The Levins model has a special significance for metapopulation ecology, as it was with this model that Richard Levins introduced the metapopulation concept in two papers published in 1969 and 1970. The Levins model represents a completely different approach to metapopulation modeling in comparison with the two-population model. The assumptions of the Levins model are: (1) The suitable habitat occurs in infinitely many patches that are equally large and of the same quality. (2) The patches have only two possible states, occupied versus empty, and hence the Levins model is an example of patch occupancy metapopulations models, in which local dynamics are ignored. (3) Local extinctions and colonizations occur independently in different patches. (4) All local populations are equally connected to other populations and patches, which is another way of saying that the model is spatially implicit.

With these assumptions, the size of the metapopulation can be described by the fraction of the currently occupied patches, denoted by p(t). Temporal changes in the value ofp(t) are given by dp(t)/dt = cp(/)(1 -p[t\) - ep(t) [1]

where c and e are colonization and extinction rate parameters. The first term describes the rate of colonization of currently unoccupied patches (fraction 1—p of all patches), while the second term gives the rate of extinction of currently occupied patches (fraction p). Local extinctions may be due to demographic and environmental stochasticity, but at the metapopulation level local stochasticity translates to a constant extinction rate parameter. Note that because the existing local populations are assumed to be identical, they all contribute equally to the rate of colonization, and hence the colonization term is proportional to p as well as to 1 —p.

The equilibrium metapopulation size is given by p* = 1— 6, where 6 = e/c is called the extinction threshold. In the Levins model a species that can invade a currently unoccupied patch network has a positive equilibrium size in the network, and vice versa (Figure 1a). This is not so in more complex models that may have alternative stable equilibria (see section titled 'Population size-structured models'). The basic reproductive number R0 in the Levins model is given by 1/ 6 , which has to exceed unity for the species to be able to spread into an empty patch network.

Colonization rate Morrison rate

Figure 1 This figure shows the metapopulation equilibria in relation to increasing colonization rate. Locally stable equilibria are shown by continuous line, unstable equilibria by broken line. (a) In the Levins model, the metapopulation may invade a patch network at the extinction threshold, at which point metapopulation extinction (p* = 0) becomes an unstable equilibrium and only the positive equilibrium is stable. (b) In population size-structured models with strong rescue effect, there are two alternative stable equilibria for some parameter values, in which a metapopulation that is originally sufficiently large may persist in a network that it cannot invade from small size (in the example colonization rate between 0.11 and 0.2). From Hanski I (1999) Metapopulation Ecology. Oxford: Oxford University Press.

Figure 1 This figure shows the metapopulation equilibria in relation to increasing colonization rate. Locally stable equilibria are shown by continuous line, unstable equilibria by broken line. (a) In the Levins model, the metapopulation may invade a patch network at the extinction threshold, at which point metapopulation extinction (p* = 0) becomes an unstable equilibrium and only the positive equilibrium is stable. (b) In population size-structured models with strong rescue effect, there are two alternative stable equilibria for some parameter values, in which a metapopulation that is originally sufficiently large may persist in a network that it cannot invade from small size (in the example colonization rate between 0.11 and 0.2). From Hanski I (1999) Metapopulation Ecology. Oxford: Oxford University Press.

The Levins model is a deterministic approximation of a model known as the stochastic logistic model, which is an example of stochastic patch occupancy models. The stochastic logistic model assumes a finite network of n patches. If a patch is occupied, it is assumed to go extinct with a fixed rate E = e, while the colonization rate of an empty patch is assumed to depend on the fraction of occupied patches, C = ck/n, where k is the number of currently occupied patches. These assumptions define a Markov process with metapopulation extinction as an absorbing state, that is, sooner or later the entire metapopulation will go extinct. A key prediction of the model is the mean time to extinction, T. Using a diffusion approximation to analyze the model, which is well justified when the number of patches is large, we obtain:

where p* is the size of the metapopulation at quasi-equili-brium. Figure 2 gives the number of patches n that the network must have to make T at least 100 times as long as the expected lifetime of a single local population. For metapopulations with large p* a modest network of around ten patches is sufficient to allow long-term persistence, but for rare species (say p* < 0.2) a large network of n > 100 is needed for long-term persistence.

The stochastic logistic model includes extinction-colonization stochasticity but no regional stochasticity, which leads to correlated extinctions and colonizations. In the presence of regional stochasticity the mean time to

1000

1000

Occupancy state p *

Figure 2 The number of habitat patches needed to make the mean time to metapopulation extinction at least 100 times longer than the mean time to local extinction. The dots show the exact result based on the stochastic logistic model, while the line is based on the approximation given by equation (2). From Ovaskainen O and Hanski I (2003) Extinction threshold in metapopulation models. Annales Zoologici Fennici 40: 81-97.

metapopulation extinction does not increase exponentially with increasing n as predicted by eqn [2] but as a power function of n, the power decreasing with increasing correlation in extinction and colonization rates. This result is analogous to the effects of demographic and environmental stochasticities on the lifetime of single populations.

The stochastic logistic model extends the Levins model by incorporating extinction-colonization stochasticity in the model. Population size-structured models represent another type of extension. These models are deterministic and retain the assumption of infinitely many identical and equally connected patches, but the models include a description of local dynamics and migration in the same manner as the two-population models. The simplest structured model divides existing local populations into just two classes, small and large, which is sufficient to model one important new process in comparison with the Levins model: migration from existing large populations (which send out many migrants because they are large) may increase the growth rate in small populations and thereby rescue them from local extinction. At the metapopulation level, the rescue effect leads to the novel prediction of alternative stable equilibria, one of which is metapopulation extinction, the other one a positive metapopulation size (Figure 1b). Therefore, a metapopulation may persist in a network that cannot be invaded from small metapopulation size.

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