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We present a framework for studying generic behaviors possible in the interaction between a resource-harvesting technological civilization (an exo-civilization) and the planetary environment in which it evolves. Using methods from dynamical systems theory, we introduce and analyze a suite of simple equations modeling a population which consumes resources for the purpose of running a technological civilization and the feedback those resources drive on the state of the host planet. The feedbacks can drive the planet away from the initial state the civilization originated in and into domains that are detrimental to its sustainability. Our models conceptualize the problem primarily in terms of feedbacks from the resource use onto the coupled planetary systems. In addition, we also model the population growth advantages gained via the harvesting of these resources. We present three models of increasing complexity: (1) Civilization-planetary interaction with a single resource; (2) Civilization-planetary interaction with two resources each of which has a different level of planetary system feedback; (3) Civilization-planetary interaction with two resources and nonlinear planetary feedback (i.e., runaways). All three models show distinct classes of exo-civilization trajectories. We find smooth entries into long-term, "sustainable" steady states. We also find population booms followed by various levels of "die-off." Finally, we also observe rapid "collapse" trajectories for which the population approaches n = 0. Our results are part of a program for developing an "Astrobiology of the Anthropocene" in which questions of sustainability, centered on the coupled Earth-system, can be seen in their proper astronomical/planetary context. We conclude by discussing the implications of our results for both the coupled Earth system and for the consideration of exo-civilizations across cosmic history. Key Words: Anthropocene-Astrobiology-Civilization-Dynamical system theory-Exoplanets-Population dynamics. Astrobiology 18, 503-518.