B757 AND DC10 ARE BOTH DEVIATING TO AVOID CONVECTIVE ACTIVITY. LTSS RESULTS WHEN THE CREWS WERE SLOW ON RESPONDING TO CTLR'S INSTRUCTIONS.

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I WAS WORKING THE RADAR CTLR POS OF A VERY SATURATED; EXTREMELY COMPLEX SECTOR; WITH EVERY ACFT DEVIATING AND NUMEROUS TFC SITS. I HAD A HDOF/COORDINATOR POS; AND AN ASSISTANT RADAR CTLR. ACR XXX (B757) CHKED ON FREQ DEVIATING AT FL370. I ASSIGNED A 100 DEG HDG FOR TFC; APPROX HIS PRESENT HDG. I WAS SURE THIS HDG WAS GOOD AROUND THE WX BECAUSE I HAD SEVERAL ACFT GOING THROUGH THE HOLE AT THE SAME PLACE -- IN FACT; ANOTHER ACFT AT FL370; 35 MI STRAIGHT AHEAD. THE PLT SAID HE WAS UNABLE DUE TO WX. I THEN WENT TO PLAN B; WHICH WAS TO MOVE THE DC10 HEADED OPPOSITE DIRECTION AT FL370. THE DC10 WAS THERE BECAUSE OF A 3-WAY TIE AT FL390 (HIS ORIGINAL ALT) AND TFC BELOW AT FL350; WHICH HE WAS JUST CLRING BY 5 MI. I ISSUED THE DSCNT CLRNC AND THE PLT ACKNOWLEDGED. A COUPLE MINS LATER; THE DC10 STILL SHOWED LEVEL AT FL370. AT THIS POINT; THE SIT WAS BECOMING IMMINENT; AND I ISSUED A TFC ALERT WITH AN 'IMMEDIATE' DSCNT CLRNC; AND ONCE AGAIN WENT BACK TO ACR XXX AND ISSUED A TFC ALERT WITH A R TURN TO 100 DEGS. BOTH PLTS ACKNOWLEDGED. APPROX 1 MIN LATER; THE DC10 STILL SHOWED LEVEL AT FL370 AND THE CONFLICT ALERT WAS FLASHING WITH BOTH PLANES HEAD-ON. I ISSUED A R TURN OF APPROX 40 DEGS; AND RESTATED MY TFC ALERT PHRASEOLOGY WITH ADDED EMPHASIS. THIS TIME AS THE PLTS ACKNOWLEDGED; I COULD HEAR THE TCASII IN THE BACKGND GOING OFF. FINALLY; THE DC10 DSNDED AND TURNED; AND THE B757 APPEARED TO TURN ALSO. AS AN ADDED NOTE; THE BELLIGERENT PLT OF THE B757 PROCEEDED TO BADGER ME REPEATEDLY ON THE FREQ ABOUT 'MY CAUSING HIM TO HAVE A TCASII RA;' AND DEMANDING TO KNOW WHAT THAT WAS ALL ABOUT. HE PROCEEDED TO BLOCK SEVERAL OF MY XMISSIONS AND THE XMISSIONS OF OTHER ACFT ON FREQ DURING A SATURATED SESSION WITH ABOUT 35-40 TARGETS. LATER; BOTH PLTS CALLED THE SUPVR; AT HIS INSTRUCTION. APPARENTLY; THE DC10 AUTOPLT HAD 'DISENGAGED WITHOUT HIS KNOWLEDGE' AND HE DIALED IN THE ALT OF FL350 AND NOTHING HAPPENED. ON MY THIRD CALL TO HIM; HE SAID HE REALIZED WHAT HAD HAPPENED AND PUSHED THE NOSE DOWN MANUALLY INTO A 6000 FPM DSCNT AND TOOK MY TURN AT A 45 DEG BANK ANGLE. THE PLT OF THE B757 ADMITTED THAT HE DID NOT TAKE MY TURN (EVEN THOUGH HE ACKNOWLEDGED THE VECTOR) UNTIL HE GOT A TCASII RA TELLING HIM TO DO SO. HE ARGUED WITH THE SUPVR ABOUT HIS INVOLVEMENT IN THE INCIDENT; BUT FINALLY CONCEDED THAT THE ACTION I TOOK 'PROBABLY SAVED THE DAY.'