Spatially realistic models

The Levins model assumed that local populations were identical; extinctions occurred independently in different patches; all patches were equally likely to be found; and patches were equally connected to each other. He did not try to describe how individuals move from one population to another, nor did he allow for differences in patch size or quality, in spite of the importance assigned to island size by MacArthur and Wilson.

Most metapopulation models also assume unconditional emigration. That is, there are no consequences to the source population from losing individuals. This seems to be a reasonable assumption in most populations since emigrants are often pictured as "extra" individuals when a population has reached or exceeded its carrying capacity. In many species non-breeding males venture away from the family group looking for an empty territory or an existing group that they may join in order to become breeders. Females of butterflies and other insects emigrate in order to find newly available host plants for egg deposition. Male and female dung beetles emigrate when the dung pile they were born

Population Growth

Distance units

Figure 5.6 Number of colonists arriving at different distances, based on a negative exponential model, Cj = p e-adj, and using different a values. Cj = the number of colonists arriving at a distance dj from the source population. In all cases p = 500.

Distance units

Figure 5.6 Number of colonists arriving at different distances, based on a negative exponential model, Cj = p e-adj, and using different a values. Cj = the number of colonists arriving at a distance dj from the source population. In all cases p = 500.

into has become depleted. If the mortality rate of dispersers, however, is much higher than that of individuals who remain "home," a very high population dispersal rate (large percentage dispersing) can lead to local, and sometimes metapopulation, extinction (Hanski 2002, J. Mickelberg, personal communication).

The rate and the scale of re-colonization of an empty habitat depend on the shape of the dispersal curve of the population. A simple model of animal movement is based on a random walk, using a coefficient of diffusion (D) and a normal distribution for movements, the variance of which increases with time (2Dt) (Okubo 1980). Studies of animal emigration, however, indicate that more individuals move very short and very long distances than predicted by random walks (Johnson and Gaines 1990). In metapopulation models emigration distances are usually modeled using a negative exponential function, which is a reasonable approximation of reality (Hanski 2002) (Eqn. 5.7, Fig. 5.6). Two characteristics determine the dispersal efficiency of a species. One is the number of individuals dispersing (here equal to 3) and the dispersal ability of each individual (a). Species differ a great deal in the amount of energy invested in reproduction each year. Obviously the more energy invested in reproduction, the greater the number of dispersal units. But for a given reproductive effort, species also differ in whether they produce a smaller number of large offspring or a larger number of small offspring. The smaller the dispersing unit, the greater distance it is likely to travel. Many highly dispersed organisms are so small that the wind can carry them hundreds or even thousands of miles. On the other hand, acorns only move as far as gravity or squirrels will take them. In Equation 5.7 a is directly associated with a greater colonizing ability per unit of dispersal, while 3 is associated with the number of colonists produced per individual from the source population. Cj is the colonization probability per unit time for population i and di is the distance from the source population.

where a and p are site- and species-specific parameters.

What is most important to a metapopulation is the proportion of individuals that move long distances. The exact shape of the distribution of migration distances among emigrants is therefore of great importance, but it is very hard to estimate in most populations (Hanski 2002).

We will discuss one approach that has been proposed to make metapopulation models more realistic, the incidence function model (IFM) championed by Hanski and his colleagues (Hanski 1994a, 1994b, 1999, Hanski and Gilpin 1997). The IFM is a stochastic patch model in which the population in each patch has one of two states, presence or absence. The IFM includes: (i) a finite number of habitat patches; (ii) patches of different sizes (sometimes including differences in quality and shape); and (iii) each patch having a unique spatial coordinate so that interactions among patches are localized in space. Since habitat patches are simply occupied or not, there is usually no estimation of population sizes or dynamics within patches. The major virtue of the IFM is that it is constructed so that parameters can be estimated from field data. This allows the application of this model to real populations.

The IFM begins with the assumption that for an empty habitat patch, i, there is a constant probability, Ci, of re-colonization per unit time. If a patch is occupied, there is a constant probability, Ei, of extinction per unit time. One event, either colonization or extinction, is allowed per time period. The long-term probability of the patch being occupied is called the "incidence" or Ji (Eqn. 5.8). The incidence function model is based on discrete time intervals, and is a stochastic rather than a deterministic model (Hanski 2001).

There are a number of difficulties if we are modeling a true metapopulation with no "mainland" source of species. With no external mainland, metapopulation extinction is the only true steady state (Hanski 1999). However, a metapopulation may theoretically persist for very long periods of time.

Recall that in the rescue effect the probability of extinction on a habitat patch is reduced through immigration of individuals from other patches. In order to allow for the rescue effect, Hanski modified the probability of extinction between times t and t + 1 by substituting (1 - Ci)Ei for Ei. This modified equation is:

Hanski (1999) then derived a relationship between extinction probability, E, and the size of the patch area, A, using the basic reasoning of the species-area curve. That is, extinction probability depends on population size, which is a function of patch area. The general relationship is as expressed below (Eqn. 5.10) in which e and X are estimated from the data. The value e is a parameter related to the probability of extinction per unit time in a patch of a given size. The parameter X is a measure of environmental stochasticity (Hanski 1999).

Colonization probability, Q, is a function of the number of immigrants, M. In a simple mainland-island metapopulation, the colonization probability is a function of distance (d) from the mainland, which was expressed above as Equation 5.7.

For a true metapopulation, however, Mt is the sum of individuals arriving from all of the surrounding habitat patches. Mt can be written as a summation for all patches:

In this equation dj represents the distance between patches i and j; pj is 0 for an unoccupied patch and 1 for an occupied patch; and Aj is the size of the patch. The summation term is represented by Si, which becomes a measure of patch isolation or, to put it positively, patch connectivity. If population sizes (N) are known for each patch, Si can be written as:

As above, the term a describes how fast the immigration rate from patch j declines with distance. Hanski suggests that this term can be found through mark-recapture data.

If interactions among immigrants are negligible, C increases exponentially with M. However, there is often a sigmoid relationship between the number of immigrants and successful re-colonization by a given species. Therefore Q can be written as:

where y is a parameter fitted from the data. In the sections below the terms y and p are combined simply as y (Hanski 1999).

Once equations were developed for the dependence of extinction on patch size and colonization on patch connectivity, Hanski combined them into the usual form of the incidence function model:

S2 AX

As before, Ji is the probability that a patch, i, is occupied; y is a parameter related to successful immigration; Ai represents the area of the patch i; X is the rate of change of extinction per unit time with increasing patch size (a measure of environmental stochas-ticity); and St describes the connectivity between patches, that is, the effect of distance on immigration rate. Equation 5.14 can be rewritten as:

If we manipulate and take natural logs of both sides of this equation we come up with a linear relationship between the expected patch occupancy (J,) and the two independent variables, connectivity (S,) and size (A,) of the habitat patches:

Hanski (1999) has described how to estimate all of these parameters from field data, and has applied the model to simulate metapopulation dynamics in butterflies (Hanski et al. 1995, Wahlberg et al. 1996), the American pika (Ochotona princeps) (Moilanen et al. 1998), and a number of other species. The basic information needed in order to apply this model in the field is simply the area of each habitat patch and the inter-patch distances (d) Subsequently, model parameters must be estimated. First a is estimated from mark-recapture data or estimated from patch-occupancy data. The parameters y and e are fitted to empirical data using nonlinear regression techniques. The value of X is fitted from the data and can be modified to include the rescue effect.

In Table 5.2, from Kindvall (2000), are the results of fitting the incidence function model to field data gathered from a fragmented population of the bush cricket Metrioptera bicolor. Kindvall (1995) found that using occupancy data from a single year did not result in realistic predictions about the metapopulation. When parameters for the IFM were estimated from patch occupancy over a five-year period, better results were obtained.

Table 5.2 Results of fitting the incidence function model to occupancy data of the bush cricket Metrioptera bicolor for two areas of Sweden during the period 1989-94. P is mean proportion of available habitats actually occupied from 1990 to 1994. Predicted mean proportion of patches occupied is based on 100 replicates. Adapted from Kindvall (2000).

Parameters Western area Eastern area

Number of available patches 66 50

Actual P 0.82 0.71

Predicted P 0.80 0.62 Multiplier of X

for the rescue effect 0.05 0.001

Table 5.3 The predicted and observed annual extinction and colonization rates for three species of shrews (Sorex) on small islands. Parameters for this mainland-island incidence function model were from 68 islands and applied to a different set of 17 islands. X and e are annual extinction parameters. C = annual colonization rate; E = annual extinction rate. Adapted from Peltonen and Hanski (1991), Hanski (1993).

Species Body size Parameter Predicted Observed

(g) estimates

Table 5.3 The predicted and observed annual extinction and colonization rates for three species of shrews (Sorex) on small islands. Parameters for this mainland-island incidence function model were from 68 islands and applied to a different set of 17 islands. X and e are annual extinction parameters. C = annual colonization rate; E = annual extinction rate. Adapted from Peltonen and Hanski (1991), Hanski (1993).

X

e

C

E

C

E

S. araneus

9

2.30

0.20

0.26

0.04

0.20

0.04

S. caecutiens

5

0.91

0.53

0.03

0.28

0.05

0.33

S. minutus

3

0.46

0.73

0.18

0.53

0.13

0.46

Kindvall (2000) compared the incidence function model to three other spatially realistic models, including a logistic regression model (Sjogren-Gulve and Ray 1996), and found the logistic regression model performed best in its ability to predict regional occupancy, local occupancy, and the number of colonizations and extinctions. Nevertheless he found the IFM did a reasonable job of predicting the actual outcomes for the bush cricket in Sweden.

The application of the IFM to mainland-island populations can be simpler since there are fewer parameters to estimate. One example is a study on the occurrence of small mammal populations on islands in lakes and in the sea (Peltonen and Hanski 1991, Hanski 1993). The IFM was based on the occurrence of three species of shrew (Sorex) on 68 islands. Using parameters estimated from this study, Hanski (1993) predicted the annual colonization and extinction probabilities on 17 additional islands. The observed and the predicted rates were well matched (Table 5.3). In this study only the extinction parameters (X and e) were estimated since colonization was from the mainland and was assumed not to differ among islands (Hanski 1999). The value X is inversely related to the strength of environmental stochasticity: a large value of X means weaker environmental stochasticity. The value of X is directly correlated with body mass in the shrews described in Table 5.3, as well as in birds from four different areas (Cook and Hanski 1995). What this implies is that species with larger mass (such as Sorex araneus) are less affected by environmental stochasticity as compared with species with a smaller mass (Hanski 1999). Table 5.3 also suggests that only Sorex araneus has a long-term metapopulation survivorship, since it is the only species with a C > E.

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