Darren J. Bender and Lenore Fahrig. 2005. Matrix spatial structure obscures the relationship between interpatch movement and patch size and isolation. Ecology 86:1023–1033.

Appendix A. A description of the simulation model.

The spatially-explicit simulation model was designed to examine the influence of landscape structure on the inter-patch movements of animals in realistic landscapes.  The model is capable of importing landscapes from a geographic information system (GIS), and stores the landscape information as a raster grid.  Although a landscape was represented as a grid of cells, movement was modeled in a vector-based fashion.  At each time step, an animal could move up to five cells in distance in any (360º) randomly chosen direction. The five-cell distance (about 1% of the landscape size) was chosen to ensure that an animal could move freely, i.e., was not constrained to move in cardinal directions, and that it could not make unrealistically large "leaps" across the landscape. The animal would move the full five-cell distance if the landscape along the route was homogeneous. However, if a boundary was encountered, the animal would stop at that boundary and make a "decision" about whether to continue on its current path. All boundary-crossing decisions were determined by the relative "perceived hospitability" of the next cell to be entered (expressed in the model as a value between 0 and 1).  Breeding habitat always had the highest perceived hospitability, and movement into these cells from the matrix was automatic (i.e., an animal would never change its movement path if it were moving into a habitat patch from the matrix).  Similarly, an animal would never change its movement path if it took it from a matrix cover type of lower perceived hospitability to a matrix cover type of higher perceived hospitability. However, if the trajectory of a movement path took the animal into a matrix cell of lower perceived hospitability, then a "decision" was made by the animal. 

The decision to cross a patch boundary was calculated in a stochastic fashion by drawing a uniform random number between 0 and just less than 1.  If the number exceeded the difference in perceived hospitability between current cover type and the new cover type to be entered, then the movement would proceed unhindered.  If the random number was less than the difference, then movement would stop at the boundary, and a new direction for movement would be drawn (either from a uniform random distribution or the directed-movement algorithm discussed below). The remainder of the animal's five-cell movement distance could be used to continue moving in its new direction, depending on the outcome of the boundary-crossing decision. Animals always continued to move in successive time-steps until they encountered an unoccupied cell in a habitat patch.  

To simulate directed movement, we used a correlated random-walk routine (Kareiva and Shigesada 1983). For each movement step, the direction of movement was based on the previous step's direction plus some angle drawn at random from an approximately normal distribution (with a mean of zero and a standard deviation of approximately /2).

For each run of the model, the population began with 250 animals placed randomly in habitat cells and the population size remained constant throughout the run. Any animal that moved beyond the boundaries of the landscape was replaced by a new animal that appeared at a randomly selected location along the landscape boundary. A patch immigration event was recorded when an animal moved out of its patch of origin and entered an unoccupied habitat cell in another patch. After an animal immigrated to an unoccupied territory in a new patch, it stopped moving, produced one offspring and then died. Therefore, each individual could only immigrate to a new patch once in its lifespan. There was no mortality in the matrix, i.e., no dispersal mortality. Immigration rate for a patch in a simulation run was calculated as the total number of immigration events into that patch during the run. Therefore, as individuals spent more time within a simulation in the matrix, there tended to be lower rates of immigration to patches in that run. The less time individuals spent in the matrix (the sooner they found new habitat patches), the higher the patch immigration rates, because these individuals were replaced by offspring, which then had a chance to immigrate to a new patch and add to the immigration rates. This simple demographic algorithm ensured that the landscape pattern was the sole influence on immigration rate.