Recreational fisheries, often recognized as fishing activities undertaken by individuals for sport and leisure, can be conceptualized as integrated systems where changes in human activities have significant implications for the interconnected components of the fishery ecosystems. These systems often exhibit nonlinear dynamics and are susceptible to abrupt and irreversible shifts in function and structure, referred to as critical transitions. Most existing studies on critical transitions in fishery ecosystems focus on the magnitude of changes in fishing practices. In this study, we examine a recreational fishery model that incorporates the prey and its predators, both of which are subject to fishing by anglers. Our analysis demonstrates that the fishery ecosystem is sensitive not only to the magnitude of changes in fishing pressure but also to the rate at which these changes are implemented. Surprisingly, a slow increase in fishing pressure, even when aiming for a relatively high fishing target, can sustain the prey fish stock at a sustainable biomass level. Additionally, we identify critical thresholds for the rate of increase in fishing pressure for different initial population sizes and accordingly assign warning levels to these values. Furthermore, to evaluate the risk of rate-induced tipping, we introduce two key concepts: tipping time and rate-tipping probability, where the latter is defined as the proportion of initial conditions that induce rate-tipping relative to the total number of selected initial conditions. Interestingly, we find that as the rate of parameter change increases, the tipping probability first increases rapidly, then slowly, and finally saturates, while tipping time decreases gradually, that is, the increase of $\lambda$ accelerates the collapse of prey population. Our study offers valuable insights for fisheries management decision-making, not only emphasizing the importance of considering the magnitude of management adjustments but also highlighting the temporal dynamics of implementing these changes to achieve desired outcomes.