoring them in models developed for impact assessment may cause underestimation of impacts.

Too many (or too few) populations

When a species exists in a metapopulation (e.g., in a fragmented habitat), determining the number of discrete subpopulations of the metapopulation in an arbitrary manner (e.g., by the number of sampling locations, etc.) may lead to the wrong number of populations.

Considering the difficulty of defining a species, a much more fundamental concept, it is perhaps not surprising that the definition and delineation of a population presents problems. A biological population can be defined as a group of interbreeding (i.e., panmictic) individuals. Assuming that the distribution of a species is more-or-less continuous across parts of the landscape, the question of delineating a population can be rephrased as: how far apart must two individuals be in order to be considered to be in different populations? This depends on the movement distance, home range, or some other measure related to the possibility of interbreeding. This approach, combined with modelling and prediction of suitable habitat, is used in RAMAS GIS to delineate populations (Akçakaya et al. 1995; Akçakaya 2000b).

The main parameter used to address the above question in RAMAS GIS is the Neighborhood distance, which is used in the "Spatial Data" subprogram of RAMAS GIS to find patches in the habitat map. The value is specified in units of cell lengths, and should be consistent with the biology of the species you are modeling. This parameter represents the spatial scale at which the population can be assumed to be panmictic. If you are modeling an animal, you might think of it as the foraging distance of the species modeled. Technically, suitable cells that are separated by a distance of less than or equal to the neighborhood distance are regarded to be in the same patch. The unit of distance is one cell, and the distances are measured from the centers of the cells. Thus, a Neighborhood distance of 1.0 means that two cells that are touching only in one corner are in two different patches (because the distance between their centers is about 1.4 cells). In many cases, the resolution of the maps are high enough that the minimum value of 1.0 is not appropriate.

For more information and examples, see the User Manual, Akçakaya & Raphael (1998) and Akçakaya & Atwood (1997).

Ignoring spatial correlation

Spatial correlation refers to the similarity (synchrony) of environmental fluctuations in different parts of the landscape and, in the case of a metapopulation, in different populations. In many metapopulations, fluctuations in the environment are at least partially correlated (Liebhold et al. 2004). In these cases, models that ignore spatial correlations (assuming independence) may be misleadingly optimistic in their estimation of risks of extinction and decline (e.g. Harrison & Quinn 1989; Akçakaya & Ginzburg 1991; LaHaye et al. 1994).

This is especially important when modeling impacts of habitat fragmentation. In a model that ignores spatial correlations, when a population is fragmented into several smaller populations, the model assumes that new fragments have independent dynamics, even though they were part of the same population before fragmentation. Although the new fragments may have partially independent dynamics (e.g., because some threats such diseases, fires, etc. may become less likely to spread to the whole population), making the extreme assumption of no correlation is unrealistic and may severely underestimate the true impacts of fragmentation, even making the fragmented populations appear as being more viable than the single population they formed before fragmentation.

For modeling spatial correlation in RAMAS Metapop, select "Correlations" from the Model menu, and then press F1 for help.

Wrong dispersal rates

There are several ways dispersal rates can be incorrectly estimated. Please read the help file and the manual for details of how dispersal is modeled, and how dispersal parameters should be estimated.

One important point to keep in mind is that in RAMAS Metapop dispersal is modeled in terms of the proportion of individuals moving (successfully dispersing) from one population to another. Thus, the program does not make a distinction among dispersers that die during dispersal, those which die before they leave the source patch, and those that die after they reach the target patch (but before the next census). In other words, it assumes that dispersal mortality is incorporated into the vital rates (e.g., specified in a stage matrix). This assumption does not mean that the dispersers necessarily have the same mortality as residents; it means that mortality during dispersal is accounted for by both the vital rates and the dispersal rates. This point is a pragmatic assumption because, in most field studies, one cannot really measure the proportion of individuals leaving a patch (unless all individuals are radio-tagged and continuously monitored) but can only count those that arrive in another patch. The distinction (as to where exactly dispersers die) is usually not only difficult to quantify but is also mostly irrelevant to the estimation of extinction risk. For a simple example that illustrates this point, see the program manual or Akçakaya (2000b, page 47-48).

Symmetric dispersal rates

In some cases, assuming that the dispersal rate is the same in both directions between two populations (i.e., assuming a symmetric dispersal matrix) may be problematic. This is especially true if one population is much larger than the other. If dispersal rate is the same between a large and a small population in both directions, the number of dispersers from the large to the small population would be much larger than the number in the other direction. The large number of migrants from a large to a small population will overshoot the small population's ceiling or carrying capacity (and thus not contribute much to its persistence), whereas the small number of migrants from the small population to large population will not compensate for the number that leaves the large population (Akçakaya and Baur 1996; Akçakaya & Raphael 1998). Thus, dispersal from the large to small population will drain the large population and cause N>K in the small population, and as a result, the small population will act as a pseudo-sink.

Although this may be realistic scenario in some metapopulations, in most models, it is just an artifact of assuming symmetric dispersal between all pairs of populations.

In RAMAS GIS, this problem can be corrected in several ways:

  1. Modify dispersal rates as proportional to the carrying capacities of the source and the target populations (see Akçakaya & Raphael 1998; page 885). There is not an automatic way of doing this, and it may be complicated when carrying capacities have a temporal trend. It can be done by carefully modifying the *.DCH files (for dispersal rates) based on the *.KCH files (for K).
  2. Use center-to-edge distances in the Spatial Data program. When the patch area is proportional to population size (so that there is a big asymmetry in area among close-by populations), center-to-edge distances make the large-to-small patch distance longer than the small-to-large distance, resulting in asymmetrical dispersal rates, with larger dispersal rate from small to large populations than from large to small populations. This would be realistic since most of the large patch lies quite far away from the small patch, whereas the entire small patch is about the same distance from the edge of the large patch.
  3. Change the "Density-dependent dispersal as a function of target population K" parameter to a large number (using the same or similar value for all populations). This parameter is in the "Dens. Dep." tab of the Populations dialog, and is the threshold K under which dispersal rates (into this population) decrease linearly as a function of the carrying capacity of this population. (It is available only if the "Dispersal depends on Target pop. K" parameter in the Dispersal dialog is checked.) Using a large value for this parameter would resolve the problem, because it would reduce immigration populations with small K much more than immigration into larger populations.

Too high (or too low) map resolution

When the spatial structure of the metapopulation is based on habitat maps (using the "Spatial Data" subprogram of RAMAS GIS), the map resolution (cell length or pixel size) should be consistent with the biology of the species being modeled. If the resolution is too high (cell size is too small), then the maps will be unnecessarily large and some program operations will take a very long time or may fail due to memory shortage. If, on the other hand, the resolution is too low (cell size is too large), the resulting patch structure will not reflect the spatial structure of the species' habitat and populations very precisely.

When determining the appropriate resolution, it is important to keep in mind that RAMAS is for population-level modeling, not for individual-based modeling. Thus, a patch is defined as an area that supports one subpopulation of a metapopulation, not as a territory or home range of an individual or a pair. In many cases, a cell area that is close to or slightly smaller than average territory or home range size may provide sufficient resolution.

Another guide is the Neighborhood distance parameter. You must specify this parameter in the program in units of cells (cell lengths), based on the biology of the species. If the biology of the species you are modeling suggests a neighborhood distance that is larger than about 4 or 5 cells, then the resolution of your maps may be too high (you may need to increase cell length, and decrease the number of rows and columns). If it suggests a neighborhood distance that is smaller than 1 cell, then the resolution of your maps may be too low.

For tips on how to change the resolution of maps, see the help topic titled "About Map Size and Resolution."

List of common mistakes in population modeling

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