Exponential stability of pseudo almost periodic delayed Nicholson‐type system with patch structure

Abstract

In this paper, we consider the existence and global exponential stability of positive pseudo almost periodic solutions for a delayed Nicholson‐type system with patch structure. With the aid of ergodic theory and differential inequality technique, we prove the existence and global exponential stability of positive pseudo almost periodic solutions for the addressed system, which generalize and improve some known results. Specially, an example and its numerical simulations are provided to manifest the effectiveness of theoretical results.

Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure

“…n as the existence domain of almost periodic solutions, we establish the existence and global exponential stability of positive almost periodic solutions for Nicholson's blowflies systems involving patch structure and nonlinear density-dependent mortality terms. Our results improve and complement some existing ones in the recent publications [5][6][7]12], and its effectiveness is demonstrated by some numerical examples. This paper is organized as follows: In Sect.…”

“…. , n}, (1.2) have been extensively investigated in previous studies [5,6] and [7], respectively. It is easy to know that scalar Nicholson's blowflies Eq.…”

“…It is easy to know that scalar Nicholson's blowflies Eq. (1.1) is a special case of Nicholson's blowflies system (1.2), where x i (t) denotes the density of the ith-population at time t, a ij (t) (i = j) is the rate of the population moving from class j to class i at time t, a ii (t) is the coefficient of instantaneous loss (which integrates both the death rate and the dispersal rates of the population in class i moving to the other classes), β ij (t)x i (tτ ij (t))e -γ ij (t)x i (t-τ ij (t)) is the birth function, β ij (t) is the birth rate for the species, 1 γ ij (t) is the ith-population reproducing at its maximum rate, and τ ij (t) is the generation time of the ith-population at time t. For the feedback function xe -x and its derivative 1-x e x , the author in [8] It is worth noting that the global exponential stability of almost periodic solutions of (1.1) has been shown in [5,6] under the restriction that the almost periodic solution exists in a small interval [κ, κ] ≈ [0.7215355, 1.342276], and the global exponential stability of (1.2) has been established in [7] where the authors adopted the restraint that the almost periodic solution exists in a small domain Obviously, the above restriction and restraint do not accord with the biological significance of the population models. On the other hand, γ ij (t) ≥ 1 for all t ∈ R, i ∈ Q, j ∈ I := {1, 2, .…”

“…. , m}, (1.5) has been made as the crucial assumption in [5][6][7]. It should be mentioned that the stability of a class of delayed nonlinear density-dependent mortality Nicholson's blowflies model has been investigated in [9][10][11][12] without assumption (1.5), when the maximum reproducing rate is not limited (i.e.…”

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

“…De Albuquerque et al established existence of ground state solutions for a fractional Kirchhoff system on the real line without compactness because of the unboundedness of the domain. Wang et al have studied the existence of solutions and exponential stability for a delay Nicholson system and provided a numerical example to manifest their results. Peng et al studied existence of solutions for a time fractional Navier‐Stokes model with the help of Faedo‐Galerkin approximations.…”