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Winking patches

Levins [8] introduced the concept of a metapopulation with the continuous-time patch-occupancy model


where tex2html_wrap_inline2256 is the set of possible equilibria. In this model empty patches are infected by occupied patches due to mass action, and local extinction is a random event independent of the metapopulation state. Extinction and migration are processes at a time scale exceeding that of local dynamics. Hanski [7] tried to make migration act on a faster time scale by assuming a rescue effect on patch extinction, leading heuristically to the model


Both the Levins and Hanski models require m;SPMgt;e for a non-trivial equilibrium. Due to our assumption that migration is such a frequent event that it influences local dynamics, empty patches do not exist in our model unless they are unsuitable for habitation. Therefore the adjustment to the Levins model made by Hanski can only be valid if habitat becomes unsuitable after a disaster.

Gotelli [3] argued that re-colonization might be due to migration out of a mainland, resulting in the model


This model does not have a trivial equilibrium unless m=0. Note that this system resembles an open system for population size. Since in our model habitat recovery is independent of dispersers, re-colonization is described by tex2html_wrap_inline2262 which is similar to tex2html_wrap_inline2264 in the Gotelli model. However, in our model re-colonization is part of a closed system, and reflects habitat recovery instead of migration from an external pool.

Patch-occupancy is influenced by the metapopulation through the extinction rate tex2html_wrap_inline2194 only. In case tex2html_wrap_inline2194 is constant, equation (29) for tex2html_wrap_inline2212 is exactly that of Gotelli. However, it only gives information about habitat quality, and none about the state of the species living in it. The existence of a viable population then depends on the dynamics of the local populations on a random pattern of suitable habitat patches. If a population exist then tex2html_wrap_inline2212 is indeed the fraction of occupied patches.

In many field experiments a positive relation between tex2html_wrap_inline2210 and tex2html_wrap_inline2212 is found. Does our model lead to this conclusion? To answer this question we simplify the model further by assuming, as in the Hanski [7] model, that the dynamics on patch level and disperser level are faster than the extinction dynamics, and such fast that the local population is in steady-state, i.e. let v(x) be such that tex2html_wrap_inline2280 , tex2html_wrap_inline2282 for some tex2html_wrap_inline2284 . Then all habitable patches will have density tex2html_wrap_inline2286 , and tex2html_wrap_inline2288 , with tex2html_wrap_inline2290 the Dirac delta function. This results in


and so the equation for patch occupancy based on fast local dynamics becomes


in which tex2html_wrap_inline2286 is computed from


With this simple model we can deal analytically with the relation between occupancy and population size. The steady-state values, tex2html_wrap_inline2286 and tex2html_wrap_inline2256 , are fully determined by the parameters and their directional changes are given by the equations


Whenever tex2html_wrap_inline2194 is an increasing function of x, the directional change of tex2html_wrap_inline2286 and tex2html_wrap_inline2256 are opposite. An increase in tex2html_wrap_inline2286 does not necessarily result in an increase of patch occupancy and mean population size tex2html_wrap_inline2210 , and tex2html_wrap_inline2210 might increase while tex2html_wrap_inline2256 is decreasing. If tex2html_wrap_inline2194 is a decreasing function of x, both patch occupancy and mean population size tex2html_wrap_inline2210 are increased. Since many field experiments show a positive relation between tex2html_wrap_inline2210 and tex2html_wrap_inline2212 our model indicates that these natural populations behave as model tex2html_wrap_inline2324 , a disaster model put forward to mimic extinction due to demographic stochasticity.

Figure 6: Probability density functions of unsafe (near bifurcation) (left picture) and safe (far from bifurcation) (right picture) meta-populations for the models in fig.1a-5a.

The assumption that empty patches are unsuitable for habitation is put in the model rather artificially in order to mimic a Levins-type model, in a situation where empty patches are impossible due to a high migration pressure. Let us return to the full model and suppose that patches are always habitable. Our model then bears a result equivalent to the migration-extinction balance of the Levins model. If the difference in time scale between the local dynamics and extinction is enlarged (Levins: m increased relative to e, our model: r increased relative to tex2html_wrap_inline2192 ), the fraction of high density patches is increased (fig. 6). So, if we consider patches with a density smaller than an arbitrary small number as empty (cf. [5]), the fraction of empty patches becomes smaller if either the extinction rate is decreased or the local dynamics increased (fig. 7).

Figure 7: Solid line: Fraction of patches above detection threshold 0.025 for example (II,i,a,q=1). Parametervalues as in fig. 1a. Dashed line: Fraction of occupied patches in the Levins (1970) model 1-e/m redefined as tex2html_wrap_inline1950 .

Another analogy is found in the requirements for the existence of a non-trivial equilibrium. In the Levins model it is required that colonization exceeds extinction, i.e. m;SPMgt;e . In our model the trivial equilibrium is unstable when at low density the net per capita reproduction within a patch exceeds extinction (Eq. (28)), i.e. in the case of logistic growth and any extinction rate from our examples,


In other words, at low densities the patch extinction rate is equal to the death rate of an individual, and migration can only take place when population density can grow away from zero locally.

next up previous
Next: Single population Up: Model comparison Previous: Model comparison

John Val
Wed Feb 26 07:30:07 EST 1997