Introduction
The existence of a diversity gradient, where the maximums are present in equatorial regions and the minimums in polar regions, is observed in many groups of organisms(Willig, Kaufman, and Stevens 2003; Hawkins et al. 2003) This gradient seems to be explained by the Energetic, Water-Energy, Altitude, Climatic Heterogeneity, Primary Productivity, and Metabolic hypotheses(Wright 1983; Hawkins and Porter 2003a; Colwell and Lees 2000). The Water-Energy hypothesis presents the greatest causal effect on species richness distribution(Hawkins et al. 2003) and predicts richness as a product of evapotranspiration and quantity of water available on the considered place(O’Brien and Road 1998).
The relationship between species richness and the variables that represent each of these hypotheses is considered linear and stationary(Angermeier and Schlosser 1989; Wylie and Currie 1993; Tedesco et al. 2005). Stationary relationships are characterized by an equal relation of cause and effect throughout the whole geographical extension of the data considered (Osborne et al. 2007). In this way, the global model (obtained utilizing all the data) is the same as local models (obtained from a subgroup of the data). In fact, stationarity is an assumption for the application of global models(Zar 2010). In many cases this assumption is neither tested nor observed in diversity gradients (Foody 2004; Cassemiro et al. 2007), consequently producing imprecise global models with few representatives of local estimates(Foody 2004).
An example of a non-stationary relationship is assumed by the Water-Energy hypothesis, that is, in lower latitudes the quantity of water available in the system is the limiting factor of diversity, whereas in higher ones, energy (evapotranspiration) determines the richness gradient (Gaston 2000; Hawkins and Porter 2003b; Eiserhardt et al. 2011). This occurs because areas of low latitudes suffer little influence from Earth’s precession movements, because of they are located near the equator. Consequently, the energy input (from the Sun) occurs in a uniform way throughout the year, making water the limiting factor of diversity. On the other hand, areas located at high latitudes suffer greater influence from Earth’s precession, so there are periods of the year when the hemispheres receive grater or smaller amounts of energy.
Non-stationarity is the main cause of non-significative relationships(Osborne et al. 2007), since the variation of regression’s coefficients is ignored and the relationships are described by their average along the geographic space (Foody 2004). Morphological terrain variations and environmental heterogeneity conditions, at the spatial or temporal level, are the main causes of non-stationarity in the relationships of diversity with the environmental variables considered by each hypothesis(O’Brien and Road 1998; Bickford and Laffan 2006). Modifications of the topographical relief can isolate or connect isolated populations, promoting or even avoiding speciation. Heterogeneity of environmental conditions also creates an environment with high habitat diversity, allowing the coexistence of species(Bickford and Laffan 2006). Therefore, a non-stationarity relationship between species richness and environment is a characteristic that cannot be overlooked in studies about diversity gradients(Foody 2004).
For tropical stream fishes it was shown that diversity distribution is a product of the interactions of three hypotheses: (i) Water-Energy; (ii) Terrestrial Primary Productivity and (iii) Climatic Temporal Heterogeneity(Vieira et al. 2018). Despite this, the Brazilian geographical area presents high morphologic and climatic heterogeneity, possessing regions with elevated temperatures and pluviosity (north/northeast regions), regions with low temperatures and high pluviosity (south and part of the southeast regions) and regions with elevated temperatures and long dry periods (midwest and northeast regions(Marengo and Valverde 2007)). Therefore, our objective is to quantify the stationarity of the relationship between stream fish richness and the following hypotheses: (i) Water-Energy; (ii) Terrestrial Primary Productivity and (iii) Climatic Temporal Heterogeneity.