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Little is known about how mutualistic interactions affect the distribution of species richness on broad geographic scales. Because mutualism positively affects the fitness of all species involved in the interaction, one hypothesis is that the richness of species involved should be positively correlated across their range, especially for obligate relationships. Alternatively, if mutualisms involve multiple mutualistic partners, the distribution of mutualists should not necessarily be related, and patterns in species distributions might be more strongly correlated with environmental factors. In this study, we compared the distributions of plants and vertebrate animals involved in seed‐dispersal mutualisms across the United States and Canada. We compiled geographic distributions of plants dispersed by frugivores and scatter‐hoarding animals, and compared their distribution of richness to the distribution in disperser richness. We found that the distribution of animal dispersers shows a negative relationship to the distribution of the plants that they disperse, and this is true whether the plants dispersed by frugivores or scatter‐hoarders are considered separately or combined. In fact, the mismatch in species richness between plants and the animals that disperse their seeds is dramatic, with plants species richness greatest in the in the eastern United States and the animal species richness greatest in the southwest United States. Environmental factors were corelated with the difference in the distribution of plants and their animal mutualists and likely are more important in the distribution of both plants and animals. This study is the first to describe the broad‐scale distribution of seed‐dispersing vertebrates and compare the distributions to the plants they disperse. With these data, we can now identify locations that warrant further study to understand the factors that influence the distribution of the plants and animals involved in these mutualisms.
Introduction
A central problem in ecology is to understand the patterns and processes shaping the distribution of species. There is a preponderance of studies of species richness at broad geographic scales (Hawkins et al. 2003, Rahbek et al. 2007, Stein et al. 2014, Rabosky and Hurlbert 2015) that has facilitated our understanding of why species are found where they are, a central tenet within the domain of ecology (Scheiner and Willig 2008). Most commonly, these studies find species distributions to be correlated with resource availability and use environmental variables (e.g. temperature and productivity; Rabosky and Hurlbert 2015) to explain putative determinants of the distributions. Environmental variables are only one determinant of species’ distributions. Another, species interaction, is a key and understudied determinant of species’ distributions (Cazelles et al. 2016). In fact, in some cases species interactions may be more important for determining distribution than environmental variables (Fleming 2005).
When species interact, we expect their geographic distributions to be correlated – either positively or negatively – depending on the effect (or sign of the interaction) of one species on the other (Case et al. 2005). For pairwise interactions, where one species benefits from another species, a positive relationship is expected between the distribution and abundance due to the increase in the average fitness of the benefitting species where they overlap (Svenning et al. 2014). Furthermore, most species interactions are not simply pairwise, but diffuse, consisting of multiple interacting species, here referred to as guilds (with guilds referring to species that use the same resource). It therefore follows that where one guild benefits from another guild, a positive relationship is expected between the distribution and richness of the guids. This should be true in the case of mutualisms, where both sides of the interaction share an increase in average fitness from being together (Bronstein 2015), and there is some evidence for correlated geographic distributions of mutualists in the New World (Fleming 2005). One example of a mutualism where both sides of the interaction have a fitness advantage in each other's presence is animal‐mediated seed dispersal. Because both interacting species and guilds in seed dispersal mutualism benefit from the relationship we would predict that the richness of animal‐dispersed plants ought to be correlated with the richness of their animal dispersers and vice versa. To our knowledge, this prediction has never been tested on a large geographic scale.
Answer: The answer is: f(x) = 2x^2 + 40x + 200
Explanation:
Given: x is the width of the Artwork
Let W be the Width of the Frame
Let L be the Length of the Frame
Width = x + 10
Length = Width * 2, so by substitution, Length = 2(x + 10)
Area = Width * Length
By substitution:
Area = (x + 10) * 2(x + 10)
f(x) = (x + 10) * (2x + 20)
f(x) = 2x^2 + 20x + 20x + 200
f(x) = 2x^2 + 40x + 200
To test this, if a picture 8" wide is put in this frame, the width would be 8+10 or 18 inches. The length would be twice the width, or 36 inches. The Area is 18 x 36 = 648.
Using the formula:
f(8) = (2 * 8^2) + (40 * 8) + 200
f(8) = (2 * 64) + 320 + 200
f(8) = 128 + 320 + 200
f(8) = 648
