Gains from switching and evolutionary stability in multi-player matrix games.

In this paper we unify, simplify, and extend previous work on the evolutionary dynamics of symmetric N-player matrix games with two pure strategies. In such games, gains from switching strategies depend, in general, on how many other individuals in the group play a given strategy. As a consequence, the gain function determining the gradient of selection can be a polynomial of degree N-1. In order to deal with the intricacy of the resulting evolutionary dynamics, we make use of the theory of polynomials in Bernstein form. This theory implies a tight link between the sign pattern of the gains from switching on the one hand and the number and stability of the rest points of the replicator dynamics on the other hand. While this relationship is a general one, it is most informative if gains from switching have at most two sign changes, as is the case for most multi-player matrix games considered in the literature. We demonstrate that previous results for public goods games are easily recovered and extended using this observation. Further examples illustrate how focusing on the sign pattern of the gains from switching obviates the need for a more involved analysis.

Maximal operators and differentiation theorems for sparse sets

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Two weight inequalities for discrete positive operators

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Two weight inequalities for maximal truncations of dyadic Calderón-Zygmund operators

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Coefficients of Drinfeld modular forms and Hecke operators

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The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

The snail hosts of schistosomiasis: some evolutionary and ecological perspectives in relation to control

Despite opportunities for radiation provided by spatio-temporal isolation, the basic morphological plan of pulmonate snails has remained conservative. In consequence of the resulting dearth of morphological characters and their plasticity, there is a case for using biochemical characters such as exogenous chemicals released by the snails (e.g. amino acids) and their chemoreception niche as taxonomic aids to classify snails of medical importance. As these same chemicals are used by snails to distinguish conspecifics they could also be used as "environmental antibodies" in controlled release formulations (CRF's) designed to remove target snails in a specific, cost-effective and ecologically acceptable manner. The snails, surface-living bacteria, algae and macrophytic plants are considered as co-evolved, interactive modular systems with strong mutualistic elements. Recently, anthropogenic perturbations such as deforestation, and damming of flowing waters, have benefited these modules whereas others such as river canalization, acid deposition, accumulation of pesticide residues and eutrophication have harmed them. Research is needed to elucidate the factors which limit the growth of snails in primitive habitats, uninfluenced by man, as well as in those subject to harmful anthropogenic factors. The understanding thus gained could be applied to develop cost-effective primary health care strategies to reduce or prevent transmission of schistosomiasis and other water related diseases.

Evolutionary and convergence stability for continuous phenotypes in finite populations derived from two-allele models.

The evolution of a quantitative phenotype is often envisioned as a trait substitution sequence where mutant alleles repeatedly replace resident ones. In infinite populations, the invasion fitness of a mutant in this two-allele representation of the evolutionary process is used to characterize features about long-term phenotypic evolution, such as singular points, convergence stability (established from first-order effects of selection), branching points, and evolutionary stability (established from second-order effects of selection). Here, we try to characterize long-term phenotypic evolution in finite populations from this two-allele representation of the evolutionary process. We construct a stochastic model describing evolutionary dynamics at non-rare mutant allele frequency. We then derive stability conditions based on stationary average mutant frequencies in the presence of vanishing mutation rates. We find that the second-order stability condition obtained from second-order effects of selection is identical to convergence stability. Thus, in two-allele systems in finite populations, convergence stability is enough to characterize long-term evolution under the trait substitution sequence assumption. We perform individual-based simulations to confirm our analytic results.

Immunological system and Schistosoma mansoni: co-evolutionary immunobiology. What is the eosinophil role in parasite-host relationship?

Schistosomes, ancestors and recent species, have pervaded many hosts and several phylogenetic levels of immunity, causing an evolutionary pressure to eosinophil lineage expression and response. Schistosoma mansoni adult worms have capitalized on the apparent adversity of living within the mesenteric veins, using the dispersion of eggs and antigens to other tissues besides intestines to set a systemic activation of several haematopoietic lineages, specially eosinophils and monocytes/macrophages. This activation occurs in bone marrow, spleen, liver, lymph nodes, omental and mesenteric milky spots (activation of the old or primordial and recent or new lymphomyeloid tissue), increasing and making easy the migration of eosinophils, monocytes and other cells to the intestinal periovular granulomas. The exudative perigranulomatous stage of the periovular reaction, which present hystolitic characteristics, is then exploited by the parasites, to release the eggs into the intestinal lumen. The authors hypothesize here that eosinophils, which have a long phylogenic story, could participate in the parasite - host co-evolution, specially with S. mansoni, operating together with monocytes/ macrophages, upon parasite transmission.

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Index theory for basic Dirac operators on Riemannian foliations

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.

Hankel operators on holomorphic Hardy-Orlicz spaces

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Evolutionary Control of Infectious Disease: Prospects for Vectorborne and Waterborne Pathogens

Evolutionary theory may contribute to practical solutions for control of disease by identifying interventions that may cause pathogens to evolve to reduced virulence. Theory predicts, for example, that pathogens transmitted by water or arthropod vectors should evolve to relatively high levels of virulence because such pathogens can gain the evolutionary benefits of relatively high levels of host exploitation while paying little price from host illness. The entrance of Vibrio cholerae into South America in 1991 has generated a natural experiment that allows testing of this idea by determining whether geographic and temporal variations in toxigenicity correspond to variation in the potential for waterborne transmission. Preliminary studies show such correspondences: toxigenicity is negatively associated with access to uncontaminated water in Brazil; and in Chile, where the potential for waterborne transmission is particularly low, toxigenicity of strains declined between 1991 and 1998. In theory vector-proofing of houses should be similarly associated with benignity of vectorborne pathogens, such as the agents of dengue, malaria, and Chagas' disease. These preliminary studies draw attention to the need for definitive prospective experiments to determine whether interventions such as provisioning of uncontaminated water and vector-proofing of houses cause evolutionary reductions in virulence