# The Levins or classical metapopulation

According to the Levins model, metapopulation persistence is due to a stochastic balance between local extinction and re-colonization of empty habitat patches. The rate of change in occupied habitat patches is a function of colonization rates (c) and extinction rates (e) as shown in Equation 5.2 (Levins 1969). P is the proportion of patches occupied by the population under consideration.

As described by Hanski (2001), if we define P' as the number of habitat fragments occupied by the species (rather than the proportion), and define T as the total number of habitat patches available, the equation can be modified as follows:

Both of these models are deterministic descriptions of the rate of change of metapopulation size, even though the models are based on stochastic events. Assumptions include:

1 The local populations are identical and have the same behavior;

2 extinctions occur independently in different patches and therefore local dynamics are asynchronous;

3 colonization spreads across the entire patch network and all patches are equally likely to be "encountered;"

4 furthermore, all patches are equally connected to all other patches.

In the Levins model we are not concerned with population dynamics within each population. We do not attempt to assess the number of individuals in each patch; we simply record a patch as occupied or not occupied. For this reason we also do not assess the size or quality of the patches.

The equilibrium value of P can be obtained by setting dP/dt = 0. This produces the expected proportion of patches to be occupied and amounts to a carrying-capacity term such as is found in the logistic equation.

Since P = 0 is not an interesting solution, we have:

The equilibrium value of P, defined as P and found by solving the above for P, is shown in Equation 5.4:

The implication here is that colonization must be greater than extinction or the equilibrium proportion of patches occupied will be zero, and the colonization rate must be greater than the extinction rate for persistence of the metapopulation. If we consider colonization a "birth" event and extinction a "death" event (thereby using c - e as the equivalent of the growth rate, r, in the logistic equation) and we use 1 - e/c to represent a "carrying capacity" term (equivalent to K in the logistic equation) as mentioned above, we can model metapopulation dynamics as a modification of the logistic (Equation 5.5). No matter what the starting patch frequency is (assuming 1 > P > 0), over time it moves to the expected value based on P = 1 - e/c (see Fig. 5.2).

This simple model has helped ecologists develop insights into the consequences of habitat destruction and fragmentation. For example, imagine a fragmented landscape in which a fraction of the habitat patches is destroyed. The extinction rate is not affected, but the colonization rate is. This is because there are fewer local populations and fewer empty patches. If the patch connectivity is reduced, it can be modeled by reducing the value of c. Habitat destruction can therefore lead to a reduction in the proportion of patches that are occupied. Alternatively, if no patches are destroyed but they are reduced in area, this would result in lower average population sizes, which would increase the extinction rate. Simultaneously, colonization rate would be reduced due to the smaller population sizes in the occupied patches. The net result again is a reduction in the fraction of occupied patches (Fig. 5.3).

Figure 5.2 Expected proportion of habitat patches occupied, based on Equation 5.5. In this example P0 = 0.50 and 0.15, the colonization rate c = 0.75, and the extinction rate e = 0.55. The expected proportion of patches occupied at equilibrium = 1 - e/c = 0.27.

Figure 5.2 Expected proportion of habitat patches occupied, based on Equation 5.5. In this example P0 = 0.50 and 0.15, the colonization rate c = 0.75, and the extinction rate e = 0.55. The expected proportion of patches occupied at equilibrium = 1 - e/c = 0.27.

original

Original colonization rate Reduced colonization rate Extinction rate

original

### Proportion of habitat patches occupied

Figure 5.3 (a) The effect on patch occupancy of a lowered colonization rate due to a reduction in the number of habitat patches. (b) Expected changes in patch occupancy with lower colonization rate and increased extinction rate. Colonization rate is reduced by loss of habitat number; extinction rate is increased by reduction in patch area. Adapted from Hanski (1999).

Original colonization rate Reduced colonization rate New extinction rate Original extinction rate original

### Proportion of habitat patches occupied

Figure 5.3 (a) The effect on patch occupancy of a lowered colonization rate due to a reduction in the number of habitat patches. (b) Expected changes in patch occupancy with lower colonization rate and increased extinction rate. Colonization rate is reduced by loss of habitat number; extinction rate is increased by reduction in patch area. Adapted from Hanski (1999).

 Local extinction Metapopulation extinction Stochastic processes (a) Demographic (a) Extinction-colonization interaction (b) Environmental (b) Regional processes Extrinsic causes Habitat loss Habitat loss and fragmentation