First-order Difference Equations and the Cobweb Model | 一阶差分方程与蛛网模型

📚 First-order Difference Equations and the Cobweb Model | 一阶差分方程与蛛网模型

In IB Mathematics, first-order difference equations provide a powerful framework for modelling discrete-time dynamical systems. They describe how a quantity evolves from one period to the next, with applications ranging from population growth to economics. The cobweb model, a classic application, explains how prices fluctuate in markets where supply decisions are based on previous prices. This article explores the theory, solution methods, stability analysis, and the construction of cobweb diagrams, equipping you with the skills needed for exam success.

在 IB 数学中，一阶差分方程为离散时间动态系统提供了强有力的建模框架。它们描述一个量如何从一个周期演变到下一个周期，其应用范围从人口增长到经济学。蛛网模型作为一个经典应用，解释了供给决策基于前一时期价格时，市场价格如何波动。本文将探讨其理论、求解方法、稳定性分析以及蛛网图的绘制，帮助你掌握考试所需的技巧。

1. Introduction to Difference Equations | 差分方程导论

A difference equation relates the value of a variable at time n+1 to its value at time n (and possibly earlier periods). We write xₙ₊₁ = f(xₙ) for a first-order equation, where xₙ is the state at step n. Unlike differential equations modelling continuous change, difference equations are suited to situations where events occur at regular intervals—annual interest, daily website hits, or harvest cycles.

差分方程将变量在时刻 n+1 的值与其在时刻 n（以及可能更早时期）的值联系起来。一阶差分方程可写作 xₙ₊₁ = f(xₙ)，其中 xₙ 为第 n 步的状态。与模拟连续变化的微分方程不同，差分方程适用于事件按固定间隔发生的情形——例如年利率、每日网站访问量或收获周期。

2. First-order Linear Difference Equations | 一阶线性差分方程

The simplest and most important type is the first-order linear difference equation with constant coefficients: xₙ₊₁ = a xₙ + b, where a and b are constants. If b = 0, it is homogeneous; if not, it is inhomogeneous. The coefficient a controls the dynamics: growth if a > 1, decay if 0 < a < 1, and oscillation if a < 0.

最简单且最重要的类型是常系数一阶线性差分方程：xₙ₊₁ = a xₙ + b，其中 a 和 b 为常数。当 b = 0 时，方程为齐次的；否则为非齐次的。系数 a 控制动态行为：若 a > 1 则增长，若 0 < a < 1 则衰减，若 a < 0 则振荡。

3. Solution by Iteration | 迭代求解法

Starting from an initial value x₀, we can repeatedly apply the equation to generate x₁, x₂, x₃, … . For example, x₁ = a x₀ + b, x₂ = a x₁ + b = a² x₀ + b(a+1), and so on. This iterative approach is fundamental to understanding the long-term behaviour and is precisely how cobweb diagrams are constructed.

从初始值 x₀ 出发，我们可以反复应用方程生成 x₁、x₂、x₃……。例如，x₁ = a x₀ + b，x₂ = a x₁ + b = a² x₀ + b(a+1)，依此类推。这种迭代方法是理解长期行为的基础，也正是构建蛛网图的原理。

4. General Solution and Equilibrium | 通解与平衡点

The general solution to xₙ₊₁ = a xₙ + b with initial condition x₀ is derived by recognising a geometric series. For a ≠ 1 we have:

在初始条件 x₀ 下，xₙ₊₁ = a xₙ + b 的通解可通过识别几何级数得到。对于 a ≠ 1，有：

xₙ = aⁿ x₀ + b (1 – aⁿ)/(1 – a)

If a = 1, then the solution simplifies to xₙ = x₀ + n b. The equilibrium (or fixed point) x* satisfies x* = a x* + b, giving:

若 a = 1，则解简化为 xₙ = x₀ + n b。平衡点（不动点）x* 满足 x* = a x* + b，由此得出：

x* = b/(1 – a) for a ≠ 1

When a = 1 and b = 0, every value is an equilibrium; if a = 1 and b ≠ 0, there is no finite equilibrium.

当 a = 1 且 b = 0 时，每个值都是平衡点；若 a = 1 且 b ≠ 0，则不存在有限平衡点。

5. Stability Analysis | 稳定性分析

The long-term fate of xₙ depends entirely on |a|. If |a| < 1, then aⁿ → 0 as n → ∞, so xₙ converges to the equilibrium x* = b/(1 - a), regardless of the starting value x₀. If |a| > 1, the solution diverges away from x* (unless x₀ = x*). If a = -1, the system oscillates indefinitely between two values. If a = 1 and b ≠ 0, the sequence grows linearly without bound. This algebraic condition is the foundation for understanding cobweb stability.

xₙ 的长期命运完全取决于 |a|。如果 |a| < 1，当 n → ∞ 时 aⁿ → 0，因此无论起始值 x₀ 为何，xₙ 都收敛到平衡点 x* = b/(1 - a)。如果 |a| > 1，解会从 x* 发散（除非 x₀ = x*）。如果 a = -1，系统会在两个值之间无限振荡。如果 a = 1 且 b ≠ 0，序列将线性增长而无界。这一代数条件是理解蛛网稳定性的基石。

6. Introduction to the Cobweb Model | 蛛网模型简介

The cobweb model is an economic application of first-order difference equations used to explain price movements in markets where supply responds to price with a time lag. For instance, farmers decide how much to plant based on this year’s price

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