The engine power pulses at higher RPM and sometimes the engine stalls after starting. The problem is intermittent and seems like it might be fuel pump related.

With intermittent issues, have to test when problem is ongoing. How long since a tune-up?
If it cranks good but won't start, have a helper crank it while you check for spark at the plugs, if spark everywhere, use a gage and check proper fuel pressure. Also, check fuel injector pulse. If your missing all three, check for rpm signal, comes from crank sensor.
If you can narrow it down to one system, we can test that one system. Any applicable trouble codes?

The Car comes to an stall when accelerating,pressing the gas pedal ...

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Feb 7, 2007 - Sometimes when I press the gas pedal seems like nothing happens car doesn't accelerate at all only it loses the power and finally comes to a Stop, and sometimes it recovers (shakily) and keeps on truckin'. ... Does the engine shake andstall when you put the transmission into ... car stalls if gas pedal not pressed.?
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I tend to agree more with the last suggestion it seems like fuel is the problem. Would do the fuel filter and if no change check fuel pump pressure. If all checks out then would start looking at power (electrical).

I had the exact problem with my 1993 chevy c1500. I found that the IAC was sticking so the engine would die at idle after warming up and the ignition module was pulsing the ECM intermittently after the engine warmed up so the fuel pump wasn't being turned on after the engine died....I replaced the ignition module and the IAC valve and all is good. I don't know if testing the ignition module reveals that it isn't sending pulses to the ECM.

Check fuel pressure test and air intake system.

Hard to start and stall at higher RPM. So you are stalling at load. What the engine needs at higher RPM and load is more fuel.The fuel pressure at idle should be at 30 psi to 45 psi. A plugged fuel filter will also do this. so you need to address the fuel pump and filter

I'd have to test when problem is ongoing. When you first turn on key, no crank, you should hear the pump run for a few seconds then stop. For initial prime, the computer doesn't need rpm signal. Engine cranking, the computer needs to see rpm signal or it won't ground the fuel pump relay. W/o rpm signal, no spark at the plugs and no fuel injector pulse. I know your talking about fuel, but, when problem is ongoing, check spark and fuel injector pulse, engine cranking.
Intermittent problems are a pain, have to test when problem is ongoing.

DTC P1599 Engine Stall or Near Stall Detected

Circuit Description
The diagnostic test checks for an engine stall or near stall and sets DTC P1599 if a stall condition is detected. A partial start where engine RPM exceeds 400 RPM and the engine does not start, will also cause DTC P1599 to set. This DTC is designed to capture snapshot data of the stall condition.

Since many things can cause a stalling condition use the snapshot taken of DTC P1599 to compare the values taken when the DTC set to the Typical Scan Data Value table. Look for conditions that could have caused the stall and how long ago the stall occurred.

3
Are any other DTCs set?
--
YES - Go to DTCs that are set
NO - Go to Step 4
4
Using a scan tool locate the Stall Data. This is the snapshot taken when the vehicle stalled and is not located with the Failure Records.
Check the MAP sensor display taken at the time this DTC was set.
Is the display more than the value specified?
95 kPa
YES - Go to Step 5
NO - Refer to Hesitation, Sag, Stumble diagnosis chart
5
One or more of the following systems may have caused the stalling condition.
• Fuel System.
• Idle Speed Control.
• Emissions System.
• Wiring and connections.
Perform Fuel System Check chart, check Test Descriptions.
Was a fault found and repaired?
--
YES - Go to Powertrain Control Module Diagnosis for Verify Repair
NO - Go to Step 6
6
Check for TCC improperly applied by a grounded TCC solenoid circuit or by transaxle hydraulic problems. Refer to Section 7 to diagnose.
Check ignition switch circuits for intermittent open to the PCM or ignition control module. Manipulate the key switch while driving to check for ignition switch intermittent or out of adjustment.
Check PCM ground and ignition control module ground.
Check for intermittent injector circuit operation.
Check the fuel PUMP ground for a loose or poor connection.
Did any of the checks above require a repair?
--
Go to Powertrain Control Module Diagnosis for Verify Repair
Refer to Diagnostic Aids

That code is indicating an intermittant crank shaft or cam shaft sensor fault. Because the engine is starting then stalling, check to see if you have spark after the stall. If spark is still present, you are probobley looking at a cam sensor circuit issue. The cam sensor controls the fuel injector pulse. If the cam sensor fails on a chrysler product, the fuel injectors will not function.

check fuel pressure sounds like ur fuel pump may be going out