Problem SE13

Consider a simple metapopulation consisting of 2 patches linked by migration. The area of patch 1 is 1 km$^2$, that of patch 2 is 4 km$^2$. The probability that a propagule released by patch i reaches patch j is $l_{ij}$ = 0.37 exp( $-\alpha d_{ij}$) where $\alpha$ = 0.3 km$^{-1}$ and $d_{12} = d_{21}$ = 5 km. The colonization rate is c = 0.15 km$^{-2}$ year$^{-1}$, while the extinction rate is e = 0.1 km$^{2}$ year$^{-1}$.

Write down the two Levins-like equations describing the dynamics of $p_1$ and $p_2$, namely the probabilities that patch 1 and patch 2 are occupied. Analyze the equations via the isocline method and find out the fate of the metapopulation. If it can persist, calculate $p_1$ and $p_2$ at equilibrium. Calculate the metapopulation capacity and verify the condition for metapopulation persistence.

Consider a 3-patch metapopulation in which patch 2 and patch 3 are linked to patch 1, but patch 2 and 3 are not connected by migration (Fig. 9).

Figure 9: Scheme of the 3-patch metapopulation.

More precisely, the probability $l_{ij}$ that a propagule released by patch i reaches patch j is detailed as follows: $l_{23}$ = $l_{32}$ = 0, $l_{12}$ = $l_{21}$ = 0.027 exp( $-\alpha d_{12}$), $l_{13}$ = $l_{31}$ = 0.09 exp( $-\alpha d_{13}$) with $\alpha$ = 0.4 km$^{-1}$. The areas of the patches are $A_1$ = 10 km$^2$, $A_2$ = 7 km$^2$, $A_3$ = 4 km$^2$. The colonization rate is c = 0.1 km$^{-2}$ year$^{-1}$, while the extinction rate is e = 0.08 km$^{2}$ year$^{-1}$. Calculate the metapopulation capacity and check whether the condition for metapopulation persistence is verified. Assume that ecological corridors are built to improve the migration between 1 and 2 and between 1 and 3. This implies a decrease of $\alpha$ from 0.4 km$^{-1}$ to 0.25 km$^{-1}$. Check the condition for persistence again.