Article
Statistics & Probability
Jiri Cerny, Alexander Drewitz, Lars Schmitz
Summary: This paper investigates the uniform boundedness property of the fronts of solutions for the randomized Fisher-KPP equation and its linearization model, the parabolic Anderson model. It is known that this property holds for the deterministic Fisher-KPP equation and a specific case of the randomized Fisher-KPP equation with ignition type nonlinearity. However, we find that this property fails to hold for the general randomized Fisher-KPP equation. In contrast, we establish this property for the parabolic Anderson model under certain assumptions.
ANNALS OF APPLIED PROBABILITY
(2023)
Article
Mathematics, Applied
Ge Tian, Zhi-Cheng Wang, Guo-Bao Zhang
Summary: This paper focuses on the stability of traveling wave solutions of the nonlocal Fisher-KPP equation, and establishes the asymptotic stability of traveling wave solutions with large wave speeds using the anti-weighted method and energy estimates.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Environmental Sciences
Jingyun Guan, Junqiang Yao, Moyan Li, Jianghua Zheng
Summary: Human activities have significant impacts on vegetation dynamics in the ecologically fragile region of Xinjiang, China. Improved NDVI prediction models and residual analysis methods were used to quantitatively assess these impacts, revealing that human activities mainly contribute to the improvement of vegetation, particularly for crops. Factors such as cultivated area, primary industry GDP, and population promote NDVI increase, while factors like animal husbandry population, agricultural population, and livestock number lead to NDVI decrease. The evolutionary trend of anthropogenic impacts on vegetation dynamics in Xinjiang shows a dominance of anti-persistence, with positive impacts continuing to increase, especially for crops, shrubs, grasslands, and alpine vegetation.
Article
Mathematics, Applied
Dmitri Finkelshtein, Yuri Kondratiev, Pasha Tkachov
Summary: The study focuses on the propagation of solutions to a doubly nonlocal reaction-diffusion equation of Fisher-KPP-type with anisotropic kernels. Necessary and sufficient conditions are presented to ensure linear time propagation in a specific direction. For kernels with a finite exponential moment, front propagation is proven to occur in all directions with a general class of initial conditions decaying faster than any exponential function.
APPLICABLE ANALYSIS
(2021)
Article
Statistics & Probability
Yan-Xia Ren, Renming Song, Fan Yang
Summary: In this paper, we investigate the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. We then demonstrate the existence of pulsating traveling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by probabilistically representing the solutions in terms of the limits of the additive and derivative martingales. Additionally, we prove the absence of pulsating traveling wave solutions in the subcritical case. The main techniques employed are spine decomposition and martingale change of measures.
ELECTRONIC JOURNAL OF PROBABILITY
(2023)
Article
Mathematics
Leonid Mytnik, Jean-Michel Roquejoffre, Lenya Ryzhik
Summary: The study focuses on the limiting extremal process of particles in binary branching Brownian motion. It demonstrates the convergence of rescaled particle density to a multiple of exponential, and density fluctuations to a 1-stable random variable. The research is motivated by the connection between the Bramson shift of Fisher-KPP equation solutions and specific initial conditions.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Christopher Griffin
Summary: In this paper, we introduce a finite population variation of the Fisher-KPP equation by utilizing the replicator dynamic to generate the reaction term. Based on previous research, we demonstrate that the resulting system of partial differential equations possesses a travelling wave solution, which can be expressed in closed form. Surprisingly, this closed form solution is obtained by reversing the sign of the known closed form solution of the classic Fisher equation. Additionally, we develop an approximate closed form solution for the corresponding equilibrium problem on a finite interval with specific boundary conditions, and propose two conjectures on these equilibrium problems, which are then analyzed numerically.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics
Ardak Kashkynbayev, Durvudkhan Suragan, Berikbol T. Torebek
Summary: This paper considers the Fisher-KPP equation on the Heisenberg group, discussing the existence of global solutions, their asymptotic behavior, and blow-up solutions. Furthermore, the obtained results are extended to the time-fractional Fisher-KPP equation on the Heisenberg group.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics
Bendong Lou, Jinzhe Suo, Kaiyuan Tan
Summary: In this paper, the advective Fisher-KPP equation on the half line with Dirichlet boundary conditions is considered. By constructing new types of entire solutions under different assumptions and analyzing the properties based on the value of beta, it is found that the essential property for such entire solutions is the asymptotically flat property as t approaches negative infinity, rather than concavity.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Nikos I. Kavallaris, Evangelos Latos, Takashi Suzuki
Summary: The current paper aims to uncover the key mechanism behind the occurrence of Turing-type instability in a nonlocal Fisher-KPP model. It is demonstrated that the solution of the considered equation is destabilized near a constant stationary solution due to diffusion-driven blow-up. The complete blow-up of the observed diffusion-driven instability is classified in terms of its blow-up rate. Furthermore, the detected diffusion-driven instability leads to the formation of unstable blow-up patterns, which are identified by analyzing the blow-up profile of the solution.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Christos Sourdis
Summary: The study focuses on the Fisher-KPP reaction-diffusion equation in the whole space and proves that a solution must coincide with a planar traveling wave if it exhibits the same exponential decay with a speed larger than the minimal speed.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Interdisciplinary Applications
Arsalan Rahimabadi, Habib Benali
Summary: This study investigates diffusion or reaction-diffusion processes on complex structures, such as brain networks, and establishes the existence and uniqueness of solutions for generalized Fisher-KPP reaction-diffusion equations on undirected and directed networks. The model has potential applications in modeling and studying neurodegenerative diseases.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Applied
Jingjing Li, Ningkui Sun
Summary: In this paper, we investigate a reaction-diffusion model in a one-dimensional river network and find that the long-term survival of a species is closely related to the water flow speed. The species can survive when the water flow speed is small, but it will be washed away when the speed is large.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Marzieh Bakhshi, Anna Ghazaryan, Vahagn Manukian, Nancy Rodriguez
Summary: This work investigates the existence of nonmonotone traveling wave solutions in a reaction-diffusion system modeling social outbursts, focusing on a bandwagon effect in the unrest when tension is high and a tension-inhibitive regime where unrest negatively impacts tension. The study uses Geometric Singular Perturbation Theory to analyze the existence of such solutions in two scenarios.
STUDIES IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Julien Berestycki, Eric Brunet, Cole Graham, Leonid Mytnik, Jean-Michel Roquejoffre, Lenya Ryzhik
Summary: We study the distance between the two rightmost particles in branching Brownian motion. Using a representation related to the Fisher-KPP equation, we determine the sharp asymptotics of the long-time limit of this random variable. We discover an algebraic correction to the previously known exponential behavior of the tail asymptotics.
