  We investigate in this paper a scalar reaction diffusion equation with a
nonlinear reaction term depending on x-ct. Here, c is a prescribed parameter
modelling the speed of climate change and we wonder whether a population will
survive or not, that is, we want to determine the large-time behaviour of the
associated solution. This problem has been solved recently when the
nonlinearity is of KPP type. We consider in the present paper general reaction
terms, that are only assumed to be negative at infinity. Using a variational
approach, we construct two thresholds determining the existence and the
non-existence of travelling waves. Numerics support the conjecture that the two
thresholds are equal. We then prove that any solution of the initial-value
problem converges at large times, either to 0 or to a travelling wave. In the
case of bistable nonlinearities, where the steady state 0 is assumed to be
stable, our results lead to constrasting phenomena with respect to the KPP
framework. Lastly, we illustrate our results and discuss several open questions
through numerics.
