  We study the long-time behavior of some McKean-Vlasov stochastic differential
equations used to model the evolution of large populations of interacting
agents. We give conditions ensuring the local stability of an invariant
probability measure. Lions derivatives are used in a novel way to obtain our
stability criteria. We obtain results for non-local McKean-Vlasov equations on
$\mathbb{R}^d$ and for McKean-Vlasov equations on the torus where the
interaction kernel is given by a convolution. On $\mathbb{R}^d$, we prove that
the location of the roots of an analytic function determines the stability. On
the torus, our stability criterion involves the Fourier coefficients of the
interaction kernel. In both cases, we prove the convergence in the Wasserstein
metric $W_1$ with an exponential rate of convergence.
