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Cognitive radio (CR) technology is crucial for enabling dynamic spectrum management. CR allows opportunistic spectrum sharing with licensed primary radio (PR) users by allowing the so-called secondary users to dynamically access the under-utilized parts of the frequency bands that are allocated/licensed to the PR networks (PRNs). The CR system pays a utilization-dependent price to PRNs for utilizing the unused spectrum. An essential challenge in this domain is how to characterize the economic implications of spectrum sharing between the CR system and PRNs. Specifically, we consider a CR system with two competing CR service providers. This market model is known as the "duopoly model," a market structure premise that consists of two companies providing the same type of service. For such a model, we investigate the problem of maximizing the overall achieved profit in the CR system. Specifically, we mathematically formulate the profit-maximization spectrum assignment problem for the two CR providers subject to spectrum sharing, power distribution, spectrum pricing, and quality of service constraints. We demonstrate that this optimization problem is an NP-hard binary nonlinear programming problem. Therefore, we adopt a meta-heuristic optimization method based on Antlion Optimization to solve the problem suboptimally. Simulation results revealed that our proposed profit-aware optimization significantly outperforms traditional CR-based spectrum access mechanisms. [ABSTRACT FROM AUTHOR]