We study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under which a nonlocal dispersal behavior is favored.
In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can become successful even in an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse.
Indeed, the example considered presents a high variance of the distribution of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic ingredients that favor nonlocal strategies.
In this paper, we study the global dynamics of a general $2\times 2$ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in \cite{BaiLi2017}, where competition models with symmetric nonlocal operators are considered.
In this paper, the global dynamics of two-species Lotka-Volterra competition models with nonlocal dispersals is studied. Under the assumption that dispersal kernels are symmetric, we prove that except for very special situations, local stability of semi-trivial steady states implies global stability, while when both semi-trivial steady states are locally unstable, the positive steady state exists and is globally stable. Moreover, our results cover the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.
In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case th…
Seasonality frequently occurs in population models, and the corresponding seasonal patterns have been of great interest to scientists. This paper is concerned with traveling waves to a time-periodic bistable Lotka-Volterra competition system with nonlocal dispersal. We first establish the existence, uniqueness and stability of traveling wave solutions for this system. Then, by utilizing comparison principle and the stability property, the relationship among the bistable wave speed, the asymptotic propagation speeds of the associated monotone subsystems and the speed of upper/lower solutions is…
For a competition-diffusion blow-up system involving the fractional Laplacian of the form \begin{equation*}\label{syst1} -(-Δ)^su=uv^2,\quad-(-Δ)^sv=vu^2,\quad u,v>0 \ \mathrm{in} \ \mathbb{R}^N, \end{equation*} whith $s\in(0,1)$, we prove that the maximal asymptotic growth rate for its entire solutions is $2s$. Moreover, since we are able to construct symmetric solutions to the problem, when $N=2$ with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when …
We consider a population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition. We introduce theclimate shift due to {\it Global Warming} and discuss the dynamicsof the population by studying the long time behavior of thesolution of the Cauchy problem. We consider three sets ofassumptions on the growth function. In the so-called {\it confinedcase} we determine a critical climate change speed for theextinction or survival of the population, the latter case taking place by "strictly following the climate shift". In the so-calle…
This paper is concerned with a Lotka-Volterra type competition model with free boundaries in time-periodic environment. One species is assumed to adopt nonlocal dispersal and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We show that this free boundary problem with more general growth functions admits a unique solution defined for all time. A spreading-vanishing dichotomy is obtained and criteria for spreading and vanishing are provided. Moreover, under the weak competition condition we provide the long-time asymptotic behavior of…
We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria f…
This paper deals with the solution of following chemotaxis system with competitive kinetics and nonlocal terms \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=d_1Δu-χ_1\nabla\cdot(u\nabla w)+u\left(a_0-a_1u-a_{2}v-a_3\int_Ωu-a_4\int_Ωv\right), &x\in Ω, t>0,\\ v_t=d_2Δv-χ_2\nabla\cdot(v\nabla w)+v\left(b_0-b_1u-b_{2}v-b_3\int_Ωu-b_4\int_Ωv\right), &x\in Ω, t>0,\\ w_t=d_3Δw-λw+k u+l v, &x\inΩ, t>0, \end{array} \right. \end{eqnarray*} in a smoothly bounded domain $Ω\subset \mathbb{R}^N, N\geq1$, where $a_0, a_1, a_2, b_0, b_1, b_2>0$ and $a_3, a_4, b_3, b_4\in\mathbb{R}$. The purpose of this pa…
In this paper we study a nonlocal diffusion problem on a manifold. These kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we st…
Kernel-Dependent Pattern Formation in a Population Model with Nonlocal Facilitation and Competition
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Carl R. Woese Institute for Genomic Biology, University of Illinois Urbana-Champaign（卡尔·R·沃斯基因组生物学研究所，伊利诺伊大学厄巴纳-香槟分校）；Colby College（科尔比学院）
To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals? We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth …
We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition b…
We study a class of free boundary problems of ecological models with nonlocal and local diffusions, which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal, with the free boundary representing the spreading front of the species. We prove that such kind of nonlocal and local diffusion problems has a unique global solution, and then show that a spreading-vanishing dichotomy holds. Moreover, criteria of spreading and vanishing, and long time behavior of solution when spreading happens are establis…
In this article we prove the existence of classical solutions to a system of mean field games arising in the study of exhaustible resource production under market competition. Individual trajectories are modeled by a controlled diffusion process with jumps, which adds a nonlocal term to the PDE system. The assumptions on the Hamiltonian are sufficiently general to cover a large class of examples proposed in the literature on Bertrand and Cournot mean field games. Uniqueness also holds under a sufficient restriction on the structure of the Hamiltonian, which in practice amounts to a small upper…
We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. It is a new mechanism of pattern formation in population dynamics that can explain emergence of biological species due to intra-specific competition and random mutations.Travelling waves connecting an unstable homogeneous equilibrium and a periodic in space stationary solution are studied numerically.
We consider a population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition. This population is facing an {\it environmental gradient}: to survive at location $x$, an individual must have a trait close to some optimal trait $y_{opt}(x)$. Our main focus is to understand the effect of a {\it nonlinear} environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with $y_{opt}(x) = \varepsilonθ(x)$, $0<\vert \varepsilon \vert \ll 1$. We construct steady states solutions and, w…
This paper is concerned with volume-constrained minimization problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on $\mathbb{R}^n$, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrarily to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, …
We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. [Hastings, Theor. Pop. Biol. 24, 244-251, 1983] and [Dockery et al., J. Math.…
In this paper we consider the diffuse interface generalized antiferromagnetic model with local/nonlocal attractive/repulsive terms in competition studied in Daneri-Kerschbaum-Runa arXiv:1907.06419. The parameters of the model are denoted by $τ$ and $\varepsilon$: the parameter $τ$ represents the relative strength of the local term with respect to the nonlocal one, while the parameter $\varepsilon$ describes the transition scale in the Modica-Mortola type term. Restricting to a periodic box of size $L$, with $L$ multiple of the period of the minimal one-dimensional minimizers, in Daneri-Kerschb…
Fitness is asymptotically irrelevant under uniform competition in phenotype-structured populations
Instituto de Matemática e Computação, Universidade Federal de Itajubá（伊塔朱巴联邦大学数学与计算研究所）；Carl Gustav Carus School of Medicine, Technische Universität Dresden（德累斯顿工业大学卡尔·古斯塔夫·卡鲁斯医学院）
