A delay suppression model with sterile mosquitoes release period equal to wild larvae maturation period

Abstract

Based on the idea that only sexually active sterile mosquitoes are included in the modeling process, we study the dynamics of the interactive wild and sterile mosquito model with time delay, which consists of three sub-equations. Due to the fact that the maturation period of sterile mosquitoes bred in the lab or mosquito factories is almost the same time period of wild adult mosquitoes matured from larvae, we particularly assume that the waiting period for two consecutive releases of sterile mosquitoes equals the maturation period of wild mosquitoes, as a new practical sterile mosquito release strategy. We first ingeniously solve the delay model with the initial functions that are solutions of the corresponding equation without delay and we call them “good” solutions. Using these “good” solutions, we then surprisedly obtain sufficient and necessary conditions for the trivial solution and a unique periodic solution of the delay model to be globally asymptotically stable, respectively. We provide a numerical example to demonstrate the model dynamics and brief discussions of our findings as well.

Appendices

Appendixes

Proof of Lemma 3.3

Proof

(1). We prove inequality (3.7) by contradiction. For convenience, we omit $t_0$ in $w_u(t;t_0)$. Assume that (3.7) is not true. Since $w_u(t_0)=\phi _u(t_0) > \phi _u(t_0-\tau )$ from Lemma 3.2, there is $\bar{t}>t_0$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} w_u(t)\ge &{} w_u(t-\tau ),\ \forall \ t\in [T, \bar{t}], \\ w_u(\bar{t})= &{} w_u(\bar{t}-\tau ),\\ w_u(t)< &{} w_u(t-\tau ),\ \forall \ \bar{t}<t<\bar{t}+\delta , \end{array}\right. } \end{aligned}$$

(A.1)

where $\delta >0$ is sufficiently small. Then we can choose $t_1\in (\bar{t},\bar{t}+\delta )$ such that $w'_u(t_1) < w'_u(t_1-\tau )$.

Now we show $\bar{t}\ge 2T$. In fact, if $\bar{t}\in (T,T+\bar{T})$, then we have, from (2.6) and (3.2),

$$\begin{aligned} w'_u(t_1) = \frac{a w_u^2(t_1-\tau )}{w_u(t_1-\tau )+c} - w_u(t_1)(\mu +\xi (w_u(t_1)+c)), \end{aligned}$$

and

$$\begin{aligned} w'_u(t_1-\tau )= \frac{a w_u^2(t_1-\tau )}{w_u(t_1-\tau )+c}-w_u(t_1-\tau ) (\mu +\xi (w_u(t_1-\tau )+c)), \end{aligned}$$

which yields $w_u(t_1) > w_u(t_1-\tau )$, a contradiction to (A.1).

If $\bar{t}\in [T+\bar{T},2T)$, then we have, from (2.7) and (3.3),

$$\begin{aligned} w'_u(t_1) = a w_u(t_1-\tau ) -w_u(t_1)(\mu +\xi w_u(t_1)), \end{aligned}$$

and

$$\begin{aligned} w'_u(t_1-\tau ) = a w_u(t_1-\tau ) -w_u(t_1-\tau )(\mu +\xi w_u(t_1-\tau )), \end{aligned}$$

which also yields $w_u(t_1) > w_u(t_1-\tau )$, a contradiction to (A.1) again. Therefore, we obtain $\bar{t} \ge 2T$, and thus, there is a positive integer $k\ge 2$ such that $\bar{t}\in [kT, (k+1)T)$.

If $\bar{t}\in [kT, kT+\bar{T})$, then from Eq. (2.6), we have

$$\begin{aligned} w'_u(t_1) = \frac{a w_u^2(t_1-\tau )}{w_u(t_1-\tau )+c} - w_u(t_1)(\mu +\xi (w_u(t_1)+c)), \end{aligned}$$

and

$$\begin{aligned} w'_u(t_1-\tau ) = \frac{a w_u^2(t_1-2\tau )}{w_u(t_1-2\tau )+c} - w_u(t_1-\tau ) (\mu +\xi (w_u(t_1-\tau )+c)), \end{aligned}$$

which yields $w_u(t_1-\tau ) < w_u(t_1-2\tau )$, a contradiction.

If $\bar{t}\in [kT+\bar{T}, (k+1)T)$, then from equation (2.7), we have

$$\begin{aligned} w'_u(t_1) = a w_u(t_1-\tau ) - w_u(t_1)(\mu +\xi w_u(t_1)), \end{aligned}$$

and

$$\begin{aligned} w'_u(t_1-\tau ) = a w_u(t_1-2\tau ) - w_u(t_1-\tau )(\mu +\xi w_u(t_1-\tau )), \end{aligned}$$

which also yields $w_u(t_1-\tau ) < w_u(t_1-2\tau )$, a contradiction to (A.1). Hence, (3.7) is true.

Next, we show that sequence $\{w_u(nT)\}$ is strictly increasing. Assume that $\{w_u(nT)\}$ is not strictly increasing. Since $w_u(T)>w_u(0)$, there is a positive integer $l\ge 2$ such that

$$\begin{aligned} w_u((i+1)T) > w_u(iT), i=0,1,\cdots ,l-1,\;\;\text{ and }\;\; w_u((l+1)T)\le w_u(lT), \end{aligned}$$

which follows from (3.7) that $w_u((l+1)T) = w_u(lT)$ and $w'_u((l+1)T)\le w'_u(lT)$. However, from (2.7), we have

$$\begin{aligned} w'_u((l+1)T) = a w_u(lT) -w_u((l+1)T)(\mu +\xi w_u((l+1)T)), \end{aligned}$$

and

$$\begin{aligned} w'_u(lT) = a w_u((l-1)T) -w_u(lT)(\mu +\xi w_u(lT)), \end{aligned}$$

which yields $w_u(lT)\le w_u((l-1)T)$, a contradiction, and so sequence $\{w_u(nT)\}$ is strictly increasing.

(2). From Lemma 3.2 (2), $\phi _u(T)= u$ implies that $\phi _u(t)$ is a T-periodic solution of equations (3.2)-(3.3) and so is of equation (2.4).

(3). The proof is similar to that of (1) and is omitted.

The proof of Lemma 3.3 is complete. $\square$

Proof of Lemma 3.4

Proof

We prove the conclusion by contradiction. Assume that (3.9) is not true. Then there is $\bar{s}\ge t_0$ such that

$$\begin{aligned} w_1(t) \le w_2(t),\;\;\text{ for }\;\; t\in [0,\bar{s}],\;\;\text{ and }\;\; w_1(t)>w_2(t),\;\;\text{ for }\;\;t\in (\bar{s},\bar{s}+\delta _1), \end{aligned}$$

(B.1)

where $\delta _1>0$ is sufficiently small. Choose $s_1\in (\bar{s},\bar{s}+\delta _1)$ such that $w'_1(s_1)>w'_2(s_1)$. Then there exists a nonnegative integer k such that $s_1\in (kT, (k+1)T]=(kT,kT+\bar{T}]\cup (kT+\bar{T},(k+1)T]$. If $s_1\in (kT,kT+\bar{T}]$, then we have, from (2.6),

$$\begin{aligned} w'_i(s_1) = \frac{a w_i^2(s_1-\tau )}{w_i(s_1-\tau )+c} - w_i(s_1)(\mu +\xi (w_i(s_1)+c)),\;\;i=1,2, \end{aligned}$$

which implies $w_1(s_1-\tau ) > w_2(s_1-\tau )$, a contradiction to (B.1). Thus (3.9) is true. For the case when $s_1\in (kT+\bar{T},(k+1)T]$, the proof is similar and is omitted. The proof is complete. $\square$

Proof of Lemma 3.5

Proof

Clearly, we see that $[t_0-\tau , t_0] \subseteq [0,t_0]$ .We then have

$$\begin{aligned} \phi _{u_1}(t) \le \max \{u_1, h(u_1), h^2(u_1)\} \le m_1 \le \phi (t),\;\;\text{ for } \;\; t\in [0, t_0] \end{aligned}$$

and

$$\begin{aligned} \phi _{u_2}(t) \ge \min \{{\bar{h}}(u_2), \bar{h}(h(u_2))\} \ge m_2 \ge \phi (t), \;\;\text{ for }\;\; t\in [0, t_0]. \end{aligned}$$

Thus, it follows from Lemma 3.4 that

$$\begin{aligned} w_{u_1}(t;t_0)=w(t;t_0,\phi _{u_1}) \le w(t;t_0,\phi ) \le w(t;t_0,\phi _{u_2}) = w_{u_2}(t;t_0), \;\;\text{ for }\;\;t\ge t_0, \end{aligned}$$

which shows that (3.10) is true. The proof is complete. $\square$