Yes, there are ways of increasing serotonin in the body, though it is less certain they will add any in the brain. Most serotonin, 98%+, is produced and used in the gut, blood vessels/platelets and skin so most, probably all, of a serotonin precursor supplement you take will end up being used by those organs and not the brain.
Moreover, back in the 1980s 33 people died and 1,500 were permanently disabled with Eosinophilia-myalgia syndrome after taking one of these supplements, L-Tryptophan, proving that natural isn't necessarily safer.
However, all this is immaterial because despite what you have have read, fluoxetine (Prozac, Seraphim, Sarafem) and other antidepressants don't work because they raise brain serotonin levels. In fact except in the first few weeks, antidepressants actually cause a reduction in brain serotonin levels. See:
Serotonin: The 'chemical imbalance' myth
If you wish to stop taking antidepressants the only alternative with a proven track record is psychotherapy, specifically one of the cognitive and/or behavioral therapies. Your physician should be able to recommend a therapist in your area.
Be aware that you should not stop taking antidepressants abruptly. When depression medications are discontinued suddenly, sudden worsening of depression, anxiety and flu-like symptoms may occur. While not life-threatening these may be very uncomfortable. These symptoms can be reduced and even eliminated by tapering off the medication over a period of weeks or months. Ask the prescribing doctor for a weaning schedule if you intend to discontinue treatment.
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