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The final growth step, road-influenced growth, is determined by the existing transportation infrastructure as well as the most recent urbanization done under steps i, ii and iii. With a probability defined by breed_coefficient, newly urbanized cells (at time t+1) are selected, and the existence of a road is sought in their neighborhoods. If a road is found within a given maximal radius (determined by road_gravity_coefficient) of the selected cell, a temporary urban cell is placed at the point on the road that is closest to the selected cell. Next, this temporary urban cell conducts a random walk along the road (or roads connected to the original road) where the number of steps is determined by the parameter dispersion_coefficient. The final location of this temporary urbanized cell is then considered as a new urban spreading nucleus. If a neighboring cell to the temporary urbanized cell (on the road) is available for urbanization, it will happen (randomly picked among possible candidates). If two adjacent cells to this newly urbanized cell are also available for urbanization it will happen (randomly picked among candidates). Thus the creation of the temporary urbanized cell on the road is defined by
(4.1) U'(k,l,t+1) = f4.1[ U(i,j,t+1), road_gravity_coefficient, R(m,n), random ]
where i,j,k,l,m, and n are cell coordinates, and R(m,n) defines a road cell. The random walk on the road may be expressed by
(4.2) U''(i,j,t+1) = f4.2[ U'(k,l,t+1), dispersion_coefficient, R(m,n), random ].
where (i,j) are road cells neighboring (k,l). If we define the location of the temporary urbanized cell at the end of the random walk by (p,q), the new adjacent urban spreading center will be defined by
(4.3) U'''(i,j,t+1) = f4.3[ U''(p,q,t+1), R(m,n), slope_coefficient, random ],
and two additional adjacent urbanized cells may be added using
(4.4) U''''(i,j,t+1) = f4.4[ U'''(p,q,t+1), slope_coefficient, random ],
where (i,j) and (k,l) belong to the nearest neighborhood of (p,q). Note how this step is similar to notation 3.
The four steps above are collectively referred to as a road trip. Each attempt to select a newly urbanized pixel to move to a road is a new road trip. The number of attempted road trips in any given growth cycle is determined by the breed_coefficient.