Background Irresponsible dog ownership in urban areas is a public health concern with significant implications for human, animal, and environmental welfare. Factors such as abandonment, variations in adoption, insufficient supervision, emerging identification initiatives, and collective feeding impact the growth of stray dog populations and the transmission of diseases. Developing a modeling tool to understand the dynamics of canine population growth and the effect of human behavior on this phenomenon is essential. Methods An ordinary differential equation model was developed to depict the growth dynamics and movements of urban dog populations, distinguishing between those with owners (restricted and semi-restricted) and those without (stray and community dogs). Two equilibrium states of the system were analyzed: with and without the presence of individually owned dogs. An increase rate for the population of individually owned dogs was calculated, and a local sensitivity analysis was conducted to assess the impact of parameters on the reduction of this population. Additionally, two global sensitivity analysis methods were used to evaluate the simultaneous influence of the parameters. Results Findings indicate that system equilibrium depends on various dog categories. Although total eradication of stray and community dogs is unlikely, equilibrium levels are directly related to subpopulation growth rates, responsible ownership practices, and adoption and abandonment rates. The growth rates of the population of dogs without individual owners have a direct and proportional influence on their regulation, while adoption rates have an inverse and proportional effect. The study, through global sensitivity analysis, identifies key parameters for each dog subpopulation. For restricted dogs, environmental carrying capacity is the most variable factor; for semi-restricted dogs, awareness of responsible ownership is crucial. The abandonment of restricted dogs significantly impacts stray dog dynamics, while the transition from stray to community status is an important variable factor for community dogs. Conclusion Addressing the situation of unowned dogs requires a collective effort to reduce risks associated with the spread of zoonotic diseases, environmental pollution, and biodiversity loss, thus contributing to public health and environmental conservation.