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If our actions determine our next body, how do souls in animal bodies, performing totally animalistic activities, evolve to higher species?

Animals don't generate Karma. They simply "evolve" through various species in a predetermined order until they reach the human form of life.

Although they don't generate good and/or bad karma they do benefit, however, from eating prasadam (sanctified food), touching the water from the lotus feet of the Lord or His devotees and hearing the Holy Name. So, by putting other living beings in contact with these very potent items of bhakti even lower species are relieved of karmic reaction and sometimes take a human birth immediately, skipping the general progression of their evolution to the human form of life.

I hope this is helpful.
Your servant,
Laxmimoni dasi