2025.6.1
原文
What should the proof entail? Which definitions are relevant? What is the overall strategy? Is one particular proof similar to something already done? Whenever there is a choice, efficiency is traded for an opportunity to reinforce some previously learned technique. Especially familiar or predictable arguments are usually sketched as exercises
翻译
这些证明应该在那些地方写的详细？哪些定义是相关的？整体学习的策略应该是怎样的？有跟已经写出来的证明相似的证明吗？ 不管有没有选择，效率相比之前学的技巧来说，都是一个值得谨记的。尤其是那些被布置为练习的熟悉的或者可预测的论述。
证明应该包含哪些内容？哪些定义是相关的？整体策略是什么？某个特定的证明是否类似于已经完成的某些证明？每当面临选择时，效率往往会为了巩固之前学到的某些技巧而被牺牲。尤其是那些熟悉或可预测的论证，通常被作为练习略去。
这些证明在哪些地方需要写得更详细？又有哪些定义是相关的？整体的学习策略应该如何安排？是否存在与已经完成的证明类似的情况？无论选择何种方式，都要在效率与复习巩固之间做好权衡。尤其是对于那些被安排为练习的熟悉或可预测的论证。
转写
What should the proof provide? Which definitions are related? What is the thorough suggestion to study this text? Are all the proofs independent of something already finished? No matter what techniques we can choose, it is important to give eifficiency an opportunity to emphasis some previouly learned methods. In particular, analogous or existed content are always arranged as practices.
What should a complete proof include? Which definitions and concepts are relevant? What is the overall approach or strategy? Is there a proof that closely resembles one we’ve already encountered? Often, when deciding how detailed a proof should be, we may trade off efficiency in favor of reinforcing previously learned techniques. Especially familiar or predictable arguments are typically left as exercises for the reader.
What should a proof include? Which definitions are relevant? What is the best strategy for studying this text? Are any of these proofs similar to those we have already completed? Regardless of the techniques we choose, it is important to balance efficiency with opportunities to reinforce previously learned methods. In particular, similar or predictable arguments are often assigned as exercises.
2025.6.2
原文
Especially familiar or predictable arguments are usually sketched as exercises so that students can participate directly in the development
of the core material. The search for recurring ideas exists at the proof-writing level and also on the larger expository level.
翻译
尤其是那些熟悉或者可预测性的论述被设定为练习，这样的话可以方便那些学生参与到核心课程的发展当中。反复出现观点的探索既在证明书写的层面上出现，也在更大的探索层面出现。
那些特别熟悉或可预见的论证通常以习题的形式概述，以便学生可以直接参与核心内容的展开。 在证明写作层面以及更大范围的阐述层面，都存在对重复出现的思想的探索。
尤其是那些熟悉或可预测的论证，通常以练习的形式呈现，以便学生能够直接参与到核心内容的展开中去。对反复出现的观点的探索，既体现在证明写作层面，也体现在更高层次的阐述层面。
转写
Particularly familiar or predictable arguments are typically presented as exercises, thereby enabling students to actively engage in the development of the core material. The exploration of recurring ideas manifests both at the level of proof construction and at the broader expository level.
Especially familiar or predictable arguments are usually sketched as exercises so that students can participate directly in the development of the core material. The search for recurring ideas exists at the proof-writing level and also on the larger expository level.
In particular, those familiar or predictable arguments are typically left as exercises so that students can directly participate in the development of the core material. The exploitation of recurring ideas appears both at the proof-writing level and at the broader expository level.
2025.6.3
原文
I have tried to give the course a narrative tone by picking up on the unifying themes of approximation and the transition from the
finite to the infinite. To paraphrase a passage from the end of the book, real numbers are approximated by rational ones; values of continuous functions are approximated by values nearby; curves are approximated by straight lines; areas are approximated by sums of rectangles; continuous functions are approximated by polynomials.
翻译
我已经尽力尝试给这门课程奠定一个相对容易被人接受的基调，通过把最大化和从有限转移到无限的主题统一起来的方式。出于文章末尾一篇文章转写的目的，实数将会用有理数去逼近；连续函数的在某一点的值将会用附近的值去逼近；曲线会用直线去逼近；区域的面积会用矩形的总和去估计；连续函数将会用多项式去逼近。
我试图通过把“近似”和“从有限到无限的过渡”作为贯穿全书的主题，使课程呈现叙述性的基调。借用书末的一段话来说，实数可以用有理数近似；连续函数的值可以用邻近的值近似；曲线可以用直线近似；面积可以用矩形的和近似；连续函数可以用多项式近似。
我努力尝试为这门课程营造一个更具叙述性的基调，通过将“近似”和“从有限到无限的过渡”这两个主题统一起来。借用书末的一段话来表达：实数可以用有理数近似；连续函数在某一点的值可以用附近的值近似；曲线可以用直线近似；区域的面积可以用矩形的和来近似；连续函数可以用多项式近似。
转写
I’ve put all my efforts to assign the course a expository tone by selecting content of approximation and the changes from the finite to infinite. To rewrite a passage from the end of the book, real numbers are presented by rational ones; values of continous functions are ilustrated by adjasint values; curves are graphed by straight lines; areas are calculated by sums of rectangles; continuous functions are described by polynomials.
I’ve aimed to give the course a more narrative feel by weaving together the central themes of approximation and the progression from the finite to the infinite. As the book concludes, in essence: real numbers can be approximated by rationals; the values of continuous functions can be approximated by nearby values; curves can be approximated by straight lines; areas can be approximated by sums of rectangles; and continuous functions themselves can be approximated by polynomials.
I’ve tried to give the course an expository tone by focusing on the themes of approximation and the transition from the finite to the infinite. To paraphrase a passage from the end of the book: real numbers can be approximated by rational ones; values of continuous functions can be approximated by nearby values; curves can be approximated by straight lines; areas can be approximated by sums of rectangles; and continuous functions can be approximated by polynomials.
2025.6.4
原文
In each case, the approximating objects are tangible and wellunderstood, and the issue is when and how well these qualities survive the limiting process. By focusing on this recurring pattern, each successive topic builds on the intuition of the previous one.
翻译
在每一种情况下，这个近似的项目是可预见的并且也是被良好理解的，并且问题重点是什么时候以及如何才能让这些性质在去极限的过程中保留下来。通过关注反复出现的思想，每一个接替的章节都建立在之前章节的直觉上。
在每一种情况下，所采用的逼近对象都是具体且易于理解的。问题在于，当这些对象进入极限过程时，它们的这些特性何时以及在多大程度上得以保留。通过专注于这一反复出现的模式，每一个连续的话题都建立在前一个话题的直观理解之上。
在每一种情况下，所采用的近似对象都是具体且易于理解的。关键问题是：在极限过程中，这些性质何时以及如何能够保留下来。通过聚焦于这一反复出现的模式，每个后续主题都在前一个主题的直观基础上加深理解。
转写
In every situation, the approxiamting objects are illusrtative and easy to understand, and the point is when and how these qualities presever after taking limits. By emphasizing on this repeated idea, each subsequent section construct on the impart of the previous one.
In each case, the approximating objects are tangible and easily understood. The central question concerns when, and to what extent, these qualities persist under the limiting process. By focusing on this recurring pattern, each successive topic builds upon the intuition developed in the preceding one.
In each case, the approximating objects are illustrative and easy to understand, and the key question is when and how these qualities persist after taking limits. By emphasizing this recurring idea, each subsequent section builds on the intuition developed in the preceding one.
2025.6.5
原文
The questions seem more natural, and a method to the madness emerges from what might otherwise appear as a long list of theorems and proofs.This book always emphasizes core ideas over generality, and it makes no effort to be a complete, deductive catalog of results.
翻译
这些问题似乎更自然一些，并且能作为一种方法，来解决长列表里的定理和证明罗列产生的混乱。相比于一般性，这本书更强调核心观点，并且作为一个完全的，一些结果的推导性目录也完全没有好处。
这些问题显得更自然，而在原本可能显得只是冗长的定理与证明清单中，显现出一种条理。这本书始终强调核心思想而非一般性，它并没有试图成为一个全面、演绎的结果目录。
这些问题显得更自然，也让原本可能显得冗长、杂乱的定理与证明清单更具条理。与其追求一般性，这本书更注重核心思想，它并不试图成为一个全面、演绎的结果目录。
转写
It seems that the questions are more natural, and a method to the madness arising from what might otherwise manifest as a long list of theorems and proofs. This book always foucus on central ideas over generality, and it is not necessary to be a thorough, intuitive catalog of results.
The questions feel more natural, and a coherent thread emerges from what might otherwise seem like just a long list of theorems and proofs.This book consistently focuses on core ideas rather than generality, and it makes no attempt to be an exhaustive, deductive compendium of results.
It seems that the questions are more natural, and there is a method to the madness that might otherwise appear as a long list of theorems and proofs. This book always focuses on central ideas rather than generality, and it makes no attempt to be a complete, deductive catalog of results.
2025.6.6
原文
It is designed to capture the intellectual imagination. Those who become interested are then exceptionally well prepared for a second course starting from complex-valued functions on more general spaces, while those content with a single semester come away with a strong sense of the essence and purpose of real analysis.
翻译
本书被设计用来捕捉这些智力想象。那些真正感兴趣的人意外的为第二门课程做好了准备，这门课程开始于广义空间上的复值函数，那些只有一个学期课程的实分析，往往伴随着更强的直觉性和目的。
这门课程旨在激发智力想象力。对于那些对这门课程产生兴趣的学生，他们将为以复值函数和更一般空间为起点的后续课程做好充分准备；而对于那些只打算学习一个学期的学生，他们也将深刻理解实分析的本质和目的。
本书的设计旨在激发智力想象。对于那些真正感兴趣的学生，他们会为后续课程（包括研究更一般空间上的复值函数）做好充分的准备。而对于只修一学期课程的学生来说，他们也能够对实分析的核心概念和目的形成深刻的理解。
转写
It aim to fullfill the intellectual imagination. Those becoming interested then expceptionally well prepare for a second course beginning from complex-valued functions on more general spaces, though those topics with a single semester always bring a strong sense of the essence and purpose of real analysis.
It is designed to stimulate intellectual curiosity. Those who develop an interest are then exceptionally well-prepared for a subsequent course focusing on complex-valued functions in more general settings. Meanwhile, those who choose to complete only one semester will still gain a deep appreciation of the core concepts and purpose of real analysis.
It aims to stimulate the intellectual imagination. Those who become interested are then exceptionally well prepared for a second course that begins with complex-valued functions on more general spaces. Meanwhile, even a single-semester course offers a strong sense of the essence and purpose of real analysis.
2025.6.7
原文
Turning once more to the concluding passages of Chapter 8, “By viewing the different infinities of mathematics through pathways crafted out of finite objects, Weierstrass and the other founders of analysis created a paradigm for how to extend the scope of mathematical exploration deep into territory previously unattainable.”
翻译
这里引用第八章的总结段落，“通过对由有限情况刻画的无穷情况的数学的学习，Weierstrass和其他数学的奠基者创造了一个如何将数学领域拓展到前人未曾涉及领域的蓝图。”
再一次回到第八章的结尾部分，“通过用有限对象构造的路径来看待数学中各种无限性，魏尔施特拉斯和其他分析学的奠基者创造了一种范式，阐明了如何将数学探索的范围扩展到此前难以企及的领域。”
这里引用第八章的总结段落：“通过以有限对象为路径来研究数学中的各种无限性，Weierstrass和其他分析学的奠基者创造了一种范式，阐明了如何将数学探索拓展到此前未曾触及的领域。”
转写
In light of the summary passages of Chapter 8, “By investigation of the different infinities of mathematics via methods developed in finite objects, Weierstrass and the other poineers of analysis constructed an intruction on how to enlarge the scope of mathematical study deep into area preceedingly unreachable.”
Turning once more to the concluding passages of Chapter 8, “By viewing the different infinities of mathematics through pathways crafted out of finite objects, Weierstrass and the other founders of analysis established a paradigm demonstrating how to expand the reach of mathematical inquiry deep into realms previously inaccessible.”
In light of the summary passages of Chapter 8: “By investigating the different infinities of mathematics through methods crafted from finite objects, Weierstrass and the other pioneers of analysis created a paradigm for extending the scope of mathematical study deep into areas previously unreachable.”
2025.6.8
原文
This exploration has constituted the major thrill of my intellectual life. I am extremely pleased to offer this guide to what I feel are some of the most impressive highlights of the journey. Have a wonderful trip!
翻译
这些研究已经覆盖了我智慧生活的所有动机。我非常高兴可以提供我在这一路学习旅程上感受最深的一些启发。希望你学习愉快。
这次探索成为了我智力生涯中的最大乐趣。我非常高兴能提供这份指南，分享我认为是旅途中最令人印象深刻的亮点。祝你旅途愉快！
这些研究是我智识生涯中最激动人心的部分。我非常高兴能为你提供一份指南，分享这段旅程中我认为最值得注意的亮点。祝你学习之旅愉快！
转写
This investigation has provided the major motivation of my academic life. I am very glad to share this instruction about what I feel are some of the most enlighted thoughts of the journey. Enjoy this trip!
This journey has represented the most significant source of intellectual excitement in my life. I am delighted to present this guide, highlighting what I consider to be some of the most remarkable features of the journey. Wishing you an inspiring and rewarding trip!
This investigation has represented the most significant source of intellectual motivation in my academic life. I am delighted to share this guide, highlighting what I feel are some of the most enlightening insights from this journey. I hope you find it a rewarding and inspiring experience.
2025.6.9
原文
Biological invasion refers to the phenomenon of alien species growing and reproducing in a new location under natural conditions or the influence of media. This phenomenon is a natural process, and invasive species are often introduced into native ecosystems intentionally or accidentally.
翻译
生物入侵通常指的是那些在自然条件下或者媒介影响下外来物种在新地区生长和繁殖的现象。这中现象是自然发生的，并且入侵物种经常会被有意或无意的引入当地的生态系统。
生物入侵是指外来物种在新的地点，在自然条件或媒介的影响下生长和繁殖的现象。这是一种自然过程，而入侵物种往往是被有意或无意地引入本土生态系统的。
生物入侵通常指在自然条件或媒介的传播作用下，外来物种在新的地区定居、繁殖并扩散的现象。这是一种自然过程，且入侵物种往往会被有意或无意地引入本地生态系统。
转写
Biological invasion manifests as the phenomenon of invasive speices growing and reproducing in a new environment due to some natrual changes or the human activities. This happens naturally, or the alien species is introduced to the native ecosytems on purpose or accidentally by human.
Biological invasion refers to the phenomenon in which alien species grow and reproduce in a new location under natural conditions or with the help of a vector. This phenomenon is a natural process, and invasive species are often introduced into native ecosystems either intentionally or accidentally.
Biological invasion refers to the phenomenon of alien species growing and reproducing in a new environment due to natural changes or human activities. This process may occur naturally, but invasive species are often intentionally or accidentally introduced into native ecosystems by humans.
2025.6.10
原文
With the introduction of species, these alien species may benefit humans on the one hand, and but also have a certain impact on the local ecological environment and even economic development on the other hand [14,28].
翻译
随着物种的引入，这些外来物种或许会在一方面给人类带来好处，但是也会在另一方面给生态环境和经济发展带来一定的影响。
“随着外来物种的引入，这些外来物种一方面可能对人类有益，但另一方面也可能对当地生态环境甚至经济发展产生一定的影响。”
“随着外来物种的引入，这些物种可能在一定程度上造福人类，但也可能对生态环境和经济发展带来潜在的负面影响。”
转写
By the invasion of species, these invasive species may bring humans benifits on the one hand, but also have a side effect on the local ecological system and even economic development on the other hand.
“With the introduction of alien species, these species may, on the one hand, bring benefits to humans, but on the other hand, they may also impact the local ecological environment and even affect economic development.”
“With the introduction of invasive species, these species may, on the one hand, benefit humans, but on the other hand, they may negatively impact the local ecosystem and even economic development.”
2025.6.11
原文
In order to reduce the harm caused by biological invasion and rationally utilize biological invasion to make positive contributions to social and economic development, it is necessary and challenging to take the in-depth research on the mechanism of biological invasion.
翻译
为了减少由于生物入侵带来的危害和随机的利用生物入侵给社交或者经济发展创造一些价值，在生物入侵的机制进行深入研究就变得很有必要性和挑战性。
为了减少生物入侵带来的危害，并合理利用生物入侵对社会和经济发展作出积极贡献，有必要且具有挑战性地深入研究生物入侵的机制。
为了减少生物入侵带来的危害，并合理利用生物入侵为社会和经济发展创造价值，对生物入侵机制的深入研究变得既必要又具有挑战性。
转写
The purpose of delving into the mechanism of biological invasion necessarily and chanllengingly is to reduce the harmful effect produced by bilogical invasion and logically take the bilogical invasion into account to facilatate social and economic development.
To mitigate the harm caused by biological invasions and to harness their potential for making positive contributions to social and economic development, it is both necessary and challenging to conduct in-depth research into the mechanisms of biological invasion.
The purpose of delving into the mechanisms of biological invasion—which is both necessary and challenging—is to reduce the harmful effects of biological invasions and to rationally utilize them to facilitate social and economic development.
2025.6.12
原文
Since alien species may not interact with neighboring organisms around them for a long period of time, such as an invasive plant population, we can use a single-population reaction-diffusion model to describe the spatial dynamics of the introduced population.
翻译
因为外来物种可能长时间内不会跟他们周围的生物产生任何互动，例如一个入侵的植物，所以我们可以使用一个单种群的反应扩散方程模型去描述引入物种的空间动力学。
由于外来物种（如入侵植物种群）在很长一段时间内可能不会与周围的生物发生相互作用，因此我们可以使用单种群的反应扩散模型来描述其空间动态。
鉴于外来物种（如入侵植物种群）可能在相当长的一段时间内不与其周边生物发生显著互动，我们可以采用一个单种群的反应-扩散模型来描述该引入种群的空间动力学过程。
转写
Alien species may not affect the surronding organisms with the laspe of time, like an invasive plant population, so it is reasonable to use sigle-population reaction-diffusion system model the spatial dynamics of the alien population.
Since alien species—such as an invasive plant population—may remain ecologically isolated from neighboring organisms for an extended period, a single-species reaction-diffusion model can be employed to describe the spatial dynamics of the introduced population.
Given that alien species, such as invasive plant populations, may remain ecologically isolated from neighboring organisms for a considerable period, a single-species reaction-diffusion model provides a suitable framework for capturing the spatial dynamics of the introduced population.
2025.6.13
原文
The well-known logistic equation just describes the spreading process of a single population. In the real world, many species, including invasive species, reproduce only at specific times of year, whose pattern is called ``reproductive pulses’’ or “birth pulse’’ .
翻译
众所周知的逻辑式方程描述了单种群的传播过程。在现实世界里，许多物种（也包含入侵物种）都在一年里特定的时间繁殖，这种繁殖模型被称为“繁殖脉冲”或者“出生脉冲”。
著名的Logistic方程只是描述了单一种群的扩散过程。在现实世界中，许多物种（包括入侵物种）仅在一年中的特定时间繁殖，这种模式被称为“繁殖脉冲”或“出生脉冲”。
著名的Logistic方程描述了单一物种的增长过程。然而在自然界中，许多物种（包括入侵物种）只在每年特定的时间进行繁殖，这种繁殖模式被称为“繁殖脉冲”或“出生脉冲”。
转写
It is well-known that the logistic equation illustrates the spreading phenomenon of a single population. In reality, the pattern that species reproduce at specific times of the year is called “reproductive pulse” or “birth pulse”, which also holds for some invasive species.
The classical logistic equation captures the growth dynamics of a single population. However, in the real world, many species—including invasive ones—reproduce only during specific times of the year, a phenomenon known as “reproductive pulses” or “birth pulses.”
It is well known that the logistic equation describes the growth dynamics of a single population. In reality, however, many species reproduce only at specific times of the year—a phenomenon known as “reproductive pulses” or “birth pulses”, which is also observed in some invasive species.
2025.6.14
原文
Actually, many mammals and fishes only reproduce during certain seasons, so as to admit the reproductive pulse pattern. For example, the hairy crabs that invaded Europe in the early 20th century have a breeding season from June to August every year, and big brown bats usually reproduce only in late June in Colorado.
翻译
实际上，许多哺乳动物和鱼只在特定的季节繁殖，刚好符合了繁殖脉冲的模式。例如，在20世纪初入侵欧洲的长毛蟹会在每年的六月到八月进行繁殖，并且colorado的大棕蝙蝠通常只在六月末进行繁殖。
实际上，许多哺乳动物和鱼类只在特定的季节繁殖，从而表现出脉冲式的繁殖模式。例如，20世纪初入侵欧洲的毛蟹每年在6月至8月间进入繁殖期，而在科罗拉多，大褐蝙蝠通常只在6月下旬繁殖。
实际上，许多哺乳动物和鱼类仅在特定季节进行繁殖，从而呈现出典型的繁殖脉冲模式。例如，在20世纪初入侵欧洲的中华绒螯蟹每年于6月至8月进入繁殖期，而科罗拉多州的大棕蝙蝠通常仅在每年6月下旬进行繁殖。
转写
In fact, many mammals and fishes only breed within specific seasons, which conincides with the reproductive pulse pattern. For example, the hairy crabs, invading Europe in the early 20th century, have a reproductive season from June to August every year, and big brown bats in Corlorado reproduce only in the late June.
It is well known that many mammals and fish reproduce only during specific seasons, forming a characteristic pattern of reproductive pulses. For example, the hairy crabs that invaded Europe in the early 20th century breed annually from June to August, while big brown bats in Colorado typically reproduce only in late June.
In fact, many mammals and fish breed only during specific seasons, which coincides with the pattern of reproductive pulses. For example, the hairy crabs that invaded Europe in the early 20th century breed annually from June to August, and big brown bats in Colorado typically reproduce only in late June.
2025.6.15
原文
For reproductive pulse species with two different stages of reproduction and diffusion, Lewis and Li proposed the following impulse response-diffusion model to describe within-seasonal and between-seasonal dynamics: where g describes the population density at the end of a reproductive stage as a function of the population density at the beginning of the stage.
翻译
对于带有不同阶段的繁殖和扩散的繁殖脉冲物种的情况，路易斯和李提出了下列的脉冲反应-扩散模型来描述季节内和季节间的动力学，其中g描述了繁殖阶段末的种群密度，并且也作为每个繁殖阶段初始种群密度的函数。
对于具有两个不同繁殖与扩散阶段的繁殖脉冲物种，Lewis 和 Li 提出了如下脉冲反应-扩散模型，以刻画其季节内与季节间的动态：其中函数 g 描述了一个繁殖阶段开始与结束时种群密度之间的关系。
对于具有两个不同繁殖和扩散阶段的繁殖脉冲物种，Lewis 和 Li 提出了一个脉冲反应-扩散模型，用以刻画其季节内和季节间的种群动态。模型中的函数 g 描述了每个繁殖阶段末的种群密度，作为该阶段初始种群密度的函数。
转写
For the species with two diferent stages of reproduction and diffusion, Lewis and Li proposed the following impulse reaction-diffusion model to capture the within-seasonal and between-seasonal dynamics: where g is the function of the population density at the beginning of the reproductive stage describing the population density at the end of a reproductive stage.
For reproductive pulse species exhibiting two distinct stages of reproduction and diffusion, Lewis and Li proposed the following impulsive reaction-diffusion model to capture both within-seasonal and between-seasonal dynamics. In this model, the function g characterizes the population density at the end of a reproductive stage as a function of the density at its beginning.
For species exhibiting two distinct stages of reproduction and diffusion, Lewis and Li proposed the following impulsive reaction-diffusion model to describe both within-seasonal and between-seasonal dynamics. In this model, the function g characterizes the population density at the end of a reproductive stage as a function of the population density at the beginning of that stage.
2025.6.16
原文
In this model, reproduction occurs only once a year, which is described by a growth function. Outside the breeding season, population density will follow the reaction-diffusion law over time. For model (1.2), the authors concluded that the diffusion coefficient, and the combination of discrete-and continuous-time growth and mortality determine the spreading and persistence dynamics of the population in a wide variety of ecological scenarios.
翻译
在这个模型里，繁殖通常一年发生一次，我们用一个增长函数来描述。在繁殖季节之外，种群密度遵循着反应扩散的规律随时间演变。对于模型1.2，作者总结了扩散系数，离散和连续时间增长与死亡的综合考量会决定种群在更广泛的场景下发生传播和持久的动力学现象。
在该模型中，繁殖每年仅发生一次，通过一个增长函数来描述。在非繁殖季节，种群密度随时间变化遵循反应-扩散规律。对于模型 (1.2)，作者得出结论认为，扩散系数以及离散-连续时间的增长与死亡率的组合，共同决定了在各种生态情境下种群的传播与持续动态。
在该模型中，繁殖每年仅发生一次，通过一个增长函数加以描述。在非繁殖季节，种群密度按照反应-扩散规律随时间演化。对于模型1.2，作者指出，扩散系数以及离散时间与连续时间的增长和死亡过程的共同作用，决定了种群在多种生态情境下的传播与持续动态。
转写
In this model, reproduction that is described by a growth function takes place annually. After the reproductive stage, population density will grow according to the reaction-diffusion law. For model (1.2), the authors proved that the diffusion coefficient, and the combination of discrete-time and continuous-time for growth and mortality decide the spreading and persistence phenonmennon in more ecological scenarios.
In this model, reproduction occurs only once per year and is represented by a growth function. During the non-breeding season, the population density evolves according to the reaction-diffusion process. For model (1.2), the authors concluded that the diffusion coefficient, along with the interplay between discrete- and continuous-time growth and mortality, determines the spreading and persistence dynamics of the population across a wide range of ecological scenarios.
In this model, reproduction occurs annually and is described by a growth function. Outside the reproductive season, the population density evolves according to the reaction-diffusion law. For model (1.2), the authors concluded that the diffusion coefficient, along with the interplay between discrete- and continuous-time growth and mortality, determines the spreading and persistence dynamics of the population across a wide range of ecological scenarios.
2025.6.17
原文
Based on [20], Fazly et al. [11,12] further investigated a more general impulse model with advection term and high dimensional spatial domain. Recently, Wu and Zhao [29] considered a birth pulse population model with the nonlocal dispersal, and they establish a threshold-type result on the global dynamics of the system in a bounded domain, as well as the existence of invasion speed and its coincidence with the minimal speed for monotone traveling waves in an unbounded domain.
翻译
基于参考文献20，Fazly等人进一步研究了一个更一般的带有对流并且在更高维空间上的脉冲模型。最近，Wu和zhao在参考文献29中考虑了带有非局部扩散的出生脉冲的种群模型，他们得到了这个系统在有界区域上关于全局动力学的阈值结果，并且也得到了在无界情况下的入侵速度的存在性以及其和最小波速的一致性。
基于文献 [20]，Fazly 等人 [11,12] 进一步研究了一个更一般的含有平流项和高维空间区域的脉冲模型。近期，Wu 和 Zhao [29] 考察了一个具有非局部扩散的出生脉冲种群模型，并在有界区域中建立了关于系统整体动力学的阈值类型结果，同时还证明了在无界区域中入侵速度的存在性及其与单调行波最小传播速度的一致性。
在参考文献 [20] 的基础上，Fazly 等人进一步研究了一个更一般的脉冲模型，该模型包含对流项并扩展到更高维的空间域。最近，Wu 和 Zhao 在文献 [29] 中考察了一个具有非局部扩散的出生脉冲种群模型，并在有界区域内建立了该系统的全局动力学的阈值型结果。同时，他们还证明了在无界域中入侵速度的存在性，并证明该速度与单调行波的最小传播速度相一致。
转写
Motivated by [20], Fazly et al. [11,12] proposed a more general impulsive model by considering advection term and high dimensional spatial domain. Recently, Wu and Zhao [29] studied a impulsive model with nonlocal dipersal in which they construct a threshold-type result for the global dynamics of this model in a boudned domain, along with the existence of invasion speed and its conincidence iwth the minimal wave speed for the traveling wave in an unbounded domain.
Building on [20], Fazly et al. [11,12] further studied a more general impulsive model incorporating an advection term and a high-dimensional spatial domain. More recently, Wu and Zhao [29] investigated a birth-pulse population model with nonlocal dispersal. They established a threshold-type result characterizing the global dynamics of the system in a bounded domain, as well as the existence of an invasion speed that coincides with the minimal speed of monotone traveling waves in an unbounded domain.
Motivated by [20], Fazly et al. [11,12] proposed a more general impulsive model incorporating an advection term and a high-dimensional spatial domain. Recently, Wu and Zhao [29] studied an impulsive model with nonlocal dispersal. They established a threshold-type result characterizing the global dynamics of the model in a bounded domain, and proved the existence of an invasion speed in an unbounded domain, which coincides with the minimal speed of monotone traveling waves.
2025.6.18
原文
As a follow-up of [29], they [30] investigated the evolution dynamics for an impulsive hybrid population model with nonlocal dispersal operator in a heterogeneous landscape by overcoming the difficult induced by the landscape heterogeneity and non-compactness. More recently, Zhang et al. [32] studied a birth pulse population model with a shifting habitat, in which both the dispersal stage and reproduction stage have been assumed to undergo a climate change, and discussed the effects of this kind of shifting environment on the spread and invasion with birth pulse.
翻译
作为参考文献29的续作，他们在参考文献30中研究了在异质环境下带有非局部扩散的杂交脉冲模型演化动力学，并且克服了由环境异质性和非紧性带来的困难。特别的最近一段时间，zhang等人研究了带有移动环境的出生脉冲模型，其中扩散和繁殖阶段都考虑了气候变化，并且也讨论了这种移动环境和出生脉冲对传播和入侵的影响。
作为对文献 [29] 的后续研究，文献 [30] 探讨了一个具有非局部扩散算子的脉冲混合种群模型在异质生境中的演化动力学，克服了由生境异质性和非紧性所带来的困难。最近，Zhang 等人 [32] 研究了一个具有繁殖脉冲的种群模型，其栖息地处于移动之中。在该模型中，扩散阶段和繁殖阶段都被假设受到气候变化的影响，并讨论了这种迁移环境对具有繁殖脉冲的扩散与入侵行为的影响。
在文献[29]的基础上，文献[30]研究了异质环境下具有非局部扩散项的脉冲混合型种群模型的演化动力学，克服了由生境异质性与非紧性所带来的困难。最近，Zhang 等人研究了一个处于移动栖息地中的繁殖脉冲模型，其中扩散阶段和繁殖阶段均考虑了气候变化的影响，并进一步探讨了移动环境与繁殖脉冲对种群传播与入侵的影响。
转写
Based on [29], they [30] delved into the evolution dynamics for an impulsive hybrid population model with nonlocal dispersal operator in a heterogeneous landsacape, in which they overcome the difficulty caused by the landscape heterogeneity and non-compactness. More recently, Zhang et al. [32] incorporated a birth pulse population with a shifting habitat, where they consider both the dispersal stage and reproduction stage and assume theses two stages undergo a climate change, and they discusse the infuluence of the shifting environment and birth pulse on the spread and invasion of populations.
As a continuation of [29], the authors in [30] investigated the evolutionary dynamics of an impulsive hybrid population model with a nonlocal dispersal operator in a heterogeneous landscape, addressing the challenges posed by landscape heterogeneity and the lack of compactness. More recently, Zhang et al. [32] examined a birth-pulse population model in a shifting habitat, where both the dispersal and reproduction stages are subject to climate change. They analyzed how such environmental shifts influence the spread and invasion dynamics under the birth-pulse framework.
Building on [29], the authors in [30] investigated the evolutionary dynamics of an impulsive hybrid population model with a nonlocal dispersal operator in a heterogeneous landscape, where they overcame the challenges arising from landscape heterogeneity and the lack of compactness. More recently, Zhang et al. [32] studied a birth-pulse population model in a shifting habitat, in which both the dispersal and reproduction stages were assumed to be affected by climate change. They further discussed how such environmental shifts, together with the birth-pulse structure, influence the spread and invasion dynamics of populations.
2025.6.19
原文
In mathematical models of biological populations, most impulse reaction-diffusion models are derived in a fixed region that does not change with time [6,33]. However, it is clear that in nature, the habitats in which almost all biological organisms live are subject to change owing to seasons, climate, and even human activities.
翻译
在生物种群的数学模型中，大多数脉冲反应扩散方程都是在不随时间变化的定区域推导出来的。然而，在自然界中，显然生物生存环境会由于季节，气候，甚至人类活动而改变。
在生物种群的数学模型中，大多数脉冲反应扩散模型是在固定区域内建立的，该区域不随时间变化。然而，显然在自然界中，几乎所有生物生存的栖息地都会因为季节、气候甚至人类活动而发生变化。
在生物种群的数学建模中，大多数脉冲反应扩散方程都是在一个随时间保持不变的固定区域中构建的。然而，在自然界中，生物所依赖的栖息环境通常会因季节更替、气候变化甚至人类活动而发生变化。
转写
For mathematical models of biological populations, most impulse reaction-diffusion models are induced in a fixed domain that does not evlove with time [6,33]. In fact, in nature, it is clear that the habitats where biological organisms live are varying with time due to seasons, climate, and human activities.
In mathematical models of biological populations, most impulsive reaction-diffusion models are formulated in fixed spatial domains that remain unchanged over time [6,33]. However, it is evident that in nature, the habitats of nearly all biological organisms are dynamic, varying with seasons, climate change, and even human activities.
In mathematical modeling of biological populations, most impulsive reaction-diffusion models are developed in fixed, time-invariant domains [6,33]. However, in nature, the habitats of biological organisms are dynamic and change over time due to seasons, climate variation, and human activities.
2025.6.20
原文
This paper investigates the spreading speeds and generalized traveling wave solutions for a timedependent epidemic system with nonlocal dispersal. In this nonautonomous epidemic system, both the nonlocal dispersal kernel and the coefficients in reaction terms are general time heterogeneous and possess uniform mean values. To characterize the spreading speed of the system, we first establish the spreading speed of a scalar nonautonomous KPP equation.
翻译
这篇文章研究了带有非局部项且时间依赖的传染病系统的传播速度和广义行波解。在这个非自治的传染病系统，非局部核函数和反应项里的系数都是一般的时间异质并且由一致平均值。为了描述这个系统的传播速度，我们首先得到了一维KPP非自治方程的传播速度。
本论文研究了具有非局部扩散的时变传染病系统的传播速度和广义行波解。在这个非自治的传染病系统中，非局部扩散核以及反应项中的系数均为一般的时间非齐性函数，并具有一致的平均值。为了刻画该系统的传播速度，我们首先建立了一个标量非自治 KPP 方程的传播速度。
本文研究了一类具有非局部扩散项和时间依赖性的传染病模型的传播速度与广义行波解。在该非自治系统中，非局部扩散核函数与反应项中的系数均具有一般性的时间异质性，且共享统一的时间平均值。为刻画该系统的传播速度，我们首先分析了一维非自治 KPP 型方程，并确定其传播速度。
转写
This paper studies the spreading speeds and generalized traveling wave solutions for a nouautonomous epidemic system with nonlocal dispersal. In this time-dependent system, both the nonlocal dispersal kernel and the coefficients in reaction terms are genral time heterogenoes and admit uniform mean values. To describe the spreading pseed of this system, we begin with the spreading speed of a scalar nonautonomous KPP equation.
This paper explores the spreading speeds and generalized traveling wave solutions of a time-dependent epidemic model with nonlocal dispersal. In this nonautonomous system, both the dispersal kernel and the coefficients in the reaction terms exhibit general temporal heterogeneity while sharing uniform time-averaged values. To characterize the spreading speed of the system, we first derive the spreading speed of a scalar nonautonomous KPP-type equation.
This paper investigates the spreading speeds and generalized traveling wave solutions in a nonautonomous epidemic model with nonlocal dispersal. In this time-dependent system, both the nonlocal dispersal kernel and the reaction coefficients exhibit general temporal heterogeneity and possess uniform time-averaged values. To characterize the spreading speed of the system, we first establish the spreading speed of a scalar nonautonomous KPP equation.
2025.6.21
原文
For instance, the depth and water area of some rivers and lakes may increase in summer or the rainy season, while decrease in winter or the dry season. This will cause the habitats of various organisms living in them to change periodically. To address this, Meng and Lin [22] introduced the periodic evolution of the region into the pulsed reaction-diffusion single population model, and found that the larger periodic evolution rate of domain may improve the survival ability of population.
翻译
例如，一些河流和湖泊的深度和水域面积会在夏天或者雨季增加，但是会在冬天和旱季缩减。这种变化会导致有很多生物生存其中的栖息地进行周期性的变化。为了解决这个问题，Meng和Li在[22]中为单种群的脉冲反应扩散方程引入了周期演变的区域，且发现栖息地过快的周期演化速率会提高种群生存的能力。
例如，一些河流和湖泊的深度与水域面积在夏季或雨季可能会增加，而在冬季或旱季则会减少。这种周期性的水文变化将导致其内部多种生物的栖息地发生周期性变化。为描述这一现象，Meng 和 Lin [22] 在单种群的脉冲反应扩散模型中引入了周期演化的区域，并指出，当域的周期演化速率较大时，种群的生存能力可能会得到提升。
例如，一些河流和湖泊的水深与水面面积在夏季或雨季可能会增加，而在冬季或旱季则会减少。这种水文变化将导致生物栖息地发生周期性变化。为研究该现象，Meng 和 Li [22] 在单种群的脉冲反应扩散模型中引入了周期演化的空间区域，并发现，较大的周期演化速率有助于提升种群的生存能力。
转写
For example, the depth and water area of some reivers and lakes may expand in summer or the rainy season and shrink in winter or the dry season. As a result, the habitas of biological organisms may change periodically. Therefore, Meng and Lin [22] proposed a impulsive reaction-diffusion single population model on the periodcally evolvoing domain and concluded that the large periodic evolution rate of the domain may benifit the survial of population.
For example, the depth and surface area of certain rivers and lakes may increase during the summer or rainy season and decrease during the winter or dry season, leading to periodic changes in the habitats of resident organisms. To capture this phenomenon, Meng and Lin [22] incorporated periodic domain evolution into a pulsed reaction-diffusion model for a single species. Their results indicate that a higher rate of periodic domain change can enhance the survival potential of the population.
For example, the depth and surface area of certain rivers and lakes may increase during the summer or rainy season and decrease during the winter or dry season. As a result, the habitats of these organisms may undergo periodic changes. Accordingly, Meng and Lin [22] proposed an impulsive reaction-diffusion model for a single population in a periodically evolving domain and concluded that a higher rate of periodic domain evolution may enhance the survival potential of the population.
2025.6.22
原文
In addition to cyclical changes in habitats due to seasonal reasons, the habitats of invasive species may also continue to expand due to human activities, global warming, and other reasons. For example, after the Aedes albopictus mosquito, originally from Southeast Asia, invaded the United States [23], its habitat spread from southern Texas to New Jersey and Illinois, spanning 14 latitudes from 1985 to 2012 alone [31].
翻译
因为季节变化导致栖息地的周期性变化，入侵物种的栖息地或许会由于人类活动，全球变暖或者其他的原因持续扩张。例如，原产于东南亚的Aedes albopictus蚊子入侵美国之后，其栖息地从南texas扩展到new jersey和illiois，从1985年到2012年独自横跨14个州。
除了由于季节性因素导致栖息地的周期性变化外，入侵物种的栖息地还可能由于人类活动、全球变暖等因素持续扩张。例如，原产于东南亚的白纹伊蚊（Aedes albopictus）入侵美国后，其栖息地从德克萨斯州南部扩展至新泽西州和伊利诺伊州，仅在1985年至2012年间就跨越了14个纬度 [31]。
除了因季节变化引起的栖息地周期性变动外，入侵物种的栖息地还可能因人类活动、全球变暖等因素而持续扩张。例如，原产于东南亚的 Aedes albopictus 入侵美国后，其栖息地从德克萨斯州南部扩展至新泽西州和伊利诺伊州，仅在1985年至2012年间就跨越了14个纬度。
转写
It is obvious that there would be cyclical changes in habitats raised by seasonal reasons and the habitats of invasive species may also expand because of human activities, global warming, and other reasons. For instance, the Aedes albopictus mosquito spread from southern Texas to New Jersey and Illinois accross 14 latitutes from 1985 to 2012 [31] after its invasion from Southeast Asia to United States [23].
In addition to cyclical habitat changes driven by seasonal variation, the geographic range of invasive species may also undergo continuous expansion due to anthropogenic activities, global warming, and other contributing factors. For instance, following its introduction from Southeast Asia, the Aedes albopictus mosquito expanded its habitat in the United States from southern Texas to as far north as New Jersey and Illinois, spanning 14 degrees of latitude between 1985 and 2012 alone [31].
Habitats are subject to cyclical changes driven by seasonal variation, and the geographic range of invasive species may continue to expand due to anthropogenic activities, global warming, and other environmental factors. For instance, following its introduction from Southeast Asia into the United States [23], Aedes albopictus expanded its range from southern Texas to as far north as New Jersey and Illinois, spanning 14 degrees of latitude between 1985 and 2012 [31].
2025.6.23
原文
In order to reveal the impact of the continuous expansion of habitat on the survival of a single population, the regional limited growth condition was incorporated into the logistic model on a single invasive population in [24], which reveals that the growth of domain gives rise to a positive effect on the asymptotic stability of positive steady state solution of the model, while has a negative effect on the asymptotic stability of the trivial solution.
翻译
为了揭示栖息地的连续扩张对种群生存的影响，在[24]中他们考虑了在单个入侵物种的logistic模型中引入区域局限增长条件，其揭示了栖息地的增长有助于分析模型正稳态的解渐近稳定性，但是不利于平凡解渐近稳定性的分析。
为了揭示栖息地持续扩张对单一种群存活的影响，文献 [24] 在对单一入侵种群的Logistic模型中引入了区域限制增长条件。研究表明，区域的增长对模型的正稳态解的渐近稳定性具有正向影响，而对零解的渐近稳定性则产生负面影响。
为了揭示栖息地持续扩张对种群存活的影响，文献 [24] 在单一入侵种群的Logistic模型中引入了区域有限增长条件。研究表明，栖息地的扩张有助于增强正稳态解的渐近稳定性，而对平凡解的渐近稳定性则产生不利影响。
转写
To capture how the continous expansion of habitat affect the survivla of a single population, the authors in [24] incorporated the regional limited growth condition into the logistic model on a single invasive population. They showed that the gorwth of domain leads to a positive effect on the asyptotic satbility of positive steady state solution of the model, but a negative effect on the asymptotic stability of the trivial solution.
To investigate the effect of continuous habitat expansion on the persistence of a single species, the study in [24] incorporated regionally limited growth conditions into a logistic model for a single invasive population. The results indicate that domain growth enhances the asymptotic stability of the positive steady-state solution, while it undermines the asymptotic stability of the trivial solution.
In order to examine the effect of continuous habitat expansion on the persistence of a single population, the study in [24] introduced a regionally limited growth condition into a logistic model describing an invasive species. It was demonstrated that domain growth promotes the asymptotic stability of the positive steady state, while it diminishes the stability of the trivial equilibrium.
2025.6.25
原文
More recently, the global dynamics has been studied in [18] for reaction-diffusion systems of multi-species interaction in a time-varying domain of three types. It is worth noting that the population models with an evolving domain are quite different from those with a free boundary, which have been studied extensively in recent years (see, e.g., [9,10] and references therein).
翻译
近来，建立在三种类型时变区域上的多个种群互动的反应扩散方程的全局动力学已经被深入研究。值得注意的是，在时变区域上的种群模型跟那些带有自由边值区域的模型是不一样的，自由边值问题已经在近些年被研究的很深入（可以在[9,10]和其中的参考文献中找到对应的例子）。
最近，文献 [18] 研究了三类时间变化区域中多种群相互作用反应-扩散系统的整体动力学。值得注意的是，具有演化区域的种群模型与近年来广泛研究的自由边界模型有本质区别（参见文献 [9,10] 及其中的参考文献）。
转写
Especiall lately, it has been studied that the global dynamics of reaction-diffusion systems for multi-species interaction in a time-varying domain of three type. Significantly, the population model with an evolving domain is quite different with those with a free boundary, which have been delved during the past years(see. e.g., [9,10] and references therein).
More recently, the global dynamics of reaction-diffusion systems describing multi-species interactions in three types of time-varying domains has been investigated in [18]. It is worth emphasizing that population models defined on evolving domains are fundamentally different from those with free boundaries, which have been extensively studied in recent years (see, e.g., [9,10] and the references therein).
Recently, the global dynamics of reaction-diffusion systems describing multi-species interactions in three types of time-varying domains has been investigated. It is worth noting that population models defined on evolving domains are fundamentally different from those with free boundaries, which have been extensively studied in recent years (see, e.g., [9,10] and the references therein).
2025.6.26
原文
The evolution of habitat usually results from environmental changes due to some objective factors (such as climate and seasons), and these factors are irrelevant to the population itself. In the free boundary problem, however, the change of domain is caused by population activities, and the outward expanding range of habitat is determined by its activity ability.
翻译
栖息地的演化通常是由客观因素（如气候或者天气）导致的环境变化引起的，且这些因素是跟种群本身无关的。但是在自由边值问题中，区域的变化通常是由种群活动引起的，且区域向外的扩张的范围通常是由种群的活动能力决定的。
栖息地的演化通常源于某些客观因素（如气候和季节）所引起的环境变化，这些因素与种群本身无关。然而，在自由边界问题中，区域的变化是由种群活动引起的，栖息地向外扩展的范围取决于其自身的活动能力。
栖息地的演化通常源于由气候或天气等客观因素引发的环境变化，这类变化独立于种群自身的特性。然而，在自由边值问题中，空间区域的演变通常由种群活动驱动，且其向外扩张的范围主要取决于种群的扩散能力。
转写
Environmental changes subject to some objective factors (such as climate and seasons) usually casue the evolution of habitat, which are irrelevant to the population itself. However, in the free boundanry problems, population activities are the main factor to give rise to the change of domain, and the outward expanding area of habitat is determined by the population itself.
The evolution of habitat is typically driven by environmental changes caused by external, objective factors such as climate and seasonal variation, which are independent of the population itself. In contrast, within the framework of free boundary problems, the spatial domain evolves as a result of population dynamics, and the extent of habitat expansion is governed by the species’ dispersal capability.
The evolution of habitat is typically driven by environmental changes caused by objective factors such as climate and seasonal variation, which are independent of the population. In contrast, in free boundary problems, the change in the spatial domain is primarily driven by population dynamics, and the extent of outward habitat expansion is determined by the species’ own dispersal ability.
2025.6.27
原文
In this paper, we will introduce the limited growth of the region and birth impulse simultaneously into the single population model, as well as adopt the general reaction mechanism, and then explore the dynamical behavior under the dual action of these two factors. We will derive an impulsive reaction-diffusion model with asymptotically bounded domain, and establish a threshold type result on persistence and extinction of population for such a class of evolution systems.
翻译
在这篇文章中，我们将会对单种群模型同时引入区域限制增长和出生脉冲，同时也考虑反应扩散机制，并且研究在这两个因素共同影响下的动力学行为。我们会推导出一个带有渐近有界区域的脉冲反应扩散方程，并且也得到对于这一类演化方程持久和灭绝的阈值结果。
在本文中，我们将同时引入区域的有限增长与种群的脉冲出生机制，并结合一般形式的反应项，探讨这两种因素共同作用下模型的动力学行为。我们将建立一个具有渐近有界区域的脉冲反应扩散模型，并给出一个关于种群持续存在与灭绝的阈值型判别结果，适用于该类演化系统。
在本文中，我们将同时向单种群模型引入区域的有限增长机制与出生脉冲效应，并结合反应扩散机制建立模型，以探讨这两种因素共同作用下系统的动力学特征。我们构建了一个具有渐近有界区域的脉冲反应扩散模型，并建立了一个关于该类演化系统中种群持久性与灭绝的阈值判据。
转写
In this paper, we consider both asympotic growth of the domain and birth impulse for the reaction-diffusion single population model, and investigate the global dynamics with the interaction of these two factors. We deduce an impulsive reaction-diffusion model with asymptotically bounded domain, and construct a threshold type result with respect to the persistence and extinction of the population for such a class of evolution systems.
In this paper, we incorporate both the limited growth of the spatial domain and impulsive birth effects into a single-species population model, while adopting a general reaction term. We aim to investigate the dynamical behavior of the system under the combined influence of these two mechanisms. An impulsive reaction-diffusion model with an asymptotically bounded domain is derived, and a threshold-type criterion for population persistence and extinction is established for this class of evolutionary systems.
In this paper, we incorporate both asymptotic domain growth and impulsive birth effects into a single-species reaction-diffusion model, and investigate its global dynamics under the combined influence of these two factors. We derive an impulsive reaction-diffusion model with an asymptotically bounded domain, and establish a threshold-type result on the persistence and extinction of the population for this class of evolutionary systems.
2025.6.28
原文
Mathematical models have long been central to the development of spatial theory in ecology (e.g., Murray 2002a, 2002b; Okubo and Levin 2001; Tilman and Kareiva 1997; Shigesada and Kawasaki 1997; Skellam 1951; Cantrell and Cosner 2003). A large portion of the mathematical literature on spread and persistence is couched in terms of reaction–diffusion equations, which often yield appealingly tractable and compact models of spread and persistence.
翻译
数学模型已经在生态空间理论的发展过程中占据了很长一段时间的中心位置（可以在这些文献中找到相关的例子， Murray 2002a, 2002b; Okubo and Levin 2001; Tilman and Kareiva 1997; Shigesada and Kawasaki 1997; Skellam 1951; Cantrell and Cosner 2003）。 关于传播速度和持久性讨论的一大部分数学文献都是反应扩散方程，也能导出关于传播和持久性的简洁和紧凑的模型。
数学模型长期以来一直是生态学空间理论发展的核心工具（例如，Murray 2002a, 2002b；Okubo 和 Levin 2001；Tilman 和 Kareiva 1997；Shigesada 和 Kawasaki 1997；Skellam 1951；Cantrell 和 Cosner 2003）。在关于物种扩散与持久性的大量数学文献中，反应–扩散方程占据了重要地位，因为它们常常能够产生结构简洁、易于分析的扩散与持久性模型。
数学模型长期以来一直是生态空间理论发展的核心工具（参见 Murray 2002a, 2002b；Okubo 和 Levin 2001；Tilman 和 Kareiva 1997；Shigesada 和 Kawasaki 1997；Skellam 1951；Cantrell 和 Cosner 2003）。在关于物种扩散速度与持久性的数学研究中，反应–扩散方程占据了重要地位。这类方程通常能导出形式紧凑、易于分析的模型，从而有效刻画物种的空间传播和长期行为。
转写
Mathematical models play an important role in the development of spatial theory in ecology (e.g., Murray 2002a, 2002b; Okubo and Levin 2001; Tilman and Kareiva 1997; Shigesada and Kawasaki 1997; Skellam 1951; Cantrell and Cosner 2003). Mathematical literature on spread and persistence mainly foucus on reaction-diffusion equations, which often give rise to appealingly tractable and compact models of spread and persistence.
Mathematical models have played a foundational role in shaping spatial theory in ecology (e.g., Murray 2002a, 2002b; Okubo & Levin 2001; Tilman & Kareiva 1997; Shigesada & Kawasaki 1997; Skellam 1951; Cantrell & Cosner 2003). Among these, a substantial body of work on species spread and persistence has been formulated in terms of reaction–diffusion equations, which offer mathematically tractable and elegantly concise frameworks for capturing spatiotemporal dynamics.
Mathematical models have played a central role in the development of spatial theory in ecology (e.g., Murray 2002a, 2002b; Okubo and Levin 2001; Tilman and Kareiva 1997; Shigesada and Kawasaki 1997; Skellam 1951; Cantrell and Cosner 2003). A substantial body of mathematical literature on species spread and persistence mainly focuses on reaction–diffusion equations, which often yield mathematically tractable and structurally concise models for describing spatiotemporal dynamics.
2025.6.29
原文
Reaction–diffusion equations assume that dispersal is governed by random diffusion and that dispersal and growth take place continuously in time and space. They have had remarkable success in explaining the rates at which species have invaded large open environments as well as spatial patterns that species have had established in bounded-patch habitats.
翻译
反应扩散方程假设扩散是自由扩散，且扩散和增长在时间和空间上是连续发生的。这些方程在解释物种入侵环境速率和物种在有限栖息地建立的空间模式上发挥着良好的作用。
反应扩散方程假设个体的扩散行为遵循随机扩散机制，且扩散与种群增长在时间和空间上是连续进行的。此类方程在解释物种入侵广大开放环境的速率，以及物种在有限斑块生境中形成空间格局方面，取得了显著成效。
反应扩散方程假设个体扩散遵循随机扩散过程，且扩散与种群增长在时间和空间上连续进行。此类方程在解释物种入侵开放环境的速率以及揭示物种在有限栖息地中形成的空间格局方面，具有重要的理论意义和应用价值。
转写
In reaction-diffusion equations, it is assumed that dispersal is ranom diffusion and that dispersal and growth occur continuously in time and space. This is powerful tool in understanding the rates where species have invaded new environment along with spatial patterns that species have had estabilished in bounded-patch habitats.
Reaction–diffusion equations are based on the assumption that dispersal follows a process of random diffusion and that both dispersal and population growth occur continuously over time and space. These equations have proven remarkably effective in capturing the invasion speeds of species into expansive open habitats, as well as in elucidating the spatial patterns established within bounded patchy environments.
Reaction–diffusion equations are based on the assumption that dispersal follows a process of random diffusion and that both dispersal and population growth occur continuously in time and space. They have proven to be a powerful tool in understanding the rates at which species invade new environments, as well as the spatial patterns established within bounded patch habitats.
2025.6.30
原文
It has been well documented that the spatial theory about species spread and persistence matches the field observations well in a number of cases (Murray 2002a, 2002b; Shigesada and Kawasaki 1997, and Cantrell and Cosner 2003). Many species such as fishes or large mammal populations exhibit what Gaughley termed a birth pulse growth pattern (Caswell 2001).
翻译
在已有的记录中，关于种群传播和共存的空间理论在一些情况中很好的解释了田野中的观察(Murray 2002a, 2002b; Shigesada and Kawasaki 1997, and Cantrell and Cosner 2003). 许多物种例如鱼或者大型哺乳动物种群展现出一种被Gaughley称为出生脉冲增长的模式(Caswell 2001).
已有大量研究表明，关于物种扩散与持续存在的空间理论在多个实际案例中与实地观测结果高度一致（Murray 2002a, 2002b；Shigesada 和 Kawasaki 1997；Cantrell 和 Cosner 2003）。许多物种，如鱼类或大型哺乳动物种群，表现出 Gaughley 所称的“出生脉冲式”增长模式（Caswell 2001）。
已有研究表明，关于种群扩散与共存的空间理论在多个案例中能够较好地解释实地观测结果（Murray 2002a, 2002b；Shigesada 和 Kawasaki 1997；Cantrell 和 Cosner 2003）。许多物种，如鱼类或大型哺乳动物种群，呈现出 Gaughley（见 Caswell 2001）所称的出生脉冲式增长模式。
转写
In mathematical literature, the spatial theory about species spread and persistence corresponds to the field observations well in substantial boby of references (Murray 2002a, 2002b; Shigesada and Kawasaki 1997, and Cantrell and Cosner 2003). Many species such as fishes or large mamal populations follows the birth growth pattern proposed by Gaughley (Caswell 2001).
It has been extensively documented that spatial theories concerning species spread and persistence correspond well with empirical field observations in various cases (Murray 2002a, 2002b; Shigesada and Kawasaki 1997; Cantrell and Cosner 2003). Numerous species, including fishes and large mammalian populations, exhibit what Gaughley (as cited in Caswell 2001) referred to as a birth-pulse growth pattern.
Spatial theories of species spread and persistence have been shown to align well with field observations, as supported by a substantial body of literature (Murray 2002a, 2002b; Shigesada and Kawasaki 1997; Cantrell and Cosner 2003). Many species, such as fish or large mammal populations, exhibit the birth-pulse growth pattern described by Gaughley (as cited in Caswell 2001).