The geometry of the Pareto front in biological phenotype space

When organisms perform a single task, selection leads to phenotypes that maximize performance at that task. When organisms need to perform multiple tasks, a trade-off arises because no phenotype can optimize all tasks. Recent work addressed this question, and assumed that the performance at each task decays with distance in trait space from the best phenotype at that task. Under this assumption, the best-fitness solutions (termed the Pareto front) lie on simple low-dimensional shapes in trait space: line segments, triangles and other polygons. The vertices of these polygons are specialists at a single task. Here, we generalize this finding, by considering performance functions of general form, not necessarily functions that decay monotonically with distance from their peak. We find that, except for performance functions with highly eccentric contours, simple shapes in phenotype space are still found, but with mildly curving edges instead of straight ones. In a wide range of systems, complex data on multiple quantitative traits, which might be expected to fill a high-dimensional phenotype space, is predicted instead to collapse onto low-dimensional shapes; phenotypes near the vertices of these shapes are predicted to be specialists, and can thus suggest which tasks may be at play. We studied the effect of trade-offs between tasks on the suite of variation of phenotypes. We find that, for a wide range of performance function shapes, Pareto optimal phenotypes lie in morphospace on low-dimensional polygons with mildly curved edges, whose vertices are phenotypes specialized for a single task, known as archetypes. We also present bounds on the Pareto front for general, non-monotonic performance functions, showing that the suite of variation is restricted to a region spanning the archetypes.