scholarly journals Populations facing a nonlinear environmental gradient: Steady states and pulsating fronts

2021 ◽  
pp. 1-82
Author(s):  
Matthieu Alfaro ◽  
Gwenaël Peltier
Keyword(s):  
Fourier Series ◽  
Environmental Gradient ◽  
Space Variable ◽  
Steady States ◽  
Series Expansions ◽  
Phenotypic Trait ◽  
Implicit Function ◽  
Perturbation Techniques ◽  
Variable Formula ◽  
Nonlocal Parabolic Equation

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets

2010 ◽  
Vol 61 (5) ◽  
pp. 1151-1181 ◽  
Author(s):  
Huo-Jun Ruan ◽  
Robert S. Strichartz
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Sierpinski Gasket ◽  
Series Expansions ◽  
Periodic Functions ◽  
Sierpiński Gasket ◽  
Higher Dimensional ◽  
Sierpinski Gaskets

Abstract.We construct covering maps from infinite blowups of the$n$-dimensional Sierpinski gasket$S{{G}_{n}}$to certain compact fractafolds based on$S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case$n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

scholarly journals Fourier Coefficients of a Class of Eta Quotients of Weight 4

2016 ◽  
Vol 4 (1) ◽  
pp. 4
Author(s):  
Barış Kendirli
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Series Expansions ◽  
Explicit Formulas

<p>Recently, there have been several works on the coefficients of the Fourier series expansions of a class of eta quotients by Williams, Yao, Xia and Jin, Kendirli, and Alaca. Some important explicit formulas have been discovered. Williams expressed all coefficients of one hundred and twenty-six eta quotients in terms of  and  and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of one hundred and four eta quotients in terms of  and  The author has expressed the even and odd coefficients of the Fourier series expansions of a class of eta quotients in terms of  and  for  Meanwhile, Alaca has obtained the coefficients of the Fourier series expansions of a class of eta quotients in  in terms of  and  Here, we will express the coefficients of the Fourier series expansions of a class of eta quotients in  in terms of  and Fourier coefficients of the four eta quotients.</p>

scholarly journals Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 276 ◽  
Author(s):  
Taekyun Kim ◽  
Dae Kim ◽  
Lee-Chae Jang ◽  
Gwan-Woo Jang
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials ◽  
Bernoulli Polynomials ◽  
Series Expansions ◽  
Linear Combinations ◽  
Lucas Polynomials ◽  
Finite Products ◽  
Derive Fourier Series

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials.

scholarly journals Fourier series of functions involving higher-order ordered Bell polynomials

2017 ◽  
Vol 15 (1) ◽  
pp. 1606-1617 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Gwan-Woo Jang ◽  
Lee Chae Jang
Keyword(s):  
Number Theory ◽  
Fourier Series ◽  
Higher Order ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Enumerative Combinatorics ◽  
Bernoulli Functions ◽  
Series Of Functions

AbstractIn 1859, Cayley introduced the ordered Bell numbers which have been used in many problems in number theory and enumerative combinatorics. The ordered Bell polynomials were defined as a natural companion to the ordered Bell numbers (also known as the preferred arrangement numbers). In this paper, we study Fourier series of functions related to higher-order ordered Bell polynomials and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.

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