arXiv subjects

Dongyuan Xiao

Publications and source records attributed to Dongyuan Xiao.

14 recordsLinked to original sources

arXiv2026

Critical Three-Species Competition-Diffusion: Traveling-Wave Rigidity and the Fastest-Species Selection

We study the rigidity of traveling waves and long-time species selection in a one-dimensional critical three-species competition-diffusion system. We establish several rigidity results for traveling waves connecting distinct equilibria on the critical simplex. In the case where one species has a strictly larger Fisher-KPP spreading speed than the other two, an entropy method, combined with Gagliardo-Nirenberg and Nash inequalities yieldsuniform extinction of the slower species and convergence to the fastest-species equilibrium throughout every cone with speed below its KPP speed.

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arXiv2026

Propagation Direction of Bistable Traveling Fronts in the Lotka--Volterra Competition--Diffusion System

We study the propagation direction of bistable traveling fronts in the two-species Lotka--Volterra competition--diffusion system under strong competition. A complete characterization of the sign of the wave speed has remained a long-standing unsolved problem. We establish the first global necessary and sufficient criterion for zero wave speed by combining a Maxwell-type identity with a phase-plane rigidity argument. This criterion yields a unique zero-speed threshold, identifies its value in the symmetric case and its limiting values in two extreme regimes, and consequently determines the propagation direction throughout the strong-competition parameter region.

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arXiv2026

Sharp logarithmic corrections for a strong-weak Lotka-Volterra competition system

We study the one-dimensional strong-weak Lotka-Volterra competition-diffusion system \[ u_t=u_{xx}+u(1-u-av),\qquad v_t=dv_{xx}+rv(1-v-bu), \] with compactly supported initial data under the parameter condition $0<a<1<b$. Existing literature only gives leading-order spreading speed asymptotics without refined logarithmic corrections for wave fronts over the full parameter space. We convert the competitive system into an equivalent cooperative parabolic system and establish moving-domain comparison principles. Based on heat-kernel estimates, we derive sharp logarithmic asymptotic expansions for the rightmost level set of the stronger species $u$. According to the magnitudes of three characteristic speeds, five long-time dynamical regimes are classified, including pulled, nonlocally pulled, pushed and critical transition fronts, with explicit logarithmic and double-logarithmic phase corrections. Our results capture delicate long-time phase offsets of wave profiles neglected in previous leading-order theories.

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arXiv2025

Complete Classification of Traveling Waves and Resolution of Linear Conjecture in Monostable Systems

In this paper, we present a complete classification of traveling wave solutions for monostable systems within a unified framework. To this end, we introduce a novel technique, referred to as the slicing method, which is based on the construction of suitable super- and sub-solutions.

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arXiv2025

Linear vs. nonlinear speed selection of the front propagation into unstable states

In this paper, we mainly consider the speed selection problem for the classical Lotka-Volterra competition system. For the first time, we propose a sufficient and necessary condition for this long-standing problem from a new point of view. Moreover, our results can also reveal the essence of the linearly selected problem for the monostable dynamical system from the observation of the decay rate of the minimal traveling wave solution.

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arXiv2024

The Bramson correction in the Fisher-KPP equation: from delay to advance

We consider the solution to the scalar Fisher-KPP equation with front-like initial data, focusing on the location of its level sets at large times, particularly their deviation from points moving at the known spreading speed. We consider an intermediate case for the tail of the initial data, where the decay rate approaches, up to a polynomial term, that of the traveling wave with minimal speed. This approach enables us to capture deviations of the form $-r \ln t$ with $r \< \frac{3}{2}$, which corresponds to a logarithmic delay when $0 \< r \< \frac{3}{2}$ and a logarithmic advance when $r \< 0$. The critical case $r=\frac 32$ is also studied, revealing an extra $\mathcal O(\ln \ln t)$ term. Our arguments involve the construction of new sub- and super-solutions based on preliminary formal computations on the equation with a moving Dirichlet condition. Finally, convergence to the traveling wave with minimal speed is addressed.

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arXiv2024

On the propagation speed of the single monostable equation

In this paper, we first focus on the speed selection problem for the reaction-diffusion equation of the monostable type. By investigating the decay rates of the minimal traveling wave front, we propose a sufficient and necessary condition that reveals the essence of propagation phenomena. Moreover, since our argument relies solely on the comparison principle, it can be extended to more general monostable dynamical systems, such as nonlocal diffusion equations.

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arXiv2024

Sufficient conditions for determining the sign of the wave speed in the Lotka-Volterra competition system

This paper mainly focuses on the sign of the wave speed in the Lotka-Volterra competition system of bistable type, also known as the strong-strong competition case. The traveling wave solution of the system is crucial for understanding the long-time behavior of solutions to the Cauchy problem. Specifically, the sign of the wave speed is key to predicting which species will prevail in the competition. In this paper, by studying a degenerate Lotka-Volterra competition system, we propose two sufficient conditions for determining the sign of the wave speed.

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arXiv2024

The long-time behavior of solutions of a three-component reaction-diffusion model for the population dynamics of farmers and hunter-gatherers: the different motility case

In this paper, we investigate the spreading properties of solutions of the Aoki-Shida-Shigesada model. This model is a three-component reaction-diffusion system that delineates the geographical expansion of an initially localized population of farmers into a region occupied by hunter-gatherers. By considering the scenario where farmers and hunter-gatherers possess identical motility, Aoki et al. previously concluded, through numerical simulations and some formal linearization arguments, that there are four different types of spreading behaviors depending on the parameter values. In this paper, we concentrate on the general case for which farmers and hunter-gatherers possess different motility. By providing more sophisticated estimates, we not only theoretically justify the spreading speed of the Aoki-Shida-Shigesada model, but also establish sharp estimates for the long-time behaviors of solutions. These estimates enable us to validate all four types of spreading behaviors observed by Aoki et al..

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arXiv2022

Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: the strong-weak type

We consider the classical two-species Lotka-Volterra competition-diffusion system in the strong-weak competition case. When the corresponding minimal speed of the traveling waves is not linear determined, we establish the precise asymptotic behavior of the solution of the Cauchy problem in two different situations: (i) one species is an invasive one and the other is a native species; (ii) both two species are invasive species.

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arXiv2021

Lotka-Volterra competition-diffusion system: the critical competition case

We consider the reaction-diffusion competition system in the so-called {\it critical competition case}. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the non-existence of {\it ultimately monotone} traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the "faster" species excludes the "slower" one (with a known {\it spreading speed}), but also provide a sharp description of the profile of the solution, thus shedding light on a new {\it{bump phenomenon}}.

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arXiv2019

Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers

In this paper, we investigate the spreading properties of solutions of farmer and hunter-gatherer model which is a three-component reaction-diffusion system. Ecologically, the model describes the geographical spreading of an initially localized population of farmers into a region occupied by hunter-gatherers. This model was proposed by Aoki, Shida and Shigesada in 1996, and by numerical simulations, they classified the spreading behaviors into four different types depending on the parameter values. Despite such intriguing observations, no mathematically rigorous studies have been made to justify these numerical observations. The main difficulty comes from the fact that the comparison principle does not hold for the whole system. In this paper, we give theoretical justification to all the four types of spreading behaviors. Furthermore, we show that a logarithmic phase drift occurs as in the scalar KPP equation.

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arXiv2019

On a three-component degenerate reaction-diffusion model for the population of farmers and hunter-gatherers

In this paper, we investigate the expanding patterns and spreading speed of solutions of farmer and hunter-gatherer model which is a three-component degenerate reaction-diffusion system. Ecologically speaking, since the lifestyle of agriculture and settlement allows for a larger population, after an initially localized population of farmers migrated into a region occupied by hunter-gatherers, the Neolithic transition from hunter-gatherers to farmers happened. This model was proposed by J. Elias, K. Humayun and M. Mimura in 2018. By numerical simulations, a travelling wave solution with a certain speed, depending on the parameter values, is observed. Despite such observation and studying on the well-posedness of the system, no mathematically rigorous studies on the spreading properties have been made. The main difficulty comes from the fact that the comparison principle does not hold for the whole system. In this paper, we give theoretical proof of the existence of two different expanding patterns. Furthermore, we show a lower estimate for the spreading speed.

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arXiv2017

A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations

We consider a variational problem associated with the minimal speed of pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1-u)u$, $x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$ is the so-called minimal speed of pulsating traveling waves. In this paper, we study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$ under some constraints. We prove the existence of the maximizer under a certain assumption of the constraint and derive the Euler--Lagrange equation which the maximizer satisfies under $L^2$ constraint $\int_0^L b(x)^2dx=β$. The limit problems of the solution of this Euler--Lagrange equation as $L\rightarrow0$ and as $β\rightarrow0$ are also considered. Moreover, we also consider the variational problem in a certain class of step functions under $L^p$ constraint $\int_0^L b(x)^pdx=β$ when $L$ or $β$ tends to infinity.

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