  The observations of radio emission from SNR~1987A can be accounted for on the
basis of diffusive shock acceleration of electrons by the supernova blast wave.
However, with this interpretation the observed spectral index implies that the
compression ratio of the gas subshock is roughly $2.7$ rather than the value of
$4$ expected of a strong shock front. We propose that in SNR~1987A, ions also
undergo diffusive acceleration at the shock, a process that is likely to be
rapid. Unlike the electron population, the accelerated ions can have an
important effect on the gas dynamics. We calculate this coupled gas and
energetic particle dynamics on the basis of the two-fluid model, in which the
accelerated ions provide an additional component to the total pressure acting
on the fluid. By accelerating and possibly heating the upstream plasma, the
initially strong shock is modified and a weaker subshock with an upstream
precursor results. The electrons behave as test particles. They are accelerated
at the evolving subshock, escape downstream, and emit synchrotron radiation in
the swept up magnetic field. Two models are considered for the surroundings of
the progenitor: that of a freely expanding wind of number density $n\propto
r^{-2}$, and that of a wind confined by a shell of denser material, creating a
stagnation zone of roughly constant density beyond the standing shock which
terminates the free wind. We model the observed radio light curves and the
relatively steep spectrum of SNR~1987A using similar values for the ion
acceleration parameters to those used in models of cosmic ray acceleration in
older SNRs which can also contain high Mach number shocks, and find a good fit
for the case in which the termination shock is located at about $2\times
10^{15}\,$m from the progenitor.
