## 数学代写|数学生态学作业代写Mathematical Ecology代考| Modeling of Technological Change

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Major Concepts of Technological Change

In mathematical economics, technological change (technical change, technical progress) refers to a combination of all effects that lead to an increasing production output without increasing the amounts of used productive inputs (capital, labor, resources). Such a concept of technological change includes the acquisition of new superior technologies as well as a progress in production management methods.
Major types of technological change include the following:

• Exogenous technological change is introduced into an economic system from outside.
• Endogenous technological change is a consequence of focused economic activities, such as research and development (R\&D) efforts of profit-maximizing firms and governmental policies.
• Embodied (investment-specific) technological change is introduced into the economic system with more efficient capital or better qualified labor.
• Autonomous (disembodied) technological change impacts the entire production process evenly.
• Output-augmenting technological change increases the labor productivity.
• Resource-saving technological change increases the efficiency of converting resources into useful work.
• Induced technological change is a result of previous economic development and is caused by other economic processes or regulations.
• Technological change as a separate sector of economy, whose product is the technological change.

Different categories from this classification use various modeling tools and lead to different conclusions because of different understanding of sources, causes, and effects of the technological change $[4,6,9,10]$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Embodied and Disembodied Technological Change

The embodied (also known as investment-specific) technological change focuses on relations between the dynamics of technological change and capital investments. It takes into account the heterogeneity of capital assets (vintages) under improving technology and assumes that the technological change is introduced into an economic system with more efficient capital or better qualified labor.

In economic reality, both autonomous and embodied changes are presented simultaneously. The autonomous technological change is also referred to as the disembodied technological change to emphasize the fact that it affects all capital vintages and workers in the same way. It describes a progress in management techniques and methods, e.g., installing new enterprise-wide software. More than half $(52 \%)$ of the growth of the US economy during the post-war time was due to the embodied technological change, so the rest can be attributed to the disembodied change.

The models of economic growth under embodied technological change are known as the vintage capital models. Vintage capital models provide a united description of separate processes of investing in new efficient capital and scrapping (disinvestment) of the capital vintages with low efficiency. In many vintage models, the improving efficiency of capital vintages is given as a function of time. So the embodied technological change can be exogenous, where the source of technological change is still unclear. The vintage capital models are explored in Chaps. 4 and $5 .$

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Endogenous Technological Change

Models of endogenous technological change were introduced to explain the driving forces behind technological change.

The majority of technological improvements results from research and development $(R \& D)$ activities carried out and financed by government and/or private firms. The concept of endogenous technological change attempts to explain economic reasons and sources of technological change. Corresponding economic models describe technological innovations as determined by economic actors and suggest economic reasons for firms to innovate, specific mechanisms and directions of inventive activity, drivers of incremental improvements that occur during technology diffusion, and so on. These mechanisms are endogenous with respect to economic activities and, thus, are determined inside the model. Some classic models of endogenous technological change are explored in Sect. 3.4.
Induced Technological Change
An early concept of the endogenous technological change is known as the induced technological change that links technological change to previous economic development. It was the result of incorporating technological change into the

neoclassical growth framework. The description of induced technological change was based on various hypotheses about relations between the technological change intensity and other aggregated economic characteristics, see Sect. 3.4.1. However, the early hypotheses of induced technological change could not explain the need of purpose-directed investments into science and technology.

In modern economics, the induced technological change commonly refers to additional technological improvements caused by other economic processes or governmental regulations, for instance by more restrictive environmental policies.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Major Concepts of Technological Change

• 外生技术变革是从外部引入经济系统的。
• 内生技术变革是集中经济活动的结果，例如利润最大化企业的研发（R\&D）努力和政府政策。
• 体现的（特定于投资的）技术变革以更有效的资本或更合格的劳动力被引入经济体系。
• 自主（非实体）技术变革均匀地影响整个生产过程。
• 增加产出的技术变革提高了劳动生产率。
• 资源节约型技术变革提高了将资源转化为有用工作的效率。
• 诱发的技术变革是先前经济发展的结果，是由其他经济过程或法规引起的。
• 技术变革作为一个单独的经济部门，其产品就是技术变革。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

A dynamic analysis of such problems is more complex and requires sophisticated mathematical tools. The results provided below are obtained employing the max= imum principle from Sect. 2.4.

Necessary Condition for an Extremum: If the function $s(t), t \in[0, T]$, is a solution of the optimization problem $(2.45)-(2.47)$, then:
(a) There exists a continuous function $\hat{\lambda}(t), t \in[0, T]$, called the dual or adjoint variable, that satisfies the dual equation
$$\hat{\lambda}^{\prime}(t)=(\mu+r) \hat{\lambda}(t)-[1-s(t)+\hat{\lambda}(t) s(t)] f^{\prime}(k(t)),$$
with the terminal transversality condition
$$\left[k(T)-k_{T}\right] \mathrm{e}^{-r T} \hat{\lambda}(T)=0,$$
where the corresponding state variable $k(t), t \in[0, T]$, is found from (2.46).
(b) $s(t)$ maximizes $[1-s(t)+\hat{\lambda}(t) s(t)]$ at each point $t \in[0, T]$.
The proof of this result follows from Corollary $2.1$ of Sect. 2.4. Namely, the current-value Hamiltonian (2.69) for the optimal control problem (2.45)-(2.47) is constructed as
$$\hat{H}(s, k, \hat{\lambda})=f(k)(1-s)+\hat{\lambda}[s f(k)-\mu k],$$
and, then, the dual equation (2.48) is obtained from $(2.70)$ as $\hat{\lambda}^{\prime}=r \hat{\lambda}-\partial H / \partial k$, the state equation (2.46) fits $k^{\prime}=\partial H / \partial \hat{\lambda}$, and $s(t)$ maximizes $H(s, k, \hat{\lambda})$.

Extremum Condition for an Interior Solution. The maximum principle is constructed specifically to handle the case of boundary solutions: $s(t)=0$ or $s(t)=1$ in the domain $0 \leq s(t) \leq 1$ at some instants $t$. The possibility of boundary (or corner) solutions essentially complicates the optimal control dynamics. If a solution is known to be interior in the domain, then the optimality conditions

become simpler. Namely, by Corollary $2.2$ from Sect. $2.4$, if $0<s(t)<1$, then the optimal $s(t)$ satisfies $\partial H / \partial s=0$.

Let us utilize this optimality condition for the optimization problem (2.45)-(2.47). Taking the derivative of $(2.50)$ in $s$, we obtain $\partial \hat{H} / \partial s=f(k)(\hat{\lambda}-1)$. If a priori $0<s(t)<1$ for $t \in[0, T]$, then $\partial H / \partial s=0$ and, therefore, $\hat{\lambda}(t)=1$. Substituting $\hat{\lambda}$ to $(2.48)$, we obtain
$$0=\mu+r-f^{\prime}(k(t))$$
which is the same golden rule of capital accumulation $(2.40)$ as obtained during static optimization in the Solow-Swan model of Sect. 2.2.

Structure of Solution: Using the extremum condition (2.48) and (2.49) and rewriting $(2.50)$ as $\hat{H}(s, k, \hat{\lambda})=s(\hat{\lambda}-1) f(k)-\hat{\lambda} \mu k+f(k)$, we can show that $s(t)=0$ maximizes $\hat{H}(s, k, \hat{\lambda})$ at $\hat{\lambda}(t)<1$ and $s(t)=0$ maximizes $\hat{H}(s, k, \hat{\lambda})$ at $\hat{\lambda}(t)>1$. If $\hat{\lambda}(t)=1$, then $\hat{H}(s, k, \hat{\lambda})$ does not depend on $s$ and the optimal $k^{}$ is found from (2.48), which is the same as the golden rule of capital accumulation (2.40). Thus, the solution $s(t), t \in[0, T]$, of the optimization problem $(2.45)-(2.47)$ is $$s(t)=\left{\begin{array}{ccl} 0 & \text { when } & \hat{\lambda}(t)<1 \\ s^{*} & \text { when } & \hat{\lambda}(t)=1, \\ 1 & \text { when } & \hat{\lambda}(t)>1 \end{array}\right.$$ where $0}<1$ is the optimal (golden-rule) saving rate $(2.41)$ in the Solow-Swan model. When $s(t)=s^{}$, the corresponding trajectory is $k(t)=k^{}$, where the unique $k^{*}$ is found from (2.40).

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Long-Term and Transition Dynamics

Because of the specifics of economic optimization problems, their dynamic analysis is usually split into two steps: the investigation of a long-term dynamics and the investigation of the transition dynamics. In many problems, the long-term dynamics is independent of initial conditions of the problem and coincides with the steady state solution of the model. Then, the transition dynamics describes how the optimal trajectory approaches the steady state.

The solution $s(t), k(t), t \in[0, T]$, of the optimization problem $(2.45)-(2.47)$ in the case $k_{0}<k^{*}<k_{\mathrm{T}}$ is illustrated in Fig. 2.3.

The transition (short-term) dynamics of the problem (2.45)-(2.47) is common for well-formulated economic problems. The optimal trajectory $s(t), k(t)$ approaches the best steady state solution $\left(s^{}, k^{}\right)$ on the initial interval $\left[0, \theta_{1}\right]$ and the transition dynamics ends at the instant $\theta_{1}$ such that $k\left(\theta_{1}\right)=k^{*}$.

The optimal trajectory $s(t), k(t)$ leaves the steady state solution $\left(s^{}, k^{}\right)$ at some instant $\theta_{2}<T$ near the right end of the planning horizon $[0, T]$. This behavior illustrates the so-called end-of-horizon effect and is also common in economic problems. Even if the terminal condition is absent, such effects still take place and even become more substantial. In particular, if $k_{T}=0$, then there is no investments at the end $\left[\theta_{2}, T\right]$ of planning horizon.

Mathematically, this end-of-horizon effect appears because the optimal trajectory $k(t)$ must satisfy the transversality condition $(2.49)$. This condition becomes less restrictive at $T=\infty$. Non-importance of the transversality condition for the infinite-horizon problem $(2.31)-(2.35)$ was pointed out by $K$. Shell in [8]. It will be shown in the next section that the end-of-horizon effect is absent in the infinitehorizon problem.

The trajectory $s(t) \equiv s^{}, k(t) \equiv k^{}$ over $\left[\theta_{1}, \theta_{2}\right]$ represents the long-term dynamics of the optimization problem. The optimal saving rate $s(t)$ coincides with the constant golden-rule saving rate $s$ in the Solow-Swan model on a certain interior part $\left[\theta_{1}, \theta_{2}\right]$ of the planning period $[0, T]$. The length of $\left[\theta_{1}, \theta_{2}\right]$ becomes larger when $T$ increases. It means that a turnpike property holds for the optimization problem $(2.45)-(2.47)$, where the turnpike trajectory is $s_{T} \equiv s^{*}$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

The dynamic analysis of the Solow-Ramsey model includes new mathematical challenges such as the convergence of the improper integral (2.53) along the optimal trajectory $c$. The condition for this convergence in our model $(2.31)-(2.34),(2.53)$ is simply
$$r>\eta,$$
where $\eta$ is the given growth rate of labor in (2.35). However, finding such conditions becomes more complicated in more advanced models (see for example Chap. 3).

Under (2.54), the extremum conditions remain the same, (2.48)-(2.52), as in the Solow-Shell model. Using (2.48) and (2.52), we can show that the solution $s(t)$, $t \in[0, \infty)$, of the problem (2.53) in the model (2.31)-(2.34) is
$s(t)=\left{\begin{array}{ccc}1 & \text { at } & 0 \leq t<\theta_{1} \ s^{} & \text { at } & \theta_{1} \leq t<\infty\end{array},\right.$, with the golden-rule saving rate $s^{\circ}$ and capital per capita $k^{}$ in the Solow-Swan model. Also, it can be shown that the transversality condition (2.49) is reduced to the inequality (2.54). So the transversality condition is less important in the infinitehorizon problem in the sense that it does not directly affect the solution dynamics.
On the qualitative side, the behavior of the optimal trajectories appears to be simpler than in the finite-horizon Solow-Shell model (2.45)-(2.47). The solution $(s(t), k(t)), t \in[0, \infty)$, of the optimization problem (2.53) in the case $k_{0}<k^{*}$ is illustrated in Fig. $2.3$ by blue curves.

The transition dynamics of the problem (2.53) over the interval $\left[0, \theta_{1}\right]$ is the same as for the Solow-Shell model. The optimal trajectory $(s(t), k(t))$ approaches the steady state $\left(s^{}, k^{}\right)$ and the transition dynamics ends at the instant $\theta_{1}$ such that $k\left(\theta_{1}\right)=k^{*}$.

The long-term dynamics is $s(t) \equiv s^{}, k(t) \equiv k^{}$ over $\left[\theta_{1}, \infty\right)$, i.e., the optimal saving rate $s(t)$ coincides with the constant golden-rule saving rate $s^{*}$ in the Solow-Swan model starting with the time $\theta_{1}$. As shown in Fig. 2.3, the optimal

trajectory $s(t), k(t)$ does not leave the steady state $\left(s^{}, k^{}\right)$ because the end-of-horizon effects are absent in infinite-horizon problems.

The considered optimization versions of the Solow-Swan, Solow-Shell, and Solow-Ramsey models, are classified by the economic theory as the models of exogenous growth because they cannot generate an endogenous growth when the labor $L(t)$ is constant. However, even small modifications of these models can lead to the endogenous growth. For instance, if we replace the neoclassic production function in the model equation (2.31) with the CES or $A K$ production function, then the corresponding models are able to generate an endogenous growth. Models with endogenous growth are discussed in Sect. 3.4.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

(a) 存在一个连续函数λ^(吨),吨∈[0,吨]，称为对偶或伴随变量，满足对偶方程
λ^′(吨)=(μ+r)λ^(吨)−[1−s(吨)+λ^(吨)s(吨)]F′(ķ(吨)),

[ķ(吨)−ķ吨]和−r吨λ^(吨)=0,

(二)s(吨)最大化[1−s(吨)+λ^(吨)s(吨)]在每个点吨∈[0,吨].

H^(s,ķ,λ^)=F(ķ)(1−s)+λ^[sF(ķ)−μķ],

0=μ+r−F′(ķ(吨))

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Dynamic Analysis

Solow-Ramsey 模型的动态分析包括新的数学挑战，例如沿最优轨迹的不正确积分 (2.53) 的收敛C. 我们模型中这种收敛的条件(2.31)−(2.34),(2.53)简直就是
r>这,

$s(t)=\left{1 在 0≤吨<θ1 s 在 θ1≤吨<∞，\对。,在一世吨H吨H和G这ld和n−r在l和s一种在一世nGr一种吨和s^{\circ}一种ndC一种p一世吨一种lp和rC一种p一世吨一种k^{}一世n吨H和小号这l这在−小号在一种n米这d和l.一种ls这,一世吨C一种nb和sH这在n吨H一种吨吨H和吨r一种ns在和rs一种l一世吨是C这nd一世吨一世这n(2.49)一世sr和d在C和d吨这吨H和一世n和q在一种l一世吨是(2.54).小号这吨H和吨r一种ns在和rs一种l一世吨是C这nd一世吨一世这n一世sl和ss一世米p这r吨一种n吨一世n吨H和一世nF一世n一世吨和H这r一世和这npr这bl和米一世n吨H和s和ns和吨H一种吨一世吨d这和sn这吨d一世r和C吨l是一种FF和C吨吨H和s这l在吨一世这nd是n一种米一世Cs.这n吨H和q在一种l一世吨一种吨一世在和s一世d和,吨H和b和H一种在一世这r这F吨H和这p吨一世米一种l吨r一种j和C吨这r一世和s一种pp和一种rs吨这b和s一世米pl和r吨H一种n一世n吨H和F一世n一世吨和−H这r一世和这n小号这l这在−小号H和ll米这d和l(2.45)−(2.47).吨H和s这l在吨一世这n(s(t), k(t)), t \in[0, \infty),这F吨H和这p吨一世米一世和一种吨一世这npr这bl和米(2.53)一世n吨H和C一种s和k_{0}<k^{*}一世s一世ll在s吨r一种吨和d一世nF一世G.蓝色曲线为 2.3 美元。 问题（2.53）在区间内的转换动态[0,θ1]与 Solow-Shell 模型相同。最优轨迹(s(吨),ķ(吨))接近稳态$\left(s^{ }, k^{ }\right)一种nd吨H和吨r一种ns一世吨一世这nd是n一种米一世Cs和nds一种吨吨H和一世ns吨一种n吨\theta_{1}s在CH吨H一种吨k\left(\theta_{1}\right)=k^{*}$。 长期动态是s(吨)≡s,ķ(吨)≡ķ超过[θ1,∞)，即最优储蓄率s(吨)与恒定的黄金法则储蓄率相吻合s∗在从时间开始的 Solow-Swan 模型中θ1. 如图 2.3 所示，最优 弹道s(吨),ķ(吨)不会离开稳定状态$\left(s^{ }, k^{ }\right)$，因为在无限视野问题中不存在视野结束效应。 所考虑的 Solow-Swan、Solow-Shell 和 Solow-Ramsey 模型的优化版本被经济理论归类为外生增长模型，因为它们不能在劳动时产生内生增长大号(吨)是恒定的。然而，即使这些模型的微小修改也可能导致内生增长。例如，如果我们将模型方程（2.31）中的新古典生产函数替换为 CES 或一种ķ生产函数，则相应的模型能够产生内生增长。具有内生增长的模型在第 3 节中讨论。3.4. 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## MATLAB代写 MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 ## 数学代写|数学生态学作业代写Mathematical Ecology代考| Steady-State Analysis 如果你也在 怎样代写数学生态学Mathematical Ecology这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。 数学生态学通过将理论和方法的发展与生态学应用联系起来，继续推动我们的领域向前发展. statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。 我们提供的数学生态学Mathematical Ecology及其相关学科的代写，服务范围广, 其中包括但不限于: • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等楖率论 • Advanced Mathematical Statistics 高等数理统计学 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 • Statistical Inference 统计推断 • Statistical Computing 统计计算 • Advanced Probability Theory 高等楖率论 • Advanced Mathematical Statistics 高等数理统计学 • (Generalized) Linear Models 广义线性模型 • Statistical Machine Learning 统计机器学习 • Longitudinal Data Analysis 纵向数据分析 • Foundations of Data Science 数据科学基础 ## 数学代写|数学生态学作业代写Mathematical Ecology代考|Steady-State Analysis The goal of a steady-state analysis is to find possible steady states, which can be • A stationary trajectory (unknown variables are constant in time) or • A balanced growth path (all variables grow at the same constant rate). The steady-state analysis plays an important role in economics and is mathematically simpler than a complete dynamic analysis. Let us find and analyze possible balanced growth paths in the model$(2.31)-(2.35)$. It is easy to see that the original variables$Q(t), C(t), I(t)$, and$K(t)$of the model grow with the same rate only if the capital-labor ratio$k(t)$is constant. Indeed, substituting$k=$const into$(2.33),(2.32), and (2.35), we obtain \begin{aligned} &K(t)=k L(t), \quad I(t)=(\mu+\eta) K(t), \quad Q(t)=(\mu+\eta) K(t) / s \ &C(t)=Q(t)-I(t), \end{aligned} i.e., all these functions increase with the same rate\eta$as the labor$L(t)=L_{0} \exp$($\eta t)$. Therefore, to find steady states, we should assume$k(t)=$const. Then$k^{\prime}(t)=0$and the equation ($2.36$) produces the equation $$s f(k)=(\mu+\eta) k$$ for possible steady states$k \equiv$const. Because$f(0)=0, f^{\prime}(k)>0, \lim {k \rightarrow 0} f^{\prime}(k)=\infty$, and$\lim {k \rightarrow \infty} f^{\prime}(k)=0$, the equation$(2.38$) has a unique solution$\hat{k}=\hat{k}(s)=$const$>0$for any given value$s>0$. The steady-state capital-labor ratio$\hat{k}(s)$increases when the saving rate$s$increases. ## 数学代写|数学生态学作业代写Mathematical Ecology代考|Static Optimization For a given saving rate$s$, the steady-state consumption per capita$c=C / L$is determined by the formula $$c(s)=f(\hat{k}(s))-(\mu+h) \hat{k}(s)$$ where the corresponding steady-state capital-labor ratio$\hat{k}(s)$is determined by (2.38). Because$f(0)=0, f^{\prime}(k)>0$and$f^{\prime \prime}(k)<0$, the composite function (2.39) increases for smaller values of$s$and decreases for larger$s$. Then, we can determine the saving rate$s^{}=$const and the corresponding steady-state$k^{}=\hat{k}\left(s^{}\right)that maximizes the consumption per capita (2.39): $$\max _{0} should satisfy$$ f^{\prime}\left(k^{*}\right)=\mu+\eta $$The relation (2.40) is known as the golden rule of capital accumulation. It implies that the marginal product of capital should be equal to the sum of the depreciation and labor growth rates. After determining the optimal k^{} from (2.40), the corresponding golden-rule saving rate is found from (2.38) as$$ s^{}=(\mu+\eta) k^{} / f\left(k^{}\right)=k^{} f^{\prime}\left(k^{}\right) / f\left(k^{}\right), $$i.e., the optimal saving rate s^{} is equal to the output elasticity of the capital \varepsilon_{K} (2.6) for the corresponding k^{}. The formulas (2.40) and (2.41) for the optimal s^{} and k^{*} are known as the golden rule of economic growth. In the case of the Cobb-Douglas production function (2.22) F(K, L)= A K^{\alpha} L^{1-\alpha}, 0<\alpha<1, the function f(k)=A k^{\alpha} and the golden rule is$$ s^{}=\alpha, \quad k^{}=\left[A s^{} /(\mu+\eta)\right]^{1 /(1-\alpha)} . $$At the optimal steady state \left(s^{}, k^{}\right) and the given labor L(t)=\bar{L} \mathrm{e}^{m t}, the original variables Q(t), C(t), I(t), and K(t) of the model (2.31)-(2.35) grow with the given rate \eta as$$ \begin{aligned} K(t)=\bar{K}^{\eta t^{t}}, \quad I(t)=\bar{I} \mathrm{e}^{\eta t^{t}}, \quad Q(t)=\bar{Q} \mathrm{e}^{\eta t}, \quad C(t)=\bar{C} \mathrm{e}^{\eta t} \ \bar{K}=\bar{L} k^{}, \bar{I}=(\mu+\eta) \bar{L} k^{}, \quad \bar{Q}=\frac{(\mu+\eta) \bar{L} k^{}}{s} \ \bar{C}=\frac{(1-s)(\mu+\eta) \bar{L} k^{*}}{s} \end{aligned} $$The constants \bar{K}, \bar{I}, \bar{Q}, \bar{C} in exponential functions of the form (2.43) are often called in economics the level variables. In the case of constant labor L(t)=\bar{L} (i.e., \eta=0 ), the steady state is given by (2.44) and known as a stationary point. Because the aggregate output Q, consumption C, investment I and capital K increase with the same rate \eta as the exogenous labor L, the Solow-Swan model is classified in the economic theory as the exogenous growth model. ## 数学代写|数学生态学作业代写Mathematical Ecology代考|Optimization over Finite Horizon The Solow-Shell model is the Solow-Swan model (2.31)-(2.34) considered on a finite planning horizon [0, T] in the case when the saving rate s=I / Q depends on the time t and is endogenous [7]. To determine this rate, we consider the following one-sector optimization problem: • Maximize the present value$$ \int_{0}^{T} \mathrm{e}^{-r t} c(t) \mathrm{d} t $$of the consumption per capita c=C / L over a given finite horizon [0, T], subject to (2.31)-(2.35) and certain initial and terminal conditions. In this problem, the given discount rate r>0 reflects the planner’s subjective rate of the decreasing utility of the output produced in more distant future. We still use the same aggregate variables of the Solow-Swan model: the output Q, consumption C, capital K, labor L, and investment I. For simplicity, let the labor L(t) be constant, that is, \eta=0 in (2.34). Switching the model (2.31)-(2.34) to the per capita variables k=K / L, q=Q / L, c=C / L, i=I / L, and excluding q, c, and i, the optimization problem under study becomes: • Find the function s(t), 0 \leq s(t) \leq 1, and the corresponding k(t), k(t) \geq 0, t \in[0, T], which maximize$$ \max {s, k} \int{0}^{T} \mathrm{e}^{-r t}(1-s(t)) f(k(t)) \mathrm{d} t $$under the equality-constraint:$$ k^{\prime}(t)=s(t) f(k(t))-\mu k(t), $$and the initial and terminal conditions:$$ k(0)=k_{0}, \quad k(T) \geq k_{T} $$The value of k(T) cannot be arbitrary because the economy will continue after the end of the planning period. The terminal condition k(T) \geq k_{\mathrm{T}} keeps a minimal acceptable level of capital at the end of the finite horizon. The problem (2.45)-(2.47) is an optimal control problem, in which the function s(t), t \in[0, T], is unknown (rather than the scalar s= const as in the static optimization of Sect. 2.2). In the optimal control terminology, the independent unknown function s(.) is referred to as the control variable and the corresponding dependent unknown k(.) is the state variable. ## 数学建模代写 ## 数学代写|数学生态学作业代写Mathematical Ecology代考|Steady-State Analysis 稳态分析的目标是找到可能的稳态，它可以是 • 静止轨迹（未知变量在时间上是恒定的）或 • 平衡的增长路径（所有变量以相同的恒定速率增长）。 稳态分析在经济学中起着重要作用，并且在数学上比完整的动态分析更简单。 让我们在模型中找到并分析可能的平衡增长路径(2.31)−(2.35). 很容易看出原来的变量问(吨),C(吨),一世(吨)， 和ķ(吨) 只有当资本-劳动力比率ķ(吨)是恒定的。确实，换ķ=常量成(2.33),(2.32), 和 (2.35), 我们得到 ķ(吨)=ķ大号(吨),一世(吨)=(μ+这)ķ(吨),问(吨)=(μ+这)ķ(吨)/s C(吨)=问(吨)−一世(吨), 即，所有这些功能都以相同的速度增加这作为劳动大号(吨)=大号0经验 ( 这吨). 因此，为了找到稳定状态，我们应该假设ķ(吨)=常量。然后ķ′(吨)=0和方程（2.36) 产生方程 sF(ķ)=(μ+这)ķ 对于可能的稳态ķ≡常量。因为F(0)=0,F′(ķ)>0,林ķ→0F′(ķ)=∞， 和林ķ→∞F′(ķ)=0, 方程(2.38) 有一个独特的解决方案ķ^=ķ^(s)=常量>0对于任何给定的值s>0. 稳态资本-劳动力比率ķ^(s)当储蓄率增加s增加。 ## 数学代写|数学生态学作业代写Mathematical Ecology代考|Static Optimization 对于给定的储蓄率s, 人均稳态消费C=C/大号由公式确定 C(s)=F(ķ^(s))−(μ+H)ķ^(s) 其中相应的稳态资本-劳动力比率ķ^(s)由 (2.38) 确定。因为F(0)=0,F′(ķ)>0和F′′(ķ)<0, 复合函数 (2.39) 对于较小的值会增加s并减少更大s. 然后，我们可以确定储蓄率s=const 和相应的稳态ķ=ķ^(s)最大化人均消费（2.39）：$$ \max _{0 }sH这在lds一种吨一世sF是F′(ķ∗)=μ+这

s=一种,ķ=[一种s/(μ+这)]1/(1−一种).处于最佳稳态(s,ķ)和给定的劳动大号(吨)=大号¯和米吨, 原始变量问(吨),C(吨),一世(吨)， 和ķ(吨)模型的 (2.31)-(2.35) 以给定的速率增长这作为ķ(吨)=ķ¯这吨吨,一世(吨)=一世¯和这吨吨,问(吨)=问¯和这吨,C(吨)=C¯和这吨 ķ¯=大号¯ķ,一世¯=(μ+这)大号¯ķ,问¯=(μ+这)大号¯ķs C¯=(1−s)(μ+这)大号¯ķ∗s

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Optimization over Finite Horizon

Solow-Shell 模型是 Solow-Swan 模型(2.31)−(2.34)在有限的规划范围内考虑[0,吨]在储蓄率的情况下s=一世/问取决于时间吨并且是内生的[7]。为了确定这个比率，我们考虑以下单扇区优化问题：

• 最大化现值
∫0吨和−r吨C(吨)d吨
人均消费C=C/大号在给定的有限范围内[0,吨], 受限于 (2.31)-(2.35) 和某些初始和终止条件。

• 查找功能s(吨),0≤s(吨)≤1, 和对应的ķ(吨),ķ(吨)≥0, 吨∈[0,吨], 最大化
最大限度s,ķ∫0吨和−r吨(1−s(吨))F(ķ(吨))d吨
在等式约束下：
ķ′(吨)=s(吨)F(ķ(吨))−μķ(吨),
以及初始和终止条件：
ķ(0)=ķ0,ķ(吨)≥ķ吨
的价值ķ(吨)不能随意，因为经济将在计划期结束后继续。终端条件ķ(吨)≥ķ吨在有限期限结束时保持最低可接受的资本水平。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考| Two-Factor CES Production Function

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Two-Factor CES Production Function

$$\begin{gathered} Q=A\left[\alpha(b K)^{\rho}+(1-\alpha)((1-b) L)^{\rho}\right]^{1 / \rho}, \quad \rho<1 \\ \text { or } q=A\left[\alpha(b k)^{\rho}+(1-\alpha)(1-b)^{\rho}\right]^{1 / \rho} . \end{gathered}$$ Here, the marginal product of capital (2.19) is $$\begin{gathered} \partial Q / \partial K=A \alpha b^{\rho}\left[\alpha b^{\rho}+(1-\alpha)(1-b)^{\rho} k^{-\rho}\right]^{(1-\rho) / \rho}, \\ h=(1-\alpha)(1-b)^{\rho} k^{1-\rho} /\left(a b^{\rho}\right), \quad \sigma=1 /(1-\rho) \end{gathered}$$ The CES production function is not neoclassical because the Inada conditions are violated. It is visible in Fig. 2.1. At a low degree of substitution $\sigma<1(\rho<0)$, its graph has a horizontal asymptote (see the brown curve in Fig. 2.1). When $\rho \rightarrow 0$, the CES production function approaches the Cobb-Douglas production function. At a high degree of substitution $\sigma>1(0<\rho<1)$, this function increases faster than the Cobb-Douglas one (see the red curve in Fig. 2.1). At $\sigma=\infty(\rho=1)$, the CES function becomes linear: $Q=A \alpha b K+A(1-\alpha)(1-b) L$. When $\rho \rightarrow-\infty$ $(\sigma \rightarrow 0)$, this production function approaches the Leontief production function $Q=\min [b K,(1-b) L]$ discussed next. There is essential economic evidence that the CES production function better fits many economic processes than the Cobb-Douglas production function. For this reason, the CES production function currently dominates in applied economic research.

We shall notice that some textbooks introduce the CES production function in a slightly different form as $Q=A\left[\alpha K^{\rho}+(1-\alpha) L^{\rho}\right]^{1 / \rho}$ and/or with the parameter $\rho$ replaced by $-\rho$ (then the new $\rho>-1$ ).

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Model Description

Let us consider an economy described by the following dynamic characteristics in the continuous time $t$ :
$Q(t)$-the total output produced at time $t$,
$C(t)$-the amount of consumption,
$I(t)$ the amount of gross investment,
$L(t)$ – the amount of labor,
$K(t)$-the amount of capital.
The Solow-Swan model is described by the following equations:
$$Q(t)=F(K(t), L(t))$$
i.e., the output $Q$ is determined by a neoclassical production function $F(K, L)$,
$$Q(t)=C(t)+I(t)$$
i.e., the output $Q$ is distributed between the consumption $C$ and the investment $I$,
$$K^{\prime}(t)=I(t)-\mu K(t), \quad \mu=\mathrm{const}>0,$$
i.e., the capital $K$ depreciates at a constant rate $\mu>0$ (a constant fraction of the capital leaves a production process at each point of time),
$$L^{\prime}(t)=\eta L(t), \quad \eta=\text { const } \geq 0$$
i.e., the labor $L(t)=L_{0} \exp (\eta t)$ grows at a constant exogenous rate $\eta$.
The structure of the Solow-Swan model is shown in Fig. 2.2. The part of the investment in the total product is known as the saving rate:
$$s(t)=I(t) / Q(t)$$
The saving rate is assumed to be constant in the classic Solow-Swan model:
$$I(t)=s Q(t), \quad 0<s<1, \quad s=\text { const. }$$
This assumption simplifies the investigation of the model and leads to a number of essential economic results. More advanced economic models (see next sections) consider the saving rate $s(t)$ as an endogenous control function.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Fundamental Equation of Model

Because the production function $F(K, L)$ is neoclassical and, therefore, linearly homogeneous, then $F(K, L)=L f(k)$ and the equation (2.33) leads to
$$K^{\prime}(t) / L(t)=f(k(t))-\mu k(t)$$
where the capital-labor ratio $k=K / L$ is defined as in (2.18). On the other side,
$$k^{\prime}(t)=K^{\prime}(t) / L(t)-\eta k(t)$$
by (2.34). Combining the last two equalities, we obtain the fundamental equation of the Solow-Swan model
$$k^{\prime}(t)=s f(k)-(\mu+\eta) k(t)$$
Thus, the dynamics of the model $(2.31)-(2.35)$ is reduced to one autonomous (not dependent on $t$ explicitly) differential equation (2.36) with respect to $k$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Model Description

C(吨)- 消费量，

ķ(吨)- 资本数额。
Solow-Swan 模型由以下等式描述：

ķ′(吨)=一世(吨)−μķ(吨),μ=C这ns吨>0,

Solow-Swan 模型的结构如图 2.2 所示。投资在总产品中的部分称为储蓄率：
s(吨)=一世(吨)/问(吨)

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Fundamental Equation of Model

ķ′(吨)/大号(吨)=F(ķ(吨))−μķ(吨)

ķ′(吨)=ķ′(吨)/大号(吨)−这ķ(吨)

ķ′(吨)=sF(ķ)−(μ+这)ķ(吨)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考| Major Types of Production Functions

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Cobb–Douglas Production Function

$$y=A x_{1}^{a_{1}} \times \ldots \times x_{n}^{a_{n}}$$
has the following characteristics:
$$\begin{array}{cl} \partial f / \partial x_{i}=\alpha_{i}\left(y / x_{i}\right), & h_{i j}=\alpha_{j} x_{i} / \alpha_{i} x_{j}, \quad \varepsilon_{i}=\alpha_{i}, \quad \varepsilon=\alpha_{1}+\ldots+\alpha_{n}, \ \sigma_{i j}=1, \quad i, j=1, \ldots, n, \end{array}$$
i.e., the elasticity of substitution for any pair $(i, j)$ of inputs is equal to one. The returns to scale are increasing in the case $\alpha_{1}+\ldots+\alpha_{n}>1$, decreasing in the case $\alpha_{1}+\ldots+\alpha_{n}<1$, and constant at $\alpha_{1}+\ldots+\alpha_{n}=1$.
By taking the logarithm of both sides of $(2.9)$, we obtain a linear expression
$$\ln y=\ln A+\sum_{i=1}^{n} \alpha_{i} \ln x_{i}$$
that after differentiation becomes
$$y^{\prime} / y=\alpha_{1}\left(x_{1}^{\prime} / x_{1}\right)+\ldots+\alpha_{n}\left(x_{n}^{\prime} / x_{n}\right)$$
i.e., the growth rate of the output in the Cobb-Douglas production function is equal to the weighted sum of the growth rates of the inputs.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Two-Factor Production Functions

Production functions with two inputs, called two-factor production functions, are the most common in economics and are usually written as
$$Q=F(K, L)$$
where
$Q$ is the output,
$K$ is the amount of capital used,
$L$ is the amount of labor used.
The capital $K$ reflects the total cost of the equipment, machines, buildings, etc., used in production process. Such production functions are characterized by single values of the marginal rate of substitution $h$ and the elasticity of substitution $\sigma$ between capital and labor.

A two-factor production function is called the neoclassical production function, if it satisfies the following properties:

1. Essentiality of inputs:
$$F(K, 0)=F(0, L)=0$$
2. Positive and diminishing returns:
$$\partial F / \partial K>0, \quad \partial F / \partial L>0, \quad \partial^{2} F / \partial K^{2}<0, \quad \partial^{2} F / \partial L^{2}<0 .$$
3. Constant returns to scale: $F(K, L)$ is a linearly homogeneous function,
$$F(l K, l L)=l F(K, L) \quad \text { for } l>0 .$$
4. The Inada conditions: the marginal products of capital and labor satisfy
$$\lim {K \rightarrow 0} \frac{\partial F}{\partial K}=\infty, \quad \lim {L \rightarrow 0} \frac{\partial F}{\partial L}=\infty, \quad \lim {K \rightarrow \infty} \frac{\partial F}{\partial K}=0, \quad \lim {L \rightarrow \infty} \frac{\partial F}{\partial L}=0$$
The Inada conditions mean that the production increases very fast if the production input (capital or labor) is low and increases slowly, whereas the production increase is very slow if the production input has been already abundant and more is added. Property 1 holds if the other three properties hold.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Two-Factor Cobb–Douglas Production Function

$$Q=A K^{a} L^{\beta}, \quad \alpha>0, \quad \beta>0$$
where the total factor productivity A reflects the level of technology. In the general case when $\alpha+\beta \neq 1$, the Cobb-Douglas production is not neoclassical because it does not satisfy Property 3 of constant returns. The Cobb-Douglas production at $\alpha+\beta=1$ is neoclassical and can be presented in the standard and intensive forms as
$$Q=A K^{\alpha} L^{1-a} \text { or } q=A k^{a}, \quad 0<\alpha<1$$
Then, the marginal products of capital and labor of $(2.22)$ are
$$\partial Q / \partial K=\alpha A k^{a-1}, \quad \partial Q / \partial L=(1-\alpha) A k^{a},$$
the marginal rate of substitution is $h=k(1-\alpha) / \alpha$, the output elasticity of capital is $\varepsilon_{K}=\alpha$, the total output elasticity is $\varepsilon=1$, and the elasticity of substitution is $\sigma=1$. The graph of the Cobb-Douglas production function (2.23) in the intensive form is shown Fig. 2.1 with a black curve and is typical for the neoclassical production functions. The output $f(k)$ increases indefinitely when the capital per capita $k \rightarrow \infty$, which reflects the Inada condition (2.17). Some economists consider such an increase to be unrealistic.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Cobb–Douglas Production Function

∂F/∂X一世=一种一世(是/X一世),H一世j=一种jX一世/一种一世Xj,e一世=一种一世,e=一种1+…+一种n, σ一世j=1,一世,j=1,…,n,

ln⁡是=ln⁡一种+∑一世=1n一种一世ln⁡X一世

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Two-Factor Production Functions

ķ是使用的资本金额，

1. 输入的本质：
F(ķ,0)=F(0,大号)=0
2. 正收益和递减收益：
∂F/∂ķ>0,∂F/∂大号>0,∂2F/∂ķ2<0,∂2F/∂大号2<0.
3. 规模报酬不变：F(ķ,大号)是一个线性齐次函数，
F(lķ,l大号)=lF(ķ,大号) 为了 l>0.
4. 稻田条件：资本和劳动的边际产品满足
林ķ→0∂F∂ķ=∞,林大号→0∂F∂大号=∞,林ķ→∞∂F∂ķ=0,林大号→∞∂F∂大号=0
稻田条件意味着如果生产投入（资本或劳动力）低并且缓慢增加，则生产增加非常快，而如果生产投入已经充足并且增加更多，则生产增加非常缓慢。如果其他三个属性成立，则属性 1 成立。

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Two-Factor Cobb–Douglas Production Function

∂问/∂ķ=一种一种ķ一种−1,∂问/∂大号=(1−一种)一种ķ一种,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考| Production Functions and Their Types

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Production Functions and Their Types

A production function describes a relationship
$$y=f\left(x_{1}, \ldots, x_{n}\right)$$
between the aggregate product output $y$ and the productive inputs $x_{1}, \ldots, x_{n}$ that can include labor, capital, knowledge (human capital), energy consumption, raw materials, natural resources (land, water, minerals), and others. The output $y$ and inputs $x_{i}$ are assumed to be identical. For example, the labor is the quantity of workers indistinguishable in a productive sense.

Henceforth, we will often use the following definition. The function $r(t)=f^{\prime}(t) /$ $f(t)$ is the relative rate of the function $f(t)$ and is often referred to as the growth rate of $f(t)$. If $r \equiv$ const, then $f(t)=C \exp (r t)$.

Economists often use the notation $\dot{f}$ for the derivative of a function $f$ in time. We will keep the standard notation $f^{\prime}$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Properties of Production Functions

Commonly accepted properties of production functions are:

1. Essentiality of inputs: If at least one $x_{i}=0$, then $y=0$, i.e., production is not possible without any of the inputs.
2. Positive returns: $\partial f / \partial x_{i}>0, i=1, \ldots, n$, i.e., the output increases if an input increases.
3. Diminishing returns: The Hessian matrix
$$H=\left[\begin{array}{ccc} \partial^{2} f / \partial x_{1}^{2} & \ldots & \partial^{2} f / \partial x_{1} \partial x_{n} \ \ldots & \ldots & \ldots \ \partial^{2} f / \partial x_{n} \partial x_{1} & \ldots & \partial^{2} f / \partial x_{n}^{2} \end{array}\right]$$
is negatively definite. It means that if only one input $x_{i}$ increases and the other inputs $x_{j}, j \neq i$, remain constant, then the efficiency of using the input $x_{i}$ decreases.
4. Proportional returns to scale: $f(\mathbf{x})$ is a homogeneous function of degree $\gamma>0$, i.e.,
$$f(l \mathbf{x})=l^{\eta} f(\mathbf{x}), \quad l \in R^{1}, \quad l>0, \quad \mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$$
The production function $f(\mathbf{x})$ exhibits increasing returns to scale at $\gamma>1$, decreasing returns to scale at $\gamma<1$, and constant returns to scale at $\gamma=1$. The increasing returns mean that a $1 \%$ increase in the levels of all inputs leads to a greater than the $1 \%$ increase of the output $y$.

In the case of constant returns to scale, the function $f(\mathbf{x})$ is linearly homogeneous: $f(l \mathbf{x})=l f(\mathbf{x})$, and the output increases linearly with respect to a proportional increase of all inputs: a $1 \%$ increase of all inputs produces exactly the $1 \%$ increase of the output. Then, the condition $(2.2)$ is reduced to
$$\partial^{2} f / \partial x_{i}^{2}<0, \quad i=1, \ldots, n$$

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Characteristics of Production Functions

The major characteristics of production functions are

• The average product $f\left(x_{1}, \ldots, x_{n}\right) / x_{i}$ of the $i$-th input is the output per one unit of the input $x_{i}$ spent, $i=1, \ldots, n$.
• The marginal product $\partial f / \partial x_{i}$ of the $i$-th input describes the additional output obtained due to the increase of the $i$-th input quantity by one unit.
• The isoquant is the set of all possible combinations of inputs $\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)$ that yield the same level of the output $y=f(x)$. Along an isoquant, the differential of the function $f(\mathbf{x})$ is zero: $\sum_{i=1}^{n}\left(\partial f / \partial x_{i}\right) \mathrm{d} x_{i}=0$.The marginal rate of substitution between the inputs $i$ and $j$
• $$• h_{i j}=\left(\partial f / \partial x_{i}\right) /\left(\partial f / \partial x_{j}\right) •$$
• shows how many units of the $j$-th input are required to substitute one unit of the $i$-th input in order to produce the same level of the output $y$.
• The partial elasticity of output with respect to the input $i$
• $$• \varepsilon_{i}(\mathbf{x})=\left(\partial f(\mathbf{x}) / \partial x_{i}\right) /\left(f(\mathbf{x}) / x_{i}\right)=\partial \ln f(\mathbf{x}) / \partial \ln x_{i} •$$
• is the ratio between the marginal product and the average product of the $i$-th input. It describes the increase of the output $y$ when the $i$-th input increases by $1 \%$.
• The total output elasticity $\varepsilon(\mathbf{x})=\sum_{i=1}^{n} \varepsilon_{i}(\mathbf{x})$ describes the output increase under a proportional production scale extension. For a homogeneous production function $(2.3), \varepsilon(\mathbf{x})=\gamma$.
• The elasticity of substitution is a quantitative measure of a possibility of changes in the input combination to produce the same output. It is equal to the relative change in the ratio of the $i$-th and $j$-th inputs divided by the relative change in their marginal rate of substitution $h_{i j}$ :
• $$• \sigma_{i j}=\frac{\mathrm{d}\left(x_{i} / x_{j}\right)}{\left(x_{i} / x_{j}\right)} \times \frac{h_{i j}}{\mathrm{~d} h_{i j}}=\frac{\mathrm{d} \ln \left(x_{i} / x_{j}\right)}{\mathrm{d} \ln h_{i j}} •$$
• This characteristic shows the percentage change of the ratio $x_{i} / x_{j}$ of these inputs along an isoquant in order to change their marginal substitution rate by one percent. The larger the $\sigma_{i j}$, the greater the substitutability between the two inputs. The inputs $i$ and $j$ are perfect substitutes at $\sigma_{i j}=\infty$ and they are not substitutable at all at $\sigma_{i j}=0$. The elasticity of substitution is used for classification of various production functions.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Properties of Production Functions

1. 输入的必要性：如果至少有一个X一世=0， 然后是=0，即没有任何投入，生产是不可能的。
2. 正回报：∂F/∂X一世>0,一世=1,…,n，即，如果输入增加，则输出增加。
3. 收益递减：Hessian 矩阵
H=[∂2F/∂X12…∂2F/∂X1∂Xn ……… ∂2F/∂Xn∂X1…∂2F/∂Xn2]
是负定的。这意味着如果只有一个输入X一世增加和其他投入Xj,j≠一世，保持不变，则使用输入的效率X一世减少。
4. 比例回报：F(X)是度的齐次函数C>0， IE，
F(lX)=l这F(X),l∈R1,l>0,X=(X1,…,Xn)
生产函数F(X)表现出规模报酬递增C>1, 规模报酬递减C<1, 规模报酬不变C=1. 收益递增意味着1%所有投入水平的增加导致大于1%增加产量是.

∂2F/∂X一世2<0,一世=1,…,n

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Characteristics of Production Functions

• 平均产品F(X1,…,Xn)/X一世的一世-th 输入是每单位输入的输出X一世花费，一世=1,…,n.
• 边际产品∂F/∂X一世的一世-th 输入描述了由于一世-第一个单位的输入数量。
• 等量线是所有可能的输入组合的集合X=(X1,…,Xn)产生相同水平的输出是=F(X). 沿等量线，函数的微分F(X)为零：∑一世=1n(∂F/∂X一世)dX一世=0. 投入品之间的边际替代率一世和j
• $$• h_{ij}=\left(\partial f / \partial x_{i}\right) /\left(\partial f / \partial x_{j}\right) •$$
• 显示有多少个单位j-th 输入需要替换一个单位一世-th 输入以产生相同级别的输出是.
• 产出相对于投入的部分弹性一世
• $$• \varepsilon_{i}(\mathbf{x})=\left(\partial f(\mathbf{x}) / \partial x_{i}\right) /\left(f(\mathbf{x}) / x_ {i}\right)=\partial \ln f(\mathbf{x}) / \partial \ln x_{i} •$$
• 是边际产量与平均产量的比值一世-th 输入。它描述了输出的增加是当。。。的时候一世-th 输入增加1%.
• 总产出弹性e(X)=∑一世=1ne一世(X)描述了按比例生产规模扩展下的产量增加。对于同质生产函数(2.3),e(X)=C.
• 替代弹性是对投入组合发生变化以产生相同产出的可能性的定量测量。它等于比率的相对变化一世-th 和j-th 投入除以其边际替代率的相对变化H一世j :
• $$• \sigma_{ij}=\frac{\mathrm{d}\left(x_{i} / x_{j}\right)}{\left(x_{i} / x_{j}\right)} \times \ frac{h_{ij}}{\mathrm{~d} h_{ij}}=\frac{\mathrm{d} \ln \left(x_{i} / x_{j}\right)}{\mathrm{ d} \ln h_{ij}} •$$
• 此特性显示比率的百分比变化X一世/Xj沿等量线将这些投入品的边际替代率改变 1%。越大的σ一世j，两个输入之间的可替代性越大。输入一世和j是完美的替代品σ一世j=∞而且它们根本不可替代σ一世j=0. 替代弹性用于对各种生产函数进行分类。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考| Optimization Problems with Constraints

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Optimization Problems with Constraints

Solving an optimization problem with constraints is more difficult compared to the unconstrained optimization. One of powerful techniques for finding local maxima and minima of a function subject to constraints is the method of Lagrange multipliers [5]. To illustrate it for a problem with equality-constraints, let us consider the following simple discrete optimization problem: find $x_{1}, x_{2}$ that
$$\text { maximize } f\left(x_{1}, x_{2}\right)$$
subject to the equality-constraint
$$f\left(x_{1}, x_{2}\right)=c,$$
where both functions $f$ and $g$ have continuous first partial derivatives.
The Lagrange function (or Lagrangian) of the optimization problem (1.42)-(1.43) is determined as
$$L\left(x_{1}, x_{2}, \lambda\right)=f\left(x_{1}, x_{2}\right)+\lambda\left(g\left(x_{1}, x_{2}\right)-c\right),$$
where the unknown variable $\lambda$ is called the Lagrange multiplier (dual variable, or adjoint variable). The second term in (1.44) is zero along a solution to the problem. Thus in order to solve (1.42)-(1.43), we can find the maximum of (1.44). The maximization of the Lagrangian (1.44) includes one more unknown variable but does not involve the equality-constraint (1.43). By construction of (1.44), if $\left(x_{10}\right.$, $x_{20}$ ) brings a maximum to the original problem (1.42)-(1.43), then there exists $\lambda_{0}$ such that $\left(x_{10}, x_{20}, \lambda_{0}\right)$ is a stationary point $(\partial L / \partial \lambda=0)$ of the Lagrange function (1.38). Note that $\partial L / \partial \lambda=0$ implies (1.37).

The method of Lagrange multipliers yields necessary conditions for optimality. Sufficient conditions for optimality are also possible but are more difficult to obtain. The Lagrangian can be reformulated in the terms of Hamiltonian for many specific optimization problems. In particular, the method of Lagrange multipliers can be used to derive the maximum principle for the optimal control of differential equations provided in Sect. 2.4.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Continuous Optimization

Optimization problems in the continuous models of Sect. $1.2 .2$ are known as continuous-time optimization problems or the optimal control problems. The control variables in such problems are scalar-or vector-valued functions of a continuous independent variable and the objective function is a functional that depends on the control variables.

Historically, calculus of variations is the first classic technique for the continuous-time optimization developed over 200 years mainly for geometric and physical applications. A variational problem minimizes a certain functional on a set of smooth functions in an open domain. Further extension of the variational techniques to the non-smooth unknown functions and closed domains leads to the modern optimal control theory and its main tools, the principle of maximum of $\mathrm{L}$. Pontryagin and the dynamic programming method of R. Bellman.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Aggregate Models of Economic Dynamics

This chapter explores aggregate optimization models of the neoclassic economic growth theory, which are based on the concept of production functions. The models are described by ordinary differential equations and involve static and dynamic optimization. Section $2.1$ analyzes production functions with several inputs, their fundamental characteristics, and major types (Cobb-Douglas, CES, Leontief, and linear). Special attention is given to two-factor production functions and their use in the neoclassic models of economic growth. Sections $2.2$ and $2.3$ describe and analyze the well-known Solow-Swan and Solow-Ramsey models. Section $2.4$ contains maximum principles used to analyze dynamic optimization problems in this and other chapters.

最大化 F(X1,X2)

F(X1,X2)=C,

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Continuous Optimization

Sect 连续模型中的优化问题。1.2.2被称为连续时间优化问题或最优控制问题。此类问题中的控制变量是连续自变量的标量或向量值函数，目标函数是取决于控制变量的函数。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考| Integral Equations

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Integral Equations

Integral equations contain integrals with unknown functions in their integrands. This textbook uses the following types of the integral equations:

• Volterra integral equation of the first kind with respect to the unknown function $x(t)$ of the one-dimensional independent variable $t \in[a, b]$ :
$$\int_{a}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau=f(t), \quad t \in[a, b]$$
where the function $f(t)$ and the kernel $K(t, \tau)$ are given functions.
• Volterra integral equation of the second kind
$$x(t)=\int_{a}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau+f(t), \quad t \in[a, b],$$
is more common as compared to (1.30) of the first kind.
Equations $(1.30)$ and (1.31) are named after Vito Volterra (1860-1940), a famous Italian mathematician and physicist, who introduced them and developed their theory and applications. The Volterra integral equations are widely used in population biology, physics, engineering, economics, and demography [3].

These equations are well suited for the description of dynamic processes. Indeed, if the variable $t$ is time, then the current state of a dynamic system (process) depends on the past states and cannot depend on the future. Hence, $K(t, \tau) \equiv 0$ at $\tau>t$ for dynamic systems, which is reflected in the integrals in (1.30) and (1.31).

Another major type of the linear integral equations is the Fredholm integral equations, which are described by the same expressions (1.31) and (1.31) where the upper integration limit $t$ is replaced with $b$. This small change causes significant differences in the qualitative dynamics of their solutions.

Despite the similarity of the Fredholm and Volterra integral equations, their properties are quite different. The Volterra integral equations (1.30) and (1.31) generalize the initial value problems for differential equations considered in Sect. 1.3.3, whereas the Fredholm integral equations correspond to boundary problems (not considered in this textbook).

Linear integral equations have well-developed theories. There is a variety of theorems about the existence and uniqueness of solutions for these equations [2]. The theorems differ in smoothness requirements and forms of the equation. In particular, the Volterra integral equation of the second kind has a unique solution under natural assumptions. A classic existence result is as follows:

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Existence and Uniqueness Theorem for Volterra Integral

If $K(t, \tau)$ is measurable on $[a, b] \otimes[a, b]$ and $f(t)$ is continuous on $[a, b]$, then a unique continuous solution $x(t)$ of the Volterra integral equation (1.31) of the second kind exists on $[a, b]$ and can be determined as
$$x(t)=\int_{a}^{t} R(t, \tau) f(\tau) d \tau+f(t)$$
where the resolvent kernel $R(t, \tau), \tau \in[a, b], t \in[a, b]$, is a solution of the following linear Volterra integral equation:
$$R(t, \tau)=\int_{\tau}^{t} R(t, u) K(u, \tau) \mathrm{d} u+K(t, \tau) .$$
In particular, if $K(t, \tau)=K=$ const, then
$$R(t, \tau)=K \mathrm{e}^{K(t-\tau)}$$
After discretization by the variable $t$, the linear integral equations (1.30) and (1.31) are reduced to systems of linear algebraic equations (1.4). The analogy between continuous integral models and their discrete analogues (1.4) is useful for better understanding and interpretation of the linear integral equations.

However, theory of linear continuous models is much more complex as compared to linear discrete models. In particular, a significant difference exists between the integral equations of the first and the second kind.
Nonlinear Volterra integral equation of the second kind
$$x(t)=\int_{a}^{t} F(t, \tau, x(\tau)) \mathrm{d} \tau+f(t), \quad t \in[a, b],$$
is the generalization of the linear integral equation (1.31) with the integrand $K(t, \tau) x$ replaced with a nonlinear function $F(t, \tau, x)$ of $x$.

Hammerstein-Volterra integral equation is the special case of the nonlinear equation (1.35) with the nonlinearity $F(t, \tau, x)=K(t, \tau) G(x)$ :
$$x(t)=\int_{a}^{t} K(t, \tau) G(x(\tau)) \mathrm{d} \tau+f(t), \quad t \in[a, b] .$$
A key condition for the existence and uniqueness of the solution $x$ to the nonlinear integral equation (1.35) is the Lipschitz condition for the function $F(t, s, x)$ with respect to $x:|F(t, s, x)-F(t, s, y)| \leq L(t, s)|x-y|$. If it holds, then (1.35) possesses a unique solution $x$, at least, for continuous $f$ and $L$.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Volterra Integral Equations with Variable Delay

The classic integral equations ( $1.30),(1.31),(1.35)$ take into account the distributed delay (after-effect, hereditary effects) on the interval $[a, t]$. The distributed delay means that a continuous sequence of the past states of a dynamical system affects the future evolution of the system. The integral equations with variable delay occur if the distributed delay exists at the initial time $t=a$ as well. Specifically, the linear integral equation with variable delay
$$x(t)=\int_{a(t)}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau+f(t), \quad t \in\left[t_{0}, T\right]$$
with the initial condition
$$x(\tau)=x_{0}(\tau), \quad \tau \in\left[\tau_{0}, \quad t_{0}\right]$$
means that the solution $x$ depends on its known behavior $x_{0}$ over a certain prehistory interval $\left[\tau_{0}, t_{0}\right]$. The lower integration limit $a(t)$ in $(1.37)$ is a given function such as $\tau_{0} \leq a(t)<t .$

Equation (1.37) can be solved by reducing it to the standard Volterra integral equation

$1.3$ Review of Selected Mathematical Tools
19
$$x(t)=\int_{t_{0}}^{t} K(t, \tau) x(\tau) \mathrm{d} \tau+f_{1}(t), \quad t \in\left[t_{0}, t_{1}\right],$$
on some interval $\left[t_{0}, t_{1}\right]$ such that $a(t) \leq t_{0}, t \in\left[t_{0}, t_{1}\right]$. Then,
$$f_{1}(t)=\int_{a(t)}^{t_{0}} K(t, \tau) x_{0}(\tau) \mathrm{d} \tau+f(t)$$
is a given function on $\left[t_{0}, t_{1}\right]$. Next, this solution process is repeated on a new interval $\left[t_{1}, t_{2}\right]$ with the updated initial condition $x(\tau)=x_{0}(\tau), \tau \in\left[\tau_{0}, t_{1}\right]$, and so on.

The integral equations with variable delay are used for the modeling of economic development in Chaps. 4 and 5. Moreover, in some economic applications, the lower integration limit $a(t)$ can be an unknown control. Then, the model (1.39) leads to nonlinear integral equations with controllable delay (see Chap. 5).

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Integral Equations

• 关于未知函数的第一类沃尔泰拉积分方程X(吨)一维自变量的吨∈[一种,b] :
∫一种吨ķ(吨,τ)X(τ)dτ=F(吨),吨∈[一种,b]
函数在哪里F(吨)和内核ķ(吨,τ)被赋予功能。
• 第二类沃尔泰拉积分方程
X(吨)=∫一种吨ķ(吨,τ)X(τ)dτ+F(吨),吨∈[一种,b],
与第一种（1.30）相比更常见。
方程(1.30)和 (1.31) 以意大利著名数学家和物理学家 Vito Volterra (1860-1940) 的名字命名，他介绍了它们并发展了它们的理论和应用。Volterra 积分方程广泛用于人口生物学、物理学、工程学、经济学和人口学 [3]。

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Existence and Uniqueness Theorem for Volterra Integral

X(吨)=∫一种吨R(吨,τ)F(τ)dτ+F(吨)

R(吨,τ)=∫τ吨R(吨,在)ķ(在,τ)d在+ķ(吨,τ).

R(吨,τ)=ķ和ķ(吨−τ)

X(吨)=∫一种吨F(吨,τ,X(τ))dτ+F(吨),吨∈[一种,b],

Hammerstein-Volterra 积分方程是具有非线性的非线性方程 (1.35) 的特例F(吨,τ,X)=ķ(吨,τ)G(X) :
X(吨)=∫一种吨ķ(吨,τ)G(X(τ))dτ+F(吨),吨∈[一种,b].

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Volterra Integral Equations with Variable Delay

X(吨)=∫一种(吨)吨ķ(吨,τ)X(τ)dτ+F(吨),吨∈[吨0,吨]

X(τ)=X0(τ),τ∈[τ0,吨0]

1.3回顾选定的数学工具
19
X(吨)=∫吨0吨ķ(吨,τ)X(τ)dτ+F1(吨),吨∈[吨0,吨1],

F1(吨)=∫一种(吨)吨0ķ(吨,τ)X0(τ)dτ+F(吨)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考| Continuous and Discrete Models

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Continuous and Discrete Models

Depending on the type of available data and process description, the two major categories of mathematical models are continuous and discrete models. The continuous models operate with continuous variables, while the discrete models operate with discrete variables. More specifically, a discrete model involves a finite number $n, n \geq 1$, of the unknown (endogenous, sought-for) scalar variables $y_{1}, y_{2}$, $\ldots, y_{n}$. A general form of a discrete model is
$$F_{j}\left(y_{1}, y_{2}, \ldots, y_{n}\right)=0, \quad j=1, \ldots, m$$
where $F_{j}(.)$ are some functions of $n$ scalar variables. In this textbook, we assume that each variable $y_{i}$ is a real number: $y_{i} \in \mathbf{R}^{1}$. Models with the integer-valued variables $y_{i}$ are less common and harder to analyze.

A continuous model uses a continuous (scalar or vector) independent variable $x$ defined on some domain $D \subset \mathbf{R}^{n}, n \geq 1$, and operates with scalar-or vectorvalued functions $y(x)$. Continuous dynamic models include time as one of the independent variables. A general form of continuous models is
$$\Phi_{j}(y)=0, \quad j=1, \ldots, m$$

where $\Phi_{j}(y)$ is a functional that sets a real value for each function $y$ from a certain functional space $\Omega$. Common examples of the functional space $\Omega$ are:

• the space $\boldsymbol{C}[a, b]$ of all continuous functions defined on the interval $[a, b]$
• the space $\boldsymbol{L}^{\infty}[a, b]$ of all measurable functions bounded almost everywhere on $[a, b]$.

A discrete analogue can usually be constructed for a continuous model, and vice versa. Discrete analogues are known for the most of continuous models of economic and ecological systems considered in this textbook. Computer simulation commonly uses discrete models or discrete analogues of continuous models in numeric algorithms. The choice between continuous and discrete models, and among their particular types, depends on the specifics of the real-life process under study. Models that combine discrete and continuous variables are known as hybrid models.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Linear and Nonlinear Models

The choice between linear and nonlinear models depends on the nature of the process under study and/or on the desired level of the process approximation. Many real-life processes are nonlinear but are commonly described by approximate linear models because the latter are simpler and have better theory and investigative techniques. Other processes are substantially nonlinear and their linearization leads to oversimplified description and incorrect modeling outcomes.

Linear discrete model is a system of linear algebraic equations:
$$\sum_{i=1}^{n} a_{i j} y_{j}=b_{i}, \quad i=1, \ldots, m, \quad \text { or } \quad A \mathbf{y}=\mathbf{b},$$
where
$\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right) \in \mathbf{R}^{n}, \mathbf{b}=\left(b_{1}, b_{2}, \ldots, b_{m}\right) \in \mathbf{R}^{m}$, and $A=\left{a_{i j}\right}$ is an $m \times n$ matrix.

Model (1.3) represents a convenient and well-investigated mathematical object. If $m=n$ and the determinant $\operatorname{det} A \neq 0$, then the system (1.3) has a unique solution $\mathbf{y}$ (under the given $A$ and $\mathbf{b}$ ).

Linear continuous model is the model (1.2) with linear functionals $\Phi_{j}, j=1, \ldots, m$. The linear functional $\Phi$ keeps the linear operations of addition and scalar multiplication for any elements $y$ and $z$ from a functional space $\Omega$ :
$$\Phi(y+z)=\Phi(y)+\Phi(z), \cdots \Phi(\alpha y)=\alpha \Phi(y) \quad \text { for } \quad a \in \mathbf{R}^{1}$$
Theories of the linear differential and integral equations are well developed and provide a good background for modeling many real systems and processes.

Nonlinear continuous model is the model (1.2) when at least one functional $\Phi_{j}(.)$ is nonlinear. There is no complete general theory for such equations, although fundamental breakthroughs are obtained for many specific nonlinear problems. The most studied categories of such models are nonlinear differential and integral equations. The theory of such equations is intensively investigated and possesses numerous essential results. Some of these results are reviewed in Sect. 1.3..

Nonlinear discrete models of the form (1.1) with nonlinear functions $F_{j}$ also do not possess a general theory, and investigation of a specific system of nonlinear equations often runs into great theoretical or numeric difficulties. The solution may be nonunique or not existing in the nonlinear models, both discrete and continuous. The famous example is the polynomial equation $a_{n} x^{n}+a_{n-1} x^{n}+$ $\ldots+a_{1} x+a_{0}=0$ of one scalar variable $x$, which allows for a complete analytic solution at $n=2,3$, and 4 , but not for $n$ larger than 4 . However, there are special classes of nonlinear discrete models, for example difference equations [4], which have well-developed theory and applications.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Vector Algebra and Calculus

Let us consider the Cartesian coordinate system $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$ in the threedimensional space $\mathbf{R}^{3}$. The vectors $\mathbf{i}=(1,0,0), \mathbf{j}=(0,1,0)$, and $\mathbf{k}=(0,0,1)$ are called the fundamental vectors or the basis.

The dot product (scalar product, inner product) of two three-dimensional vectors $\mathbf{x}$ and $\mathbf{y}$ is a scalar
$$\mathbf{x} \cdot \mathbf{y}=(\mathbf{x}, \mathbf{y})=x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3} .$$
The dot product is used to find the angles between the two vectors, determine an orthogonal basis, find a normal to a plane, find work done by a force, and for others purposes (see Chap. 9).

The cross product (vector product, outer product) of two three-dimensional vectors $\mathbf{x}$ and $\mathbf{y}$ is the vector
$$\mathbf{x} \times \mathbf{y}=\left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ x_{1} & x_{2} & x_{3} \ y_{1} & y_{2} & y_{3} \end{array}\right|$$
Applications of the cross product are to find the moment of a force, the velocity of a rotating body, the volume of solids, and others.

The gradient of a scalar differentiable function $f\left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{R}^{\mathbf{1}}$ is the vector
$$\nabla f=\operatorname{grad} f=\frac{\partial f}{\partial x_{1}} \mathbf{i}+\frac{\partial f}{\partial x_{2}} \mathbf{j}+\frac{\partial f}{\partial x_{3}} \mathbf{k}$$
It defines the direction and magnitude of the maximum rate of increase of the function $f$ at the point $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$. The gradient is a normal vector to the surface $f\left(x_{1}, x_{2}, x_{3}\right)$ at point $\mathbf{x}$.
The differential operator $\nabla$ (nabla) is $\nabla=\frac{\partial}{\partial x_{1}} \mathbf{i}+\frac{\partial}{\partial x_{2}} \mathbf{j}+\frac{\partial}{\partial x_{3}} \mathbf{k}$.
The Laplace operator $\Delta$ (delta) is $\Delta=\nabla^{2}=\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{\partial x_{2}^{2}}+\frac{\partial^{2}}{\partial x_{3}^{2}}$.
The Laplacian of a scalar function $S\left(x_{1}, x_{2}, x_{3}\right)$ is the scalar
$$\Delta S=\operatorname{div} \operatorname{grad} S=\nabla \cdot(\nabla S)=\nabla^{2} S=\frac{\partial^{2} S}{\partial x_{1}^{2}}+\frac{\partial^{2} S}{\partial x_{2}^{2}}+\frac{\partial^{2} S}{\partial x_{3}{ }^{2}}$$

Let $x_{1}=x_{1}(t), x_{2}=x_{2}(t), x_{3}=x_{3}(t)$. Then, the total derivative of a scalar function $S\left(x_{1}, x_{2}, x_{3}, t\right)$ with respect to $t$ is
$$\frac{\mathrm{d} S}{\mathrm{~d} t}=\frac{\partial S}{\partial t}+\frac{\partial S}{\partial x_{1}} \frac{\mathrm{d} x_{1}}{\mathrm{~d} t}+\frac{\partial S}{\partial x_{2}} \frac{\mathrm{d} x_{2}}{\mathrm{~d} t}+\frac{\partial S}{\partial x_{3}} \frac{\mathrm{d} x_{3}}{\mathrm{~d} t} .$$
The partial derivative of a vector-function $\mathbf{V}(\mathbf{x})=V_{1} \mathbf{i}+V_{2} \mathbf{j}+V_{3} \mathbf{k} \in \mathbf{R}^{3}$ with respect to $x_{i}$ is the vector
$$\frac{\partial \mathbf{V}}{\partial x_{i}}=\frac{\partial V_{1}}{\partial x_{i}} \mathbf{i}+\frac{\partial V_{2}}{\partial x_{i}} \mathbf{j}+\frac{\partial V_{3}}{\partial x_{i}} \mathbf{k}$$
The divergence of a vector function $\mathrm{V}\left(x_{1}, x_{2}, x_{3}\right)$ is the scalar
$$\operatorname{div} \mathbf{V}=\nabla, \mathbf{V}=\frac{\partial V_{1}}{\partial x_{1}}+\frac{\partial V_{2}}{\partial x_{2}}+\frac{\partial V_{3}}{\partial x_{3}} .$$

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Continuous and Discrete Models

Fj(是1,是2,…,是n)=0,j=1,…,米

• 空间C[一种,b]在区间上定义的所有连续函数[一种,b]
• 空间大号∞[一种,b]几乎处处有界的所有可测量函数[一种,b].

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Linear and Nonlinear Models

∑一世=1n一种一世j是j=b一世,一世=1,…,米, 或者 一种是=b,

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Vector Algebra and Calculus

X⋅是=(X,是)=X1是1+X2是2+X3是3.

X×是=|一世jķ X1X2X3 是1是2是3|

∇F=毕业⁡F=∂F∂X1一世+∂F∂X2j+∂F∂X3ķ

Δ小号=div⁡毕业⁡小号=∇⋅(∇小号)=∇2小号=∂2小号∂X12+∂2小号∂X22+∂2小号∂X32

d小号 d吨=∂小号∂吨+∂小号∂X1dX1 d吨+∂小号∂X2dX2 d吨+∂小号∂X3dX3 d吨.

∂在∂X一世=∂在1∂X一世一世+∂在2∂X一世j+∂在3∂X一世ķ

div⁡在=∇,在=∂在1∂X1+∂在2∂X2+∂在3∂X3.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Principles and Tools of Mathematical Modeling

statistics-lab™ 为您的留学生涯保驾护航 在代写数学生态学Mathematical Ecology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数学生态学Mathematical Ecology代写方面经验极为丰富，各种代写数学生态学Mathematical Ecology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Role and Stages of Mathematical Modeling

Mathematical modeling is a vital component of scientific research and policy making. Its effectiveness has been proven for centuries. The modeling provides an explanation and prediction of the behavior of complex economic and environmental systems and helps to obtain new theoretical knowledge about the nature and society. The concept of the economic-environmental system assumes the influence of both the economy and environment on each other and the possibility of human control in the system [7]. The importance of modeling of such systems increases proportionally to the scale of human impact on the environment.

Mathematical modeling and computer simulation have a special place among scientific methods. The advantages of modeling as compared to experimentation are as follows:

• Universal availability and applicability of modeling tools.
• Low costs and short timeline of the modeling process.
• Multiple simulations on a wide range of model parameters (“what-if” analysis).
• Possibility of making various model modifications and improvements.
• Evading negative outcomes of real experiments, and others.

Modeling should begin at the early stage of a study, just after initial observation or experimentation. It can take decades to notice visible changes in environmental systems, by which time the changes may have already become irreversible. Mathematical modeling can predict negative changes in such systems and recommend measures to prevent them. The analysis of early modeling results can also suggest what kind of additional information is required and what model modifications can be made to achieve a better correspondence with the real-life picture.
A mathematical model is not a copy of the real world: it is always a simplification of the reality, which assists in revealing principal features of real phenomena. In theoretical research or decision-making practice, people use models because they do not possess an absolute knowledge of reality. The models initially emerge in the human brain. Scientific research improves and justifies such mental models, which become conceptual models in corresponding areas of science. Mathematical and computer modeling methods are based on the conceptual models and, therefore, cannot be more informative than these models. Formal mathematical models are secondary with respect to the conceptual models; however, they allow for finding new insights that are impossible to obtain by other scientific methods.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Mathematical Modeling and Computer Simulation

Computer modeling complements and extends traditional analytic forms of mathematical modeling. Modern computers are able to process vast amounts of data, including various choices of system evolution in a quick and efficient manner. Therefore, computer simulation has become a common additional or even primary modeling technique, especially when analytic solution is challenging or impossible to obtain. Computers are widely used in interactive modeling of complex environmental problems, such as weather prediction and global climate change.
Modeling gives a quantitative description of a real system and its connections with the external environment in the presence of unpredicted or inaccessible factors. Both traditional and computer models meet major challenges related to principal impossibility to obtain complete ecological information for modeling. At the same time, the increasing capabilities and reasonable prices of modern computers lead to the appearance of new modeling concepts entirely based on computer information processing, such as the agent-based modeling. Such models are populated by millions of computer-simulated agents that act as predicted for

living organisms. In economics, the agent-based models try to simulate elementary transactions that occur in an actual economy. This area of research is emerging, but it has not yet delivered convincing breakthroughs.

The possibilities of computer modeling and simulation should not be overestimated because computer models are also based on original conceptual models of specific disciplines. In any scenario, traditional mathematical modeling keeps its place and relevance in the predictable future, as both a learning and decision-support tool.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Choice of Models

Deterministic models operate with certain quantitative characteristics of systems and processes without assuming their probabilistic nature. Deterministic models are helpful in many realistic situations that involve relatively few sources of

uncertainty inside the system. In modeling practice, deterministic models can deal with the averaged probabilistic characteristics of processes under study (an average “concentration of pollutant” instead of the real concentration, the expected value of “equipment lifetime” instead of the real equipment lifetime, and so on) and are based on the approximation of a real process.

Economic and environmental systems belong to complex systems with high dimensionality and uncertainty of the relationships inherent in them. Nevertheless, the subsequent chapters demonstrate that deterministic models are commonly used for their description. In many cases, increasing complexity of mathematical description using stochastic factors does not lead to substantial insight into the nature of a problem.

Stochastic models describe connections among stochastic (probabilistic) characteristics of systems and processes under study. They are useful for the analysis of repetitive processes and usually require a large amount of data to start modeling. Implementation of economic and environmental processes is unique and accompanied by a shortage of data (especially for large-scale systems). A comprehensive analysis of all available information should be the first step of the system analysis. A majority of the models in this textbook are deterministic. Stochastic models are used in Sect. $10.2$ due to a substantially stochastic nature of the natural resource discovery.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Role and Stages of Mathematical Modeling

• 建模工具的普遍可用性和适用性。
• 建模过程成本低且时间短。
• 对各种模型参数进行多次模拟（“假设”分析）。
• 可以进行各种模型修改和改进。
• 逃避真实实验和其他实验的负面结果。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。