Men normally pursue women. It is like this in all species. Men hunt, women are "caught".
This belief was taught to you. Can you guess by whom? (It was probably one or both of your parents.) It is an idea that they gave you about gender -concerning your "object" orientation rather than "self" direction. "Self" directed thinking utilizes free will and decision making. Being hunted and "caught" romantically would not happen.
Many cultures still disallow "self" directed thought for women- but if you are living in the Western World, the thought that a woman is caught by a Man is now somewhat outdated unless it continues to be taught in the family's belief system. Women who are raised with this type of belief become encouraged by girlishness and are what the Greeks call a "Kore;" the eternal girl.
In Greek mythology she was represented by the Goddess Persephone, who presented herself as eternally youthful in a way that caused her not to act- but to be acted upon by others. Hence, she is known as a kidnap victim who gets "caught" by Men. Most women go through this stage in adolescence, but learn to grow into themselves and make and *keep* commitments to the Self. Persephone represents a time when a young woman waits for someone or something to come along and shape her life before any other possibility occurs to her that she can do this herself. She may have something very girlish in her personality that translates to outsiders as a “take care of me” element that allows others to make decisions for her rather than to make decisions for herself. Consequently she attracts people who wish to control her.
In my other relationships, whenever I, as a woman, ended up in a "pursuit" mode or "crazy about" someone, I would normally be in one down position and this r/s would not go anywhere.
Since object relations is a theory- there is no difference in the way that gender plays a part. However, Men who *do not* allow themselves to be objectified may cause shame to a woman who projects objectification. It's not the Woman who is at fault, it's the belief. If objectification is all she understands about her value, then she also values others this way. In doing so, she may push away any meaningful ability to be seen as her real "self." The ability to step outside of the projective identification with being objectified may be necessary to see a pattern in relationships and begin a new autonomy that doesn't rely on gender roles or their objectification.