Journal of Advances in Mathematics and Computer Science

Motivation: Temporal delays and response saturation fundamentally shape system dynamics in computational networks, biological systems, and distributed algorithms, yet their combined effects on global convergence remain incompletely understood despite extensive separate study of each phenomenon.

Aims: To analyze four-compartment dynamical systems combining delay distributions with saturating nonlinearities, establishing threshold conditions for convergence to equilibrium states and providing quantitative design criteria for practical systems.

Study Design: Mathematical analysis integrating spectral methods, Lyapunov functional techniques, sensitivity analysis, and computational simulations with quantitative comparisons to existing approaches.

Methodology: The study formulates functional differential equations with gamma-distributed delay kernels and saturating transfer functions. Spectral analysis identifies a threshold parameter R0 governing system behavior. Lyapunov techniques establish convergence properties for subcritical and supercritical regimes, while numerical simulations explore parameter sensitivity and transient dynamics across different kernel distributions. Formal sensitivity analysis identifies critical parameters affecting threshold dynamics Findings: The threshold parameter R₀ characterizes long-term dynamics: when R₀ ≤ 1, solutions converge to trivial equilibrium; when R₀ > 1, a unique nontrivial equilibrium exists and attracts solutions from the interior of the feasible region. Kernel variance modulates transient dynamics, while saturation stabilizes the system by preventing oscillatory instabilities common in purely linear models. The survival probability φ during transitions influences the threshold throughR₀ \(\frac{βΛφ}{μ(μ+γ)}\), Sensitivity analysis reveals transfer coefficient β and survival probability φ as most influential parameters. Quantitative comparison shows 23% faster convergence and 15% larger stability regions versus models treating delays or saturation separately.

Conclusion: Delay distributions and saturating nonlinearities jointly determine system outcomes through a single threshold parameter, with unified Lyapunov framework enabling global convergence analysis impossible with existing separate approaches. This framework applies to computational task processing pipelines (MapReduce, gradient descent), networked control systems, and epidemic models where heterogeneous processing times and resource limitations jointly govern dynamics. The techniques extend to broader classes of functional differential equations with nonlinear coupling, with identified extensions to stochastic and spatially heterogeneous systems.