8  Quiz 1

8.1 Strategy for solving the fixed point problem

Suppose we are solving the fixed point equation \[ H(N^*;p)=N^*. \] Here \(N^*\) is the fixed point value and \(p\) represents model parameters. In the flow chart below we represent \(N^*\) by \(x\) for notational convenience.

flowchart LR
  A[Know variables and parameters]-->B[Identify FP equation for unknown x]  
  B--> C(Linear)
  C --> D(Solve)
  B --> E(Quadratic)
  E --> F(ax^2+bx+c=0)
  F -->G(Solve for x using quadratic formula)
  E --> H(ax^2+bx=0)
  H -->I(Factorise x=0 and solve)
  B -->J(Terms involving x on denominator)
  J -->K(Multiply by denominators and gather terms in powers of x)
  K-->O(Linear or quadratic)
  K-->Q(Do something else)
  B-->L(Other form)
  L-->M(Write as some 'gx=a')
  M-->P(Apply inverse function to isolate x)
  L-->N(Do something else)

Spotting the quadratic with a zero root

Suppose the fixed point were a quadratic equation of the form \[ a{N^*}^2+bN^*+c \] where \[a,b,c \in \Re^+.\]

The solution is \[ N^*=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]

However, in the case you are trying to solve \(c=0\).

In this case, factorise instead, i.e. \[ a{N^*}^2+bN^*=0. \] Hence \[ N^*(aN^*+b)=0. \] Hence one root is \[ N^*=0. \] The other is \[ N^*=-b/a. \]

flowchart LR
  A[Know variables and parameters]:::foo-->B[Identify FP equation for unknown x]:::foo 
  B--> C(Linear)
  C --> D(Solve)
  B --> E(Quadratic)
  E --> F(ax^2+bx+c=0)
  F -->G(Solve for x using quadratic formula)
  E --> H(ax^2+bx=0)
  H -->I(Factorise x=0 and solve)
  B -->J(Terms involving x on denominator):::foo
  J -->K(Multiply by denominators and gather terms in powers of x):::foo
  K-->O(Linear or quadratic):::foo
  K-->Q(Do something elsse)
  B-->L(Other form)
  L-->M(Write as some 'gx=a')
  M-->P(Apply inverse function to isolate x)
  L-->N(Do something else)

  classDef foo stroke:#f00

Figure 8.1: The strategy for this problem is highlighted.

In Quiz 1 the fixed point problem takes the form \[ N^*=\frac{rN^*}{1+N^*}-hN^*. \tag{8.1}\]

We want to solve Equation 8.1 for the unknown \(N^*\). \(r\) and \(h\) are the model parameters (these are fixed constants).

To isolate the \(N^*\) we need to deal with the denominator.

Note

There is at least one term involving the variable in a denominator.

Hence multiply every term in Equation 8.1 by the denominator \(1+N^*\).

Therefore \[ N^*(1+N^*)=rN^*-hN^*(1+N^*). \]

Now gather coefficients of powers of \(N^*\).

Hence \[ {N^*}^2 (1+h) + N^*(1+h-r)=0. \]

Note

This is of the form \[ ax^2+bx=0. \]

Factorise out an \(N^*\). Hence \[ N^*((1+h)N^*+1+h-r)=0. \] Hence one solution is \[ N^*=0. \] The other is \[ N^*=\frac{r}{1+h}-1. \]

8.2 Derivatives

\[ N_{t+1}=H(N_t)=\frac{rN_t}{1+N_t}-hN_t. \]

To differentiate see below.

Note

Suppose \[ f(x)=\frac{u(x)}{v(x)}. \]

Write as \[ f(x)={u(x)}{v(x)}^{-1}. \]

Apply the product rule \[ \frac{df}{dx}=\frac{du}{dx}v(x)^{-1} +u(x)\left(-v(x)^{-2}\frac{dv}{dx}\right). \]

Hence \[ H'(N_t)= r (1+N_t)^{-1} +rN_t\left(-(1+N_t)^{-2}\right) -h. \]

Tidying \[ H'(N_t)= \frac{r}{(1+N_t)^2} -h. \]