  In this paper, we prove a uniqueness result in the inverse problem of
determining several non-constant coefficients of one-dimensional
reaction-diffusion equations. Such reaction-diffusion equations include the
classical model of Kolmogorov, Petrovsky and Piskunov as well as more
sophisticated models from biology. When the reaction term contains an unknown
polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our
result gives a sufficient condition for the uniqueness of the determination of
this polynomial part. This sufficient condition only involves pointwise
measurements of the solution $u$ of the reaction-diffusion equation and of its
spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a
time interval $(0,\epsilon).$ In addition to this uniqueness result, we give
several counter-examples to uniqueness, which emphasize the optimality of our
assumptions. Finally, in the particular cases N=2 and $N=3,$ we show that such
pointwise measurements can allow an efficient numerical determination of the
unknown polynomial reaction term.
