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

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.


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][2][3][4][5][6][7][8][9][10][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 where  1 () and  2 () denote the densities of competing species at time  and  3 () denotes the density of cooperating species at time .  ,   ∈ (, [0, ∞)) and   ∈ (, ) are -periodic functions ( > 0).The parameters   () ≥ 0 ( = 1, 2, 3;  = 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][14][15][16][17][18][19][20][21][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   ( = 1, 2) to the growth of the species itself and the feedback time delay of cooperating species  3 to the growth of the species itself exist and are the same.Then system (1) can be described as the form (2) 2

Complexity
In this paper, we consider the effect of time delay  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.

Stability of the Positive Equilibrium and Local Hopf Bifurcations
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  0 ( ) . ( 3 and still denote   () ( = 1, 2, 3) by   () ( = 1, 2, 3), and then (2) takes the form where The linearization of (6) near (0, 0, 0) is given by whose characteristic equation takes the form det That is, where Multiplying   on both sides of (8), it is easy to obtain We need the following lemma to discuss the stability of the positive equilibrium.
For  = 0, (10) becomes Obviously,  3 > 0. By the Routh-Hurwitz criteria, it follows that all eigenvalues of (12) have negative real parts if and only if the condition is fulfilled.
For  > 0,  is a root of (10) if and only if Separating the real and imaginary parts, we get It follows from ( 14) that According to sin  = ± √ 1 − cos 2 , then (15) takes the form It is easy to see that ( 16) is equivalent to where where Let cos  =  and denote It is easy to obtain that   ( Let  =  +  2 /4 1 .Then (21) becomes where . By ( 22), then we obtain By the discussion above, we can obtain the expression of cos , say where  1 () is a function with respect to .Substitute ( 24) into (15); then we can easily get the expression of sin , say where  2 () is a function with respect to .Thus we obtain If all the coefficients of system (2) are given, it is easy to use computer to calculate the roots of (26) (say ).Then from (24), we derive Let () = () + () be a root of (10) near  =   , (  ) = 0, and (  ) =   .Due to functional differential equation theory, for every   ,  = 0, 1, 2, 3, . .., there exists  > 0 such that () is continuously differentiable in  for | −   | < .Substituting () into the left hand side of (10) and taking derivative with respect to , we have Then where In order to obtain the main results in this paper, it is necessary to make the following assumption: In view of Lemma 1, it is easy to obtain the following result on stability and bifurcation of system (2).
By the representation theorem, there is a matrix function with bounded variation components (, ),  ∈ [−1, 0], such that In fact, we can choose where  is the Dirac delta function.