For surface erosion, the Revised Universal Soil Loss Equation has a slope and length of slope factors that could be helpful in addressing this general subject area and perhaps identify a critical slope.  There is a difference between ground slope and rainfall type erosion and gradient that concentrated flow follows in rills, channels or gullies and the resulting erosion of concentrated flow network.  There is on line some information on Water Erosion Prediction Project (WEPP) developed in USA that may be helpful to consider.  The critical gradient is going to be in part based on soil type, erodibility, cover, gradient, disturbance, and storm severity, but these would consider the elements, but you may have to read up on model details and assumptions, methods, and collect some data to use the models.  

The identification of the location of inception of channels from Digital Elevation Model based on a critical threshold is a long debated issue. For instance, you may find some references in an old paper of mine: Pilotti, M., Gandolfi C., Bischetti G.B. Identification and Analysis of Natural Channel Networks from Digital Elevation Models , Earth Surface Processes and Landforms, 21, 1007-1020, 1996.

I do not think you'll find a critical value of slope for water erosion or tropiales environments or anywhere else. Water erosion depends on the involvement of many fatores to establish critical thresholds of any of them, regardless of others, it would not help much.