Among the great apes, chimpanzees and gorillas live in very hierarchical, male-dominated clans that are often in violent conflict with other clans. Bonobos, on the other hand, lead very peaceful lives, and are female-dominated, using sexual contact as a manner of communication to reduce tension within and between groups. Orangutans are largely solitary animals, with heavy parental investment (7 years, longest among non-human primates), wherein the mother lives separately with her child during the early years of development. Among the lesser apes, gibbons and siamangs are also highly territorial and pair-bonded, though notably, polyandrous siamang groups have been shown to display higher degrees of cooperation.
Though Anarchism varies greatly between different definitions, the most popular form of contemporary Anarchism values non-hierarchical organization, centered around mutual aid, rather than the kind of isolated individualism that you seem to associate with the term. By this definition, bonobos would be more "Anarchist" than chimpanzees. By your definition, orangutans might be the most "Anarchist."
Your definition of "Communism" is closer to the contemporary definition of Anarchism above, due mainly to the historical trajectory of Communism, which in practice, did not work out so well, leaving contemporary utopian thinkers largely in the Anarchist camp. By your definition of "Communism," bonobos may be the most "Communist"; however, if you mean the state-based gulag Communism of popular culture (ie portrayals of the Soviet Union), then you might say that chimps were more Communist. This is because individual chimps sacrifice individual goal attainment for group ends under the coercion of the alpha male chimp. Of course, some people may call that Fascism.
You may want to take a look at primatologist Franz De Waal's research on bonobos vs. chimpanzees. As a pacifist, he has a bit of an agenda, in terms of trying to prove that female-dominated bonobo "societies" are inherently more cooperative and less hierarchical. Another primatologist that agrees with this view is Harvard primatologist Richard Wrangham, who looks at hierarchical organization and violent behaviors in chimpanzees.
Robert Sapolsky's research on baboons may also be of interest to you. He found a clan of baboons, in which all the alpha males had died from tuberculosis due to exposure to an infected meat found in a neighboring human village. He noted, then, that the remaining baboons lived much more peaceably and cooperatively than other observed clans, and furthermore, that this clan retained its "culture" and passed on its more cooperative ways of behaving to new members in the clan, therefore demonstrating that perhaps baboons (and other primates) are capable of having and passing on "culture." - Perhaps a more "Anarchist" culture than nature normally prescribes.
However, as a caveat, I must say: it is tenuous to make direct associations between ape behavior and human societies. Human culture, social intelligence, and symbolic reasoning facilitate human political systems. Though Sapolsky's work suggests primates may also be capable of some form of "culture," biological descriptions of ape social organization is insufficient to explain human behavior.