We consider double-diffusive convection between two parallel plates and compute bounds on the flux of the unstably stratified species using the background method. The bound on the heat flux for Rayleigh-Bénard convection also serves as a bound on the double-diffusive problem (with the thermal Rayleigh number equal to that of the unstably stratified component). In order to incorporate a dependence of the bound on the stably stratified component, an additional constraint must be included, like that used by Joseph (Stability of Fluid Motion, 1976, Springer) to improve the energy stability analysis of this system. Our bound extends Joseph's result beyond his energy stability boundary. At large Rayleigh number, the bound is found to behave like R

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A distinctive transition in reaction-diffusion systems is the creation of travelling fronts from stationary fronts in a pitchfork bifurcation. We explore how this bifurcation is modified when the systems are made spatially discrete. We consider two model systems: a chain of coupled Lorenz equations, and a discretized Fitz-Hugh-Nagumo model. In the former, the pitchfork bifurcation of the corresponding continuum model is replaced by a supercritical Hopf bifurcation to a pulsating front that is, on average, stationary, which is then followed by a heteroclinic bifurcation that glues the pulsation cycles together into an unsteadily propagating front. In the second model, the Hopf bifurcation is subcritical, the heteroclinic bifurcation glues together unstable pulsation cycles, and there is a saddle-node bifurcation in which the unstable moving front turns around into a stable one. In the vicinity of the discrete version of the bifurcation we derive an amplitude equation that captures qualitatively the aspects of the two different bifurcation sequences. However, the amplitude equation is quantitatively in error, apparently as a result of the beyond-all-orders nature of the effect of discreteness.