Paul R. Levitt

Group selection on the boundary of a stable population

A model of group selection is constructed for the case of differential extinction acting on small boundary populations of a large, fixed population. Consideration is restricted to extinction operators acting at or near to carrying capacity. Under the assumption that the extinction rate is large relative to individual genetic parameters affecting gene frequencies in boundary populations, we discuss the conditions under which differential extinction is most likely to produce a significant effect. In particular, a condition for bimodality in the distribution of gene frequencies in boundary populations (population polymorphism) is that there be some critical allele frequency at which the extinction rate jumps from high to low. An extinction operator linear in allele frequency produces no qualitative effect of this kind. In consequence, we are able to make precise rather limited circumstances under which group selection is likely to have a significant effect.

Blockmodeling complex statutes : Mapping techniques based on combinatorial optimization for analyzing economic legislation and its stress point over time

Blockmodeling, a combinatorial technique for relational data analysis, is applied to studying texts of complex economic legislation. By making this area a subject for mathematical modeling, using methods related to combinatories, logic, and discrete optimization, we describe a new type of frontier between law and economics.

The network matching principle: A model of efficient resource allocation by informal social networks in non-profit and other non-market social structures☆

Abstract A new mathematical model related to the theory of order statistics is introduced to show how informal social networks promote efficient (second-best) resource use in non-market settings. Such networks may play extremely important — if frequently little recognized — roles in ‘matching’ resource packages to their best users in complex social structures subject to many legal and administrative constraints.

The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model.

The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if E(0) − E(1)s > B(1 − h), where s ⪡ 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit.