A stochastic differential equation model for pest management
- Xuewen Tan^{1, 2}Email author,
- Sanyi Tang^{1},
- Xiaozhou Chen^{2},
- Lianglin Xiong^{2} and
- Xinzhi Liu^{3}
https://doi.org/10.1186/s13662-017-1251-x
© The Author(s) 2017
Received: 8 December 2016
Accepted: 20 June 2017
Published: 11 July 2017
Abstract
In order to comprehend the effects of the duration of pesticide residual effectiveness on successful pest control, a stochastic integrated pest management (IPM) model with pesticides which have residual effects is proposed. Firstly, we show that our model has a global and positive solution and give its explicit expression when pest goes extinct. Then the sufficient conditions for pest extinction combined with the ones for the global attractivity of the pest solution only chemical control are established. Moreover, we also derive sufficient conditions for weak persistence which show that the solution of stochastic IPM models is stochastically ultimately bounded under some conditions.
Keywords
1 Introduction
Integrated pest management (IPM) was introduced and was widely practised recently [1–11]. IPM is a long time management strategy that uses a combination of cultural, biological and chemical tactics that reduce pests to tolerable levels with little cost to the grower and a minimal effect on the environment.
Especially, spraying of pesticides to crops is intended to reduce the population of pest, but can under some circumstances exacerbate a pest problem. This phenomenon, frequently called insecticide-induced resistance, has several possible mechanisms including reduction in herbivore-herbivore competition, physiological enhancement of pest fecundity, altered host plant nutrition, changes in pest behavior, and the killing of natural predators and parasites of the pest [12–18]. As a result, considerable emphasis has been placed on tactics other than chemical controls, including cultural, biological, and genetic methods and the deployment of crop varieties that are resistant to pests. Researchers have found that the cultural and biological pest controls are ecologically sound and could provide solutions that are sustainable in the long term [19, 20].
However, chemical pesticides may have long-term residual effects which reduce the populations of pest for several weeks, months or years. On the other hand, population dynamics in the real world is inevitably affected by environmental noises. So it is very important to study stochastic population systems to analyze the effect of environmental noises on population systems [21–27].
Based on the above discussions, this paper is to construct a stochastic integrated pest management (IPM) model with pesticides that have residual effects firstly. In the first place, we derive that the model has a global positive solution by using the mathematical induction method. On the other hand, the explicit expression of the pest-eradication solution is given. Moreover, in order to show residual effects, the sufficient condition for pest extinction and the global attractivity of the pest solution when the natural enemy disappears are established. Furthermore, we establish sufficient conditions for weak persistence in the mean and also show that the solution of stochastic IPM models is stochastically ultimately bounded under some conditions. The methods may have applications for a wider range of aspects of theoretical biology.
2 The killing efficiency rate with pulses of chemical control
It is universally acknowledged that the residual effects of pesticides exert desirable impacts on pest control. Therefore, a more realistic and appropriate method for modeling chemical control in such a case is to use continuous or piecewise-continuous periodic functions which affect the growth rate in the logistic model [1–3]. Such periodic functions make the growth rate fluctuate so that it decreases significantly, or even takes on negative values when the effects of pesticides have disappeared, but increases again during the recovery stage.
3 The stochastic IPM model with residual effects of pesticides
The complex interactions amongst pests, natural enemies and pesticides can be enhanced with mathematical modeling [4, 5]. Modeling advances regarding IPM strategies include analyses of continuous and discrete predator-prey models [6–10]. In addition, discrete host-parasitoid models have been used to study four cases involving the timing of pesticide applications when these also led to the death of the parasitoids [16].
On the other hand, in the real world, the growth of species usually suffers some discrete changes of relatively short time interval at some fixed times with some natural and man-made factors, such as drought, harvesting, fire, earthquake, flooding, deforestation, hunting and so on. These phenomena cannot be considered with certainty, so in this case, the determined model (3.1) cannot be applicable. In order to describe these phenomena more accurately, some authors considered the stability of stochastic differential equation (SDE) [21]. However, as far as our knowledge is concerned, there is very little amount of work on the impulsive stochastic population model. By now, there have been no results related to the stochastic IPM model with residual effects of pesticides with impulsive effects.
3.1 Mathematical analysis of the pest-free solution
Throughout this paper, let \((\Omega , \mathscr{F}, \{ \mathscr{F}_{t} \}_{t\geq 0},\mathscr{P}) \) be a complete probability space with a filtration \(\{\mathscr{F}_{t}\}_{t\geq 0}\) satisfying the usual conditions. Let \(B(t)\) denote a standard Brownian motion defined on this probability space. Moreover, we always assume that a product equals unity if the number of factors is zero.
Definition 1
[21]
- (i)
\(X(t)\) is \(\mathscr{F}_{t}\)-adapted and continuous on \((0, T)\) and each interval \((nT, (n+1)T)\subset R_{+}\), \(n \in \mathcal{N}\); \(f(t, X(t))\in\mathscr{L}^{1}(R_{+}; R_{n})\), \(\sigma (t,X(t))\in \mathscr{L}^{2}(R_{+}; R_{n})\), where \(\mathscr{L}^{n}(R_{+};R_{n})\) is all \(R^{n}\)-valued measurable \(\mathscr{F}_{t}\)-adapted processes \(g(t)\) satisfying \(\int _{0}^{\zeta }\vert g(t) \vert ^{n}\,dt<\infty \) a.s. (almost surely) for every \(\zeta >0\);
- (ii)
for each nT, \(n\in \mathcal{N}\), \(X(nT^{+})=\lim_{t\rightarrow nT^{+}}X(t)\) and \(X(nT^{-})=\lim_{t\rightarrow nT^{-}}X(t)\) exist and \(X(nT)=X(nT^{-})\) with probability one;
- (iii)for almost all \(t\in (0, T]\), \(P(t)\) obeys the integral equationAnd for almost all \(t\in (nT, (n+1)T]\), \(n\in \mathcal{N}\), \(X(t)\) obeys the integral equation$$ X(t)=X(0) + \int _{0}^{t} f \bigl(s,X(s) \bigr)\,ds + \int _{0}^{t} \sigma \bigl(s, X(s) \bigr)\,dB(s). $$Moreover, \(X(t)\) satisfies the impulsive conditions at each \(t=nT, n\in \mathcal{N}\) with probability one.$$ X(t)=X \bigl(nT^{+} \bigr) + \int _{nT}^{t} f \bigl(s,X(s) \bigr)\,ds + \int _{nT}^{t} \sigma \bigl(s, X(s) \bigr)\,dB(s). $$
Let us state and prove the following result.
Theorem 1
Proof
3.2 Mathematical analysis of chemical control only
Definition 2
If \(N_{1}(t)\), \(N_{2}(t)\) are two arbitrary solutions of model (3.3) with the initial values \(N_{1}(0)\), \(N_{2}(0)\)>0, respectively. If \(\lim_{t\rightarrow \infty } \vert N_{1}(t)-N_{2}(t) \vert =0\) a.s., then Eq. (3.3) is globally attractive.
Lemma 1
If \(N(t)\) is a solution of (3.3) without impulsive effects for any initial value \(N(0)=N_{0}>0\). If Assumption 1 holds, then almost every sample path of \(N(t)\) is uniformly continuous for \(t\geq 0\).
Lemma 2
If f is a non-negative function defined on \(R_{+}\) such that f is integrable on \(R_{+}\) and is uniformly continuous on \(R_{+}\), then \(\lim_{t\rightarrow \infty }f(t)=0\).
In the following, we will give our main result.
Theorem 2
If \(r-0.5\alpha _{1}^{2}-\frac{m\tau }{T}<0\), then the solution \(P(t)\) of SDE (3.5) with any positive initial value has \(\lim_{t\rightarrow \infty } P(t) = 0\).
Proof
Theorem 3
The solution of model (3.5) is globally attractive.
Proof
3.3 The persistence and stochastically ultimate boundedness
In order to investigate the persistence and stochastically ultimate boundedness, we need to guarantee the existence and uniqueness of the positive solution. In the following, we illustrate the definitions of solutions of system (3.2).
Theorem 4
Model (3.2) has a unique solution \((P(t), N(t))\) on \(t > 0\) for any given initial value \((P(0),N(0))\in R_{+}^{2}=\{(P(t), N(t))\in R_{+}^{2} \mid P(t)>0, N(t)>0\}\) and the solution will remain in \(R_{+}^{2}\) a.s.
Proof
Theorem 4 indicates that model (3.2) has a unique global positive solution. This main result allows us to further examine how the solution varies in \(R_{+}^{2}\) in more detail. Now let us further examine how this solution pathwisely moves in \(R_{+}\).
Assumption 1
There are two positive constants L and U such that \(L \leq \prod_{0< nT< t}(1+q(nT))\leq U\).
Theorem 5
Proof
For better discussion later, we give several definitions, then try to explore sufficient conditions for them.
Definition 3
If \(\lim_{t\rightarrow \infty }\sup \frac{\int _{0}^{t}P(s)\,ds}{t}>0\) a.s., then the species \(P(t)\) is weakly persistent in the mean.
Definition 4
We are now in a position to prove persistence. For the pest \(P(t)\) and the natural enemy \(N(t)\), we have the following results.
Theorem 6
- (i)
If \(r-0.5\alpha _{1}^{2}-\frac{m\tau }{T}>0\), \(\lim_{t\rightarrow \infty }\sup \frac{\sum_{0< nT< t}\ln (1+q(nT))}{t}<-d+0.5\alpha _{2}^{2}\), then the pest \(P(t)\) is weakly persistent in the mean.
- (ii)
If \(\lambda \beta (r-0.5\alpha _{1}^{2}-\frac{m\tau }{T}) +b \lim_{t\rightarrow \infty }\sup \frac{\sum_{0< nT< t}\ln (1+q(nT))}{t}+d-0.5\alpha _{2}^{2}>0\), and \(\lim_{t\rightarrow \infty }\sup \frac{\sum_{0< nT< t}\ln (1+q(nT))}{\ln t}<\infty \), then the natural enemy \(N(t)\) will be weakly persistent in the mean.
Proof
Theorem 7
The solution \((P(t),N(t))\) of the equation system (3.2) is stochastically bounded.
Proof
4 Discussion
In this paper, a simple stochastic mathematical model of IPM systems with pesticides that have residual effects is proposed and studied. Theorem 4 shows that our model has a global positive solution for any given positive initial value, and we obtain its explicit expression under the pest-eradication condition. Firstly, we consider chemical control only, with the other parameters left the same as in model (3.2). Secondly, under \(r-0.5\alpha _{1}^{2}-\frac{m\tau }{T}<0\), which implies that the pest will die out eventually, we also prove the global attractivity of the pest solution when the natural enemy disappears.
It follows from Theorem 6 that we prove the persistence of pest and natural enemy when they satisfy the conditions respectively. Fortunately, Theorem 7 proves that the solution of model (3.2) is stochastically bounded with \(L \leq \prod_{0< nT< t}(1+q(nT))\leq U\) (L and U are positive constants).
Integrated pest management is a complex process. In fact, farmers and other pest managers usually control pests so that they cannot exceed the economic injury level. How to model an IPM strategy with residual effects of pesticides, taking account of the economic injury level, will also be studied in future research to include analyses of dynamical behavior of the models and the biological implications of the results.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No.11601268, 11461082, 11461083, 31460297), the Educational Commission of Yunnan Province (2015Y223), and the Educational Commission of Hubei Province (Q20161212). Moreover, this work is partly supported by Key Laboratory of IOT Application Technology of Universities in Yunnan Province.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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