Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of n nodes that is decomposed into two disjoint subsets, having m and n − m vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter α ∈ [0, 1] that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter ξ ≡ ξ(n, m, α) that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when ξ < 1/10 (ξ > 10) the eigenvectors are localized (extended), whereas the localization-to-delocalization transition occurs in the interval 1/10 < ξ < 10. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed ξ, the spectral properties of our graph model are also universal.

Water is a basic natural resource for life and the sustainable development of society. Methods to assess water quality in freshwater ecosystems based on environmental quality bioindicators have proven to be low cost, reliable, and can be adapted to ecosystems with well-defined structures. The objective of this paper is to propose an interdisciplinary approach for the assessment of water quality in freshwater ecosystems through bioindicators. From the presence/absence of bioindicator organisms and their sensitivity/tolerance to environmental stress, we constructed a bipartite network, G. In this direction, we propose a new method that combines two research approaches, Graph Theory and Random Matrix Theory (RMT). Through the topological properties of the graph G, we introduce a topological index, called J P ( G ) , to evaluate the water quality, and we study its properties and relationships with known indices, such as Biological Monitoring Working Party ( B M W P ) and Shannon diversity ( H ′ ). Furthermore, we perform a scaling analysis of random bipartite networks with already specialized parameters for our case study. We validate our proposal for its application in the reservoir of Guájaro, Colombia. The results obtained allow us to infer that the proposed techniques are useful for the study of water quality, since they detect significant changes in the ecosystem.

The studies on the evolution of tourist destinations are not a new issue, however, most of them have been focused on consolidated destinations, whereas only a minimum has been done on tourism transformation in rural areas. The objective of this work is to diagnose the evolutionary process of tourism in Coastal Rural Communities (CRCs). To do this, we propose a model which combines two research approaches, Path Dependence and Social Network Analysis. The methodological approach is divided into three parts: design, application and validation, and it is based on collecting in situ and identifying key informants. In the first part, the stages of a Path Dependence for a CRC are conceptualized and bipartite graphs are constructed to show the relationships between: (1) the identified establishments built in a period of time and (2) positive and negative lock-ins with a greater degree of influence on the evolutionary process. On this basis, the resulting graphs are associated with the stages of the Path Dependence. In the second part, the model theoretically raised is applied as an empirical case at the CRC of Playa Ventura, Guerrero, in the Southern Pacific of Mexico. Finally, we validated the model based on the results obtained, which indicate that the model is suitable for the generation of knowledge about the evolutionary process of tourism in CRC’s, and therefore, it opens the possibility of being replicated in other communities with the same characteristics.

From a bibliographic analysis of the major published research studies about the index values used to evaluate water quality, the general characteristics of the so-called Biological Monitoring Working Party (BMWP) index are described; the degree of complexity of water quality assessments is measured and the integration of new technologies is analyzed through the use of Geographic Information Systems (GIS). The BMWP index is studied, as well as its applications from 1978 to 2017, specifying its temporal and spatial distribution, its current trends and complementary values. The analysis is based on the application of the Graph Theory, particularly on the Social Networks Analysis(SNA) from a holistic approach. Impact Factor (JCC): 4.6869 NAAS Rating: 3.58 In Spain, Alba-Tercedor and Sánchez-Ortega (1988) modify the tolerance values of each macro-invertebrate family, adapt the BMWP references to local conditions (BMWP') and define five types and colors associated with the index value, to describe water quality, and in order to develop biological maps. On the other hand, the BMWP-Cu index (Muñoz-Riveaux et al., 2003), is a direct consequence of the BMWP'. Both methodologies describe how to map water quality, but do not show any representation. Additionally, Pérez (2003) suggests that the methodological theory for regionalization of water quality should be tailored to each region. On the other hand, Forero, Longo, Ramírez, Jairo, and Chalar (2014) in Colombia, develop the Multi-Metric Ecological Quality Index (ICE RN-MAE ), as a biological and physicchemical method for the evaluation of water quality. It exploits the physical and chemical parameters, environmental gradients, abundance, and macro-invertebrate genus identification. Furthermore, they compare it with the BMWP-Col index (Pérez, 2003) and prove that both indicators provide similar information on the variability of water quality.Kohlmann, Russo, Itzep, and Solís (2010) employ a cooperative methodology to assess water quality: using the BMWP'-Cr reference in two rivers of rural Costa Rican communities. Moreover, they integrate the participation of the scholar community through a survey, before and after the study of water quality. They conclude that the participants changed their minds about how they perceived their water-related environmental problems. On the other hand, the impact of anthropogenic activities on water quality in Iran is studied through the use of Geographic Information Systems (GIS) and comparative analysis between physicochemical and biological indexes -one of them the BMWP-and show that the biological indexes provide a better approach to the health level of ecosystems. (Sharifinia, Mahmoudifard, Namin, Ramezanpour, & Yap, 2016) In this way, a complete water quality assessment is governed by the monitoring of three major components: hydrological, physicochemical and biological. The latter, should target the response of species or communities to changes in their environment, and determine what the factors and activities that interfere directly...

Sustainable water management is important to ensure its availability for future generations. The study of water quality is fundamental for this purpose. Assessing the health of aquatic ecosystems through bioindicators has been shown to be reliable and inexpensive. The objective of this work was to evaluate water quality through a biomathematical model that involves environmental stress indicator organisms and their close relationship with dissolved oxygen. In this direction, a system of differential equations describing the population dynamics of aquatic macroinvertebrates under the influence of dissolved oxygen is proposed. The model is validated by its application in the Coyuca Lagoon, Mexico. Likewise, population changes over time were represented, which allowed us to deduce that the increase or decrease in the aeration/oxygenation rate significantly affects the population dynamics of the bioindicator organisms. In addition, to classify water quality, a one-to-one correspondence was established between water quality and the equilibrium points of the system of differential equations. The results obtained allow inferring that the proposed techniques are useful for the study of water quality since they can predict significant changes in the ecosystem and provide researchers and water managers with tools for decision making.