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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 958140, 18 pages
Bifurcation Analysis of a Lotka-Volterra Mutualistic System with Multiple Delays
School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
Received 18 April 2014; Revised 15 June 2014; Accepted 16 June 2014; Published 14 August 2014
Academic Editor: Yongli Song
Copyright © 2014 Xin-You Meng and Hai-Feng Huo. 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 class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.
In recent years, population models have merited a great deal of attention due to their theoretical and practical significance since the pioneering theoretical works by Lotka  and Volterra ; see [3–6] and references therein. Generally speaking, there are three kinds of fundamental forms of the interactions between two species such as competition, cooperation, and prey-predation in population biology. Among these interactions, the competition mechanism has been paying extreme attention because it possesses very significant function as a kind of restriction factor in the process of evolvement of biology.
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. The effect of the past history on the stability of the system is an important problem in population biology. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate. In 2002, Jin and Ma  investigated the competition model with four delays: where are maturation times and are called the hunting delays for the prey and the predator species, respectively. The result was that time delay in the competitive Lotka-Volterra system does not harm the boundedness and persistence. Song et al.  analyzed the stability of the interior positive equilibrium and the existence of local Hopf bifurcation for system (1) with by taking the delay as the bifurcation parameter. Their results showed that changes of hunting delays for system (1) do not lead to the occurrence of Hopf bifurcation when interspecies competition is weaker than intraspecies competition. Zhang et al.  studied the stability and Hopf bifurcation of system (1) with . Zhang  also investigated the dynamics of system (1) only when and .
In fact, predator-prey system with time delays has also been investigated by lots of authors [7–11]. Faria  considered the delayed predator-prey system of the form where and can be interpreted as the population densities of the prey and the predator at the time , respectively, and and denote the hunting delay and the time of predator maturation. In , taking the single delay, , as a parameter and assuming the ratio to be constant, the author studied the properties of the local Hopf bifurcation. Song et al.  obtained the properties of the local Hopf bifurcation as well as the global existence of periodic solutions by choosing the sum as a bifurcation parameter. Yan and Li  considered the following delayed predator-prey system: where denotes the feedback time delay of prey species to the growth of species itself, and also for the predator. They found that the unique positive equilibrium of system (3) will no longer be absolutely stable and the switches from stability to instability to stability disappear as the feedback time delay increases monotonously from zero, which had been obtained for system (3) by Song and Wei .
However, the research for cooperative systems with time delays is still relatively little because delays in mutualistic systems usually deprive the boundedness and persistence in . But time delays in prey-predator and competitive systems do not harm these properties [3, 14, 15]. Two species cohabit a common habitat and each species enhances the average growth rate of the other; this type of ecological interaction is known as facultative mutualism. Mutual phenomenon can increase viability and make species persistently multiply. As far as we know, the research on mutual system is less than prey-predator system and competitive system. At present, the known results of mutual system mainly focus on stability and persistence [16, 17]. Research on the ecologic system stability of positive equilibrium and existence of periodic solutions is very crucial, which can help us to realize the law for species quantity and predict the trend for species quantity. In 1997, He and Gopalsamy  considered the Lotka-Volterra mutualistic system with delay and obtained the stability of the positive equilibrium and the existence of Hopf bifurcation when . Meng and Wei  studied the following system with delay: where and are the rates of transmission between two species. They found that there are stability switches and Hopf bifurcation occurring when the delay passes through a sequence of critical values.
In general, the delays appearing in different terms of an ecological system are not equal. Therefore, it is more realistic to consider dynamical system with different delays. Motivated by the references [6, 9, 17], we consider the following two-species Lotka-Volterra mutualistic system with multiple delays: where and are the densities of two species at the time and and are the intrinsic growth rates of the two species. is the feedback delay of one species to its grow and is the feedback delay of the other species; and are the time delays of the benefit. The coefficients are all positive constants and . In addition, we would like to mention the work of F. Q. Zhang and Y. J. Zhang , who considered the system (6) only with and . In fact, feedback delay on different species should not be same, and the benefit period should not be equal to the feedback time delay. Thus, how does every delay or the combined two or three delays of the above four delays have an effect on the dynamics of system (6)? The purpose of this paper is to answer this question partially.
This paper is organized as follows. In Section 2, by analyzing the characteristic equation of linearized system of system (6) at the positive equilibrium, some sufficient conditions ensuring the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Some explicit formulas determining the direction and stability of periodic solutions bifurcating from Hopf bifurcations are given by applying the normal form method and center manifold theory due to Hassard et al.  in Section 3. In Section 4, we show the global existence of the periodic solutions due to a global Hopf bifurcation result of Wu  for functional differential equations. To support our theoretical predictions, some numerical simulations are included in Section 5. A brief discussion is also given in the last section.
2. Local Stability and Hopf Bifurcation
For convenience, letting , and dropping the bars for simplification of notation, system (6) is transformed into where , , , and .
From the point view of biological meaning, we are interested in the positive equilibrium. It is obvious that system (7) has a unique positive equilibrium defined by provided that the following condition(H1)is satisfied.
Next, we will consider the stability of the positive equilibrium and the existence of Hopf bifurcation.
The associated characteristic equation of system (10) with is It is obvious that is not the root of (11). All roots of (11) have negative real parts since and if (H1) holds. So, the equilibrium point is locally asymptotically stable when (H1) holds.
2.1. Only Considering and
Case 1A. Consider , . Equation (12) can be written in the form
We first introduce the following result which was proved by Ruan and Wei  using Rouch’s theorem.
Lemma 1. Consider the exponential polynomial where , are constants. As vary, the sum of the order of the zeros of on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
Let be the root of (13). Substituting it into (13) and separating the real and imaginary parts, we have It follows that Equation (16) does not have positive root since and . Thus, Hopf bifurcation does not occur at the positive equilibrium of system (7).
Case 1B. Consider , . Equation (12) is written in the form Equation (17) is similar to (13). Thus, Hopf bifurcation also does not occur at the positive equilibrium of system (7). We omit the corresponding proof.
Case 1D. Consider and . Let be a root of (12). We can obtain the following form by giving the value and regarding as a parameter: It follows that It is obvious that Hopf bifurcation does not occur at the positive equilibrium of system (7) according to (16).
2.2. Only Considering and
Case 2A. Consider , . Equation (21) is written as follows:
Define . Let be the root of (22) near satisfying , . We first check whether the transversality condition is satisfied.
Lemma 2. the following transversality condition is satisfied:
Proof. This will show that there exists at least one eigenvalue with positive real part for . Moreover, the conditions for the existence of a Hopf bifurcation [15, 22] are satisfied to yield a periodic solution. Differentiating (22) with respect to , it follows that
which implies that
For simplifying, define as and as , and we can obtain Thus, the transversality condition holds and Hopf bifurcation occurs at . We have the following theorem.
Theorem 3. For system (7), if (H1) holds, then there exists a positive number such that the coexistence equilibrium is locally asymptotically stable for and unstable for . Further, system (7) undergoes a Hopf bifurcation at the equilibrium for .
Suppose that . We have when (H1) holds. From the second equation of (32), we can get . Thus, we can get , . Define , . Let be the root of (31) near , satisfying , . Further, (31) has a pair of purely imaginary roots .
If , then . If , then the second equation of (32) becomes If , (33) has one positive root . Thus, we can get , . Define , . Let be the root of (31) near , satisfying , . Further, (31) has a pair of purely imaginary roots . If , and (33) has two positive roots . It is obvious that . From (32), we denote , . When (resp., ), then (resp., ) are a pair of purely imaginary roots of (31). Further, we can check that there exists a positive integral such that and . Let be the root of (31) near , satisfying , . If , then the second equation of (32) becomes , which does not have positive root.
Lemma 4. Suppose that (H1) holds.(i)If , then all roots of (31) have strictly negative real parts; if , then all roots of (31), except for ±, have strictly negative real parts.(ii)If and , then all roots of (31) have strictly negative real parts; if , then all roots of (31), except for , have strictly negative real parts.(iii)If , there exist two sequences and . Further, there exists , when , and all the roots of (31) have negative real part; when , (31) has at least one root with positive real part. When and , (31) has a pair of pure imaginary roots.
Lemma 5. The following transversality condition is satisfied:
Proof. Multiplying the into the both sides of (31) and differentiating (31) with respect to , we can obtain that For simplifying, define as and as , and we can obtain Thus, the transversality condition is satisfied. This ends the proof.
Remark 6. When , the transversality condition is satisfied as follows:
Theorem 7. For system (7), consider the following.(i)If (H1) (resp., (H1) and ) holds, then there exists a positive number (resp., ) such that the positive equilibrium is locally asymptotically stable for (resp., ) and unstable for (resp., ). Further, system (7) undergoes a Hopf bifurcation at the positive equilibrium for .(ii)If (H1) and hold, then there is a positive integer , such that the positive equilibrium switches times from stability to instability to stability; that is, the positive equilibrium of system (7) is locally asymptotically stable when and unstable when . Hopf bifurcation occurs at the positive equilibrium when and .
Case 2D. Consider and . Regarding as a parameter, we consider (21) with in its stable interval. Without loss of generality, we consider system (7) under Case 2A. Let be a root of (21), and then we obtain, by selecting delay , Eliminating the parameter , (38) leads to where and .
Define , . Equation (39) has a pair of purely imaginary roots for .
In the following, differentiating Equation (21) with respect to and substituting , we can get where
Theorem 8. For system (7), suppose that (H1) and (H2) hold and . Then, system (7) undergoes a Hopf bifurcation at the positive equilibrium when . That is, system (7) has a branch of periodic solution bifurcation from the zero solution near .
2.3. Only Considering and
Case 3C. Consider . Equation (43) is written in the form Suppose that is the root of (46) and we have that It obtains that It is obvious that (48) has only one positive root . From (47), we obtain that Equation (46) has a pair of purely imaginary roots . Define , . Let be the root of (46) near satisfying , .
Lemma 9. If the condition (H1) holds, then the following transversality condition is satisfied:
Proof. Differentiating (46) with respect to and noticing that is a function of , we can obtain that For simplifying, define as and as , and we can obtain Thus, if the condition (H1) holds, the transversality condition is satisfied. This ends the proof.
Theorem 10. For system (7), if (H1) holds, then there exists a positive number such that the interior equilibrium is locally asymptotically stable for and unstable for . Further, system (7) undergoes a Hopf bifurcation at the equilibrium for .
Case 3D. Consider and . Regarding as a parameter, we consider (43) with in its stable interval. Without loss of generality, we consider system (7) under Case 3A. Let be a root of (43). We can obtain the following results by selecting the value of delay : Eliminating the parameter , (53) leads to where , and .
Suppose that(H3)Equation (53) has at least finite positive roots.
If (H3) holds, the roots of (53) are defined by . From (53), for every fixed , there exists a sequence , where and then are a pair of purely imaginary roots of (43). Let , . When , (43) has a pair of purely imaginary roots for . Let be the root of (43) near satisfying , .
In the following, differentiating Equation (43) with respect to and substituting , we get where
Therefore, by the general Hopf bifurcation theorem for FDEs in Hale , we have t