Dynamics of a Stage Structured Pest Control Model in a Polluted Environment with Pulse Pollution Input

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 678762, 8 pages http://dx.doi.org/10.1155/2013/678762 Research Article Dynamics of a Stage Structured Pest Control Model in a Polluted Environment with Pulse Pollution Input Bing Liu,1 Ling Xu,2 and Baolin Kang1 1 2 Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China Correspondence should be addressed to Bing Liu; Received 18 December 2012; Revised 3 August 2013; Accepted 23 August 2013 Academic Editor: Maoan Han Copyright © 2013 Bing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using pollution model and impulsive delay differential equation, we formulate a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments and investigate the dynamics of such a system. We assume only that the natural enemies are affected by pollution, and we choose the method to kill the pest without harming natural enemies. Sufficient conditions for global attractivity of the natural enemy-extinction periodic solution and permanence of the system are obtained. Numerical simulations are presented to confirm our theoretical results. 1. Introduction Nowadays, the problem of the world’s environmental pollution is serious, which has a frustrating effect on the ecosystem damage in the direct or indirect ways. Pollution leads to the living environmental change and gene mutation. It results in not only birth defects and deformities but also population variability, which decreases the number of the population in the nature and even makes them extinct. In order to assess the risk of the populations exposed to a polluted environment, in recent years, mathematical models concerning this topic have been studied extensively including continuous pollution input and impulsive pollution input [1–11]. As we all know, the predator-prey system can be used to model the process of controlling the pests by spraying pesticides, as well as relying on their natural enemies. However, in a polluted environment, some natural enemies are affected by pollution seriously and pests almost are not affected. For example, frogs are the natural enemies of beetles, locusts, and mole cricket, but some chemical plants discard waste products into rivers for their convenience, which cause severe water contamination, seriously injures frog’s reproductive system, and significantly decreases their fertility. Moreover, water pollution also causes large quantities of the fertilized eggs and tadpoles to die, resulting in the decrease of frogs. It is shown in a Sweden’s new study that male tadpoles can eventually grow into female frogs only in the environment similar to the nature but full of pollutants with estrogen. However, some male frogs have ovaries but no fallopian tubes, and they finally turn into lifelong infertile frogs, which are called “Yin and Yang frog”, and nearly one-third of the world’s frog species may be extinct because of the environmental pollution. People must control the period and quantity of emission of pollution to prevent natural enemy from extinction. In addition, too much pesticide spraying will reduce pests significantly; meanwhile, it also causes serious environmental pollution. Therefore, when controlling pests, we had better choose the method to kill the pests without polluting the environment and harming natural enemies at regular intervals. The predator-prey models with stage structure for the predator were introduced or investigated by Hastings and Wang [12–14]. Since the immature predator takes 𝜏 (which is called maturation time delay) units of time to mature, the death toll during the juvenile period should be considered, and time delays have important biological meanings in stage structured models. Recently, many models with time delay were extensively studied [15–22]. 2 Journal of Applied Mathematics According to the above biological background, in this paper, we suggest an impulsive predator-prey pollution model with stage structured for predator by introducing a constant periodic pollutant input and proportional killing pest at different fixed moments to model the process of pest control and polluted environment. Recently, there has been quite a lot of literatures on the applications of impulsive differential equations on population [1, 2, 8, 10, 11, 20–31]. To our knowledge, there have been no results on this topic in the literature. The questions that arise here are as follows: how do we control the emission of pollution to prevent the extinction of natural enemies? Under what condition can the system be permanent? How can we control pests effectively? The organization of this paper is as follows. In the next section, we formulate our model and give several lemmas which are useful for our main results. In Section 3 and Section 4, the sufficient conditions for the global attractivity of the “natural enemy-extinction” periodic solution and permanence of the system are obtained. We give a brief discussion of our results in Section 5. Numerical simulations are presented to illustrate our theoretical results. 2. Model Formulation and Preliminaries In this paper, we assume only that the natural enemies are affected by pollution and we choose the method to kill the pest without harming natural enemies. Then a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pests at different fixed moment is formulated as follows: 𝑞𝑥 (𝑡) 𝑦2 (𝑡) 𝑥 (𝑡) 𝑑𝑥 (𝑡) , = 𝛽𝑥 (𝑡) (1 − )− 𝑑𝑡 𝐾 1 + 𝛼𝑥 (𝑡) 𝑑𝑦1 (𝑡) 𝑞𝑥 (𝑡) 𝑦2 (𝑡) =𝜆 𝑑𝑡 1 + 𝛼𝑥 (𝑡) Δ𝑐0 (𝑡) = 0, Δ𝑦2 (𝑡) = 0, Δ𝑐𝑒 (𝑡) = 0, 𝑡 = 𝑛𝑇, (1) where 0 ≤ 𝑙 ≤ 1, Δ𝑥(𝑡) = 𝑥(𝑡+ ) − 𝑥(𝑡), Δ𝑦𝑖 (𝑡) = 𝑦𝑖 (𝑡+ ) − 𝑦𝑖 (𝑡) (𝑖 = 1, 2), Δ𝑐0 (𝑡) = 𝑐0 (𝑡+ ) − 𝑐0 (𝑡), Δ𝑐𝑒 (𝑡) = 𝑐𝑒 (𝑡+ ) − 𝑐𝑒 (𝑡). 𝑥(𝑡), 𝑦1 (𝑡), and 𝑦2 (𝑡) represent the densities of prey (pest), immature, and mature predator (natural enemy) at time 𝑡, respectively; 𝑐𝑒 (𝑡), 𝑐0 (𝑡) represent the concentration of pollution in the environment and organism at time 𝑡, respectively; 𝛽 is intrinsic growth rate of the pests in the absence of natural enemies; 𝐾 > 0 is the pest capacity of environment; 𝑞 is the predation rate of natural enemy and 𝜆 represents the conversion rate at which ingested pest in excess of what is needed for maintenance is translated into natural enemy increase; 𝛼 is the saturation which represents that a certain amount of natural enemies can prey on a limited amount of pests, though the pests are numerous; 𝑑 and 𝛾 are the death rate of immature and mature natural enemies, respectively; in addition, we assume that (...truncated)