2001 Honda Odyssey

My 2001 Honda Odyssey (194,000 miles on it) has begun a strange kind of intermittent stalling. Only happens when starting the car (not while driving). Engine will turn on, then it seems to hunger for fuel (mild vibrations), threatens to cut out and sometimes does. When it cuts out, the dash lights all come on. Trying again immediately is usually a failure. Successful turnover seems to require a minute or two of rest. Today, in order to keep it from cutting out again, my wife had to start it, then keep her foot on the gas several seconds. The third time she tried it, it stayed on.