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

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• 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|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.

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