Stability and permanence of an eco-epidemiological SEIN model with impulsive biological control

Abstract

Natural enemies of insects are extremely important in preventing pest outbreaks in crop fields. Therefore, we investigate the dynamical behavior of a pest-dependent consumption pest–natural enemy (i.e., prey–predator) SEIN model concerning diseases in pest population with three classes (susceptible–exposed–infectious) and impulsive releasing of infectious pests and natural enemies at fixed moments of time. We prove that all solutions of the system are uniformly ultimately bounded. In first part of the main results, the sufficient conditions for local as well as global asymptotic stability of the susceptible and exposed pest extinction periodic solution are determined using a Floquet’s theorem of impulsive differential equations, small-amplitude perturbation skills and comparison theorem. In second part, the sufficient condition for the permanence of a system is determined. These dynamics imply that susceptible and exposed pest populations become extinct when impulse period is less than some critical value and pests coexist with natural enemies at low level when impulse period crosses the critical value. Thus, our results provide some reliable theoretical tactics for pest management and finally these are verified by performing some numerical simulations.

Appendices

Appendix 1

Proof of Theorem 2

Let (S(t), E(t), I(t), N(t)) be any solution of (2.1). Using Theorem 1 the susceptible and exposed pest eradication periodic solution (0, 0, I(t), N(t)) is locally asymptotically stable; therefore, we only need to prove its global attraction. Since $T<\frac{1}{r}(\frac{\beta _1p_1}{\mu _1(1+mK)}+\frac{\beta _2p_2}{\mu _2})$, we can choose $\epsilon _1>0$ small enough such that

$$\begin{aligned} \int _{0}^{T}{\left[ r-\frac{\beta _1\left( \widetilde{I}(t)-\epsilon _1\right) }{1+mK}-\beta _2 \left( \widetilde{N}(t)-\epsilon _1\right) \right] \mathrm{d}t}:=\sigma _1<0. \end{aligned}$$

From the third equation of the system (2.1), we have

$$\begin{aligned} \frac{\mathrm{d}I(t)}{\mathrm{d}t}\ge -\mu _1I(t),\ t\ne nT, \end{aligned}$$

Consider the comparison system

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm{d}v(t)}{\mathrm{d}t}= -\mu _1v(t),\ t\ne nT,\\ v(t^+)=v(t)+p_1,\ t=nT. \end{array} \right. \end{aligned}$$

(7.1)

$\square$

Using the Lemma 3, we obtain that the system (7.1) has a periodic solution

$$\begin{aligned} \widetilde{v}(t)=\frac{p_1\exp (-\mu _1(t-nT))}{1-\exp (-\mu _1T)},\ \ nT<t\le (n+1)T, n\in Z_+, \end{aligned}$$

which is globally asymptotically stable. In view of Lemma 3 and comparison theorem of impulsive equation (Bainov and Simeonov 1993), we have $I(t)\ge v(t)$ and $v(t)\rightarrow \widetilde{v}(t)$ as $t\rightarrow \infty .$ Then there exists an integer $k_1$ such that

$$\begin{aligned} I(t)\ge v(t)> \widetilde{I}(t)-\epsilon _1,\ t\ge k_1T. \end{aligned}$$

(7.2)

Similarly, there exists an integer $k_2 \ (k_2>k_1)$ such that

$$\begin{aligned} N(t)> \widetilde{N}(t)-\epsilon _1,\ t\ge k_2T. \end{aligned}$$

(7.3)

Thus, for $t\ge k_2 T$, from first equation of the system (2.1), we have

$$\begin{aligned} \displaystyle \frac{\mathrm{d}S(t)}{\mathrm{d}t}\le S(t)\left[ r-\frac{\beta _1\left( \widetilde{I}(t)-\epsilon _1\right) }{1+mK}-\beta _2 \left( \widetilde{N}(t)-\epsilon _1\right) \right] . \end{aligned}$$

(7.4)

Integrating (7.4) on $(k_2T,t]$, which yields

$$\begin{aligned} S(t)\le & {} \displaystyle S(k_2T)\exp \left( \int _{k_2T}^{t}\left[ r-\frac{\beta _1\left( \widetilde{I}(t)-\epsilon _1\right) }{1+mK}-\beta _2 \left( \widetilde{N}(t)-\epsilon _1\right) \right] \mathrm{d}t \right) \\\le & {} S(k_2T)\exp (k\sigma _1),\\ \end{aligned}$$

where $t\in ((k_2+k)T, (k_2+k+1)T],\ k\in Z_+$. Since $\sigma _1 <0$, we can easily obtain that $S(t)\rightarrow 0$ as $k\rightarrow +\infty$. Thus, for an arbitrary positive constant $\epsilon _2$ small enough, there exists an integer $k_3\ (k_3>k_2)$ such that $S(t)<\epsilon _2$ for all $t\ge k_3T$. Hence, we can obtain an integer $k_4\ (k_4>k_3)$ such that $\frac{\beta _1S(t)I(t)}{1+mS(t)}<\epsilon _1$ for all $t\ge k_4T$. From the second equation of the system (2.1), we have

$$\begin{aligned} \frac{\mathrm{d}E(t)}{\mathrm{d}t}\le \epsilon _1-(\alpha +\mu _1)E(t), \end{aligned}$$

which yields that $\displaystyle E(t)\le E(0^+)e^{-(\alpha +\mu _1)t}+\frac{\epsilon _1}{\alpha +\mu _1}- \frac{\epsilon _1}{\alpha +\mu _1}e^{-(\alpha +\mu _1)t}$ $\rightarrow 0$ as $t\rightarrow \infty$. Thus, there exists an integer $k_5\ (k_5>k_4)$ such that $E(t)<\epsilon _2$ for all $t\ge k_5T$.

Again from the third equation of the system (2.1), we have

$$\begin{aligned} \frac{\mathrm{d}I(t)}{\mathrm{d}t}\le \alpha \epsilon _2-\mu _1 I(t), \ \ t\ne nT. \end{aligned}$$

Consider the following impulsive system

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm{d}w(t)}{\mathrm{d}t}= \alpha \epsilon _2-\mu _1w(t),\ t\ne nT,\\ w(t^+)=w(t)+p_1,\ t=nT. \end{array} \right. \end{aligned}$$

(7.5)

Using the Lemma 3, we obtain that the system (7.5) has a periodic solution

$$\begin{aligned} \widetilde{w}(t)=\frac{\alpha \epsilon _2}{\mu _1}+\frac{p_1\exp (-\mu _1(t-nT))}{1-\exp (-\mu _1T)},\ \ nT<t\le (n+1)T, n\in Z_+, \end{aligned}$$

which is globally asymptotically stable. Again, in view of Lemma 3 and comparison theorem of impulsive equation Bainov and Simeonov (1993), we have $I(t)\le w(t)$ and $w(t)\rightarrow \widetilde{w}(t)$ as $t\rightarrow \infty .$ Then there exists an integer $k_6\ (k_6>k_5)$ such that

$$\begin{aligned} I(t)\le w(t)< \widetilde{w}(t)+\epsilon _1,\ t\ge k_6T, \end{aligned}$$

(7.6)

From the fourth equation of the system (2.1), we obtain that

$$\begin{aligned} \frac{\mathrm{d}N(t)}{\mathrm{d}t}\le -\left( \mu _2-\gamma \beta _2\epsilon _2 \right) N(t), \ \ t\ne nT. \end{aligned}$$

Using similar argument, there exists an integer $k_7 \ (k_7>k_6)$ such that

$$\begin{aligned} N(t)< N^*(t)+\epsilon _1,\ t\ge k_7T, \end{aligned}$$

(7.7)

where $\displaystyle N^*(t)=\frac{p_2\exp \left( -\left( \mu _2-\gamma \beta _2\epsilon _2 \right) (t-nT)\right) }{1-\exp \left( -\left( \mu _2-\gamma \beta _2\epsilon _2 \right) T\right) },\ \ nT<t\le (n+1)T, n\in Z_+.$ Note that $\epsilon _1$ and $\epsilon _2$ are positive constant small enough; therefore, $\widetilde{w}(t)\rightarrow \widetilde{I}(t)$, $N^*(t)\rightarrow \widetilde{N}(t)$, as $\epsilon _2\rightarrow 0$. Thus, it is obtained that $I(t)\rightarrow \widetilde{I}(t)$ and $N(t)\rightarrow \widetilde{N}(t)$, as $t\rightarrow +\infty$. Therefore, the exposed and infectious pest eradication periodic solution $(0,0,\widetilde{I}(t),\widetilde{N}(t))$ of (2.1) is globally asymptotically stable.

Appendix 2

Proof of Theorem 3

Suppose (S(t), E(t), I(t), N(t)) is a solution of the system (2.1). From the boundedness of the solution, we get that $S(t)\le M,\ E(t)\le M,\ I(t)\le M$ and $N(t)\le M$ for all $t\ge 0$. $\square$

From the Eqs. (7.2) and (7.3), it is obtained that

$$\begin{aligned} I(t)>\widetilde{I}(t)-\epsilon _1:=m_1>0, \quad \text{ for } \quad t\ge k_1T, \\ N(t)>\widetilde{N}(t)-\epsilon _1:=m_2>0, \quad \text{ for }\quad t\ge k_2T. \end{aligned}$$

Therefore, I(t) and N(t) are ultimately positively bounded below.

Since $T>T_\mathrm{max}$, therefore, we can select positive constants $\epsilon _3$ and $m_3$ $(0<m_3<K)$ small enough such that

$$\begin{aligned} \left[ r\left( 1-\frac{m_3}{K}\right) -\frac{\beta _1\alpha M}{\mu _1} \right] T-\left[ \frac{\beta _1p_1}{\mu _1}-\frac{\beta _2p_2}{\mu _2-\gamma \beta _2m_3}\right] :=\sigma _2>0. \end{aligned}$$

Now, for the permanence of the system, we only need to find $m_3$ and $m_4$ such that $S(t)\ge m_3$ and $E(t)\ge m_4$ for large enough t. First, we prove $S(t)\ge m_3$ for large enough t, and the result is proved in two steps.

Step I First, assume that $S(t)\ge m_3$ is not true, then there exists a $K_1\in Z_+$, such that $S(t)< m_3$ for all $t\ge K_1T$. Using this assumption, we get following subsystem of the system (2.1):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm{d}I(t)}{\mathrm{d}t}\le \alpha M-\mu _1 I(t),\ t\ne nT \\ I(t^+)=I(t)+p_1,\ t=nT. \end{array} \right. \end{aligned}$$

Consider the following comparison system:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm{d}{w_1}(t)}{\mathrm{d}t}=\alpha M-\mu _1 w_1(t),\ t\ne nT,\\ {w_1}(t^+)={w_1}(t)+p_1,\ t=nT. \end{array} \right. \end{aligned}$$

(8.1)

Using the Lemma 3, the system (8.1) has a periodic solution

$$\begin{aligned} \widetilde{{w_1}}(t)=\frac{\alpha M}{\mu _1}+\frac{p_1\exp (-\mu _1(t-nT))}{1-\exp (-\mu _1T)}, \end{aligned}$$

which is globally asymptotically stable. Then, there exists an $N_2\ (K_2>K_1)$ such that

$$\begin{aligned} I(t)\le \widetilde{{w_1}}<\frac{\alpha M}{\mu _1}+\frac{p_1\exp (-\mu _1(t-nT))}{1-\exp (-\mu _1T)}+\epsilon _3,\quad \text{ for } t\ge K_2T. \end{aligned}$$

Again from the system (2.1), we can obtain following impulsive system:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm{d}N(t)}{\mathrm{d}t}= \frac{\gamma \beta _2S(t)N(t)}{1+h\beta _2S(t)}-\mu _2 N(t)\le -\left( \mu _2-\gamma \beta _2m_3\right) N(t),\ t\ne nT \\ N(t^+)=I(t)+p_2,\ t=nT. \end{array} \right. \end{aligned}$$

Consider the following comparison system:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm{d}w_2(t)}{\mathrm{d}t}= -\left( \mu _2-\gamma \beta _2m_3\right) w_2(t),\ t\ne nT\\ w_2(t^+)=w_2(t)+p_2,\ t=nT. \end{array} \right. \end{aligned}$$

(8.2)

Using similar argument, we obtain that

$$\begin{aligned} N(t)\le \widetilde{{w_2}}<\frac{p_2\exp \left( -\left( \mu _2-\gamma \beta _2m_3 \right) (t-nT)\right) }{1-\exp \left( -\left( \mu _2-\gamma \beta _2m_3 \right) T\right) }+\epsilon _3,\quad \text{ for } t\ge K_2T. \end{aligned}$$

Thus, from the system (2.1) we can obtain that

$$\begin{aligned} \frac{\mathrm{d}S(t)}{\mathrm{d}t}\ge \left[ r-\frac{rm_3}{K}-\beta _1\left( \widetilde{{w_1}}+\epsilon _3\right) -\beta _2\left( \widetilde{{w_2}}+\epsilon _3\right) \right] S(t). \end{aligned}$$

which yields that

$$\begin{aligned} S((K_2+k)T)\ge S(K_2T)e^{\int _{K_2T}^{(K_2+k)T}\left[ r-\frac{rm_3}{K}-\beta _1\left( \widetilde{{w_1}}+\epsilon _3\right) -\beta _2\left( \widetilde{{w_2}}+\epsilon _3\right) \right] \mathrm{d}t }\ge S(K_2T)e^{k\sigma _2}. \end{aligned}$$

Thus, it is easily got that $S((K_2+k)T)\rightarrow +\infty$ as $k\rightarrow +\infty ,$ which is a contradiction with the boundedness of S(t). Hence, for an arbitrary $K_1\in Z_+$, there exists a $t_1>K_1T$ such that $S(t_1)\ge m_3$.

Step II If $S(t)\ge m_3$ for all $t\ge t_1$, then our aim will be fulfilled. Otherwise, $S(t)<m_3$ for some $t> t_1$. Let $t^*=\inf \ \{t\ |\ S(t)<m_3,\ t>t_1\}$, then we have, $S(t)\ge m_3$ for $t\in [t_1,t^*)$ and $t^*\in (n_1T,(n_1+1)T]$, $n_1$ is some positive integer. It is easily obtain that $S(t^*)=m_3$, since S(t) is continuous. Assume that $T_1=n_2T+n_3T,$ where $n_2$ and $n_3$ satisfy the following inequalities:

$$\begin{aligned} n_2T>\max \left\{ \frac{-\ln \frac{\epsilon _4}{2M}}{\mu _1}, \frac{-\ln \frac{\epsilon _4}{2M}}{\mu _2-\gamma \beta _2m_3} \right\} , \end{aligned}$$

$$\begin{aligned} n_3 \sigma _3>\left( \frac{r}{K}+\beta _1+\beta _2\right) M(n_2+1)T. \end{aligned}$$

Now, we claim that there exists a time $t'\in ((n_1+1)T,(n_1+1)T+T_1)$ such that $S(t')\ge m_3,$ if it is not true, then true, then $S(t)<m_3,\ t\in ((n_1+1)T,(n_1+1)T+T_1)$. If the system (8.1) is considered with initial value ${w_1}({(n_1+1)T}^+)=I({(n_1+1)T}^+)$, then using Lemma 3 for $t\in (nT,(n+1)T]$ and $n_1\le n \le n_1+n_2+n_3$, we have

$$\begin{aligned} {w_1}(t)=\left( {w_1}({(n_1+1)T}^+)-\left( \frac{\alpha M}{\mu _1}+\frac{p_1}{1-e^{-\mu _1T}}\right) \right) e^{-\mu _1(t-(n_1+1)T)}+\widetilde{{w_1}}(t), \end{aligned}$$

which evidence that

$$\begin{aligned} |{w_1}(t)-\widetilde{{w_1}}(t)|<2Me^{-\mu _1(t-(n_1+1)T)}<\epsilon _4, \end{aligned}$$

and $I(t)\le {w_1}(t)<\widetilde{{w_1}}(t)+\epsilon _4$ for all $(n_1+n_2+1)T\le t\le (n_1+1)T+T_1$.

Now, we consider the system (8.2) with initial value ${w_2}((n_1+1)T+n_2T)^+=N((n_1+1)T+n_2T)^+\ge 0$. Again using the Lemma 3, we obtain that

$$\begin{aligned} |{w_2}(t)-\widetilde{{w_2}}(t)|<2Me^{-(\mu _2-\gamma \beta _2m_3)(t-(n_1+1)T+n_2T)}<\epsilon _4, \end{aligned}$$

and $N(t)\le {w_2}(t)<\widetilde{{w_2}}(t)+\epsilon _4$ for $(n_1+n_2+1)T\le t\le (n_1+1)T+T_1.$ Thus, we have

$$\begin{aligned} \frac{\mathrm{d}S(t)}{\mathrm{d}t}\ge \left[ r-\frac{rm_3}{K}-\beta _1 \left( \widetilde{{w_1}}(t)+\epsilon _4\right) -\beta _2 \left( \widetilde{{w_2}}(t)+\epsilon _4 \right) \right] S(t), \end{aligned}$$

for all $t\in [(n_1+1)T+n_2T,(n_1+1)T+T_1]$. Integrating on $[(n_1+1)T+n_2T,(n_1+1)T+T_1]$, we get

$$\begin{aligned} S((n_1+1+n_2+n_3)T)\ge & {} S((n_1+n_2)T)\ e^{n_3\sigma _3}. \end{aligned}$$

(8.3)

Now for $t\in (t^*,(n_1+1)T]$, there are two possible cases:

Case (a) If $S(t)<m_3$ for $t\in (t^*, (n_1+1)T]$, then $S(t)<m_3$ for all $t\in (t^*, (n_1+1+n_2)T]$. Therefore, we have

$$\begin{aligned} \frac{\mathrm{d}S(t)}{\mathrm{d}t}\ge \left[ -\frac{r}{K}-\beta _1 -\beta _2 \right] MS(t):=\eta S(t). \end{aligned}$$

(8.4)

On integrating the system (8.4) in the interval $[t^*,(n_1+1+n_2)T]$, it is obtain that

$$\begin{aligned} S((n_1+1+n_2)T)\ge S(n_1T)\ e^{\eta (n_2+1)T}, \end{aligned}$$

(8.5)

substituting (8.5) into (8.3), we get that

$$\begin{aligned} S((n_1+1+n_2+n_3)T)\ge S(n_1T)\ e^{\eta (n_2+1)T}\ e^{n_3\sigma _3}\ge m_3 \ e^{\eta (n_2+1)T}\ e^{n_3\sigma _3}>m_3, \end{aligned}$$

which is a contradiction. Let $\check{t}=\inf \{t|\ S(t)\ge m_3,\ t>t^* \}$, then $S(\check{t})=m_3$. For $t\in [t^*,\check{t})$, (8.4) holds. Integrating (8.4) on $[t^*,\check{t}$, we have

$$\begin{aligned} S(t)\ge S(t^*)\ e^{\eta (t-t^*)}\ge m_3 \ e^{\eta (1+n_2+n_3) T}:=\overline{m_3}. \end{aligned}$$

Since $S(\check{t})\ge m_3$ for $t>\check{t}$, the same argument can be continued. Hence, we have $S(t)\ge \overline{m_3}$ for all $t>t_1$.

Case (b) There exists $t''\in (t^*,(n_1+1)T$ such that $S(t'')\ge m_3.$ Let $\bar{t}=\inf \{t|\ S(t)\ge m_3,\ t>t^* \}$, then $S(t)<m_3$ for $t\in [t^*,\bar{t})$ and $S(\bar{t})=m_3$. For $t\in [t^*,\bar{t})$, (8.4) holds. Integrating (8.4) on $[t^*,\bar{t})$, we have

$$\begin{aligned} S(t)\ge S(t^*)\ e^{\eta (t-t^*)}\ge m_3 \ e^{\eta T}>\overline{m_3}. \end{aligned}$$

The same argument can be continued, since $S(\bar{t})\ge m_3$. Hence, we have $S(t)\ge \overline{m_3}$ for all $t>t_1$. Thus, in both cases, we conclude $S(t)\ge \overline{m_3}$ for all $t\ge t_1$. This proves that S(t) is ultimately positively bounded below.

Secondly, we prove that E(t) is ultimately positively bounded below. Since the function $\frac{\beta _1S(t)}{1+mS(t)}$ is monotonically increasing for $S(t)\ge 0$, from the second equation of system (2.1), we have

$$\begin{aligned} \frac{\mathrm{d}E(t)}{\mathrm{d}t}\ge \frac{\beta _1\overline{m_3}m_1}{1+m\overline{m_3}}-(\alpha +\mu _1)E(t), \end{aligned}$$

we can easily obtain that $\lim _{t\rightarrow +\infty }\ \inf E(t)\ge m_4$, where $\displaystyle m_4=\frac{\beta _1 \overline{m_3}m_1}{(1+m\overline{m_3})(\alpha +\mu _1)}$.

Now using Step I and Step II, let us define $m=\min \{m_1,m_2,\overline{m_3},m_4 \}$ and $\Omega =\{(S,E,I,N)\ :\ m\le S(t),E(t), I(t), N(t) \le M \}$. We know that $\Omega \subset int R_+^4$ is a global attractor and hence every solution of system (2.1) with positive initial values will eventually enter and remain in region $\Omega$. This evidences that the system (2.1) is permanent.