risk. The significance of our results, their consequence for general dispersal patterns and their relations to previous works are addressed in detail in Section \[sec:discussion\].

Material and Methods
====================

For ease of discussion we focus on the metapopulation model used in previous approaches [@Vuilleumier06; @Bode08; @Vuilleumier10]. This stochastic patch occupancy model connects a number of $N$ patches through a complex dispersal matrix; the model is detailed in Section \[sec:metapopulation-model-vuilleumier\]. Within the scope of this work the viability of metapopulations exposed to dispersal patterns with different degree of symmetry is investigated. A consistent definition of the degree of symmetry and details on the dispersal patterns are provided in Sections \[sec:symmetry-def\] and \[sec:viab-metap-conn\].

\[sec:metapopulation-model-vuilleumier\]Metapopulation model
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We consider a metapopulation consisting of $N$ patches of equal quality, where, at a given time, each patch is either empty ($0$) or populated ($1$). Interactions of the patches are specified by means of the $N\times N$ connectivity matrix $D$, where the elements $d_{ij}\in\{0,1\}$ determine whether patch $j$ is connected to patch $i$ ($d_{ij}=1$) or not ($d_{ij}=0$). For ease of discussion we require $d_{ii}=0$ for all $i$ implying that patches are not connected with themselves.

Building on previous works we used a stochastic discrete time model 