Metapopulation models

8.2.1 Introduction to the metapopulation concept

A metapopulation is defined as a set of local populations linked by dispersal. This could be described and modelled by cellular automata but we will focus on results arising from analytical considerations. In the original model of Levins (1969, 1970) it was assumed that all local populations were of equal size and that a local population could either become extinct or reach carrying capacity instantaneously following colonization. Therefore only two states of local population were envisaged: full (carrying capacity) or empty (extinct).

In reality, the definition of a local population, and therefore a metapopulation, is very difficult. Hanski and Gilpin (1991) defined a local population as a 'set of individuals [of the same species] which all interact with each other with a high probability'. But how high is that probability? Furthermore, 'local' may be different for different interactions. For example, two plants may show intraspecific competition over a scale of a few centimetres but be reproductively linked by pollination over hundreds of metres. It is also very difficult to say over what distance colonization of new areas, and therefore the 'birth' of new local populations, may occur. Typically the frequency of movements of propagules such as seeds over short distances is known, but longer-distance movement is poorly known, partly because it may be a rare event and partly because it is difficult to record. This excludes species which show seasonal and predictable long-distance migration.

Even when local populations can be identified, the pure Levins model of local populations with equal carrying capacity is unusual. More realistically, it is reasonable to envisage a spectrum of possibilities from mainland/island or core/satellite to pure Levins populations (Fig. 8.2). These and other possi-

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Fig. 8.2 Various types of spatial distribution of populations. Closed ovals represent occupied habitat patches and open ovals represent vacant habitat patches. Dashed lines indicate the boundaries of populations. Arrows indicate migration and colonization. (a) Levins metapopulation. (b) Core/satellite metapopulation (Boorman & Levitt 1973). (c) Patchy population. (d) Non-equilibrium metapopulation (differs from (a) in that there is no recolonization). (e) An intermediate case that combines (b) and (c) (Harrison 1991).

Fig. 8.2 Various types of spatial distribution of populations. Closed ovals represent occupied habitat patches and open ovals represent vacant habitat patches. Dashed lines indicate the boundaries of populations. Arrows indicate migration and colonization. (a) Levins metapopulation. (b) Core/satellite metapopulation (Boorman & Levitt 1973). (c) Patchy population. (d) Non-equilibrium metapopulation (differs from (a) in that there is no recolonization). (e) An intermediate case that combines (b) and (c) (Harrison 1991).

bilities have been discussed by Harrison (1991), who considered the rarity of true Levins metapopulations in the field, and by Hanski and Gyllenberg (1993), who showed how to model both mainland/island and pure Levins with related equations. Hanski (1999) provides an overview of the whole subject.

Various processes will promote something close to a metapopulation structure in the field or at least create conditions under which local extinction and colonization are integral features of the population dynamics:

• gap creation or other disturbance generating new recruitment habitat (includes habitat fragmentation);

• a mosaic of successional habitats where, for example, an annual plant must move from one transient early successional habitat to another (may be a function of the previous feature);

• sedentary and localized resources such as plants, dung or decaying logs, all of which may be colonized by various insects, fungi and other organisms. The resources may have a short colonization period allowing a maximum of one or a few generations of the attacking organism, thereby necessitating dispersal to other similar resources.

8.2.2 The metapopulation model of Levins

Despite the problems of finding Levins metapopulations in the field it is instructive to consider its dynamics before proceeding to more complex models. Levins (1969, 1970) was interested in the number of islands or island-like habitats occupied by a species. Later Levins and Culver (1971) modified the model to investigate the effect of competition on migration and extinction rates. Levins began by considering the number of local populations (N), the total number of sites (T), an extinction rate (e) and a migration rate (m'). The rate of change of N with time (t) could then be expressed as a differential equation:

This equation was simplified by using p = N/T where p represents the fraction of habitat patches occupied by a species and replacing m T by m. The rate of change in the fraction of habitat patches occupied by a species, dp/dt - that is, the rate of change in the proportion of local populations (p) at a given time - was now described by:

where m defines the colonization rate of local populations and e the extinction rate of local populations. Therefore ep represents loss (or extinction) of local populations from metapopulations. The birth rate of local populations is represented by mp (1 - p). The reason why p is multiplied by 1 - p can be conceptualized as a local neighbourhood problem. If there is one occupied patch surrounded by eight empty patches then the probability of colonization of any one empty patch is likely to be less than if there was one empty patch surrounded by eight occupied patches. Thus in determining the colonization probability the density of occupied patches and unoccupied patches needs to be combined, for example by multiplying them. In reality, the colonization (m) and extinction (e) parameters are likely to be complex functions of a set of variables. For example, m involves finding a new site, which depends on propagule dispersal (in turn dependent on the taxon and habitat under scrutiny and perhaps wind or water current speed or abundance of animal dispersers), the spatial distribution of occupied and unoccupied sites, initial establishment of propagules and subsequent population growth.

Now let us consider the dynamics of the system described by equation 8.1. What are the conditions for metapopulation increase, no change or decline? No change in p is given by dp/dt = 0:

Increase in p will occur if dp/dt > 0 and therefore p < 1 - e/m. In considering these results we need to think about the original formulation of the metapopulation concept. If extinction and colonization rates (death and birth) are balanced (e = m) there should be no change in metapopulation size. Similarly, if the extinction rate (e) is greater than the colonization rate (m) then the metapopulation size should decrease (and vice versa). These requirements are only partly supported by the manipulation of equation 8.1. The problem is that when dp/dt = 0, if e = m we are left with the result that p = 1 - 1 = 0. Thus, when extinction balances colonization there are no local populations. If colonization is greater than extinction then e/m is less than 1 but greater than 0 and therefore 1 - e/m lies between 1 and 0. Therefore a steady value of p occurs when colonization exceeds extinction. Hanski (1991) considers various refinements and developments of the basic Levins model. Despite the drawbacks of equation 8.1 we will see in the next section how it can be used to explain limits to species range and how more complex models can give similar predictions.

8.2.3 The Carter and Prince model of geographic range

The plant metapopulation model of Carter and Prince (1981, 1988) linked the ideas of Levins with models of infectious disease to provide an explanation for the geographical range limits of plant species. In particular, they challenged the view that distribution was determined solely by correlation to climate variables; for example, that the northerly distribution limit of plant species in Britain was determined by physiological intolerance of cold winters (see examples in Carter & Prince 1988). Carter and Prince used a differential equation to describe a strategic model of plant distribution:

where x is the number of susceptible sites (sites available to be colonized), y is the number of infective sites (occupied sites from which seed is produced and dispersed), b is the infection rate and c is the removal rate. b and c are essentially local population birth (colonization) and death (extinction) rates and therefore equivalent to m and e in the Levins equation (8.1). Similarly, x and y are related to 1 - p and p where p is the proportion of local populations which are occupied and potentially 'infective' and 1 - p is the proportion of vacant and therefore susceptible sites.

These comparisons show that any conclusions from the Carter and Prince model are relevant to metapopulations in general as defined by Levins. The important conclusion of Carter and Prince was that, along a climatic gradient, a very small change in, for example, temperature might tip the balance from metapopulation persistence to metapopulation extinction. In Carter and Prince's words: 'a climatic factor might lead to distribution limits that are abrupt relative to the gradient in the factor, even though the physiological responses elicited might appear too small to explain such limits.' Thus climate and physiological factors are still important but their effects are amplified and made nonlinear by the threshold properties of equation 8.2.

The results of these simple models are supported by the conclusions from more complex models. For example, the model of Herben, Rydin and Soderstrom (1991) examined the dynamics of the moss Orthodontium lineare which occurs on temporary substrates such as rotting wood. The model addressed not only the metapopulation structure of the moss but also the fact that the species was spreading throughout western and central Europe. The model included deterministic increase on occupied logs with a carrying capacity and the assumptions that dispersal by spores was in proportion to local population size and that spore dispersal distance declined exponentially from an occupied log. The results of this model suggested, like the simpler models, that there is a threshold for metapopulation persistence. In this case the percentage of logs occupied was a nonlinear function of probability of local population establishment (pest; Fig. 8.3).

At a pest value of about 0.0002 the model predicted a sudden increase in the percentage of occupied sites. So here was a threshold value above which metapopulation persistence was likely to be high. If pest is a function of climate then this would produce exactly the type of sharp break in species range predicted by Carter and Prince. It seems that such thresholds may be generated in a variety of ways. Below we consider how changes in gap frequency in grassland can generate a threshold for plant population abundance and how diffusion processes can lead to thresholds.

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Fig. 8.3 Threshold in occupied sites produced by small increments in probability of establishment (model of Herben et al. 1991).

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Fig. 8.3 Threshold in occupied sites produced by small increments in probability of establishment (model of Herben et al. 1991).

An alternative formulation for the metapopulation equation 8.1 is to use the logistic equation to model metapopulation dynamics and consider again the possibility of a threshold determining the edge of a species range. One key feature of the model needs to be retained; that is, that there is some interaction between the densities of infectives (occupied patches) and susceptibles (empty patches) in determining colonization rates. This interaction is represented by p(1 - p) in equation 8.1 and xy in equation 8.2. This can be taken further by considering the relationship between the relative or net colonization rate (m/e or m - e) and the density of susceptibles with respect to infectives (S/I). In the absence of any effect of S/I the change in infectives (dI/dt) can be described as the net colonization rate multiplied by the number of susceptible patches:

The relative colonization rate m - e will be expected to vary with S and I. If S/I is high then m - e should be low. If S/I is low then we expect m - e to be close to its maximum value. There is a clear analogy with the logistic equation. m - e can be replaced by a value r (births minus deaths) and S/I by S'. The simplest reduction of r is linear with respect to S' which is described by 1 - S'/K. The resultant equation is:

Note that, in contrast to the logistic equation, the rate of change variable (I) on the left-hand side is not the same as the variable on the right-hand side. At equilibrium, dI/dt = 0 so rS = 0, which can be interpreted as r = 0 (e = m), and/or S = 0 (no remaining susceptible sites) or 1- S'/K = 0 and therefore dI/dt = (m - e)S

S' = K (as expected from the logistic equation). If S' > K then 1 - S'/K is negative and so dI/dt is negative. Therefore I decreases and consequently S/I continues to increase. Thus K is a threshold condition. If S is too high (above K) then the metapopulation cannot persist and the number of infectives declines (and so the proportion of susceptibles, S , increases).

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