Depending on the problem definition, the eigenvalues might for example tell you whether the model is stable/unstable, a systems eigenfrequencies, preferred directions of growth or similar.
From my experience , to solve any biological system which contains a set of differential equations , you first have to find the singular points of the system since the type of the diff. eqn. will be nonlinear we have to examine the solutions of the corresponding linear system near each singular point. Then you might be using the Jacobian matrix to get the eigenvalues easier , and depending on the eigenvalues you find the corresponding eigenvectors. So they are related in this form : A (matrix) *v(scalar) = v(scalar) * lambda (eigenvalues).