Existence, Uniqueness, and Input-to-State Stability of Ground State Stationary Strong Solution of a Single-Species Model via Mountain Pass Lemma

In this study, the authors utilize mountain pass lemma, variational methods, regularization technique, and Lyapunov function method to derive unique existence of positive classical stationary solution a single-species ecosystem. Particularly, geometric characteristic saddle point in lemma guarantees that equilibrium is ground state Based on obtained uniqueness result, use globally exponential stability criterion, which illuminates under some suitable conditions, certain internal competition conducive global population, amount family planning overall population. Most notably, regularity technique weak employed study can also be applied existing literature related with time-delays reaction-diffusion systems for purpose solutions. Finally, an illuminative numerical example shows effectiveness proposed methods.