## Introduction

Biological systems often need to perform more than one task. A given design or shape – that is, a phenotype – cannot usually be optimal at all tasks at the same time. This situation gives rise to a fundamental trade-off (Arnold 1983). Such trade-offs have been widely studied in ecology; examples include life history aspects such as fertility versus offspring survival (Stearns 1992), and performance measures such as speed versus endurance in lizards (Vanhooydonck et al. 2001), foraging scale versus precision (Campbell et al. 1991) and growth versus shell robustness in snails (Trussell 2000). The broad context of this study is to ask how such trade-offs affect the range of phenotypes found in nature.

Recently, Pareto optimality was used to understand the range of phenotypes that best resolve such trade-offs (Shoval et al. 2012). To define Pareto optimality, consider a system with n traits (quantitative traits such as size and shape parameters). A phenotype *v* is a vector of trait values, and can be described as a point in morphospace – the space of trait values.

Assume that the system needs to perform *k* different tasks. The phenotype's performance *p*_{i}(*v*) at each task *i* is a function of its trait values, *v*. The fitness of the organism, *F*(*v*), is an increasing function of its performance at each task *F*(*v*) = *f*_{h}(*p*_{1}(*v*), *p*_{2}(*v*), …, *p*_{k}(*v*)) (Arnold 1983). The function *f*_{h} describes the relative importance of the performance of each task in determining the fitness in niche *h*. In the following, we do not need to know the explicit form of *f*_{h}, only that it increases with performances. Note the difference between performance and fitness: the fitness function *f*_{h} is associated with a given niche and determines which phenotype will be selected at that niche. The fitness combines the different performances in a way that is relevant to that given niche. The performance functions are global and do not depend on the niche. It is usually easier to experimentally measure performances in the lab than fitness in the wild (Arnold 1983).

Pareto optimality is usually defined in performance space (schematically shown in Fig. 1). If phenotype *v* has higher performance at all tasks than phenotype *v’*, one can erase *v’*. Eliminating all such *v’* phenotypes results in the Pareto front. Moving along the front leads to improvement in some tasks at the expense of others. The front is the set of best compromises. Note that natural selection tends to select phenotypes on the Pareto front (or close to the front, the closer the higher the selection pressure), rather than phenotypes that are off the front (Oster and Wilson 1979; Farnsworth and Niklas 1995; El Samad et al. 2005; Kennedy 2009; Warmflash et al. 2012). This is due to the fact that fitness is an increasing function of each of the performances. Each niche *h* corresponds to a different point on the front, determined by the relative importance of the different tasks in that niche, as defined by the particular form of the fitness function *f*_{h}.

Most studies of Pareto optimality in economics and engineering focus on performance space (Steuer 1986). Few studies explore the trait space (morphospace), as we do in this article. An exception is location theory that studies optimizing functions of a distance from given points (Kuhn 1967; Thisse et al. 1984; Durier and Michelot 1986).

Shoval et al. (2012) calculated the shape of the Pareto front in morphospace. To do so, three assumptions were made. The main aim of this study is to explore the effects of relaxing these assumptions. The first assumption is that for each task *i*, there is a single phenotype that maximizes the corresponding performance function *Pi*. This phenotype is called the *archetype* for task *i*. Relaxing this assumption means that performance can be maximized at multiple points.

The second assumption is that the performance of a phenotype is a decreasing function of its distance from the archetype for that task: *P*_{i}(*v*) = *p*_{i}(*d*_{i}(*v*)), where and . The important point here is the existence of a distance metric, more specifically an inner-product norm distance on the morphospace. This distance function governs the decrease of the performance functions. An inner-product norm is defined by , where *M* is a positive-definite matrix. One example for such a norm is Euclidian distance (given by *M* = *I*, the identity matrix). Relaxing this assumption means that performance decays not with a distance metric away from its maximum.

The third assumption was that all performance functions decay with the *same* norm from their maxima. Relaxing this assumption means that each performance decays with a different norm.

Under these assumptions, it was shown (Shoval et al. 2012) that the Pareto front is the convex hull of the archetypes. In other words, phenotypes on the Pareto front are linear combinations of the *k* archetypes: , with nonnegative coefficients , that sum to one . The Pareto front for two tasks is a line segment that connects the two archetypes; three tasks result in a triangle shaped Pareto front. Four tasks result in a tetrahedron, etc. (see Fig. 2). These results generalize previous theorems in location theory, such as (Kuhn 1967), which considered only Euclidean norms, and did not make a connection with biological evolution. Consequently, no matter how large the number of traits in the system – as long as they correspond to tasks that show trade-offs – the theory predicts that naturally selected phenotypes fall on a low-dimensional space, and within that space on a polytope (line, triangle, etc.). The vertices of the polytope are the archetypes. In practice, one can fit a polytope to the data, and discover the potential archetypes, which are the vertices of the polytope. The niches or behaviors of the species in the dataset closest to the archetypes give clues as to what tasks might be at play. Evidence for such lines and triangles was presented by Shoval et al. based on classic studies of animal morphology, and bacterial gene expression datasets.

Here we ask what happens if we relax these assumptions. The article is organized as follows: we first relax assumption (iii), to consider a different inner-product norm for each performance function. We then relax assumption (i) to consider cases where performance is maximized in a region and not at a single point. Finally, we relax all assumptions, and consider general performance functions that need not be monotonic or depend on a norm.

Our main conclusions are that the shape of the Pareto front for the case of different norms is composed of mildly curved hyperbolae. We also present a theorem that places bounds on the Pareto front in cases of general, non-monotonic performance functions. Generally, relaxing the assumptions of Shoval et al. changes the straight edges of the polytopes to mildly curved ones. Table 1 lists the results in this study that go beyond the study of Shoval et al. (Shoval et al. 2012).

Shoval et al. 2012 | Present study | |
---|---|---|

Shape of Pareto fronts for two tasks in a 2-dimensional morphospace with different inner-product norms | Numerical calculation showed that shape can be curved (Figs. 3D and S2) | The curve is analytically solved, found to be a hyperbola (Appendix S2) |

Shape of Pareto fronts for two tasks in an N-dimensional morphospace with different inner-product norms | Not discussed | Is calculated along with 2D projections. Axes can be rotated such that all projections on principal planes are hyperbolae. (Appendix S3) |

The maximal deviation of the Pareto front from a straight line | Was calculated numerically (Fig. S3) | Is calculated analytically. Bounds are provided. (Appendix S4) |

Pareto fronts for three tasks with different norms are curved triangles or multi-connected regions | Mentioned. | Proved (Appendix S7) |

Relaxing the assumption that the Pareto front is maximized at a single point | Discussed for the case of two tasks and performance that decays with Euclidean norm | The case of three tasks is discussed (Appendix S8) |

Inverse problem of deducing the norms from the shape of the Pareto front | Not studied | Studied for 2 and 3 tasks in 2D (Appendices S5 and S6) |

Bounds for the Pareto front in the case of general performance functions (not decaying with a norm, not necessarily monotonic) | Not studied | Proved to be restricted to a region near the archetypes (Appendix S9) |