  This paper is devoted to the analysis of some uniqueness properties of a
classical reaction-diffusion equation of Fisher-KPP type, coming from
population dynamics in heterogeneous environments. We work in a one-dimensional
interval $(a,b)$ and we assume a nonlinear term of the form $u \,
(\mu(x)-\gamma u)$ where $\mu$ belongs to a fixed subset of $C^{0}([a,b])$. We
prove that the knowledge of $u$ at $t=0$ and of $u$, $u_x$ at a single point
$x_0$ and for small times $t\in (0,\varepsilon)$ is sufficient to completely
determine the couple $(u(t,x),\mu(x))$ provided $\gamma$ is known.
Additionally, if $u_{xx}(t,x_0)$ is also measured for $t\in (0,\varepsilon)$,
the triplet $(u(t,x),\mu(x),\gamma)$ is also completely determined. Those
analytical results are completed with numerical simulations which show that, in
practice, measurements of $u$ and $u_x$ at a single point $x_0$ (and for $t\in
(0,\varepsilon)$) are sufficient to obtain a good approximation of the
coefficient $\mu(x).$ These numerical simulations also show that the
measurement of the derivative $u_x$ is essential in order to accurately
determine $\mu(x)$.
