Global asymptotical stability of almost periodic solutions for a non-autonomous competing model with time-varying delays and feedback controls

Abstract

In this paper, we are concerned with a non-autonomous competing model with time delays and feedback controls. Applying the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. An example with its numerical simulations is given to illustrate the feasibility of our results.

Remotely Almost Periodic Solutions of Ordinary Differential Equations

“…e class of vector-valued remotely almost periodic functions defined on R n was introduced by Yang and Zhang in [12], where the authors have provided several applications in the study of existence and uniqueness of remotely almost periodic solutions for parabolic boundary value problems (for some results about parabolic boundary value problems, one may refer to [13][14][15] and references cited therein). In Propositions 2.4-2.6 in [16], the authors have examined the existence and uniqueness of remotely almost periodic solutions of multidimensional heat equations, while the main results of Section 3 are concerned with the existence and uniqueness of remotely almost periodic type solutions of the certain types of parabolic boundary value problems (see also [17,18], where the authors have investigated almost periodic type solutions and slowly oscillating type solutions for various classes of parabolic Cauchy inverse problems). Concerning applications of remotely almost periodic functions, research articles [19] by Zhang and Piao, where the authors have investigated the time remotely almost periodic viscosity solutions of Hamilton-Jacobi equations, and [20] by Zhang and Jiang, where the authors have investigated remotely almost periodic solutions for a class systems of differential equations with piecewise constant argument, should be mentioned, see [21] and the research articles [22][23][24][25][26][27][28][29], for more details about the subject.…”

“…In the existing literature, we can find numerous results about the existence, uniqueness, stability, applications in biology, etc. (concerning the last issue, see, e.g., the research articles by Xu et al [16,[33][34][35][36][37][38] as well as the references cited therein). e strong motivational point for the genesis of this paper lies in the fact that, with the exception of [20] by Zhang and Liang, no one else has applied remotely almost periodic functions in the theory of ordinary differential equations.…”

“…. , F k (•) are remotely almost periodic (k ∈ N), then, for each ε > 0, the set of their common ϵ-remote-translation vectors s is relatively dense in R; see, e.g., Proposition 2.3 in [16].…”

Dynamics of a Discrete Allelopathic Phytoplankton Model with Infinite Delays and Feedback Controls

“…As we all know, due to seasonal fluctuations in the environment and hereditary factors, time delays have been introduced into the biological system (see [11,14,21,[27][28][29][30][31][32][33][34][35][36]). Zhao et al [8] further considered a discrete Lotka-Volterra competition system with infinite delays and single feedback control variable as follows:…”

Modelling Delay in Migration for Constant Predator and Predator-Density-Dependent Prey Migration

“…Time delays have mainly been incorporated in the interaction part of the predator-prey models and not in the migration of these species, for instance see [8,9,10,11]. A time delay in migration is the duration of time taken by one species before they relocate from one patch to another in a heterogeneous environment after facing an unfavorable condition in the initial patch.…”