Nonlinear Dynamic in an Ecological System with Impulsive Effect and Optimal Foraging

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 169609, 12 pages http://dx.doi.org/10.1155/2014/169609 Research Article Nonlinear Dynamic in an Ecological System with Impulsive Effect and Optimal Foraging Min Zhao1,2 and Chuanjun Dai1,2 1 2 School of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang 325035, China Zhejiang Provincial Key Laboratory for Water Environment and Marine Biological Resources Protection, Wenzhou University, Wenzhou, Zhejiang 325035, China Correspondence should be addressed to Min Zhao; Received 13 January 2014; Accepted 15 May 2014; Published 16 June 2014 Academic Editor: Weiming Wang Copyright © 2014 M. Zhao and C. Dai. 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. The population dynamics of a three-species ecological system with impulsive effect are investigated. Using the theories of impulsive equations and small-amplitude perturbation scales, the conditions for the system to be permanent when the number of predators released is less than some critical value can be obtained. Furthermore, because the predator in the system follows the predictions of optimal foraging theory, it follows that optimal foraging promotes species coexistence. In particular, the less beneficial prey can support the predator alone when the more beneficial prey goes extinct. Moreover, the influences of the impulsive effect and optimal foraging on inherent oscillations are studied using simulation, which reveals rich dynamic behaviors such as period-halving bifurcations, a chaotic band, a periodic window, and chaotic crises. In addition, the largest Lyapunov exponent and the power spectra of the strange attractor, which can help analyze the chaotic dynamic behavior of the model, are investigated. This information will be useful for studying the dynamic complexity of ecosystems. 1. Introduction In recent years, interest in studying nonlinear dynamic systems has exploded. In the 1970s, since the pioneering work of May on the relationship between food-web complexity and stability and the chaotic phenomenon [1–3], more and more researchers have become interested in dynamic behavior involving ecological mechanisms that promote species diversity [4–20]. More recently, dynamic systems’ studies have benefited from an infusion of interest and new techniques in ecology. It is known that when a predator is shared by two noncompeting species, predator-mediated apparent competition often leads to competitive exclusion of one prey population [21]. This phenomenon is related to optimal foraging and adaptive foraging. A two-prey-one-predator population model with optimal predator foraging behavior has been studied in a fine-grained environment [22–24]. On this basis, Křivan and Eisner considered a system composed of two prey types and an optimally foraging predator [25] in a system described by the following model: 𝑥̇ (𝑡) = 𝑥 (𝑡) (𝑟1 (𝑥 (𝑡)) − 𝜆 1 𝑧 (𝑡) ) 1 + ℎ1 𝜆 1 𝑥 (𝑡) + 𝑢ℎ2 𝜆 2 𝑦 (𝑡) 𝑦̇ (𝑡) = 𝑦 (𝑡) (𝑟2 (𝑦 (𝑡)) − 𝑧̇ (𝑡) = 𝑧 (𝑡) ( (1) 𝑢𝜆 2 𝑧 (𝑡) ) 1 + ℎ1 𝜆 1 𝑥 (𝑡) + 𝑢ℎ2 𝜆 2 𝑦 (𝑡) 𝑒1 𝜆 1 𝑥 (𝑡) + 𝑢𝑒2 𝜆 2 𝑦 (𝑡) − 𝑚) . 1 + ℎ1 𝜆 1 𝑥 (𝑡) + 𝑢ℎ2 𝜆 2 𝑦 (𝑡) This paper considers an impulsive differential-equation model based on model (1), which assumes that predators 2 Abstract and Applied Analysis forage according to optimal foraging theory [23, 24]. This system can be expressed by the following equations: 𝑘 − 𝑥 (𝑡) ) − 𝑏1 𝑥2 (𝑡) 𝑥̇ (𝑡) = 𝑟1 𝑥 (𝑡) ( 0 𝑘1 − 𝑥 (𝑡) − 𝜆 1 𝑥 (𝑡) 𝑧 (𝑡) 1 + ℎ1 𝜆 1 𝑥 (𝑡) + 𝑢ℎ2 𝜆 2 𝑦 (𝑡) 𝑘 − 𝑦 (𝑡) 𝑦̇ (𝑡) = 𝑟2 𝑦 (𝑡) ( 2 ) − 𝑏2 𝑦2 (𝑡) 𝑘3 − 𝑦 (𝑡) 2. Analysis of the System 𝑢𝜆 2 𝑦 (𝑡) 𝑧 (𝑡) − 1 + ℎ1 𝜆 1 𝑥 (𝑡) + 𝑢ℎ2 𝜆 2 𝑦 (𝑡) 𝑧̇ (𝑡) = 𝑧 (𝑡) ( 𝑒1 𝜆 1 𝑥 (𝑡) + 𝑢𝑒2 𝜆 2 𝑦 (𝑡) − 𝑚) 1 + ℎ1 𝜆 1 𝑥 (𝑡) + 𝑢ℎ2 𝜆 2 𝑦 (𝑡) conditions for extinction, and obtain the conditions for permanence of System (2) using the Floquet theory of impulsive equations at small-amplitude perturbation scales. In Section 3, the results of computer-based numerical analysis are shown and discussed briefly. In addition, the largest Lyapunov exponent, which also indicates the chaotic dynamic behavior of the model, is computed, and the Fourier spectra, which illustrate the qualitative nature of strange attractors, are plotted. Finally, conclusions and remarks are stated. (2) 𝑡 ≠ 𝑛𝑇 𝑥 (𝑡+ ) = 𝑥 (𝑡) 𝑦 (𝑡+ ) = 𝑦 (𝑡) 𝑧 (𝑡+ ) = 𝑧 (𝑡) + 𝑝 𝑡 = 𝑛𝑇, where 𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡) are, respectively, the densities of two prey types and one predator at time 𝑡, 𝑟𝑖 (𝑖 = 1, 2) is the per capita prey intrinsic growth rate, 𝑟1 ⋅ 𝑘0 (0 ≤ 𝑘0 /𝑘1 ≤ 1), 𝑟2 ⋅ 𝑘2 (0 ≤ 𝑘2 /𝑘3 ≤ 1) are the respective carrying capacities of the prey, and 𝑘1 , 𝑘3 are the corresponding values of available resources or in the ideal case (i.e., where no resources are wasted) the carrying capacity. However, the ideal case is impossible in reality. The ratios 𝑘0 /𝑘1 , 𝑘2 /𝑘3 express the relative efficiency of nutrient utilization in species 𝑥, 𝑦. At any time, if 𝑥 < 𝑘0 < 𝑘1 , 𝑦 < 𝑘2 < 𝑘3 , the efficiency is high as long as 𝑘0 /𝑘1 , 𝑘2 /𝑘3 are close to one; when the values are lower, this indicates that resource limitations are restricting the population increase [23]. 𝑏𝑖 (𝑖 = 1, 2) are the rate of intraspecific competition of the prey, 𝜆 𝑖 (𝑖 = 1, 2) is the cropping rate of a predator feeding on the 𝑖th prey type, 𝑒𝑖 (𝑖 = 1, 2) is the conversion factor relating predator reproduction to prey consumption, and ℎ𝑖 (𝑖 = 1, 2) is the per capita mortality rate for the forager. In this paper, it is assumed that prey type 𝑥 is more beneficial than the other and hence 𝑒1 /ℎ1 > 𝑒2 /ℎ2 [26, 27]. To study optimal foraging, a control parameter 𝑢 (0 ≤ 𝑢 ≤ 1) is introduced [25], which represents the probability that the alternative second prey type is included in the predator’s diet. 𝑇 is the period of the impulsive effect, 𝑛 ∈ 𝑁, and 𝑝 > 0 is the number of predators released at 𝑡 = 𝑛𝑇. To achieve a set of conditions which can guarantee that the system will be permanent and that the numbers of the two prey types are not so large that they go extinct because of exceeding the carrying capacity of the environment, the model will release a certain number of predators only at 𝑡 = 𝑛𝑇 because the predator is assumed to be a versatile and advanced predator. The rest of this paper is organized as follows. Section 2 will review the effect of impulsive perturbations, establish Let 𝑅+ = [0, ∞), 𝑅+ = {𝑋 ∈ 𝑅3 : 𝑋 ≥ 0, 𝑋 = (𝑥, 𝑦, 𝑧)}, Ω = int 𝑅+3 , and let 𝑁 be the set of all nonnegative integers. The map 𝑔 = (𝑔1 , 𝑔2 , 𝑔3 )𝑇 is defined by the right-hand side of the first three equations of System (2). Let 𝑉 : 𝑅+ × 𝑅+3 → 𝑅+ ; then 𝑉 is said to belong to class 𝑉0 if (1) 𝑉 is continuous in (𝑛𝑇, (𝑛 + 1)𝑇] × 𝑅+3 , and for each 𝑥 ∈ 𝑅+3 , 𝑛 ∈ 𝑁, lim(𝑡,𝑦) → (𝑛𝑇+ ,𝑥) 𝑉(𝑡, 𝑦) = 𝑉(𝑛𝑇+ , 𝑥) exists; (2) 𝑉 is locally Lipschitzian in 𝑋. Definition 1. Let 𝑉 ∈ 𝑉0 ; th (...truncated)