Mathematical Models of Gene Regulation, Predator–Prey Cycles, and Epidemics Reveal Deterministic–Stochastic Dynamics, Bistability, and Outbreak Variability

By striking a compromise between biological interpretability and analytical tractability, minimal mathematical models are effective instruments for clarifying the fundamental ideas underlying complicated biological systems. In order to study three canonical classes of biological dynamics (i) a gene regulatory toggle switch, (ii) predator prey ecological interactions, and (iii) infectious disease propagation in a host population. We present a unified computational framework in this work that combines deterministic and stochastic approaches. We guarantee that every model is not only immediately comparable across domains but also completely reusable for independent verification and extension by using a standardized simulation environment with reproducibility controls, automated parameter logging, and raw output archiving. We map bistable regimes and detect hysteresis like transitions for the gene regulatory toggle switch using equilibrium analysis, parameter sweeps, and ordinary differential equation (ODE) integration. By combining Jacobian based eigenvalue classification with numerical root identification, stable and unstable equilibria may be explicitly characterized, offering mechanistic insight into switching behavior. Noise-induced state transitions are further revealed by stochastic simulations using the Gillespie method and chemical Langevin approximations, emphasizing circumstances in which intrinsic fluctuations cause deterministic attractors to become unstable. Both phase-space visualization and time domain integration are used to evaluate the predator prey subsystem, which is represented using canonical Lotka Volterra dynamics. Analytically determined nullclines combined with simplified phase pictures show parameter dependent stability bounds, extinction trajectories, and oscillatory coexistence. This investigation demonstrates how simple ecological models can be modified to evaluate seasonal forcing or environmental perturbations and capture important aspects of trophic relationships. In order to investigate epidemic variability in small populations, we apply the susceptible infected recovered (SIR) model in both deterministic ODE and stochastic Gillespie formulations in the epidemiological domain. In order to calculate peak infection distributions and measure outbreak size variability, we compare deterministic epidemic curves with massive ensembles of stochastic realizations. According to this comparison, stochastic influences bring significant dispersion, particularly in small populations or close to epidemic thresholds, whereas deterministic predictions reflect mean-field trends. All together, our results show that minimal models, analyzed from both deterministic and stochastic viewpoints, are interpretable and accurately reproduce characteristic events in molecular, ecological, and epidemiological systems. Without the need for high dimensional or data intensive models, the underlying mathematical structure spontaneously reveals oscillatory dynamics in predator-prey interactions, bistability and hysteresis in gene regulation, and unpredictability in epidemic outcomes. In addition to facilitating quick hypothesis development and parameter sensitivity analysis, the shared computational framework described here encourages methodological uniformity across many biological fields. This work provides the systems biology, ecology, and epidemiology communities with a conceptual synthesis as well as a useful toolkit by linking molecular to population-level scales in a single reproducible setting.