Jacobian matrices
The Jacobian matrix collects the first partial derivatives of a multidimensional dynamical system.
Two-variable system
If
\[\frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y),\]then
\[J(x,y)=\begin{pmatrix}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}\\[4pt]\frac{\partial g}{\partial x}&\frac{\partial g}{\partial y}\end{pmatrix}.\]Evaluation at an equilibrium
At an equilibrium \((x^*,y^*)\), the matrix \(J(x^*,y^*)\) describes how small changes in the state alter the local rates of change.
Biological interpretation
Each entry measures the local sensitivity of one equation to one state variable. Off-diagonal terms often reflect interactions between biological components.
Key idea. The Jacobian is the local derivative of the full vector field and is the central matrix used in linear stability analysis.