When the engine starts "cold", the PCM initiates a purge cycle, which sucks residual gas from a reservoir connected to the charcoal filter; this is to keep the charcoal element from getting saturated. If you have water condensate in there, if your purge valve or solenoid are defective, or if any of the plumbing in that loop has deteriorated, the engine gets a dose of water, air, or nothing, for starting.
The PCM also looks for sparks before it will enable the fuel pump, so if ignition problems exist, the engine might sort of start with poor ignition, and then quit because the PCM sees ignition not good enough to enable fuel pump.
An odd but very important quirk of Mazdas is they will crank happily with a low battery, but their ignition systems and alternators are so damn fussy that a battery voltage below 12V, while allowing engine cranking, screws up both ignition and alternator excitation, resulting in the PCM deciding not to enable the fuel pump.
So its worth checking that the battery voltage is above 12.5v ignition off, and above 14V with the engine at fast idle.
So the member who commented that the fuel starvation turned out be electrical is right on the money. An acid test would be to do a boost start and see if the fuel starvation symptom still persists. This might be worth trying even before addressing the fuel / Evap systems at all.

 

may want to check the air flow sensor after the air filter box.
got mine dirty(water) and it was doing the same as yours.replaced with a spare i had and she fired up.

ecm see no air flow shuts down the fuel/ignition?

 

I checked the hoses and tested the solenoid valve and all are in good shape. I don't see any reservoir near the charcoal canister (or is it inside it?), anyway all appears to be in good shape, nothing deteriorating. I put on a new set of plug wires yesterday (nothing different), spark looks OK, I'll probably put in new plugs this week. Lastly I put a volt meeter on the battery and was 12.51v with key off & 14.7v with engine running. Seems OK to me. This morning it started right up ran fine for about 15 second and shut right down, almost like someone turned the key off. I attempted to restart it and it turned over for a few times then started up and stayed running (same situation).

 

I pulled the air flow sensor out this morning and did a visual check & looks really clean. I sprayed it out with "Air flow cleaner" I picked up at Auto Zone (since i had it out). I did some test to the sensor also and checked out OK. This makes sense to me seems like the ECM is not seeing air flow and shutting down the fuel pump! I'm thinking it is most likely something in this area or some other sensor.

Thanks for the comments everything I check is one step closer!

 

OK, assuming your chassis & engine grounds are good, your electrical system system is a go.
The reservoir (catch tank) is not near the canister; it's up front, usually on the firewall. If you trace the hoses from the purge valve you'll find one connected to this tank, which has two hose connections, the purge solenoid valve is one, and the other goes via a transition to metal fuel line, back to the charcoal canister located near the fuel tank.
When you go to start the engine from cold, try this trick; before turning to start, go to ignition, count slowly to ten, and then turn the key to crank. Two key players in engine idling are the MAF and the upstream O2 sensor(s); both rely on electrical heating for proper operation, so by giving them a 10 second preheat, they will be close enough to correct operating temperature that they'll be better able to do their idle mixture control job.
Another tip is to restrict cranking to 5 seconds, then wait 10 and go for a restart rather than continuously cranking. This tricks the PCM so that on the second start you won't have to worry about the purge cycle issues since the PCM has already done one, furthermore it allows the battery to recover somewhat from the aborted start, so you get better spark on the second attempt.
I don't know if you've noticed this, but during stubborn starts, it often happens that just when you give up and release the key from the crank position, the engine suddenly springs to life, because getting rid of that 70 or 80 amp. cranking load immediately buys you a couple of volts which is just what the ignition system needed.
Before I sign off, there's another thing worth doing with the season changing, and that's to dump two bottles of gas line deicer into the tank, then fill up to capacity. The alcohol will scavenge all the water vapor condensate you've collected over the summer and help a bit to prevent tank leaks, fuel line leaks, pinhole perforations of the charcoal canister and catch tank etc.
I wish the oil companies would quit blathering on about their "performance enhancing" secret ingredients, and instead wake up and realize that with the advent of closed loop pollution control systems, plus the increasing use of recycled steel in non-stressed automotive components, there is a real need to pay attention to adding corrosion inhibitors, injector cleaners and water scavengers to their gas.

 

is it running rich? might be coolant temp sensor

 

Id suspect the ECM...VERY common issue. I may have a spare.

 

i would think when you start cranking... i know it runs continuously when idling. ( due to me testing one on a 4runner though )

 

OK, Been having problems with the truck randomly running rough (like in 3 cylinders) and not dependable anymore. I had the ECU sent in to be rebuilt and it fixed the problem. The truck starts right up first time and runs great. the computer even threw a trouble code for the abs that has never worked and telling me that the rear sensor is bad. Thanks for all the input. I recommend anyone with the issues I had to spend the $80 and have the ECU rebuilt, with the price of the parts for these trucks its probably the cheapest thing to start with.

 

I have a similar problem with a miata 1990 it will start imeaditaly but dies after 3-5 seconds will restart every time but will not stay running will die imedaitly if i press aceelerator ,yesterday i cleaned air flow sensor started and ran 30 seconds or so probaly from cleaner in system now very hard to start will start only if hold dows gas pedal keep above 1500 rpm exactly oppisate of before car has just had new head ,cams ,plugs and wire never driven since replaced cam marks lined up after 3 tries ca ANYONE help