In the study of the dispersal of species across a landscape, most previous models approximate heterogeneous landscapes by a set of homogeneous patches and allow for different demographic and dispersal rates within each patch. Some work has been done designing and analyzing models which also include a patch preference at the boundaries, which is commonly referred to as a degree of bias. Individuals dispersing across a patchy landscape can detect the changes in habitat at a neighbourhood of a patch boundary, and as a result, they might change the direction of their movement if they are approaching a bad patch. This thesis is devoted to the mathematical derivation of a generalization of the classic Laplace kernel, which includes different dispersal rates in each patch as well as different degrees of bias at the patch boundaries. The simple Laplace kernel and the truncated Laplace kernel most often used in classical work appear as special cases of this general kernel. The form of this general kernel is the sum of two different terms: the classic truncated Laplace kernel within each patch, and a correction accounting for the bias at patch boundaries.
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