Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 302494, 15 pages http://dx.doi.org/10.1155/2015/302494 Research Article Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation Xinyou Meng1,2 and Qingling Zhang1 1 Institute of Systems Science, Northeastern University, Shenyang 110819, China Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China 2 Correspondence should be addressed to Qingling Zhang; Received 28 August 2015; Accepted 17 September 2015 Academic Editor: Luca Guerrini Copyright © 2015 X. Meng and Q. Zhang. 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. A singular delayed biological economic predator-prey system with and without stochastic fluctuation is proposed. The conditions of singularity induced bifurcation are gained, and a state feedback controller is designed to eliminate such bifurcation. Furthermore, saddle-node bifurcation is also showed. Next, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of roots of the corresponding characteristic equation, and the hybrid control strategy is used to control the occurrence of Hopf bifurcation. In addition, some explicit formulas determining the spectral densities of the populations and harvest effort are given when the system is considered with stochastic fluctuation. Finally, numerical simulations are illustrated to verify the theoretical results. 1. Introduction The dynamic relationship between predator and prey has long been and will still be one of the dominant themes in both biology and mathematical biology because of its universal existence and importance. In the description of dynamics interactions, a crucial element of all models is the classical definition of functional response. Lots of predator-prey models with Holling type [1], Leslie-Gower type [2], and Beddington-DeAngelis type [3, 4], and so forth have been investigated extensively by scholars. However, some predatorprey models in which prey population exhibits herd behavior, such as plankton-phytoplankton model [5], may appear in realistic world. Since the use of the square root properly accounts for the assumption that the interactions occur along the boundary of the population, Ajraldi et al. [6] proposed the following predator-prey system in which interaction terms use the square root of prey population rather than simply prey population: 𝑅 (𝑡) d𝑅 (𝑡) = 𝑟𝑅 (𝑡) (1 − ) − 𝑎√𝑅 (𝑡)𝑆 (𝑡) , d𝑡 𝑁 d𝑆 (𝑡) ̃ (𝑡) + 𝑎̃ = −𝑚𝑆 𝑒√𝑅 (𝑡)𝑆 (𝑡) , d𝑡 (1) where 𝑅 and 𝑆 denote the prey and predator, respectively. The prey population exhibits a highly socialized behavior and lives in herds as the form 𝑎√𝑅(𝑡)𝑆(𝑡); that is, the weaker individuals are being kept at the center of their herd for defensive purpose. Braza [7] also investigated the dynamics of system (1) and showed that the prey exhibits strong herd structure and the predator interacts with the prey along the outer corridor of the herd of prey. Recently, Yuan et al. [8] considered a predator-prey system as follows: 𝑋 𝛼√𝑋𝑌 d𝑋 , = 𝑟𝑋 (1 − ) − d𝑡 𝑁 1 + 𝑡ℎ 𝛼√𝑋 d𝑌 𝑐𝛼√𝑋𝑌 , = −𝑠𝑌2 + d𝑡 1 + 𝑡ℎ 𝛼√𝑋 (2) where −𝑠𝑌2 represents the quadratic mortality for predator population. Predator-prey systems with such functional response have attracted little attention (see [6–9]). It is well-known that time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time, capturing time, or other reasons. Delay differential equations often show 2 Discrete Dynamics in Nature and Society much more complicated dynamics than ordinary differential equations because a time delay can cause a stable equilibrium to become unstable and cause the population to fluctuate. Many authors have been devoted to investigating the time delay effect on the dynamics of system and obtained some results (see [10–15]). Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in system (2), Xu and Yuan [9] introduced a time delay into system (2) and obtained the local stability and the existence of Hopf bifurcation of this system. However, bifurcate oscillation is harmful in some engineering applications, which has enormous potential in many technological disciplines such as power networks protection. Bifurcation control, which refers to the aim of designing a controller to suppress or reduce some existing bifurcation dynamics of a given system, can be useful. Various disciplines are attracted to bifurcation control and various methods of bifurcation control can be found [16–18]. In addition, Gordon [19] proposed the economic theory of a common-property resource, which focuses on the effect of the harvest effort on the ecosystem from an economic perspective. If 𝐸(𝑡) and 𝑌(𝑡) represent the harvest effort and the density of harvested population, respectively, then the total revenue TR = 𝜔𝐸(𝑡)𝑌(𝑡) and the total cost TC = 𝑐𝐸(𝑡), where 𝜔 represents unit price of harvested population and 𝑐 represents the cost of harvest effort. Thus, an algebraic equation, which considers the economic interest V of the harvest effort on the harvested population, is established as follows: 𝐸 (𝑡) (𝜔𝑌 (𝑡) − 𝑐) = V. (3) Based on the Gordon [19] theory and theory of singular system, Zhang et al. [20] first proposed a class of singular biological economic systems. Some results on those systems, such as the stabilities, bifurcations, and chaos, can be found in [20–24]. Based on the previous models, we establish the following predator-prey system consisting of two differential equations and an algebraic equation as follos: 𝑋 𝛼√𝑋𝑌 d𝑋 − 𝐸𝑋, = 𝑟𝑋 (1 − ) − d𝑠 𝑁 1 + 𝑡ℎ 𝛼√𝑋 ̃ d𝑌 ̃ 2 + 𝑏𝛼√𝑋 (𝑠 − 𝜏)𝑌 , = −𝑑𝑌 d𝑠 1 + 𝑡ℎ 𝛼√𝑋 (𝑠 − 𝜏) price of harvested population and the cost of harvest effort. ̃ is harvest interest of the harvest effort on the harvested 𝑚 prey. It is important to make some simplifying assumptions to discern the basic dynamics and to make the analysis more tractable. In order to simplify system (4), we use the following dimensionless transformations: 𝑥 = 𝑋/𝑁, 𝑦 = 𝛼𝑌/𝑟√𝑁, 𝑒 = 𝐸/𝑟, and 𝑡 = 𝑟𝑠. Then system (4) can be rewritten as follows: √𝑥𝑦 d𝑥 − 𝑒𝑥, = 𝑥 (1 − 𝑥) − d𝑡 1 + 𝑎√𝑥 d𝑦 𝑏√𝑥 (𝑡 − 𝜏)𝑦 , = −𝑑𝑦2 + d𝑡 1 + 𝑎√𝑥 (𝑡 − 𝜏) (5) 0 = 𝑒 (𝑝𝑥 − 𝑐) − 𝑚, ̃ 𝑚 = where 𝑎 = 𝑡𝑛 𝛼√𝑁, 𝑏 = 𝛼√𝑁̃𝑏/𝑟, 𝑐 = 𝑐̃, 𝑑 = √𝑁𝑑/𝑟, ̃ ̃ and 𝑝 = 𝑝𝑁. 𝑚/𝑟, In the papers [6–9], the author used the simplifying assumption that 𝑎 = 0; that is, the average handling time is zero. In line with the work in [6–9], we also assume that 𝑎 = 0. Thus, system (5) takes the form: d𝑥 = 𝑥 (1 − 𝑥) − √𝑥𝑦 − 𝑒𝑥, d𝑡 d𝑦 = −𝑑𝑦2 + 𝑏√𝑥 (𝑡 − 𝜏)𝑦, d𝑡 (6) 0 = 𝑒 (𝑝𝑥 − 𝑐) − 𝑚. The initial conditions of system (6) are 𝑥 (𝜃) = 𝜓 (𝜃) ≥ 0, 𝑦 (𝜃) = (...truncated)