Different modelling approaches to the study of metapopulations

Because of the concurrence of two different spatial scales, describing the dynamics of a metapopulation in quantitative terms is far from being a simple task. To clarify the problems arising in the choice of a suitable modelling approach an example can be usefully provided.

The amphibian Rana muscosa is an endemic species of the Sierra Nevada (USA). Once, it was very common in lakes, streams and ponds of mountain areas. For a variety of co-occurring reasons, still under scientific investigation, in the past century the species has disappeared from about 90% of the sites it previously occupied in California and Nevada. One of the identified causes is the predation upon tadpoles by non-native species of trout of the genera Oncorhynchus, Salmo and Salvelinus which have been artificially introduced in mountain lakes since the middle of the past century to facilitate sport fishing (Knapp et al., 2007). Experimental removal of non-native trout (Vredenburg, 2004) have demonstrated that the abundance of tadpoles and frogs goes back rather rapidly to values statistically indistinguishable from those of adjacent lakes where the fish have always been absent. Vance Vredenburg studied the effect of fish removal on the metapopulation structure of R. muscosa for a system of 81 mountain lakes in Sixty Lake Basin, Kings Canyon National Park, California. Fig. 2 shows a simplified distribution map of frogs and trout in the major lakes of the basin. The five lakes indicated by a number are those where trout have been progressively removed over the five years from 1997 to 2001. Two more aspects deserve mentioning. First, in the year following the trout removal, each of the five lakes had a number of larvae and post-metamorphic frogs significantly different from that of lakes where fish were not removed. This is a signal that the trophic interaction played an important role in determining the disappearance of frogs. In the absence of predators, in fact, the density of both tadpoles and adult frogs have been rapidly increasing over the years, and have reached, at the end of the experiment, values ​​of the order of 2-3 post-metamorphic frogs per linear decameter of coast. Second, the lake that responded more slowly to ecosystem restoration was lake number 5. Among the possible causes, Vredenburg includes the fact that this lake is not connected by rivers to other lakes where R. muscosa was present before starting the experiment. By looking again at the territorial distribution of lakes in Fig. 2 one can notice that while lake 1, as well as the lake subsystem 2-3-4, is downstream of a lake already occupied by frogs, lake 5 is upstream of any other lake of the river network; therefore it can be colonized more difficultly. In all probability, indeed, the colonization of the lake took place because of the migration of frogs present in the little grey lake next to lake 5. Although the little lake is not connected to lake 5, successful migration of R. muscosa individuals can evidently occur by other routes (for example, terrestrial). That is why Vredenburg has connected lake 5 with the small northern grey lake by a dashed arc.

Figure 2: The fragmented territory available to the metapopulation of R. muscosa in a system of mountain lakes in the Kings Canyon National Park (California, USA) as described by Vredenburg (2004).

Suppose we want to describe the dynamics of the species metapopulation in the region of Kings Canyon National Park. We must make two important and independent choices. The first concerns the accuracy with which we want to describe the habitat in question, while the second regards the number and type of variables that we want to use to describe the population dynamics at the local scale.

As outlined by Hanski (1998), there are three qualitatively very different possibilities to describe the habitat (see Fig. 3). The simplest alternative is to imagine that the lakes available to the frogs are all equal to each other and arranged in a regular manner (Fig. 3A). This description is clearly too simplistic, although it at least preserves the key feature of the territory, that is its fragmentation. The lakes are in fact fragments of habitat that are suitable for the species and immersed in an unsuitable environmental matrix. On the other extreme of the spectrum of possibilities (Fig. 3C) one can use a geographic information system (GIS) to detail not only the spatial location and the specific shape of each of the lakes, but also the biotic and abiotic features of the lakes, as well as their riverine connections and a description of the territory that is in between them. Another spatially explicit approach, which is however of intermediate complexity (Fig. 3B), is to divide the habitat into a finite number of fragments of regular shape (e.g. circular like in Fig. 3B), each of which is characterized by ​​the same size and location as in reality, so that the distances between them can be considered explicitly. Depending on the level of detail at which habitat is modelled, even the movement of organisms between fragments can be described with different levels of accuracy. Considering the example again, if one imagines that individuals can move with the same probability and effectiveness from any of the lakes to any other (this mechanism is called propagule rain), the spatial coordinates of the fragments are not so important and therefore the metapopulation can be described with what is called spatially implicit modelling. In this approach the fragments can be accounted for without being located geographically. Whenever the fragments are instead arranged according to precise spatial coordinates, it is more spontaneous to also introduce a realistic description of the organisms dispersal. In these cases, one generally assumes that movement can take place with different probability and effectiveness according to distance and/or direction. The probability distribution of the dispersal distance is called dispersal kernel.

Similarly to what was done for the habitat, one can think of three qualitatively different alternatives to model the local demography. The simplest approach considers only the presence and absence of the species R. muscosa and produces models that are termed boolean (see Fig. 3D), because each fragment can be in only one of two states: empty ($0$) or occupied ($1$). These models are apt to describe a situation in which the demographic information is very poor, because they exclusively make a distinction between empty fragments and the other. Considering our example again, that lake number 3 is occupied by a single frog or 2-3 adults per coast decameter would not make any difference for a boolean model. Much more informative (Fig. 3F) would be an exact count of the number (necessarily discrete) of amphibians inhabiting each lake. This approach, which appears spontaneous, is however much more complex from a mathematical viewpoint because it requires the use of equations similar to those used to deal with demographic stochasticity in Chapter . An alternative of intermediate complexity may consist in using models that complement the boolean ones by using, for example, information related to the mean and variance of the abundances of frogs in the occupied fragments (Fig. 3E). Models in which there exists at least one variable measuring local abundances are called structured.

Figure 3: Different ways for describing habitat (top) or demography (bottom) in a simple model for part of the metapopulation of Fig. 2 (bottom right corner).

Since the choices pertaining to the description of habitat, dispersal mechanism and demography are relatively independent of each other, many combinations are possible and have therefore been proposed in the literature. The families of available metapopulation models range from those spatially implicit and boolean à la Levins (1969), to those that are spatially implicit but structured, such as Markov models (Casagrandi and Gatto, 1999; Chesson, 1984; Casagrandi and Gatto, 2002a) or partial derivatives models (Gyllenberg and Hanski, 1992); from models that are spatially explicit and boolean, such as cellular automata (Hiebeler, 1997; Molofsky, 1994), to the more complex structured and spatially explicit models like e.g. coupled maps (Hastings, 1993; Yakubu and Castillo-Chavez, 2002; Allen et al., 1993; Earn et al., 2000), or spatially realistic Levins models (Hanski and Ovaskainen, 2000) or Interacting Particle Systems (Casagrandi and Gatto, 2006). In this chapter, for the sake of simplicity, we will only deal with boolean models, such as the one due to Levins (1969), which still represents a conceptual basis, a sort of “null model”, against which to compare predictions deriving from any other more realistic metapopulation model.