Abstract
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 predator-prey 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 theoremwhere and denote the densities of competing species at time and denotes the density of cooperating species at time . and are -periodic functions . The parameters 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 to the growth of the species itself and the feedback time delay of cooperating species to the growth of the species itself exist and are the same. Then system (1) can be described as the form
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.
2. 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 if the condition holds, where Let , , and and still denote by , and then (2) takes the formwhereThe linearization of (6) near is given bywhose characteristic equation takes the formThat is,where Multiplying on both sides of (8), it is easy to obtainWe need the following lemma to discuss the stability of the positive equilibrium.
Lemma 1 (see [23]). For the transcendental equation as vary, the sum of orders of the zeros of in the open right half complex plane can change, only if a zero appears on or crosses the imaginary axis.
For , (10) becomesObviously, . 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 , is a root of (10) if and only ifSeparating the real and imaginary parts, we getIt follows from (14) thatAccording to , then (15) takes the formIt is easy to see that (16) is equivalent towhere where Let and denote It is easy to obtain that SetLet Then (21) becomeswhere and
Define , By (22), then we obtain By the discussion above, we can obtain the expression of , saywhere is a function with respect to . Substitute (24) into (15); then we can easily get the expression of , saywhere is a function with respect to . Thus we obtainIf 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 , , and Due to functional differential equation theory, for every , there exists 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).
Theorem 2. Suppose that – hold; then (i)for system (2), its positive equilibrium is asymptotically stable for ;(ii)system (2) undergoes a Hopf bifurcation at the positive equilibrium when ; that is, system (2) has a branch of periodic solutions bifurcating from the positive equilibrium near .
3. Direction and Stability of the Hopf Bifurcation
In this section, we shall analyze the direction, stability, and period of these periodic solutions bifurcating from the positive equilibrium at these critical value of , by using techniques from normal form and center manifold theory [24]. Throughout this section, we always assume that system (2) undergoes Hopf bifurcation at the positive equilibrium for , , and then are corresponding purely imaginary roots of the characteristic equation at the positive equilibrium .
For convenience, let and , where is defined by (27) and , drop the bar for the simplification of notations, and then system (4) can be written as an FDE in aswhere and , , , and and are given byrespectively, where and From the discussion in Section 2, we know that if , then system (31) undergoes a Hopf bifurcation at the positive equilibrium and the associated characteristic equation of system (31) has a pair of simple imaginary roots .
By the representation theorem, there is a matrix function with bounded variation components , , such thatIn fact, we can choosewhere is the Dirac delta function.
For , defineThen (31) is equivalent to the abstract differential equationwhere ,
For , define
For and , define the bilinear formwhere , , and are adjoint operators. By the discussions in Section 2, we know that are eigenvalues of , and they are also eigenvalues of .
Suppose that is the eigenvector of corresponding to ; then . It follows from definition and (32) and (35) thatThen we can obtain Similarly, let be the eigenvector of corresponding to ; then . It follows from the definition that Hence In order to assure and , we need to determine the value of . From (40), we haveThus we can choose Next, we use the same notations as those in Hassard et al. [24] and we first compute the coordinates to describe the center manifold at . Let be the solution of (31) when .
Defineon the center manifold , and we havewhereand and are local coordinates for center manifold in the direction of and . Noting that is also real if is real, we consider only real solutions. For solutions of (31)which we write in abbreviated form as where Hence we have whereThus we obtain For unknown , , , , in , we still need to compute them.
In view of (38) and (47), we havewhereComparing the coefficients, we obtainWe know that for Comparing the coefficients of (60) with (57) gives thatFrom (58), (61), and the definition of , we getNoting that , we havewhere is a constant vector.
Similarly, from (59), (62), and the definition of , we havewhere is a constant vector.
In what follows, we shall seek appropriate and in (64) and (66), respectively. It follows from the definition of , (61), and (62) thatwhere . From (58), we haveNoting that and substituting (64) and (69) into (67), we have That is, It follows thatwhere Similarly, substituting (65) and (70) into (68), we have That is, It follows thatwhereFrom (64), (66), (74), and (78), we can calculate and derive the following values: These formulae give a description of the Hopf bifurcation periodic solutions of (31) at , on the center manifold. From the discussion above, we have the following result.
Theorem 3. The direction of the Hopf bifurcation is forward (backward) if ; the bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if ; the periods of the bifurcating periodic solutions increase (decrease) if
4. Numerical Examples
In this section, numerical simulations are carried out to investigate the effect of the time delay on dynamical behaviors of system (2) as well as illustrate our theoretical results. For convenience, we assume that some parametric values of system (2) are kept as , , , , , , , , , , , and . Then system (2) takes the formwhich has a positive equilibrium and all the conditions given in Theorem 2 are satisfied. When , the positive equilibrium is asymptotically stable. Fix and by means of Matlab 7.0, we get , , and – Thus we can derive the following values: –, , , and Furthermore, it follows that and . Thus the positive equilibrium is stable when . When , we can easily plot the time series of every population and phase portrait of the system and find that the solution of (81) with initial value , , and would tend to a positive equilibrium (see Figures 1(a)–1(j), in detail); that is, the positive equilibrium is asymptotically stable. When passes through the critical value , the positive equilibrium loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from the positive equilibrium . Figures 2(a)–2(j) are plotted by fixing the time delay and initial value , , and . It is shown that a Hopf bifurcation occurs from the positive equilibrium . In view of and , we find that the direction of the Hopf bifurcation is and these bifurcating periodic solutions from at are stable.
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5. Biological Meanings and Conclusions
In this paper, the dynamics including the local stability of the positive equilibrium and the local Hopf bifurcation of a competitor-competitor-mutualist Lotka-Volterra model are investigated. It is showed that the positive equilibrium of system (2) is asymptotically stable for all when the conditions – are satisfied. This reveals that the densities of competing species and cooperating species will tend to stabilization; in other words, the densities of competing species and cooperating species will tend to , respectively, and this situation does not vary with the delay . When the delay increases, the positive equilibrium loses its stability and a sequence of Hopf bifurcations occur from the positive equilibrium ; that is, a family of periodic orbits bifurcate from the positive equilibrium . From this we can find that competing species and cooperating species can coexist and keep in periodic mode; that is, the densities of competing species and cooperating species will oscillate in the vicinity of , and , respectively. Finally, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed by applying the normal form theory and the center manifold theorem. A numerical example is also included to verify our theoretical findings.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work is supported by National Natural Science Foundation of China (no. 61673008 and no. 11261010) and Project of High-Level Innovative Talents of Guizhou Province ([2016]5651).