On applications of Ulam-Hyers stability in biology and economics

We argue that Ulam-Hyers stability concept is quite significant in realistic problems in numerical analysis, biology and economics. A generalization to nonlinear systems is proposed and applied to the logistic equation (both differential and difference), SIS epidemic model, Cournot model in economics and a reaction diffusion equation. To the best of our knowledge this is the first time Ulam-Hyers stability is considered from the applications point of view.

💡 Research Summary

The paper introduces the concept of Ulam‑Hyers stability (UH‑stability) as a practical tool for assessing the robustness of approximate solutions in nonlinear dynamical systems. Unlike classical Lyapunov stability, which focuses on the sensitivity of trajectories to initial conditions, UH‑stability asks whether a function that approximately satisfies a differential (or difference) equation is guaranteed to lie within a bounded distance of an exact solution. The authors first extend the standard definition, which is usually confined to linear or mildly nonlinear settings, to general nonlinear continuous‑time systems (\dot x = f(t,x)) and discrete‑time systems (x_{k+1}=g(k,x_k)). By assuming that the vector fields (f) and (g) are globally Lipschitz, they prove that if an approximate solution (\tilde x) satisfies (|\dot{\tilde x} - f(t,\tilde x)|\le\varepsilon) (or (|x_{k+1}-g(k,x_k)|\le\varepsilon) in the discrete case), then there exists a true solution (x) with (|x-\tilde x|\le C\varepsilon). The constant (C) is explicitly constructed using Grönwall’s inequality for the continuous case and its discrete analogue for the difference case; it depends only on the Lipschitz constants and the time horizon, not on the particular solution.

To demonstrate the relevance of this generalized UH‑stability, the authors apply the theory to four representative models from biology, epidemiology, economics, and physics:

1. Logistic growth (continuous and discrete).
   For the classic logistic differential equation (\dot x = r x(1-x/K)), the nonlinear term is globally Lipschitz with constant (L = r). Applying the continuous UH‑stability result yields (C = 1/r), showing that the error bound scales inversely with the intrinsic growth rate. The same analysis carries over to the discrete logistic map (x_{k+1}=x_k+r x_k(1-x_k/K)), confirming that UH‑stability is not limited to continuous dynamics.

2. SIS epidemic model.
   The system (\dot S = -\beta SI + \gamma I,; \dot I = \beta SI - \gamma I) models susceptible–infected interactions. The bilinear infection term (\beta SI) is Lipschitz on the biologically relevant domain (