Intraspecific Variation Promotes Coexistence Under Competition For Essential Resources

Intraspecific variation may be key to coexistence in diverse communities, with some even suggesting it is necessary for large numbers of competitors to coexist. However, theory provides little support for this argument, instead finding that intraspecific variation generally makes it more difficult for species to coexist. Here we present a model of competition where two species compete for two essential resources and individuals within populations vary in their ability to take up different resources. We found a range of cases where intraspecific variation expands the range of conditions under which coexistence can occur, which provides a mechanism that allows the ecologically neutral evolutionary stable strategy (ESS) to become ecologically stable. We demonstrate that this result relies on nonlinearity in the function that describes how traits map onto ecological function. A sigmoid mapping function is necessary in order to model essential resources because it allows for variation in a unbounded trait while maintaining biologically realistic boundaries on uptake rates, and differs from other kinds of nonlinearity, which only unidirectionally increase or decrease ecological function. The sigmoid function’s nonlinearity spreads individuals unevenly along the growth function, which allows positive growth contributions from some individuals to compensate for growth loses from others, leading to coexistence. We discuss empirical systems beyond competition for essential resources in which discontinuous functions are important.

Approximation of discontinuous functions of three variables by discontinuous interpolation splines

In this paper, discontinuous interpolation splines of three variables are constructed and a method for reconstructing of the discontinuous internal structure of a three-dimensional body by constructed splines is proposed. It is believed that a three-dimensional object, which is described by a function of three variables with discontinuities of the first kind on a given grid of nodes, is completely covered by a system of parallelepipeds. The experimental data are the one-sided value of the discontinuous function in a given grid of nodes. In the article, theorems on interpolation properties and the error of the constructed discontinuous structures are formulated and proved. Moreover, the constructed discontinuous interpolation splines include, as a special case, classical continuous splines. The developed approximation method can be applied in three-dimensional mathematical modeling of discontinuous processes, including in computed tomography.

Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

On the Influence of Interlayer Nonuniformity on Contact Characteristics during the Interaction of a Pipe and a Rigid Cylindrical Insert

Contact problem for viscoelastic aging pipe with a longitudinally nonuniform thin elastic internal coating and a rigid cylindrical insert is considered in the paper. The basic integral equation with integral operators of different types (mixed integral equation) is given. It's analytical solution for contact stresses in insert area is presented. The solution is constructed in such a way that the function describing the inner coating nonuniformity is distinguished by a separate factor. This fact allows one to perform accurate calculations even in cases where the coating properties are described by rapidly changing and even discontinuous functions. Other known analytical methods do not allow one achieving such a results.