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In order to understand how a heterogeneous habitat affects the population dynamics of the predator-prey system, a spatially explicit lattice model consisting of predators, prey and obstacles is constructed. The model includes smart pursuit (predators to prey) and evasion (prey from predators). Both species can affect their movement by visual perception within their finite sighting range. Non-conservative processes that change the number of individuals within the population, such as breeding and physiological dying, are implemented in the model. Obstacles are represented by non-overlapping lattice shapes that are randomly placed on the lattice. In the absence of obstacles, numerical simulations reveal regular, coherent oscillations with a nearly constant predator-prey phase difference. Numerical simulations have shown that changing the probabilities for non-conservative processes can increase or decrease the period of coherent oscillations in species abundances and change the relative lag between coherent components. After introducing obstacles into the model, we observe random transitions between coherent and non-coherent oscillating regimes. In the non-coherent regime, predator and prey abundances continue to oscillate, but without a well-defined phase relationship. Our model suggests that stochasticity introduced by density fluctuations of obstacles is responsible for the reversible shift from coherent to non-coherent oscillations. © 2023 IOP Publishing Ltd and SISSA Medialab srl.