We consider again a pair of coupled homogeneous linear first-order differential equations with constant coefficients of the form $$\left\{\begin{aligned} \frac{\dd x}{\dd t} &= a\, x+ b\, y\\[0.25cm] \frac{\dd y}{\dd t} &= c\, x + d\, y\end{aligned}\right.$$ and write it in the matrix-vector form $$\frac{\dd}{\dd t}\cv{x\\y}=\matrix{a & b\\ c & d}\cv{x\\y }$$ The matrix $$A=\matrix{a & b\\ c & d}$$ actually describes the linear system of differential equations. The eigenvalues and eigenvectors determine the nature of the solutions and the stability of the equilibrium $\cv{x\\y}=\cv{0\\0}$.

The eigenvalues of $A$ can be written as $$\lambda_{1,2}=\frac{\text{sp}(A)}{2}\pm\frac{1}{2}\sqrt{\bigl(\text{sp}(A)\bigr)^2-4\det(A)}$$ where $\text{sp}(A)=a+d$ is the trace of $A$ and $\det(A)=ad-bc$ is the determinant of $A$.

We have already seen that the equilibrium is attracting when $A$ has

• two different eigenvalues both of which are negative;
• one negative eigenvalue;
• complex eigenvalues with a real part less than zero.

We can also formulate the stability of the equilibrium in terms of the trace and the determinant of the $\text{matrix }A$.

We can explore all combinations of the signs of $\det(A)$ and $\text{sp}(A)$ and describe behaviour of solutions near the equilibrium $(0,0)$. The figure below comes from the book Mathematical Models in Biology of Leah Edelstein-Keshet and it summarizes the results. In this figure, $\beta=\text{sp}(A)$ and $\gamma=\det(A)$. Then $\beta^2-4\gamma$ is the discriminant of the characteristic equation of $A$. When the discriminant is equal to 0, then the stability of the equilibrium depends on the sign of $\beta$: repelling if $\beta>0$ and attracting if $\beta<0$. There are six additional cases:

1. repelling equilibrium: $\beta>0$ and $\gamma>0$.
2. saddle node (semi-stable equilibrium) $\gamma<0$.
3. attracting equilibrium: $\beta<0$ and $\gamma>0$.
4. expanding spiral: $\beta>0$ and $\beta^2<4\gamma$.
5. periodic solutions around the equilibrium: $\beta=0$ and $\beta^2<4\gamma$.
6. shrinking spiral: $\beta<0$ and $\beta^2<4\gamma$.