In this work we use the geometrical point of view of the Ronchi test and the caustic-touching theorem to describe the structure of the ronchigrams for a parabolical mirror when the point light source is on and off the optical axis and the grating is placed at the caustic associated with the reflected light rays. We find that for a given position of the point light source the structure of the ronchigram is determined by the form of the caustic and the relative position between the grating and the caustic. We remark that the closed loop fringes commonly observed in the ronchigrams appear when the grating and the caustic are tangent to each other. Furthermore, we find that the caustic locally has singularities of the purse or hyperbolic umbilic type, and the ronchigram obtained when the grating is located at certain specific positions at the caustic locally is of the serpentine type. The main motivation of this work is that nowadays a quantitative analysis of the Ronchi test is applied only when the grating is outside the caustic, and we claim that by working at the caustic, the sensitivity of the Ronchi test will be improved. Therefore, a clear understanding of the properties of the ronchigrams when the grating is placed at the caustic will be needed to extend the Ronchi test to that region.

Coexistence of apparently similar species remains an enduring paradox in ecology. Spatial structure has been predicted to enable coexistence even when population-level models predict competitive exclusion if it causes each species to limit its own population more than that of its competitor. Nevertheless, existing hypotheses conflict with regard to whether clustering favours or precludes coexistence. The spatial segregation hypothesis predicts that in clustered populations the frequency of intra-specific interactions will be increased, causing each species to be self-limiting. Alternatively, individuals of the same species might compete over greater distances, known as heteromyopia, breaking down clusters and opening space for a second species to invade. In this study we create an individual-based model in homogeneous two-dimensional space for two putative sessile species differing only in their demographic rates and the range and strength of their competitive interactions. We fully characterise the parameter space within which coexistence occurs beyond population-level predictions, thereby revealing a region of coexistence generated by a previously-unrecognised process which we term the triadic mechanism. Here coexistence occurs due to the ability of a second generation of offspring of the rarer species to escape competition from their ancestors. We diagnose the conditions under which each of three spatial coexistence mechanisms operates and their characteristic spatial signatures. Deriving insights from a novel metric — ecological pressure — we demonstrate that coexistence is not solely determined by features of the numerically-dominant species. This results in a common framework for predicting, given any pair of species and knowledge of the relevant parameters, whether they will coexist, the mechanism by which they will do so, and the resultant spatial pattern of the community. Spatial coexistence arises from complementary combinations of traits in each species rather than solely through self-limitation.

There are economic and physical limitations when applying prevention and control strategies for urban vector borne diseases. Consequently, there are increasing concerns and interest in designing efficient strategies and regulations that health agencies can follow in order to reduce the imminent impact of viruses like Dengue, Zika and Chikungunya. That includes fumigation, abatization, reducing the hatcheries, picking up trash, information campaigns. A basic question that arise when designing control strategies is about which and where these ones should focus. In other words, one would like to know whether preventing the contagion or decrease vector population, and in which area of the city, is more efficient. In this work, we propose risk indexes based on the idea of secondary cases from patch to patch. Thus, they take into account human mobility and indicate which patch has more chance to be a corridor for the spread of the disease and which is more vulnerable, i.e. more likely to have cases?. They can also indicate the neighborhood where hatchery control will reduce more the number of potential cases. In order to illustrate the usefulness of these indexes, we run a set of numerical simulations in a mathematical model that takes into account the urban mobility and the differences in population density among the areas of a city. If we label by i a particular neighborhood, the transmission risk index (TR_i) measures the potential secondary cases caused by a host in that neighborhood. The vector transmission risk index (VTR_i) measures the potential secondary cases caused by a vector. Finally, the vulnerability risk index (VR_i) measures the potential secondary cases in the neighborhood. Transmission indexes can be used to give geographical priority to some neighborhoods when applying prevention and control measures. On the other hand, the vulnerability index can be useful to implement monitoring campaigns or public health investment. A disease is called vector-borne when it is not transmitted directly among humans, but in a human-vector-human way. Examples of major importance, due to its epidemiological magnitude, are Dengue, Chikungunya and Zika tropical diseases, for which the main vector is the Aedes aegypti mosquito. Usually, some indexes are used to measure the potential damage that such a disease could cause, which concern infectivity and entomological issues, human mobility and geographical density, etc. We consider a mathematical epidemiological model which takes specifically into account these spatial factors (mobility, density), and propose a set of new risk indexes which give information about how the epidemic spread could occur, and where the outbreaks could take place. These indexes come from a preventive perspective, and they also pay attention to the epidemic dynamics. Consequently, they potentially allow to proceed in the correct places, and before troubles have arisen. We think these new tools could eventually help health agencies in the design of efficient and effective strategies on prevention and control of vector-borne diseases epidemics.