ccquad.m

This represents the $2 \times 2$ autonomous system associated with two first order, quadratic ODEs with constant coefficients. Competing species and predator-prey models are forms of this system.

$$\begin{eqnarray*} \lefteqn{\begin{array}{r*{6}{cr}} x' &=& a_1\,x &+& b_1\,y &+&... ...B = [$d_1$,$\beta_1$,$\gamma_1$; $d_2$,$\beta_2$,$\gamma_2$];} \end{eqnarray*}$$

The linearization of this system is accomplished via the four partial derivatives

$$\begin{eqnarray*} F_x(x,y) \;=\; a_1 \,+\, d_1\,y \,+\, 2\,\beta_1\,x, && F_y(x,... ..._2\,x, && G_y(x,y) \;=\; b_2 \,+\, d_2\,x \,+\, 2\,\gamma_2\,y \end{eqnarray*}$$

evaluated at any critical point $(x_0, y_0) \; \Longrightarrow \;$

$$\begin{eqnarray*} \left[\begin{array}{c}x' \\ y'\end{array}\right] &\approx& \le... ...array}{c}x \\ y\end{array}\right] \quad\mbox{near $(x_0,y_0)$} \end{eqnarray*}$$

Once the linearization has been formed, we use cclin.m with

$$\mbox{\tt A = [$F_x(x_0,y_0)$,$F_y(x_0,y_0)$; $G_x(x_0,y_0)$,$G_y(x_0,y_0)$];} \quad\mbox{\tt B = [$\;$];}$$

to investigate stability about each $(x_0, y_0)$.

The competing species and predator-prey forms are
Competing species

$$\begin{eqnarray*} \begin{array}{rcrcrcr} x' &=& \epsilon_1\,x &-& \alpha_1\,xy &... ..., & d_2 \,=\, -\alpha_2, & \beta_2 \,=\, -\sigma_2 \end{array} \end{eqnarray*}$$

$$\begin{eqnarray*} &\; \Longrightarrow \;& \mbox{\bf A}\;=\; \left[\begin{array}{... ...t B = [$-\alpha_1$,$-\sigma_1$,0; $-\alpha_2$,0,$-\sigma_2$];} \end{eqnarray*}$$

Predator-prey

$$\begin{eqnarray*} \begin{array}{rcrcr} x' &=& a\,x \,-\, \alpha\,x\,y \\ y' &=&... ... \,=\, -\alpha \\ b_2 \,=\, -c, & d_2 \,=\, \gamma \end{array} \end{eqnarray*}$$

$$\begin{eqnarray*} &\; \Longrightarrow \;& \mbox{\bf A}\;=\; \left[\begin{array}{... ... 0,$-c$];} \quad \mbox{\tt B = [$-\alpha$,0,0; $\gamma$,0,0];} \end{eqnarray*}$$

The symbolicplot option `eigenvals' is useful only for the linearization.