  We consider a certain lattice branching random walk with on-site competition
and in an environment which is heterogeneous at a macroscopic scale
$1/\varepsilon$ in space and time. This can be seen as a model for the spatial
dynamics of a biological population in a habitat which is heterogeneous at a
large scale (mountains, temperature or precipitation gradient...). The model
incorporates another parameter, $K$, which is a measure of the local population
density. We study the model in the limit when first $\varepsilon\to 0$ and then
$K\to\infty$. In this asymptotic regime, we show that the rescaled position of
the front as a function of time converges to the solution of an explicit ODE.
We further discuss the relation with another popular model of population
dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$.
Combined with known results on the Fisher-KPP equation, our results show in
particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in
general. We conjecture that an interpolating regime appears when $\log K$ and
$1/\varepsilon$ are of the same order.
