In various real-world scenarios, such as autonomous navigation, robotics, or watercraft control, the ability to navigate efficiently and avoid obstacles is crucial. By simulating avoidance strategies in a one-dimensional vector field of moving fluids, this research addresses challenges associated with fluid dynamics, providing insights into optimal trajectory planning for mobile objects. This knowledge is applicable to fields like autonomous vehicle navigation, where understanding how vehicles can navigate through fluid environments is vital for enhancing safety and efficiency. The study has broader implications for industries involving fluid-based systems, contributing to advancements in control mechanisms and avoidance strategies for mobile objects in dynamic fluid environments. The research investigates the pursuit process on a horizontal plane as one boat pursues another in a moving fluid field. The research focuses on developing a mathematical model and exploring the paradoxical characteristics inherent in the pursuit dynamics. The primary objective is to ascertain the trajectory for intercepting a fleeing boat by the pursuing boat amid the fluid motion. The mathematical model, comprising four first-order ordinary differential equations, considers the river's speed, creating a one-dimensional fluid field. The pursuit strategy involves the pursuer aligning with an instantaneous line connecting them and the fugitive, while the latter aims to cross the river swiftly toward a “lifeline”. Numerical solutions, implemented in the MathCad software, unveil the pursuit curve, revealing intriguing characteristics and paradoxical phenomena. The study identifies dependencies, including the relationship between the fugitive's horizontal movement towards the “lifeline” and the angular coefficient of its escape line in still water, pinpointing a well-defined local maximum. Furthermore, numerical analysis highlights a critical threshold for the speed ratio between pursuer and fugitive, impacting the pursuer's ability to detain the fleeing boat within a specified lane before it crosses the “lifeline”. The research on pursuit dynamics has potential applications in a wide range of practical problems involving navigation, interception, and avoidance in fluid environments, spanning from maritime operations to autonomous systems and environmental monitoring. Enhancing the effectiveness of pursuit dynamics can be crucial in search and rescue missions, where boats or autonomous vehicles need to intercept or follow a target in challenging fluid environments. Implementing optimized pursuit strategies is relevant for autonomous vehicles navigating water bodies. Understanding the dynamics of pursuing or evading objects is essential for designing efficient and safe navigation systems.