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# 数学代写|常微分方程代考Ordinary Differential Equations代写|Population dynamics

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Population dynamics

This section illustrates the use of stability analysis to investigate models of interacting populations. The main reference for our discussion is [106].
The first model of population dynamics was proposed by Thomas Robert Malthus in a simplified setting that supposes unbounded resources and a constant growth factor $R$. In this setting, the population growth (of a living system) is given by
$$y(x)=y_0 e^{R x}$$
where $y_0>0$ represents the population size at time $x=0$. However, in the presence of limited resources, this model is unrealistic. For this reason, years later, Pierre-Francois Verhulst suggested that $R$ should depend on the population size $y$ and on the carrying capacity $K>0$ of the environment where the population lives. The model of Verhulst considers the following variable growth factor:
$$R(y)=r\left(1-\frac{y}{K}\right)$$
Therefore, the resulting population dynamics is described by the following ODE model:
$$y^{\prime}(x)=r y(x)\left(1-\frac{y(x)}{K}\right),$$
which is called the logistic equation. The solution of the Cauchy problem defined by this model with the initial condition $y\left(x_0\right)=y_0$ is given by
$$y(x)=\frac{K y_0}{y_0+\left(K-y_0\right) e^{-r\left(x-x_0\right)}} .$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Lorenz model

In this section, we present the well-known Lorenz model that shows how rich the dynamics of a (at least) three-dimensional autonomous system can be. This model was proposed by Edward N. Lorenz [95] as a simplified model for atmospheric convection. It is given by
\begin{aligned} & u^{\prime}=\sigma(y-u) \ & y^{\prime}=r u-y-u z \ & z^{\prime}=u y-b z \end{aligned}
where $\sigma, r, b$ are positive constants. The model (6.27) allows us to illustrate the so-called chaotic behaviour of a dynamical system. For this reason, we call the independent variable $x$ the time coordinate, and also refer to (6.27) as the system $\underline{y}^{\prime}=\underline{f}(\underline{y})$

In the following, we investigate the equilibrium points of the Lorenz model. One of these points is the origin $P^0=(0,0,0)$. This and other equilibrium points are solutions to the following system:
\left{\begin{aligned} y-u & =0 \ r u-y-u z & =0 \ u y-b z & =0 \end{aligned}\right.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Population dynamics

$$y(x)=y_0 e^{R x}$$

$$R(y)=r\left(1-\frac{y}{K}\right)$$

$$y^{\prime}(x)=r y(x)\left(1-\frac{y(x)}{K}\right),$$

$$y(x)=\frac{K y_0}{y_0+\left(K-y_0\right) e^{-r\left(x-x_0\right)}} .$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Lorenz model

\begin{aligned} & u^{\prime}=\sigma(y-u) \ & y^{\prime}=r u-y-u z \ & z^{\prime}=u y-b z \end{aligned}

\left{\begin{aligned} y-u & =0 \ r u-y-u z & =0 \ u y-b z & =0 \end{aligned}\right.

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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。