2025.7.1
原文
That is, reproduction takes place in a relatively short period each year. In between these pulses of growth, mortality takes its toll, and the population decreases. The population dynamics consist of a within-season stage and a between-season stage.
翻译
也就是说，繁殖每年都会在短时间内发生。在这些出生脉冲间隙，死亡开始发生，种群开始衰减。种群动力学包含了一个季节内阶段和一个季节间阶段。
也就是说，种群的繁殖活动每年仅在一个相对较短的时期内发生。在两个生长脉冲之间，死亡率持续作用，导致种群数量下降。种群动态包括季节内阶段与季节间阶段两个过程。
种群的繁殖活动通常集中在每年一个相对较短的时期内。在繁殖脉冲之间，死亡率的持续作用导致种群数量下降。总体而言，种群动力学可划分为季节内与季节间两个阶段。
转写
In other words, the population will reproduce each year in a relatively short period. Between these pulsed of growth, the population will decrease due to the mortality. The population dynamics can be divided into a within-season stage and a between-season stage.
Reproduction occurs during a relatively brief period each year, forming distinct pulses of population growth. Between these reproductive events, mortality gradually reduces the population size. Thus, the population dynamics are characterized by a within-season phase and a between-season phase.
In other words, reproduction typically occurs within a relatively short period each year. Between these pulses of growth, mortality leads to a decline in population size. The population dynamics consist of a within-season phase and a between-season phase.
2025.7.2
原文
Within a season, population mortality is continuous while between seasons population growth is discrete. Models incorporating both discrete and continuous components are referred to as semidiscrete models (Singh and Nisbet 2007, Pachepsky et al. 2008). There have been extensive studies regarding the dynamics of nonspatial semidiscrete models in the form of impulsive ordinary differential equations; see, for example, Eskola and Geritz (2007), Geritz and Kisdi (2004), Eskola and Parvinen (2007), Pachepsky et al. (2008), Gyllenberg et al. (1997), Thieme (2003), Brauer and Castillo-Chávez (2001).
翻译
季节内，种群死亡率是连续的，但是在季节间，种群增长是离散的。同时考虑离散和连续部分的模型被称为半离散模型 (Singh and Nisbet 2007, Pachepsky et al. 2008). 以脉冲形式的常微分方程形式出现的非空间半离散模型的种群动力学已经被研究的很广泛，例如，skola and Geritz (2007), Geritz and Kisdi (2004), Eskola and Parvinen (2007), Pachepsky et al. (2008), Gyllenberg et al. (1997), Thieme (2003), Brauer and Castillo-Chávez (2001).
在一个季节内，种群的死亡是连续发生的，而季节之间的种群增长则是离散的。结合连续和离散过程的模型被称为半离散模型（semidiscrete models）（Singh and Nisbet 2007；Pachepsky 等 2008）。已有大量关于非空间半离散模型动力学的研究，这类模型通常以脉冲常微分方程（impulsive ordinary differential equations）的形式出现；参见如 Eskola and Geritz (2007), Geritz and Kisdi (2004), Eskola and Parvinen (2007), Pachepsky 等 (2008), Gyllenberg 等 (1997), Thieme (2003), Brauer and Castillo-Chávez (2001) 的相关文献。
在一个季节内，种群死亡过程是连续进行的，而在季节之间，种群增长则以离散方式发生。将连续过程与离散事件结合的模型被称为半离散模型（semidiscrete models）（Singh and Nisbet, 2007；Pachepsky 等，2008）。非空间半离散模型通常以脉冲常微分方程的形式表述，其种群动力学已被广泛研究，例如 Eskola 和 Geritz (2007)、Geritz 和 Kisdi (2004)、Eskola 和 Parvinen (2007)、Pachepsky 等 (2008)、Gyllenberg 等 (1997)、Thieme (2003)、Brauer 和 Castillo-Chávez (2001)。
转写
Population mortality is continuous during a season, but it is discrete between seasons. Models considering both discrete and continuous elements are termed as semidiscrete models (Singh and Nisbet 2007, Pachepsky et al. 2008). The dynamics of nonspatial semidiscrete models for impusive ordinary differential equations have been studied extensively; see, e.g., Eskola and Geritz (2007), Geritz and Kisdi (2004), Eskola and Parvinen (2007), Pachepsky et al. (2008), Gyllenberg et al. (1997), Thieme (2003), Brauer and Castillo-Chávez (2001).
During the season, population mortality proceeds continuously, whereas population growth occurs discretely between seasons. Models that integrate both continuous and discrete dynamics are referred to as semidiscrete models (Singh and Nisbet 2007; Pachepsky et al. 2008). The dynamics of nonspatial semidiscrete models—typically formulated as impulsive ordinary differential equations—have been extensively studied; see, for instance, Eskola and Geritz (2007), Geritz and Kisdi (2004), Eskola and Parvinen (2007), Pachepsky et al. (2008), Gyllenberg et al. (1997), Thieme (2003), and Brauer and Castillo-Chávez (2001).
Within a season, population mortality occurs continuously, whereas between seasons, population growth takes place in a discrete manner. Models that incorporate both continuous and discrete components are referred to as semidiscrete models (Singh and Nisbet 2007; Pachepsky et al. 2008). The dynamics of nonspatial semidiscrete models—typically formulated as impulsive ordinary differential equations—have been extensively investigated; see, for example, Eskola and Geritz (2007), Geritz and Kisdi (2004), Eskola and Parvinen (2007), Pachepsky et al. (2008), Gyllenberg et al. (1997), Thieme (2003), and Brauer and Castillo-Chávez (2001).
2025.7.3
原文
The results given by these authors show that various discrete-time population models can be derived mechanistically just by altering the patterns of reproduction and interaction. These models include classical examples such as the Ricker model (Ricker 1954), the Beverton and Holt model (Beverton and Holt 1957), the Skellam model (Skellam 1951), and others, which generate equilibrium dynamics, limit cycles, and sometimes chaos.
原文
这些作者呈现出的结果显示通过改变出生和互动的模型可以机制性地导出很多不同的离散时间的种群模型。这些模型包括例如像Ricker模型(Ricker 1954)， Beverton and Holt 模型(Beverton and Holt 1957), Skellam 模型 (Skellam 1951)和一些其他的模型，这些模型会产生平衡点的动力学，极限环，有时还会有混沌。
这些作者所给出的结果表明，仅通过改变种群的繁殖和相互作用模式，就可以从机制上推导出多种离散时间种群模型。这些模型包括一些经典的例子，如 Ricker 模型（Ricker 1954）、Beverton-Holt 模型（Beverton and Holt 1957）、Skellam 模型（Skellam 1951）等，这些模型能够产生平衡动态、极限环，甚至混沌现象。
这些作者的研究结果表明，仅通过改变繁殖和相互作用的模式，就可以从机制上推导出多种离散时间的种群模型。这些模型包括一些经典的例子，如 Ricker 模型（Ricker 1954）、Beverton-Holt 模型（Beverton and Holt 1957）、Skellam 模型（Skellam 1951）等，它们能够产生平衡态动力学、极限环，甚至混沌现象。
转写
It is shown by these authors’ results that various discrete-time population models can be deduced inheratively by modifying the patterns of reproduction and interaction. Theses models incorporate classical examples such as Ricker model (Ricker 1954), the Beverton and Holt model (Beverton and Holt 1957), the Skellam model (Skellam 1951), and others, leading to equilibrium dynamics, limit cycles, and sometimes chaos.
The results presented by these authors demonstrate that a wide range of discrete-time population models can be derived mechanistically by simply modifying the patterns of reproduction and interaction. These models include classical examples such as the Ricker model (Ricker 1954), the Beverton–Holt model (Beverton and Holt 1957), and the Skellam model (Skellam 1951), among others, which are capable of producing equilibrium dynamics, limit cycles, and even chaotic behavior.
The results presented by these authors demonstrate that various discrete-time population models can be derived mechanistically by modifying the patterns of reproduction and interaction. These models include classical examples such as the Ricker model (Ricker 1954), the Beverton–Holt model (Beverton and Holt 1957), the Skellam model (Skellam 1951), among others, which can produce equilibrium dynamics, limit cycles, and even chaotic behavior.
2025.7.6
原文
When population dynamics contain growth and dispersal, as well as continuous and discrete components, classical reaction–diffusion equations are not suitable to describe spread and persistence of the population, and impulsive reaction–diffusion equations (hybrid dynamical systems) provide a natural description of the spatial dynamics of the population.
翻译
当种群动力学包含增长和扩散，以及连续和离散元素，经典的反应扩散方程用来描述种群的传播和持久性就不是很合适,脉冲反应扩散方程(混合方程)提供了种群空间动力学的自然描述。
当种群动态同时包含增长与扩散过程，并同时具有连续性与离散性成分时，经典的反应-扩散方程已难以有效描述种群的扩散与持久性。在这种情况下，脉冲反应-扩散方程（即混合动力系统）能够自然地刻画种群的空间动态过程。
当种群动力学同时涉及增长与扩散过程，并兼具连续性与离散性特征时，传统的反应–扩散方程已难以有效描述种群的空间扩散与持久性。在此背景下，脉冲反应–扩散方程（即混合动力系统）为刻画种群空间动力学提供了一种自然且合理的数学框架。
转写
While population dynamics incorporate growth and dispersal, along with continuous and discrete components, classical reaction-diffusion equations are not capable of capturing spread and persistence of the population, and impulsive reaction-diffusion equations (hybrid dynamical systems) describe the spatial dynamics of the population more naturally.
When population dynamics involve both growth and dispersal processes, as well as a mixture of continuous and discrete components, classical reaction–diffusion equations are no longer adequate to describe the spatial spread and persistence of the population. In such cases, impulsive reaction–diffusion equations, which are regarded as hybrid dynamical systems, offer a natural and effective framework for modeling the spatial dynamics of the population.
When population dynamics involve both growth and dispersal processes, as well as a combination of continuous and discrete components, classical reaction–diffusion equations fail to adequately capture the spatial spread and persistence of the population. In such cases, impulsive reaction–diffusion equations, which are considered hybrid dynamical systems, offer a natural and effective framework for describing the spatial dynamics of the population.
2025.7.7
原文
In this paper, we propose simple impulsive reaction–diffusion equation models to study persistence and spread of species with a reproductive stage and a dispersal stage in bounded and unbounded domains. It is assumed that in a reproductive stage pulse growth occurs, and in a dispersal stage movement and mortality take place. The formulations of the models consist of discrete maps and nonlinear reaction–diffusion equations.
翻译
在本文中，我们提出一个简单脉冲反应扩散方程模型，用其来研究在有界区域和无界区域上带有繁殖阶段和扩散阶段种群的持久和传播。我们假设繁殖阶段会发生脉冲增长，且在扩散阶段会发生移动和死亡。模型的是由离散和非线性反应扩散方程构成的。
在本文中，我们提出了一类简单的脉冲反应扩散方程模型，用于研究在有界域与无界域中具有繁殖阶段与扩散阶段的物种的持久性与扩散过程。模型假设物种在繁殖阶段经历脉冲式增长，而在扩散阶段则发生运动与死亡。模型的构建基于离散映射与非线性反应扩散方程的结合。
在本文中，我们提出了一类简单的脉冲反应扩散方程模型，用于研究在有界与无界区域中，具有繁殖阶段和扩散阶段的物种的持久性与空间传播特性。模型假设物种在繁殖阶段经历脉冲式增长，而在扩散阶段则发生扩散与死亡。该模型由离散映射与非线性反应扩散方程相结合而成。
转写
In this paper, simple impulsive reaction-diffusion models are proposed to study persistence and spread of species in bounded and unbounded domains, along with a reproductive stage and a dispersal stage. We assume that pulse growth happens in a reproductive stage, and movement and mortality occurr in a dispersal stage. The models are fomulated by discrete maps and nonlinear reaction-diffusion equations.
In this study, we develop simple impulsive reaction–diffusion equation models to investigate the persistence and spatial spread of species that undergo distinct reproductive and dispersal stages in both bounded and unbounded domains. The models assume that pulsed population growth occurs during the reproductive stage, while dispersal and mortality take place during the dispersal stage. The mathematical formulations integrate discrete-time maps with nonlinear reaction–diffusion equations.
In this paper, we develop simple impulsive reaction–diffusion models to investigate the persistence and spatial spread of species that exhibit distinct reproductive and dispersal stages in both bounded and unbounded domains. In the models, pulsed population growth occurs during the reproductive stage, while dispersal and mortality take place during the dispersal stage. The models are formulated by integrating discrete-time maps with nonlinear reaction–diffusion equations.
2025.7.8
原文
The discrete maps describe pulse growth, which are allowed to be nonmonotone (i.e., there may be overcompensation in population growth). We also discuss how the model can be extended to the case of impulsive harvesting in a continuously growing and dispersing population.
翻译
离散映射用来描述出生脉冲，其可以是非单调的（也就是说，在种群增长中可能会存在过度补偿）。我们也讨论了如何讲这个模型拓展到带有脉冲收获的连续增长和扩散种群。
离散映射刻画了种群的脉冲增长过程，该过程允许非单调性，即种群增长中可能存在过度补偿现象。此外，我们还探讨了如何将该模型扩展至连续增长和扩散的种群中含脉冲式捕捞的情形。
离散映射用于描述脉冲式出生过程，该过程可能表现出非单调性，即种群增长过程中可能存在过度补偿现象。此外，我们还讨论了如何将该模型推广至在连续增长与扩散过程中存在脉冲收获的种群情形。
转写
The discrete maps formulate pulse growth, which could be nonmonotone(i.e., the population growth may overcompensate). We also investigate if the model can be generalized to the case of impulsive harvesting in a continuously growing and dispersing population.
The discrete maps characterize pulsed population growth, which may exhibit nonmonotonicity, allowing for overcompensatory dynamics. In addition, we discuss the potential extension of the model to incorporate impulsive harvesting in populations that undergo continuous growth and dispersal.
The discrete maps characterize pulsed population growth, which may exhibit nonmonotonicity (i.e., overcompensation in population growth). We further explore the possibility of generalizing the model to incorporate impulsive harvesting in populations undergoing continuous growth and dispersal.
2025.7.10
原文
We shall address two fundamental questions for the models: What are the spreading speed and traveling wave speeds when a population invades an unbounded domain? and what is the minimal domain size in which the population can persist when the spatial domain is bounded and has a lethal exterior?
翻译
关于这个模型我们会解决两个基础问题：物种入侵一个无界区域时他的传播速度和行波解的波速是多少？且当环境是有界并且外围都是致命时，最小的栖息地大小是多少？
我们将针对模型探讨两个基本问题：当种群入侵一个无界区域时，其扩散速度和行波速度是多少？当空间区域有界且外部环境致死时，种群能够持续存在的最小域大小是多少？
我们将针对该模型探讨两个基本问题：一是当物种入侵无界区域时，其扩散速度与行波解的传播速度；二是在有界空间中，当外围环境为致死区时，物种能够持续存在的最小栖息地尺度。
转写
We concern two basic questions for the models: What are the spreading speed and traveling wave speeds of an invasive population into an unbounded domain? and what is the minimal domain size where the population can survive when the bounded spatial domain has a lethal exterior?
We aim to address two fundamental questions concerning the models:
(i) What are the spreading speed and the traveling wave speed when a population invades an unbounded domain?
(ii) What is the minimal domain size required for population persistence in a bounded habitat surrounded by a lethal exterior?
We aim to address two fundamental questions concerning the models:
(i) What are the spreading speed and traveling wave speed of an invasive population in an unbounded domain?
(ii) What is the minimal domain size required for population persistence in a bounded habitat with a lethal exterior?
2025.7.11
原文
We demonstrate that, although the underlying dynamics of the models can be complicated, explicit analytical solutions to the questions can be given. We particularly show that when a species spreads into an unbounded domain, there is a spreading speed that can be formulated in terms of species vital rates (survival, fecundity, or development rates) and dispersal characteristics, and the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions.
翻译
我们证明，尽管这些模型的潜在动力学行为会相对复杂，但是这些方程的显式的解析解还是可以得出。特别地，我们会证明当种群入侵无界区域，传播速度可以用种群的重要速率（生存，繁殖，或者发展速率）来进行解释，且种群的扩散特点，传播速度可以用一类行波解的最慢的速度来解释。
我们表明，尽管这些模型的基本动态过程可能相当复杂，但对于相关问题仍然可以给出显式的解析解。特别地，我们证明了，当物种入侵一个无界区域时，其传播速度可以通过物种的关键生命参数（如存活率、繁殖率或发育率）以及扩散特性来明确表达。此外，该传播速度还可以刻画为一类行波解中的最小波速。
我们证明，尽管这些模型的潜在动力学过程可能相对复杂，但仍然可以给出相关问题的显式解析解。特别地，我们进一步证明，当物种扩散至无界空间时，其传播速度可以明确地用种群的关键生命参数（如存活率、繁殖率和发育率）以及扩散特性来表示。此外，该传播速度还可以刻画为一类行波解中的最小波速。
转写
Although the underlying dynamics of the models can be complicated, it is demonstrated that explicit analytical solutions to the questions can be solved. We prove that when a speices spreads into an unbounded domain, there is a spreading speed captured by the species vital rates(survival, fecundity, or development rates) and dispersal characteristics, and the spreading speed can be denoted by the slowest speed of a class of traveling wave solutions.
We demonstrate that, although the underlying dynamics of the models may be complex, explicit analytical solutions to the key questions can still be derived. In particular, we show that when a species invades an unbounded domain, its spreading speed can be explicitly expressed in terms of vital rates—such as survival, fecundity, and developmental rates—and dispersal properties. Moreover, this spreading speed can be characterized as the minimal speed among a family of traveling wave solutions.
We demonstrate that, although the underlying dynamics of the models may be complex, explicit analytical solutions can still be derived. In particular, we prove that when a species spreads into an unbounded domain, its spreading speed can be explicitly expressed in terms of vital rates—such as survival, fecundity, and development rates—and dispersal characteristics. Moreover, this spreading speed can be characterized as the minimal speed among a family of traveling wave solutions.
2025.7.13
原文
Loosely speaking, the spreading speed describes the asymptotic rate at which a species initially concentrated in a finite region, will expand its spatial range. A traveling wave solution describes the propagation of a species as a wave with a fixed shape and a fixed speed.
翻译
宽松地来说，传播速度描述种群在有限区域内聚集的渐近速率，并会扩张其空间范围。行波解描述种群以一种特定波形和特定速度的传播现象。
粗略地说，传播速度描述了一个最初局限于有限区域内的物种，其空间分布范围随时间扩展的渐近速率。行波解则描述了物种以固定的形状和固定的速度向外传播的过程。
宽泛地说，传播速度描述了一个最初局限于有限区域内的种群，其空间分布范围随时间扩展的渐近速率。行波解描述了种群以固定的形状和固定的速度向外传播的过程。
转写
Generaly speaking, the spreading speed characterize the asymptotic rate of a speices expanding with from a finite region. A traveling wave solution captures the propagation of a speices as a wave with a fixed shape and a fixed speed.
Loosely speaking, the spreading speed characterizes the asymptotic rate at which a species, initially confined to a finite region, expands its spatial range over time. A traveling wave solution describes the propagation of the species in the form of a wave that maintains a fixed shape and travels at a constant speed.
Generally speaking, the spreading speed characterizes the asymptotic rate at which a species, initially confined to a finite region, expands its spatial range. A traveling wave solution describes the propagation of the species in the form of a wave that maintains a fixed shape and travels at a constant speed.
2025.7.14
原文
The spreading speeds and traveling wave solutions provide important insight into the spatial patterns and rates of invading species in space. In the case that the spatial domain of a population is bounded with a lethal exterior, we prove the existence of a minimal domain size that
can be determined explicitly by the same set of model parameters used for computing the spreading speed.
翻译
传播速度和行波解对于入侵种群在空间的空间模式和速率提供了重要的认识。在有界域并伴有致命区域环绕的空间环境下，我们提供了区域最小尺寸的存在性，其可以由那些用于计算传播速度的模型参数来显示的决定。
传播速度与行波解揭示了入侵种群在空间扩张中的空间格局与传播速率。在有界空间且外部为致死区的情形下，我们证明了最小可存活区域尺寸的存在性，并表明该最小尺寸可由决定传播速度的同一组模型参数显式刻画。
传播速度和行波解为入侵种群在空间中的空间格局及传播速率提供了重要的理论认识。在种群所处空间为有界区域且外部为致死环境的情形下，我们证明了最小区域尺寸的存在性，该最小尺寸可以通过用于计算传播速度的同一组模型参数显式确定。
转写
The spreading speeds and traveling wave solutions characterize the spatial patterns and rates of invading species in space. In the case where the bounded spatial domain of a population is surrounded by a lethal environment, we show that the exsitence of a minimal domain size can be given explicitly by the same group of model parameters used for calculating the spreading speed.
The spreading speed and traveling wave solutions offer fundamental insights into the spatial dynamics and invasion rates of species. In the case where the spatial domain is bounded and surrounded by a lethal exterior, we establish the existence of a minimal domain size required for population persistence. Moreover, this minimal size can be explicitly characterized in terms of the same model parameters that determine the spreading speed.
The spreading speed and traveling wave solutions characterize the spatial patterns and invasion rates of species. In the case where the population occupies a bounded domain surrounded by a lethal environment, we establish the existence of a minimal domain size necessary for persistence. Moreover, this minimal size can be explicitly determined by the same set of model parameters used to compute the spreading speed.
2025.7.15
原文
We present simulations for the models. We observe that the numerical solutions for the unbounded domain case can exhibit oscillations, and that the numerical solutions for the bounded domain case can have different spatial patterns of species distributions.
翻译
我们对这个模型进行了数值模拟。我们观察到对于无界情况的数值解可能会显示振荡，对于有界情况的数值解在物种的空间分布上有不同的模式。
我们对所提出的模型进行了数值模拟。结果表明，在无界区域情形下，数值解可能表现出震荡行为，而在有界区域情形下，数值解可能呈现出不同的物种空间分布格局。
我们对所研究的模型进行了数值模拟。结果表明，在无界区域情形下，数值解可能表现出振荡行为，而在有界区域情形下，数值解在物种空间分布上呈现出不同的格局。
转写
We simulate the models. The numerical solutions for the unbounded domain case show oscillations, and the numerical for the bounded domain case give rise to different spatial patterns of species distributions.
We perform numerical simulations for the proposed models. The results indicate that, in the case of an unbounded domain, the numerical solutions may exhibit oscillatory behavior, while in the case of a bounded domain, the solutions can exhibit various spatial patterns of species distributions.
We perform numerical simulations of the proposed models. The results show that, in the case of an unbounded domain, the numerical solutions exhibit oscillatory behavior, while in the bounded domain case, the solutions give rise to distinct spatial patterns of species distributions.
2026.7.16
原文
This paper is concerned with traveling wave solutions of the recursive system $un+1 = Q[un] := run + h(un)\ast k, n \geq 0$, where $\ast$ stands for the convolution over the real line, $k$ is nonnegative, symmetrically nonincreasing, and $\int_\mathbb{R} k = 1$, $r$ is a real number with $0 \leq r < 1$, and $h \in C^1(\mathbb{R}_+,\mathbb{R}_+)$ is of unimodal type. By unimodal type, we mean that there exists $z^\ast > 0$ such that $h\prime (z) > 0$ for $z \in [0, z^\ast )$ and $h\prime (z) < 0$ for $z \in (z^\ast ,+\infty )$.
翻译
本文主要研究递归系统的行波解问题，系统如下 $un+1 = Q[un] := run + h(un) \ast k, n \geq 0$, 其中 $\ast $ 表示实数轴上的卷积运算, $k$ 是非负的, 对称增长, 并且满足$\int_\mathbb{R} k = 1$, $r$ 是一个满足 $0 \leq r < 1$的师叔, 且$h \in C^1(\mathbb{R}_+,\mathbb{R}_+)$ 具有单一点形式. 对于单一点形式, 我们定义，存在 $z^\ast > 0$ 使得$h\prime (z) > 0$ 对于所有 $z \in [0, z^\ast )$，且 $h\prime (z) < 0$ 对于$z \in (z^\ast ,+\infty )$.
本文主要研究递归系统的行波解问题，该系统形式如下：$u_{n+1} = Q[u_n] := r u_n + h(u_n) \ast k, \quad n \geq 0,$ 其中 $\ast\ast 表示在实数轴上的卷积运算，$k$ 为非负、关于原点对称且单调不增的函数，并满足 $\int_{\mathbb{R}} k = 1$。$r$ 为满足 $0 \leq r < 1$ 的实数。函数 $h \in C^1(\mathbb{R}_+, \mathbb{R}_+)$，且具有单峰性质。具体而言，所谓单峰性质，是指存在 $z^\ast > 0$，使得 $h’(z) > 0$ 对于所有 $z \in [0, z^\ast)$，而 $h’(z) < 0$ 对于所有 $z \in (z^\ast, +\infty)$。
本文研究递推系统的行波解问题，系统如下： $u_{n+1} = Q[u_n] := r u_n + h(u_n) \ast k, \quad n \geq 0,$ 其中 $\ast$ 表示在实数轴上的卷积，$k$ 为非负、关于原点对称且单调不增的函数，满足 $\int_{\mathbb{R}} k = 1$；$r$ 为满足 $0 \leq r < 1$ 的实数；$h \in C^1(\mathbb{R}_+, \mathbb{R}_+)$，并具有单峰性质。所谓单峰性质，是指存在 $z^\ast > 0$，使得 $h’(z) > 0$ 对于 $z \in [0, z^\ast)$ 成立，且 $h’(z) < 0$ 对于 $z \in (z^\ast, +\infty)$ 成立。
转写
This paper study the traveling wave solutions of the following recursive system $un+1 = Q[un] := run + h(un) \ast k, n \geq 0$, where $\ast $ is the convolution on the real line; $k$ is nonnegative, symmetrically nonincreasing, $\int_\mathbb{R} k = 1$; $r$ satisfies $0 \leq r < 1$; $h \in C^1(\mathbb{R}_+,\mathbb{R}_+)$ is unimodal type. $h$ is called unimodal type in the sense that there is $z^\ast > 0$ such that $h\prime (z) > 0$ for $z \in [0, z^\ast )$ and $h\prime (z) < 0$ for $z \in (z^\ast ,+\infty )$.
This paper investigates the traveling wave solutions of the recursive system $u_{n+1} = Q[u_n] := r u_n + h(u_n) \ast k, \quad n \geq 0,$ where $\ast$ denotes the convolution over the real line, $k$ is a nonnegative, symmetric, and nonincreasing function with $\int_{\mathbb{R}} k = 1$, $r$ is a real number satisfying $0 \leq r < 1$, and $h \in C^1(\mathbb{R}_+, \mathbb{R}_+)$ is a unimodal function. Specifically, by unimodal we mean that there exists $z^\ast > 0$ such that $h’(z) > 0$ for $z \in [0, z^\ast)$ and $h’(z) < 0$ for $z \in (z^\ast, +\infty)$.
This paper studies the traveling wave solutions of the following recursive system: $u_{n+1} = Q[u_n] := r u_n + h(u_n) \ast k, \quad n \geq 0,$
where $\ast$ denotes the convolution on the real line. The kernel $k$ is nonnegative, symmetric and nonincreasing, and satisfies $\int_{\mathbb{R}} k = 1$. The parameter $r$ is a real number such that $0 \leq r < 1$. The function $h \in C^1(\mathbb{R}_+, \mathbb{R}_+)$ is of unimodal type in the following sense: there exists $z^\ast > 0$ such that $h’(z) > 0$ for all $z \in [0, z^\ast)$, and $h’(z) < 0$ for all $z \in (z^\ast, +\infty)$.
2025.7.18
原文
A typical example is $h(z) = pz{\rm e}^{qz}$, $p, q > 0,$ which is often called the Ricker type function. By (1.2) and (1.4), we infer that $0$ is an unstable fixed point of $Q$ and there is a unique spatially homogeneous positive fixed point, denoted by $u^\ast$. Clearly, $u^\ast$
satisfies $ru^\ast + h(u^\ast ) = u^\ast$.
翻译
一个典型的例子就是函数$h(z) = pz{\rm e}^{qz}$, $p, q > 0,$ 经常也被称为Ricker类型函数. 通过(1.2)和(1.4), 我们得到$0$是映射$Q$的一个不稳定的不动点， 且存在一个空间均质的正不动点，用$u^\ast$来表示. 显然, $u^\ast$ 满足方程$ru^\ast + h(u^\ast ) = u^\ast$.
一个典型的例子是 $h(z) = pz{\rm e}^{qz}$，其中 $p, q > 0$，该函数通常被称为 Ricker 型函数。由(1.2)和(1.4)可知，$0$ 是映射 $Q$ 的一个不稳定不动点，且存在唯一的空间齐次正不动点，记为 $u^\ast$。显然，$u^\ast$ 满足方程 $ru^\ast + h(u^\ast) = u^\ast$。
一个典型的例子是函数 $h(z) = pz{\rm e}^{qz}$，其中 $p, q > 0$，该函数通常被称为 Ricker 类型函数。由 (1.2) 和 (1.4) 可知，$0$ 是映射 $Q$ 的一个不稳定不动点，且存在唯一的空间齐次正不动点，记为 $u^\ast$。显然，$u^\ast$ 满足方程 $ru^\ast + h(u^\ast) = u^\ast$。
转写
A protypical example is the Ricker typer fucntion that $h(z) = pz{\rm e}^{qz}$, $p, q > 0$. From (1.2) and (1.4), we obtain that $Q$ admits an unstable fixed point $0$ and that there exists a unique spatially homogeneous positive fixed point $u^\ast$ with $ru^\ast + h(u^\ast ) = u^\ast$.
A representative example is $h(z) = pz{\rm e}^{qz}$ with $p, q > 0$, which is commonly referred to as the Ricker-type function. By (1.2) and (1.4), we deduce that $0$ is an unstable fixed point of $Q$, and there exists a unique spatially homogeneous positive fixed point, denoted by $u^\ast$. It is evident that $u^\ast$ satisfies $ru^\ast + h(u^\ast) = u^\ast$.
A prototypical example is the Ricker-type function given by $h(z) = pz{\rm e}^{qz}$ with $p, q > 0$. It follows from (1.2) and (1.4) that $0$ is an unstable fixed point of $Q$, and there exists a unique spatially homogeneous positive fixed point $u^\ast$ satisfying $ru^\ast + h(u^\ast) = u^\ast$.
2025.7.19
原文
In the celebrated work [33], Weinberger introduced a systematic idea to establish the spreading speed and traveling waves for a class of recursive systems admitting a comparison principle and monostable structure. This idea has been greatly developed to vector-valued cooperative recursive systems by Lui [18, 19, 20, 21], cooperative and competitive reaction-diffusion systems [11, 14, 36], and systems in homogeneous and periodic media [34, 35].
翻译
在著名文献[33]中，Weinberger引入一种系统性的思想研究一类具有比较原理和单稳定结构的迭代系统。这种思想由Lui[18, 19, 20, 21]等人推广到向量合作迭代系统，且[11,14,36]的作者将理论推广到合作和竞争的反应扩散系统，还有[34,35]中的均质和周期介质系统。
在其著名的工作 [33] 中，Weinberger 提出了一种系统性的理论方法，用于研究具有比较原理和单稳结构的一类递推系统的传播速度与行波解。该方法随后被 Lui [18, 19, 20, 21] 显著拓展至向量值协作型递推系统，以及协作与竞争型反应扩散系统 [11, 14, 36]，并进一步推广至齐次与周期介质中的相关系统 [34, 35]。
在著名文献 [33] 中，Weinberger 引入了一种系统性的方法，用于研究一类具有比较原理和单稳结构的迭代系统。该方法随后被 Lui [18, 19, 20, 21] 等人进一步推广至向量值协作型迭代系统，[11, 14, 36] 的作者又将其扩展至协作与竞争型反应扩散系统，而 [34, 35] 则将相关理论应用于齐次与周期介质中的系统。
转写
In the well-known reference [33], Weinberger proposed a systematic idea to develop the spreading speed and traveling waves for a class of recursive systems satisfying the comparison principle and monostable structure. This idea has been greatly studied for vector-valued cooperateive recursive system by Lui [18,19,20,21], cooperative and competitive reaction-diffusion systems [11,14,36], and systems in homogeneous and periodic media [34,35].
In his celebrated work [33], Weinberger introduced a systematic framework for establishing the spreading speeds and traveling wave solutions for a class of recursive systems that possess a comparison principle and exhibit a monostable structure. This framework has been significantly extended to vector-valued cooperative recursive systems by Lui [18, 19, 20, 21], as well as to cooperative and competitive reaction-diffusion systems [11, 14, 36], and further to systems in both homogeneous and periodic media [34, 35].
In his seminal work [33], Weinberger introduced a systematic framework for establishing the spreading speeds and traveling wave solutions of a class of recursive systems satisfying the comparison principle and possessing a monostable structure. This framework has been extensively developed by Lui [18, 19, 20, 21] for vector-valued cooperative recursive systems, further extended to cooperative and competitive reaction-diffusion systems [11, 14, 36], and subsequently applied to systems in homogeneous and periodic media [34, 35].
2025.7.20
原文
The criterion for finite or infinite spreading speed was established for a general dispersal kernel function by Weinberger and Zhao [37]. Recently, Weinberger’s theory was developed for a class of monotone compact semiflows in [15] and interval-contract semiflows in [16]. Such a compactness condition was then further weakened to be a point-contraction in [6].
翻译
有关一般扩散核函数的有限和无限传播速度理论的标准是由Weinberger和zhao建立的。近些年，Weinberger的理论已经被发展到一类单调紧的半流上[15]和区间收缩半流上[16]. 这样的紧性条件进一步在[6]中被减弱为点收缩。
对于一般的扩散核函数，有限或无限传播速度的判别准则已由 Weinberger 和 Zhao [37] 建立。最近，Weinberger 的理论被推广至一类单调紧半流 [15] 和区间压缩半流 [16]。随后，这一紧性条件被进一步削弱为点压缩 [6]。
对于一般扩散核函数，有限或无限传播速度的判别准则由 Weinberger 和 Zhao 建立。近年来，Weinberger 的理论已被推广至一类单调紧半流 [15] 和区间收缩半流 [16]。此外，该紧性条件在 [6] 中进一步被削弱为点收缩。
转写
Weinberger and Zhao [37] developed the criterion for finite and infinite spreading speed with a general dispersal kernel function. Moreover, Weinberger’s theory was generalized to a class of monotone compact semiflows in [15] and interval-contract semiflows in [16]. This compactness condition was further expanded to the case of a point-contraction in [16].
The criterion for determining whether the spreading speed is finite or infinite for general dispersal kernels was established by Weinberger and Zhao [37]. Recently, Weinberger’s theory has been extended to a class of monotone compact semiflows in [15] and interval-contracting semiflows in [16]. Moreover, the compactness condition was further relaxed to a point-contraction in [6].
Weinberger and Zhao [37] established the criterion for determining whether the spreading speed is finite or infinite for general dispersal kernels. Subsequently, Weinberger’s theory has been extended to a class of monotone compact semiflows [15] and interval-contracting semiflows [16]. Moreover, this compactness condition has been further relaxed to the point-contraction case in [6].
2025.7.21
原文
The recursive system (1.1) admits the comparison principle in Weinberger’s framework if $u\ast \leq z\ast$, which is satisfied when $h\prime (0)- (1-r)$ is small. However, $h\prime (0)$ is often big when it is understood as the growth rate in an ecological scenario. Thus, in general, system (1.1) is not monotone. When $r = 0$, system (1.1) becomes $u_{n+1}(x) =\int_\mathbb{R}k(x-y)h(u_n(x))dy$, $x \in \mathbb{R}$, $n \geq 0$, which was introduced in 1986 by Kot and Schaffer [9] as a model for spatial ecological process.
翻译
在weinberger的框架下，迭代系统 (1.1)满足比较原理，如果 $u\ast \leq z\ast$, 其会自动满足当 $h\prime (0)- (1-r)$ 非常小时. 然而, $h\prime (0)$会相对较大当其作为生长速率在生态情境中讨论时. 因此，一般来讲， 系统(1.1) 时非单调的. 当$r = 0$, 系统 (1.1) 变为 $u_{n+1}(x) =\int_\mathbb{R}k(x-y)h(u_n(x))dy$, $x \in \mathbb{R}$, $n \geq 0$, 其已经在 [9]中被Kot和Schaffer当作空间生态过程的一个模型讨论.
在 Weinberger 的理论框架下，递归系统 (1.1) 满足比较原理，如果 $u\ast \leq z\ast$，而当 $h\prime (0) - (1 - r)$ 足够小时，该条件成立。然而，在生态学情景中，$h\prime (0)$ 往往被理解为增长率，因此通常较大。因此，一般而言，系统 (1.1) 并不具有单调性。当 $r = 0$ 时，系统 (1.1) 退化为 $u_{n+1}(x) = \int_\mathbb{R}k(x-y)h(u_n(x))dy$，其中 $x \in \mathbb{R}$, $n \geq 0$。该模型最早由 Kot 和 Schaffer [9] 于 1986 年提出，用于描述空间生态过程。
在 Weinberger 的框架下，迭代系统 (1.1) 满足比较原理，如果 $u\ast \leq z\ast$，该条件通常在 $h\prime (0)-(1-r)$ 足够小时自然成立。然而，在生态学情境中，$h\prime (0)$ 通常被视为种群的增长速率，往往较大。因此，一般而言，系统 (1.1) 是非单调的。当 $r = 0$ 时，系统 (1.1) 退化为 $u_{n+1}(x) = \int_\mathbb{R}k(x-y)h(u_n(x))dy$，其中 $x \in \mathbb{R}$, $n \geq 0$。该模型已在 [9] 中由 Kot 和 Schaffer 提出，用于描述空间生态过程。
转写
Recursive system (1.1) satisfies comparison principle in Weinberger’s framework if $u\ast \leq z\ast$, which holds ture whenever $h\prime (0)- (1-r)$ is small. However, $h\prime (0)$ is usually big when it is regarded as the growth rate in ecological scenarios. Therefore, to be more general, system (1.1) is alwayed assumed to be not monotone. Specifically, when $r = 0$, system (1.1) reduces to $u_{n+1}(x) =\int_\mathbb{R}k(x-y)h(u_n(x))dy$, $x \in \mathbb{R}$, $n \geq 0$, which was proposed by Kot and Schaffer [9] to investigate the spatial ecological process.
The recursive system (1.1) satisfies the comparison principle in Weinberger’s framework if $u \ast \leq z \ast$, which holds when $h\prime (0) - (1 - r)$ is sufficiently small. However, in ecological contexts, $h\prime (0)$ is often interpreted as the growth rate and tends to be large. Therefore, in general, system (1.1) is not monotone. When $r = 0$, system (1.1) reduces to $u_{n+1}(x) = \int_\mathbb{R}k(x - y)h(u_n(x))dy$, with $x \in \mathbb{R}$ and $n \geq 0$, which was introduced by Kot and Schaffer [9] in 1986 as a model for spatial ecological processes.
The recursive system (1.1) satisfies the comparison principle in Weinberger’s framework if $u\ast \leq z\ast$, which holds true provided that $h\prime (0) - (1 - r)$ is sufficiently small. However, $h\prime (0)$ is usually large when considered as the growth rate in ecological contexts. Therefore, system (1.1) is generally considered to be non-monotone. Specifically, when $r = 0$, system (1.1) reduces to $u_{n+1}(x) = \int_\mathbb{R}k(x-y)h(u_n(x))dy$ (with $x \in \mathbb{R}$ and $n \geq 0$), which was introduced by Kot and Schaffer [9] as a model for spatial ecological processes.