7  Curve sketching

7.1

In the module we will be required to graph of nonlinear function.

Consider a nonlinear function \[ f(x;p), \quad x \in \Re. \] where \(p\) is a constant.

For example, \[ f(x)=rx(1-x), \quad x \in \Re. \]

Our goal is usually to sketch \(f\) in qualitatively distinct cases. For example: the graph of \(f\) when \(r=1\) is qualitatively equivelent to the graph of \(f\) when \(r=2\). However, when \(r=-1\) it is qualitatively distinct.

The challenge is to firstly identify qualitatively distinct cases. Then sketch \(f\) for each of these cases.

T0 identify qualitatively distinct cases we test the properties of \(f\) as follows:

1. Identify the biologically relevant roots of f.  We define \(x^*\) such that \[ f(x^*;p)=0. \] Biologically relevant solutions require \(x^*\geq0\).

   Does the number of biologically relevant solutions depend on the parameter \(p\)?

2. Identify the turning points of f.
   Differentiate \(f\) with respect to \(x\). Identify turning points, \(x_c\), such that \[ \frac{df}{dx}\bigg|_{x=x_c}=0. \] Are there any turning points? Are they biologically relevant? Does the number of biologically relevant solutions depend on the parameter \(p\)?

3. Identify limiting behaviour as \(x\rightarrow \infty\). Compute the limit \[ \lim_{x\rightarrow \infty}f(x;p)=0. \]