Delay-Induced Oscillations in a Competitor-Competitor-Mutualist Lotka-Volterra Model

Hindawi Complexity Volume 2017, Article ID 2578043, 12 pages https://doi.org/10.1155/2017/2578043 Research Article Delay-Induced Oscillations in a Competitor-Competitor-Mutualist Lotka-Volterra Model Changjin Xu Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China Correspondence should be addressed to Changjin Xu; Received 14 January 2017; Accepted 30 March 2017; Published 26 April 2017 Academic Editor: Alicia Cordero Copyright © 2017 Changjin Xu. 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. This paper deals with a competitor-competitor-mutualist Lotka-Volterra model. A series of sufficient criteria guaranteeing the stability and the occurrence of Hopf bifurcation for the model are obtained. Several concrete formulae determine the properties of bifurcating periodic solutions by applying the normal form theory and the center manifold principle. Computer simulations are given to support the theoretical predictions. At last, biological meaning and a conclusion are presented. 1. Introduction The study of dynamical behaviors for tremendous predatorprey models has been a hot issue in population dynamics in the past few decades. Many results have been reported [1–11]. In the real world, any biological or environmental parameters are naturally subject to fluctuation in time. The effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Meanwhile, time delay due to gestation is common example because, generally, the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Based on all the above point, Lv et al. [12] had investigated the periodic solution of the following competitor-competitor-mutualist Lotka-Volterra model by using Krasnoselskii’s fixed point theorem 𝑥̇ 1 (𝑡) = 𝑥1 (𝑡) [𝑟1 (𝑡) − 𝑎11 (𝑡) 𝑥1 (𝑡 − 𝜏11 (𝑡)) 𝑥̇ 1 (𝑡) − 𝑎12 (𝑡) 𝑥2 (𝑡 − 𝜏12 (𝑡)) + 𝑎13 (𝑡) 𝑥3 (𝑡 − 𝜏13 (𝑡))] , 𝑥̇ 2 (𝑡) = 𝑥2 (𝑡) [𝑟2 (𝑡) − 𝑎21 (𝑡) 𝑥1 (𝑡 − 𝜏21 (𝑡)) − 𝑎22 (𝑡) 𝑥2 (𝑡 − 𝜏22 (𝑡)) + 𝑎23 (𝑡) 𝑥3 (𝑡 − 𝜏23 (𝑡))] , 𝑥̇ 3 (𝑡) = 𝑥3 (𝑡) [𝑟3 (𝑡) + 𝑎31 (𝑡) 𝑥1 (𝑡 − 𝜏31 (𝑡)) + 𝑎32 (𝑡) 𝑥2 (𝑡 − 𝜏32 (𝑡)) − 𝑎33 (𝑡) 𝑥3 (𝑡 − 𝜏33 (𝑡))] , where 𝑥1 (𝑡) and 𝑥2 (𝑡) denote the densities of competing species at time 𝑡 and 𝑥3 (𝑡) denotes the density of cooperating species at time 𝑡. 𝑟𝑖 , 𝑎𝑖𝑗 ∈ 𝐶(𝑅, [0, ∞)) and 𝜏𝑖𝑗 ∈ 𝐶(𝑅, 𝑅) are 𝜔-periodic functions (𝜔 > 0). The parameters 𝜏𝑖𝑗 (𝑡) ≥ 0 (𝑖 = 1, 2, 3; 𝑗 = 1, 2, 3) are the feedback time delay of different species. In detail, one can see [12]. It is well known that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation, is one of the most important themes on the predator-prey dynamics. There are a great deal of papers which deal with this topic [11, 13–22]. The purpose of this paper is to discuss the stability and the properties of Hopf bifurcation of model (1). To simplify the analysis for model (1), we make the following assumptions: all biological and environmental parameters are constants in time and only the feedback time delay of competing species 𝑥𝑖 (𝑖 = 1, 2) to the growth of the species itself and the feedback time delay of cooperating species 𝑥3 to the growth of the species itself exist and are the same. Then system (1) can be described as the form (1) = 𝑥1 (𝑡) [𝑟1 − 𝑎11 𝑥1 (𝑡 − 𝜏) − 𝑎12 𝑥2 (𝑡) + 𝑎13 𝑥3 (𝑡)] , 𝑥̇ 2 (𝑡) = 𝑥2 (𝑡) [𝑟2 − 𝑎21 𝑥1 (𝑡) − 𝑎22 𝑥2 (𝑡 − 𝜏) + 𝑎23 𝑥3 (𝑡)] , 𝑥̇ 3 (𝑡) = 𝑥3 (𝑡) [𝑟3 + 𝑎31 𝑥1 (𝑡) + 𝑎32 𝑥2 (𝑡) − 𝑎33 𝑥3 (𝑡 − 𝜏)] . (2) 2 Complexity In this paper, we consider the effect of time delay 𝜏 on the dynamics of system (2). We not only give the conditions on the stability of the positive equilibrium of (2) and the existence of periodic solutions but also derive the formulae for determining the properties of a Hopf bifurcation. The remainder of the paper is organized as follows. In Section 2, we investigate the stability of the positive equilibrium and the occurrence of local Hopf bifurcations. In Section 3, the direction and stability of the local Hopf bifurcation are established. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Biological explanations and some main conclusions are drawn in Section 5. 𝑥̇ 3 (𝑡) = 𝑑1 𝑥1 (𝑡) + 𝑑2 𝑥2 (𝑡) + 𝑑3 𝑥3 (𝑡 − 𝜏) + 𝑑4 𝑥1 (𝑡) 𝑥3 (𝑡) + 𝑑5 𝑥2 (𝑡) 𝑥3 (𝑡) + 𝑑6 𝑥3 (𝑡) 𝑥3 (𝑡 − 𝜏) , (4) where 𝑏1 = −𝑎11 𝑥1∗ , 𝑏2 = −𝑎12 𝑥1∗ , 𝑏3 = 𝑎13 𝑥1∗ , 𝑏4 = −𝑎11 , 2. Stability of the Positive Equilibrium and Local Hopf Bifurcations 𝑏5 = −𝑎12 , 𝑏6 = 𝑎13 , Consider the realistic implication and actual application of biological system; in this section, we shall only study the stability of the positive equilibrium and the existence of local Hopf bifurcations. It is easy to see that system (2) has a unique positive equilibrium 𝐸0 (𝑥1∗ , 𝑥2∗ , 𝑥3∗ ) if the condition sign {Δ} = sign {Δ 1 } = sign {Δ 2 } = sign {Δ 3 } 𝑐1 = −𝑎21 𝑥2∗ , 𝑐2 = −𝑎22 𝑥2∗ , 𝑐3 = 𝑎23 𝑥2∗ , (H1) (5) 𝑐4 = −𝑎21 , 𝑐5 = −𝑎22 , holds, where 𝑐6 = 𝑎23 , 𝑎11 𝑎12 −𝑎13 𝑑1 = 𝑎31 𝑥3∗ , Δ = det (𝑎21 𝑎22 −𝑎23 ) , 𝑎31 𝑎31 −𝑎33 𝑑2 = 𝑎32 𝑥3∗ , 𝑟1 𝑎12 −𝑎13 𝑑3 = −𝑎33 𝑥3∗ , Δ 1 = det ( 𝑟2 𝑎22 −𝑎23 ) , −𝑟3 𝑎31 −𝑎33 𝑎11 𝑟1 −𝑎13 𝑑4 = 𝑎31 , 𝑑5 = 𝑎32 , (3) Δ 2 = det (𝑎21 𝑟2 −𝑎23 ) , 𝑎31 −𝑟3 −𝑎33 𝑑6 = −𝑎33 . The linearization of (6) near (0, 0, 0) is given by 𝑥̇ 1 (𝑡) = 𝑏1 𝑥1 (𝑡 − 𝜏) + 𝑏2 𝑥2 (𝑡) + 𝑏3 𝑥3 (𝑡) , 𝑎11 𝑎12 𝑟1 Δ 3 = det (𝑎21 𝑎22 𝑟2 ) . 𝑥̇ 2 (𝑡) = 𝑐1 𝑥1 (𝑡) + 𝑐2 𝑥2 (𝑡 − 𝜏) + 𝑐3 𝑥3 , 𝑎31 𝑎31 −𝑟3 Let 𝑥1 (𝑡) = 𝑥1 (𝑡) − 𝑥1∗ , 𝑥2 (𝑡) = 𝑥2 (𝑡) − 𝑥2∗ , and 𝑥3 (𝑡) = 𝑥3 (𝑡) − 𝑥3∗ and still denote 𝑥𝑖 (𝑡) (𝑖 = 1, 2, 3) by 𝑥𝑖 (𝑡) (𝑖 = 1, 2, 3), and then (2) takes the form 𝑥̇ 1 (𝑡) = 𝑏1 𝑥1 (𝑡 − 𝜏) + 𝑏2 𝑥2 (𝑡) + 𝑏3 𝑥3 (𝑡) 𝑥̇ 3 (𝑡) = 𝑑1 𝑥1 (𝑡) + 𝑑2 𝑥2 (𝑡) + 𝑑3 𝑥3 (𝑡 − 𝜏) , whose characteristic equation takes the form 𝜆 − 𝑏1 𝑒−𝜆𝜏 det ( 𝑥̇ 2 (𝑡) = 𝑐1 𝑥1 (𝑡) + 𝑐2 𝑥2 (𝑡 − 𝜏) + 𝑐3 𝑥3 + 𝑐4 𝑥1 (𝑡) 𝑥2 (𝑡) + 𝑐5 𝑥2 (𝑡) 𝑥2 (𝑡 − 𝜏) + 𝑐6 𝑥2 (𝑡) 𝑥3 (𝑡) , −𝑐1 −𝑏2 −𝑏3 𝜆 − 𝑐2 𝑒−𝜆𝜏 −𝑐3 −𝑑1 + 𝑏4 𝑥1 (𝑡) 𝑥1 (𝑡 − 𝜏) + 𝑏5 𝑥1 (𝑡) 𝑥2 (𝑡) + 𝑏6 𝑥1 (𝑡) 𝑥3 (𝑡) , (6) −𝑑2 ) = 0. (7) −𝜆𝜏 𝜆 − 𝑑3 𝑒 That is, 𝜆3 + 𝑚1 𝜆 + 𝑚2 + (𝑚3 𝜆2 + 𝑚4 ) 𝑒−𝜆𝜏 + 𝑚5 𝜆𝑒−2𝜆𝜏 + 𝑚6 𝑒−3𝜆𝜏 = 0, (8) Complexity 3 where It follows from (14) that 𝑚1 = 𝑏2 𝑐1 + 𝑐3 𝑑2 − 𝑏3 𝑑1 , [𝑚2 cos 𝜔𝜏 − (𝑚1 𝜔 − 𝜔3 ) sin 𝜔𝜏 + 𝑚4 𝑚2 = −𝑏1 𝑐3 𝑑1 − 𝑏3 𝑑2 𝑐1 , 𝑚3 = − (𝑏1 + 𝑐2 + 𝑑3 ) , 𝑚4 = 𝑏3 𝑐2 𝑑1 − 𝑏2 𝑐1 𝑑3 − 𝑏1 𝑑2 𝑐3 , 2 + 𝑚5 𝜔 sin 𝜔𝜏] + [𝑚2 sin 𝜔𝜏 + (𝑚1 𝜔 − 𝜔3 ) cos 𝜔𝜏 (15) (9) 𝑚5 = (𝑏2 + 𝑐2 ) 𝑑3 + 𝑏1 𝑐2 , According to sin 𝜔𝜏 = ±√1 − cos2 𝜔𝜏, then (15) takes the form 𝑚6 = −𝑏1 𝑐2 𝑑3 . Multiplying 𝑒𝜆𝜏 on both sides of (8), it is easy to obtain (𝜆3 + 𝑚1 𝜆 + 𝑚2 ) 𝑒𝜆𝜏 + 𝑚3 𝜆2 + 𝑚4 + 𝑚5 𝜆𝑒−𝜆𝜏 + 𝑚6 𝑒−2𝜆𝜏 = 0. [𝑚2 cos 𝜔𝜏 − (𝑚1 𝜔 − 𝜔3 ) (±√1 − cos2 𝜔𝜏) + 𝑚4 (10) We need the (...truncated)