The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time are considered. New almost periodic and perfect Mittag–Leffler stability criteria are proposed. Lyapunov’s like impulsive functions, the properties of the fractional derivatives and comparison principle are the main tools in the investigation. Illustrative examples are also presented to demonstrate the proposed criteria. Our results contribute to the development of qualitative the theory of fractional-order gene regulatory networks.

The present paper introduces the concept of integral manifolds for a class of delayed impulsive neural networks of Cohen–Grossberg-type with reaction–diffusion terms. We establish new existence and boundedness results for general types of integral manifolds with respect to the system under consideration. Based on the Lyapunov functions technique and Poincarѐ-type inequality some new global stability criteria are also proposed in our research. In addition, we consider the case when the impulsive jumps are not realized at fixed instants. Instead, we investigate a system under variable impulsive perturbations. Finally, examples are given to demonstrate the efficiency and applicability of the obtained results.

Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers are different. It is observed that the intralayer synchronization state occurs in weaker intralayer couplings when using nonidentical fractional-order derivatives rather than integer-order or identical fractional orders. Furthermore, the needed interlayer coupling strength for interlayer near synchronization decreases for lower fractional orders. The dynamics of the neurons in nonidentical layers are also considered. It is shown that in lower fractional orders, the neurons’ dynamics change to periodic when the near synchronization state occurs. Moreover, decreasing the derivative order leads to incrementing the frequency of the bursts in the synchronization manifold, which is in contrast to the behavior of the single neuron.

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

In this study, we deal with an impulsive boundary value problem (BVP) for differential equations of variable fractional order involving the Caputo–Hadamard fractional derivative. The fundamental problems of existence and uniqueness of solutions are studied, and new existence and uniqueness results are established in the form of two fixed point theorems. In addition, Ulam–Hyers stability sufficient conditions are proved illustrating the suitability of the derived fundamental results. The obtained results are supported also by an example. Finally, the conclusion notes are highlighted.