Delay-dependent attractivity on a tick population dynamics model incorporating two distinctive time-varying delays

Abstract

In this paper, we aim to investigate the influence of delay on the global attractivity of a tick population dynamics model incorporating two distinctive time-varying delays. By exploiting some differential inequality techniques and with the aid of the fluctuation lemma, we first prove the persistence and positiveness for all solutions of the addressed equation. Consequently, a delay-dependent criterion is derived to assure the global attractivity of the positive equilibrium point. And lastly, some numerical simulations are presented to verify that the obtained results improve and complement some existing ones.

Global attractivity in tick population models incorporating seasonality and diapausing stages

“…It is worth noting that our results cover the best delay-independent condition of global attraction, see comments below. By exploiting some differential inequality techniques and with the help of the fluctuation lemma, Huang et al [4] and Huang & Wu [5] developed subtle criteria of global attraction of the positive equilibrium in (1.1) for the Ricker birth rate f (x) = x e −x and f (x) = x γ e −x , respectively. Compared with these papers, our approach has three strengths:…”

“…The autonomous version of model (5.3) was recently analysed by Huang et al [4]. Specifically, they considered the equation…”

“…By proposition 5.1, 1 is a global attractor in (0, +∞) for (5.12) when ( Therefore, (5.11) is more restrictive than (5.13). In figure 2, we provide an example in which (5.13) holds but we cannot apply the results in [4].…”

“…In fact, ticks are one of the major vectors of human pathogens. This remarkable epidemiological role has provoked an increasing interest in tick population dynamics and tick-borne disease transmission in recent years (see [1,[3][4][5][6][7][8][9] and references therein). Unfortunately, despite several significant advances made, many questions remain to be solved in the field.…”

“…However, if a tick is subject to a diapause period, its maturation time will be τ 2 = τ 1 + τ d with τ d the duration of the diapause. Under this assumption, Zhang & Wu [9], Zhang et al [3] and Huang et al [4,5] proposed the delay differential equation…”