Modeling Prey–Predator Populations With Noise Following the Extended Gaussian Distribution

This study examines how tourism influences the ecological balance of a protected natural park where two interacting wildlife species follow Lotka–Volterra-type prey–predator dynamics. Tourists’ decisions to visit the park depend on environmental fluctuations, species visibility, and time-varying preferences toward prey and predator populations. To capture these complex interactions, we develop a coupled deterministic–stochastic model that includes seasonal variations in tourist carrying capacity and attraction rates and models’ environmental noise using the Beta-Exponential-Gaussian (BExG) probability distribution by developing a local R package called BExGaussian for the first time, which accounts for skewed and heavy-tailed disturbances. Qualitative analysis confirms the model’s well-posedness, positivity, existence, and boundedness and identifies a bifurcation at a net tourist effect of η,γ=0.5, separating destabilizing from coexistence-promoting regimes. The numerical simulations using MATLAB and the Euler–Maruyama scheme show that deterministic thresholds accurately predict persistence boundaries under stochastic forcing and align with the analytical results clearly. The ex-Gaussian noise induces extreme oscillations useful for stress testing, while BExG noise produces realistic, moderate variability. Sensitivity analysis highlights predator recovery rate, conversion efficiency, probability distribution types, tourist net effect, and tourist turnover as key factors sustaining persistence, whereas strong predation and negative tourist effects increase extinction risk. Ecologically, the results suggest that maintaining a neutral-to-positive tourist effect η,γ>0.5 and promoting moderate tourist departure rates can stabilize population cycles, balancing ecological resilience with tourism-driven socioeconomic benefits.