Controlling Biological Invasions: A Stochastic Host–Generalist Parasitoid Model

Abstract

On a global scale, biological invasions are seriously destroying the stability of ecosystem, sharply decreasing biodiversity and even endangering human health and causing huge economic losses. However, there exist few effective measures controlling biological invasions. To more accurately examine the prevention and control effects of biological control on biological invasions in real environments of random fluctuations, we construct a stochastic host–generalist parasitoid model. We first establish, respectively, the sufficient conditions for the persistence and extinction of invasive hosts and generalist parasitoids, including (1) only the intrusive hosts go extinct; (2) only the generalist parasitoids are extinct, and (3) the intrusive hosts and generalist parasitoids are both extinct or persistent. Then, we perform a series of numerical simulations to verify the validity of the theoretical results obtained, based on which we further discuss the impacts of stochastic environmental fluctuations on the control of intrusive hosts, especially the possible changes of qualitative behavior caused by environmental noises in the bistable scenario. Our theoretical and numerical results indicate that compared with the invasive hosts, the generalist parasitoids are more vulnerable to environmental noises, and the prevention and control effects of biological control on invasive hosts are closely dependent to the initial population sizes. Thus, improving the ability of early detection of ecosystems, including the initial densities of biological populations and their dynamic characteristics, will provide effective predictive guidance for the prevention and control of alien host invasions.

Appendices

Appendix

1.1 A Proof of Theorem 1

Proof

In order to verify the existence and uniqueness of global positive solution, we divide two steps to prove the conclusion.

Step 1 The proof of the existence and uniqueness for local positive solution. In order to obtain this conclusion, we generally need to test the linear growth conditions and local Lipschitz conditions of model (3) (see (Mao 1997; Hu et al. 2008; Wang 2010)). However it is easy to see that neither of these two criteria of model (3) holds. Hence, to prove the existence and uniqueness of local positive solution, when $t\ge 0$ we let $X_{1}(t)=\ln x(t)$ and $X_{2}(t)=\ln y(t)$. Then for any given positive initial value $(x(0),y(0))\in \mathbb {R}_+^2$, we can obtain the following stochastic differential equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{d}X_{1}=\left( 1-e^{X_{1}}-\frac{be^{X_{2}}}{a+e^{X_{1}}} -\frac{\sigma _{1}^{2}}{2}\right) \textrm{d}t+\sigma _{1}\textrm{d}B_{1}(t),\\ \textrm{d}X_{2}=\left( \delta -e^{X_{2}}+\frac{ce^{X_{1}}}{a+e^{X_{1}}} -\frac{\sigma _{2}^{2}}{2}\right) \textrm{d}t+\sigma _{2}\textrm{d}B_{2}(t), \end{array}\right. } \end{aligned}$$

(34)

with initial value $X_{1}(0)=\ln x(0)$, $X_{2}(0)=\ln y(0)$. It is easy to check that system (34) satisfies the linear growth conditions and local Lipschitz conditions, which imply the system (34) exists a unique local solution $(X_{1}(t),X_{2}(t))$ for any time $t\in [0,\tau _{e})$ (Mao 1997), where $\tau _{e}$ is the explosion time. It then follows from Itô’s formula that $x(t)=e^{X_{1}(t)}$ and $y(t)=e^{X_{2}(t)}$ are the solution of model (3) with any given initial values $x(0)>0$ and $y(0)>0$. Thus, it proves the existence and uniqueness of local positive solution (x(t), y(t)) of model (3) for all $t\in [0,\tau _{e})$.

Step 2 The proof of global property. In order to testify the global property of the solution (x(t), y(t)) for model (3), we only need to prove $\tau _{e}=+\infty$. To this end, we let $n_{0}>1$ be a sufficiently large constant such that the initial values both $x(0)>0$ and $y(0)>0$ lying in $[\frac{1}{n_{0}},n_{0}]$. Thus, for each positive integer $n>n_{0}$, we define the following stopping time:

$$\begin{aligned} \tau _{n}=\inf \left\{ t\in [0,\tau _{e})\big |\min \{x(t),y(t)\}\le \frac{1}{n} ~\textrm{or}~\max \{x(t),y(t)\}\ge n\right\} . \end{aligned}$$

Obviously, $\tau _{n}$ is monotonic increase as $n\rightarrow +\infty$. We further define $\tau _{\infty }=\lim _{n\rightarrow +\infty }\tau _{n}$, then $\tau _{\infty }\le \tau _{e}$ a.s. Next, we only need to prove $\tau _{\infty }=+\infty$. By proof of contradiction, if $\tau _{\infty }<+\infty$, which implies that there are a pair of constants $T>0$ and $\varepsilon \in (0,1)$ satisfying $\mathbb {P}\{\tau _{\infty }\le T\}>\varepsilon$. Thus, there exists the positive integer $n_{1}\ge n_{0}$ such that

$$\begin{aligned} \mathbb {P}\{\tau _{n}\le T\}\ge \varepsilon ,~n\ge n_{1}. \end{aligned}$$

(35)

We further construct a $\textbf{C}^{2}$ function $V(x,y)=(x-1-\ln x)+\frac{b}{c}(y-1-\ln y)$, by Itô’s formulate one yields

$$\begin{aligned} \textrm{d}V(x,y)=\mathcal {L}V(x,y)\textrm{d}t+\sigma _{1}(x-1)\textrm{d}B_{1}(t)+\frac{b\sigma _{2}}{c}(y-1)\textrm{d}B_{2}(t), \end{aligned}$$

where $\mathcal {L}$ denotes the operator of stochastic differential equation defined by Mao in (Mao 1997). Then

$$\begin{aligned} \mathcal {L}V(x,y)=&\big (1-\frac{1}{x}\big )\left[ x(1-x)-\frac{bxy}{a+x}\right] +\frac{\sigma _{1}^{2}}{2} +\frac{b}{c}\big (1-\frac{1}{y}\big )\left[ y(\delta -y)+\frac{cxy}{a+x}\right] +\frac{b\sigma _{2}^{2}}{2c}\\ =&-(x-1)^{2}+\frac{by}{a+x}+\frac{\sigma _{1}^{2}}{2}+\frac{b\delta y}{c} -\frac{by^{2}}{c}-\frac{b\delta }{c}+\frac{by}{c}-\frac{bx}{a+x} +\frac{b\sigma _{2}^{2}}{2c}\\ \le&-\frac{by^{2}}{c}+\frac{by}{c}(\delta +1)+\frac{by}{a}-\frac{b\delta }{c} +\frac{\sigma _{1}^{2}}{2}+\frac{b\sigma _{2}^{2}}{2c}\\ \le&\frac{b[a(1+\delta )+c]^{2}}{4a^{2}c}-\frac{b\delta }{c} +\frac{\sigma _{1}^{2}}{2}+\frac{b\sigma _{2}^{2}}{2c}\\ :=&C_{3}, \end{aligned}$$

where $C_{3}$ denotes the bounded positive constant. Hence, we can obtain

$$\begin{aligned} \textrm{d}V(x,y)\le C_{3}\textrm{d}t+\sigma _{1}(x-1)\textrm{d}B_{1}(t)+\frac{b\sigma _{2}}{c}(y-1)\textrm{d}B_{2}(t). \end{aligned}$$

(36)

Integrating from 0 to $\tau _{n}\wedge T:=\min \{\tau _{n},T\}$, then taking expectation on both two sides of (36), we can get

$$\begin{aligned} \mathbb {E}V(x(\tau _{n}\wedge T),y(\tau _{n}\wedge T))\le V(x(0),y(0))+C_{3}T. \end{aligned}$$

(37)

Let $\Omega _{n}=\{\tau _{n}\le T\}$, it then follows from (35) that $\mathbb {P}(\Omega _{n})\ge \varepsilon$. By the definition of stopping time, we can know when $t=\tau _{n}$ and for any $\omega \in \Omega _{n}$, at least one of between x(t) and y(t) either is equal to $\frac{1}{n}$ or is equal to n. Thus, we can derive that $V(x(\tau _{n},\omega ),y(\tau _{n},\omega ))$ is not less than

$$\begin{aligned} (n-1-\ln n)\wedge \left( \frac{1}{n}-1+\ln n\right) \wedge \frac{b}{c}(n-1-\ln n)\wedge \frac{b}{c}\left( \frac{1}{n}-1+\ln n\right) . \end{aligned}$$

Further combining with the above conclusion and (37), we have

$$\begin{aligned} V(x(0),y(0))+C_{3}T\ge&\mathbb {E}[\textbf{1}_{\Omega _{n}}V(x(\tau _{n}\wedge T),y(\tau _{n}\wedge T))]\\ \ge&\varepsilon \big [(n-1-\ln n)\wedge \big (\frac{1}{n}-1+\ln n\big )\\&\wedge \frac{b}{c}(n-1-\ln n)\wedge \frac{b}{c}\big (\frac{1}{n}-1+\ln n\big )\big ], \end{aligned}$$

where $\textbf{1}_{\Omega _{n}}$ is the indicator function of $\Omega _{n}$. When $n\rightarrow +\infty$, one can have

$$\begin{aligned} +\infty >V(x(0),y(0))+C_{3}T=+\infty , \end{aligned}$$

which is obviously contradictory. This completes the proof of global property.

In summary, we derive the conclusion of Theorem 1. $\square$

Proof of Theorem 2

Proof

Let $Z(t)=cx(t)+by(t)$, then by simple computations we can yield

$$\begin{aligned} \textrm{d}Z(t)=&c\textrm{d}x(t)+b\textrm{d}y(t)\\ =&[cx(1-x)+by(\delta -y)]\textrm{d}t+c\sigma _{1}x\textrm{d}B_{1}(t)+b\sigma _{2}y\textrm{d}B_{2}(t)\\ =&(2cx-cx^{2}-cx+b\delta y-by^{2}+by-by)\textrm{d}t+c\sigma _{1}x\textrm{d}B_{1}(t)+b\sigma _{2}y\textrm{d}B_{2}(t)\\ \le&\left( c+\frac{(1+\delta )^{2}}{4}-Z(t)\right) \textrm{d}t+c\sigma _{1}x\textrm{d}B_{1}(t)+b\sigma _{2}y\textrm{d}B_{2}(t). \end{aligned}$$

Let N(t) be the solution of the following stochastic differential equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{d}N(t)=\left( c+\frac{(1+\delta )^{2}}{4}-N(t)\right) \textrm{d}t+c\sigma _{1}x\textrm{d}B_{1}(t)+b\sigma _{2}y\textrm{d}B_{2}(t),\\ N(0)=Z(0)=cx(0)+by(0). \end{array}\right. } \end{aligned}$$

Then we can derive the following formal solution:

$$\begin{aligned} N(t)=\left( c+\frac{(1+\delta )^{2}}{4}\right) +e^{-t}\left( N(0)-c-\frac{(1+\delta )^{2}}{4}\right) +\mathcal {M}(t), \end{aligned}$$

(38)

where $\mathcal {M}(t)=\int _{0}^{t}e^{-(t-\theta )}[c\sigma _{1}x(\theta )\textrm{d}B_{1}(\theta )+b\sigma _{2}y(\theta )\textrm{d}B_{2}(\theta )]$ is a locally continuous martingale satisfying $\mathcal {M}(0)=0$. Further, Eq. (38) can be rewritten as

$$\begin{aligned} N(t)=N(0)+A(t)-U(t)+\mathcal {M}(t), \end{aligned}$$

where $A(t)=\left( c+\frac{(1+\delta )^{2}}{4}\right) (1-e^{-t})$ and $U(t)=N(0)(1-e^{-t})$. It is obvious that A(t) and U(t) are bounded and are two continuous adapted increasing processes with $A(0)=U(0)=0$ for all $t\ge 0$. It then follows from (Mao 1997, Theorem 1.3.9) that $\lim _{t\rightarrow +\infty }N(t)$ exists and is finite. With the help of stochastic comparison theorem, we have $\lim _{t\rightarrow +\infty }Z(t)\le \lim _{t\rightarrow +\infty }N(t)<+\infty ,$ a.s. This completes the proof of the Theorem 2. $\square$

Proof of Lemma 1

Proof

With the help of Fokker-Plank equation as well as the ergodic stationary distribution, we investigate the main conclusions of (4). Let

$$\begin{aligned} f(\zeta )=\zeta (\Lambda -\zeta ),~g(\zeta )=\sigma \zeta ,~\mathrm{for~all}~\zeta >0. \end{aligned}$$

By simple computations, we can obtain that

$$\begin{aligned} \int \frac{f(\zeta )}{g^{2}(\zeta )}\textrm{d}\zeta =\int \frac{\zeta (\Lambda -\zeta )}{\sigma ^{2}\zeta ^{2}}\textrm{d}\zeta =&\frac{\Lambda }{\sigma ^{2}}\int \frac{1}{\zeta }\textrm{d}\zeta -\frac{1}{\sigma ^{2}}\int \textrm{d}\zeta =\frac{\Lambda }{\sigma ^{2}}\ln \zeta -\frac{\zeta }{\sigma ^{2}}+C_{1},\\ \int _{0}^{+\infty }\frac{1}{g^{2}(\zeta )}e^{\int _{1}^{\zeta }\frac{2f(s)}{g^{2}(s)} \textrm{d}s}\textrm{d}\zeta =&\frac{1}{\sigma ^{2}}\int _{0}^{+\infty }\zeta ^{\frac{2\Lambda }{\sigma ^{2}}-2} e^{\frac{2}{\sigma ^{2}}(1-\zeta )}\textrm{d}\zeta <+\infty , \end{aligned}$$

where $C_{1}$ denotes any constant. Based on the sufficiently criteria for the existence of invariant density (see Klebaner 2005, PP. 170–171), it then follows from the above computed results that Eq. (4) exists the stationary distribution with the following density

$$\begin{aligned} \pi (\zeta )=\frac{C_{2}}{g^{2}(\zeta )}e^{\int _{1}^{\zeta }\frac{2f(s)}{g^{2}(s)} \textrm{d}s}=\frac{C_{2}}{\sigma ^{2}}\zeta ^{\frac{2\Lambda }{\sigma ^{2}}-2} e^{\frac{2}{\sigma ^{2}}(1-\zeta )},~\zeta >0, \end{aligned}$$

(39)

where $C_{2}$ is a constant such that

$$\begin{aligned} \int _{0}^{+\infty }\pi (\zeta )\textrm{d}\zeta =\int _{0}^{+\infty }\frac{C_{2}}{\sigma ^{2}}\zeta ^{\frac{2\Lambda }{\sigma ^{2}}-2} e^{\frac{2}{\sigma ^{2}}(1-\zeta )}\textrm{d}\zeta =1. \end{aligned}$$

(40)

In order to derive the explicit expression of $\pi (\zeta )$, we use variable substitution $\theta =\frac{2\zeta }{\sigma ^{2}}$ to Eq. (40) getting

$$\begin{aligned} 2^{1-\frac{2\Lambda }{\sigma ^{2}}}C_{2}e^{\frac{2}{\sigma ^{2}}} \sigma ^{\frac{4\Lambda }{\sigma ^{2}}-4}\int _{0}^{+\infty }\theta ^{\frac{2\Lambda }{\sigma ^{2}}-2}e^{-\theta }\textrm{d}\theta =1. \end{aligned}$$

(41)

Considering the definition of Gamma function in one-dimensional real number domain, we can know that only when $\frac{2\Lambda }{\sigma ^{2}}-1>0$, Eq. (41) has practical significance. Further we can obtain $2^{1-\frac{2\Lambda }{\sigma ^{2}}}C_{2}e^{\frac{2}{\sigma ^{2}}} \sigma ^{\frac{4\Lambda }{\sigma ^{2}}-4}\Gamma \big (\frac{2\Lambda }{\sigma ^{2}}-1 \big )=1,$ that is to say,

$$\begin{aligned} \frac{C_{2}}{\sigma ^{2}}e^{\frac{2}{\sigma ^{2}}}= \left( \frac{\sigma ^{2}}{2}\right) ^{1-\frac{2\Lambda }{\sigma ^{2}}}\Gamma ^{-1} \left( \frac{2\Lambda }{\sigma ^{2}}-1\right) . \end{aligned}$$

(42)

Thus, substituting expression (42) into (39), we finally derive the desired result (5).

In what follows, we verify the Markov process z(t) for system (4) admits a unique ergodic stationary distribution with the invariant density (5) when $t>0$, which implies that for any integrable function $h(\cdot )$ possesses the conclusion (6). The specific verification processes are as follows:

With the help of the definition of ergodic stationary distribution in (Khasminskii 2012, Theorems 4.1 and 4.2), it is easy to check that the diffusion matrix of system (4) is non-degenerate for all bounded open set $S=(\frac{1}{n},n)$. Moreover, we need to construct a $\textbf{C}^{2}$ function $V(z):\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ satisfying $\mathcal {L}V(z)<0$ for all $z\in \mathbb {R}_{+}\backslash S$. Thus, under the condition $\frac{\sigma ^{2}}{2}<\Lambda$, we define

$$\begin{aligned} \overline{V}(z)=-M\ln z+z, \end{aligned}$$

where $M>0$ is a sufficiently large value such that

$$\begin{aligned} -M\big (\Lambda -\frac{1}{2}\sigma ^{2}\big )+\sup _{z\in \mathbb {R}_{+}}\{-z^{2} +(M+\Lambda )z\}\le -2. \end{aligned}$$

It is easy to see that $\overline{V}(z)$ is a continuous function and $\liminf _{n\rightarrow +\infty ,z\in \mathbb {R}_{+}\backslash S}\overline{V}(z)=+\infty$, which show that when $z\in \mathbb {R}_{+}$, the function $\overline{V}(z)$ can reach the lowest value at a point $\overline{z}$. Hence, we further construct a new $\textbf{C}^{2}$ function $V(z)=\overline{V}(z)-\overline{V}(\overline{z}):\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ as follows:

$$\begin{aligned} V(z)=-M\ln z+z-\overline{V}(\overline{z}). \end{aligned}$$

Applying Itô’s formula to V(z) along the sample path of system (4), we have

$$\begin{aligned} \textrm{d}V(z)=\mathcal {L}V(z)\textrm{d}t+(z-M)\sigma _{2}\textrm{d}B_{2}(t), \end{aligned}$$

where

$$\begin{aligned} \mathcal {L}V(z)=&\left( -\frac{M}{z}+1\right) z(\Lambda -z)+\frac{1}{2}\frac{M}{z^{2}} \sigma ^{2}z^{2}\\ =&-M\Lambda +Mz+\Lambda z-z^{2}+\frac{1}{2}M\sigma ^{2}\\ =&-M\left( \Lambda -\frac{1}{2}\sigma ^{2}\right) +\left( -z^{2}+(M+\Lambda )y\right) . \end{aligned}$$

Therefore, we easily know that

$$\begin{aligned} \mathcal {L}V(z)= {\left\{ \begin{array}{ll} \mathcal {L}V(+\infty )\rightarrow -\infty , &{}\textrm{as}~z\rightarrow +\infty ,\\ -M\big (\Lambda -\frac{1}{2}\sigma ^{2}\big )+\sup _{z\in \mathbb {R}_{+}}\{-z^{2} +(M+\Lambda )z\}\le -2, &{}\textrm{as}~z\rightarrow 0^{+}. \end{array}\right. } \end{aligned}$$

In summary, we prove $\mathcal {L}V(z)<0$ for all $z\in \mathbb {R}_{+}\backslash S$. It thus follows from the above two conditions that the system (4) exists a unique ergodic stationary distribution and the conclusion (6) holds. $\square$