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In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by using the classical technique of functional analysis. At the end, the results are verified with the help of examples.

In this paper, we study a Leslie–Gower predator-prey model with harvesting effects. We carry out local bifurcation analysis and stability analysis. Under certain conditions, the model is shown to undergo a supercritical Hopf bifurcation resulting in a stable limit cycle. Numerical simulations are presented to illustrate our theoretic results.

In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carath e ´ odory approximation. Furthermore, with the help of H o ¨ lder’s inequality, Jensen’s inequality, It o ^ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.