Some of the most challenging problems in ecology, hydrology, and geomorphology arise from processes which play out over large spatial scales and long time intervals. High-resolution global digital elevation models are becoming available. They will allow problems of broad spatial extent to be addressed, but only if we can handle the enormous volume of data. Similarly, performance gains in computers now make it feasible to test more complex theory using computational models, but only if efficient techniques are used. In this dissertation I develop and demonstrate techniques for handling large geospatial raster datasets and rapidly modeling ecological and hydrological processes, producing results that are orders of magnitude more efficient than previous work. Notably, these techniques work on both high-performance machines and laptops. Finally, I apply the techniques to a challenging problem at the interface of ecology, evolution, climate, and geology.

In the past few decades, ecology has gone through exciting breakthroughs in developing and applying quantitative methods to more accurately describe and predict patterns and dynamics. However, a theoretical framework under which models could be built and results interpreted in one language is still missing, impeding the communication and collaboration among subfields of ecology. To serve the ultimate goal of developing such a framework, in this dissertation, a theory is developed to make unified predictions of several facets of ecology that have so far been addressed mostly in independent ways: starting from a simple scenario of resource allocation, the theory simultaneously predicts species coexistence, community-level energy distribution, population demographic growth function, evolutionary tradeoffs and life history strategies. The approach of maximizing information entropy (MaxEnt) is used to make sure that the theory makes the most objective predictions from the fewest ad hoc assumptions. Previous applications of MaxEnt in ecology are reviewed in Chapter 1. In Chapter 2, the fundamental framework and a first model of the theory is introduced. Based on the same framework, several different approaches to expand model predictions are explored in Chapters 3 and 4. As is discussed in Chapters 2-4, the assumptions and parameters of the theory can be related to many important concepts in ecology (e.g. fitness equality, stabilizing effect, niche and neutrality), and the predictions reveal many previously unidentified links between patterns and processes at the population and community level to metabolism and functional traits. Combining all above, the work in this dissertation is potentially a first step towards a unified theoretical framework of ecology.