Stability and periodicity in a mosquito population suppression model composed of two sub-models

Abstract

In this paper, we propose a mosquito population suppression model which is composed of two sub-models switching each other. We assume that the releases of sterile mosquitoes are periodic and impulsive, only sexually active sterile mosquitoes play a role in the mosquito population suppression process, and the survival probability is density-dependent. For the release waiting period T and the release amount c, we find three thresholds denoted by $$T^*$$ T ∗ , $$g^*$$ g ∗ , and $$c^*$$ c ∗ with $$c^*>g^*$$ c ∗ > g ∗ . We show that the origin is a globally or locally asymptotically stable equilibrium when $$c\ge c^*$$ c ≥ c ∗ and $$T\le T^*$$ T ≤ T ∗ , or $$c\in (g^*, c^*)$$ c ∈ ( g ∗ , c ∗ ) and $$T<T^*$$ T < T ∗ . We prove that the model generates a unique globally asymptotically stable T-periodic solution when either $$c\in (g^*, c^*)$$ c ∈ ( g ∗ , c ∗ ) and $$T=T^*$$ T = T ∗ , or $$c>g^*$$ c > g ∗ and $$T>T^*$$ T > T ∗ . Two numerical examples are provided to illustrate our theoretical results.

Periodic Orbits of a Mosquito Suppression Model Based on Sterile Mosquitoes

“…., plays an important role in judging the number of T-periodic solutions of the model (3) and (4). Lemma 2 (Lemma 2.1 in [38]). For any given initial value u, the following conclusions hold.…”

“…The proof of the global asymptotic stability of ŵ(t) is similar to that of Theorem 3.4 in [38] and is omitted.…”

“…One is to study the Wolbachia spread dynamics in mosquito populations aiming to study the possible invasion of Wolbachia in mosquito populations [10][11][12][13][14][15][16][17][18][19], and the other is to study the interactive dynamics of wild and sterile mosquitoes aiming to eliminate or control the wild mosquitoes below the risk of mosquito-borne disease outbreaks. Most early studies on the interactive dynamics mainly concentrate on models where the number of sterile mosquitoes was assumed to be an independent variable satisfying an independent dynamical equation, see, for example, [20][21][22][23][24], whereas recent investigations have proposed a motivation-attention modeling idea where the number of sterile mosquitoes is treated as a control function instead of an independent variable, since the fundamental and only role of sterile mosquitoes is to mate with wild mosquitoes to induce the infertility of wild females, see, for example, [25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”

“…In designing the release strategies of sterile mosquitoes, there are three important parameters: c, the release number in each batch; T, the waiting period between two adjacent releases; T, the sexual lifespan of sterile mosquitoes. From the perspective of the relations between T and T, there are three scenarios: T > T, T = T and T < T. In [38], we studied the first case, and split Equation (2) into the following two sub-equations…”

“…However, rearing sterile mosquitoes needs huge economic input and puts the density of wild mosquitoes below a given value which is more realistic than eradicating them. Taking these facts into consideration, in this paper, we mainly discuss the dynamics of the model ( 3) and (4) under the release strategy of c ≤ g * , which concentrates on reducing the number of wild mosquitoes with a lower cost than that in [38]. The remaining of the paper is structured as follows.…”