Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system

Abstract In this paper, we consider a class of delay coupled Lotka-Volterra ring systems. Based on the symmetric bifurcation theory of delay differential equations and representation theory of standard dihedral groups, properties of phase locked periodic solutions are given. Moreover, the direction and the stability of the Hopf bifurcation periodic orbits are obtained by using normal form and center manifold theory. Finally, the research results are verified by numerical simulation.


Introduction
Mathematical models are typically used in ecology to illustrate the basic processes and dynamic mechanisms of ecosystems [1][2][3]. By the description and analysis of dynamics models, the essential characteristics of life processes can be understood more impressively. Mathematically, we usually describe the ecological mathematical model depending on the theory of functional di erential equation. Among them, Lotka-Volterra system described by the theory of functional di erential equation is one of the most famous and important ecological population dynamic models [4,5].
where g i represents the linear growth rate of species i, a ij represents the interaction between species i and j, A = (a ij ) represents the interaction matrix with a ij . It is easy to see that the model is bidirectional,that is, the growth of the species i depends on self-feedback and feedback from the species i + and i − . Among these feedbacks, the interaction between species is not necessarily symmetrical. So in general a ij ≠ a ji . Without loss of generality, we assume that all g i = a i = , which is equivalent to the carrying capacity of each population x i in the absence of other species and the unit time of the reverse growth rate of each species [5]. Golubitsky and his collaborators [6] proved that some phase relations can be modeled by coupled systems with observing the gait of animals. Under some conditions, coupled systems produce vibration, while uncoupled systems do not produce vibration. Therefore, the rich dynamic characteristics of the coupled oscillator can be understood by discussing the coupled system. Based on the practical signi cance of coupled Lotka-Volterra ring system, it has been widely used in natural science and ecology. There are many researchers who have conducted deeply studies on the dynamic characteristics of Lotka-Volterra system such as stable, unstable and oscillatory behavior [7][8][9][10][11][12][13][14][15][16][17][18][19]. In [20,21], various continuity theorems play an important role in studying the existence of periodic solutions for Lotka-Volterra systems.
Recently, symmetry has grown up to be an important subject in the study of nonlinear dynamic systems. Generally speaking, symmetry re ects some spatial invariants of dynamical systems. When the system is symmetric, it can exchange with the action of a compact Lie group Γ in Euclidean space [6]. Although symmetry makes the analysis of system more complicated, but it also imposes many special restrictions on the system. Bifurcation can occur to the system with speci c symmetry and smaller size, and in some cases, even if the system is symmetric, bifurcation will not occur. Bifurcation phenomenon refers to the qualitative change of some attributes of the object of study. In nature, bifurcation phenomenon is ubiquitous. Therefore, whether in mathematical theory or in practical application, bifurcation theory research has its great signi cance, especially in symmetric system. So the symmetric Lotka-Volterra system will produce more interesting bifurcation phenomenon, which have been studied initially by some scholars [23,24] We mainly study a class of Lotka-Volterra ring systems with coupling [25].
where i represents the number of species from to N, and assume x N+ = x for create periodic boundary conditions. The system is a closed coupled Lotka-Volterra system consisting of N identical species. Each species competes with two of the four neighbouring species for limited resources. Splott [25] studied this coupled ring system deeply and found that the system exhibits spatiotemporal chaos in a spatial dimension, and its quasi-periodic paths are chaos, bifurcation, spontaneous symmetry destruction and spatial pattern formation, however, due to its impossible connectivity, it is not a very realistic model, because the author neglected the growth cycle of species and did not consider the e ect of time delay on the model (1.1). Because time delay is an important controls parameter, it is imperative to introduce species growth time. Various scholars also incorporate time delays into the symmetric model [26][27][28][29][30][31][32][33][34]. Wu [27][28][29] used bifurcation theory to study local and global Hopf bifurcations of symmetric functional di erential equations. Zheng and Zhang [30][31][32] obtained some results of the symmetric neural network model with delay. Hu [34] and Guo [35] discussed Hopf bifurcation periodic orbits and the spatial patterns of periodic orbits respectively in neuron ring systems with delay.
Based on the original model, the dynamic behavior of three identical species connected into a ring system is considered, a new three-dimensional coupled Lotka-volterra ring system with delays is constructed by adding appropriate delay τ. We just introduce the time delay into adjacent species, assuming that any species is coupled with the nearest species and symmetrical. In the model (1.1), a simpler choice of parameters are a − = a = and a − = a = b.
where b > and τ > . Because the system is symmetry, then the characteristic equation corresponding to the linearization of the Eq. (1.2) has multiple pure imaginary roots at speci c parameter values, so the classical Hopf bifurcation theory can not be applied.
In this paper we mainly consider the dynamical properties of Eq. (1.2). The remainder of this paper is organized as follows. In section 2, we proved that a series of Hopf bifurcations will occur when the delay τ increases. In section 3, we obtained the existence and spatial pattern of multiple periodic solutions of Eq. (1.2). In section 4, the detailed calculations of the normal form on center manifold of Eq. (1.2) near to Equivariant-Hopf bifurcation points are determined. We also analysis the direction and the stability conditions of bifurcation nonsynchronous periodic solutions. In section 5, some numerical simulation are given to illustrate the results. Finally, we provides a brief conclusion of our results. The Appendix contains some detailed calculation procedures of coe cients h.

Hopf bifurcation
Consider the complex delay Eq. (1.2). In order to study the e ect of τ on the stability of equilibrium point, we need to analyze the distribution of roots of eigenvalue equation corresponding to the linear part of the system. It is clear that ( + b , + b , + b ) is an unique positive equilibrium point of Eq.(1.2). Let c = + b , then make an equilibrium transformation of Eq. (1.2), we have The linearization of Eq. (2.1) at origin as follows: regarding τ as the bifurcating parameter, the associated characteristic equation of Eq. (2.2) takes the form then we have ∆ (λ) = or ∆ (λ) = . ω and τ ( ) j are de ned by (2.5).

Lemma 2.2. The transversality conditions is
Proof By substituting λ(τ) into ∆ (λ) and taking the derivative with the respect τ from it, we get

In case τ > and b > , ∆ (λ) = has two pairs of purely imaginary roots ±iω if and only if
ω and τ ( ) j are de ned by (2.7).

Lemma 2.4. The transversality conditions is
Proof By substituting λ(τ) into ∆ (λ) and taking the derivative with the respect τ from it, we get Based on the above analysis, we have the following theorem.

Multiple periodic solutions
Next, we consider the symmetric characteristic of the Eq. (1.2). we know that the Eq.
the dihedral group D is generated by the cyclic subgroup subgroup of Z acts with the generator ρ and the ip of κ. Let T = π ω , P T represents the set of all continuous T-periodic function x(t) : R → R . According to maximum norm, P T is a Banach space. Apply the action of D × S on P T with Let SP T be a subspace of P T , which consist of all T-periodic solution of Eq. (1.2) when parameter τ = τ ( ) j , then for each closed subgroup of D × S is called the isotropy group Σ, and Σ = {(r, θ) ∈ D × S ; (r, θ)x(t) = x(t)}. Under usual non-resonance and transversality conditions, the Σ-xed-point subspace of SP T as follows. [6], symmetric delay di erential equations have a bifurcation of periodic solution whose spatiotemporal symmetry can be completely characterized by Σ. We consider the following subgroups of D × S to describe the symmetry of periodic solution of system Eq.

Normal form for equivariant-Hopf bifurcation
In this part, we only consider the case that the Eq. (2.3) has double characteristic values ±iω , where the equivariant-Hopf bifurcation occurs. Center manifold theory and normal form method [36,37] are used to study Hopf bifurcation. Firstly, rescale the time by t → t τ , Eq. (2.1) can be written aṡ Suppose that the Eq. (4.1) undergoes Equivariant-Hopf bifurcation at τ = τ j = τ * . Choosing the phase The linearized equation of Eq. (4.1) at zero as followṡ is the characteristic equation of the linearizaion of Eq. (2.1). Since ∆ ( , iω )ν j = , j = , , the center space at τ = τ * and in complex coordinates is . It is easy to check that a basis of the adjoint space of P * is with φ ∈ C, ψ ∈ C * , and a = + cτ * − iω τ * Introducing new parameter variables µ = τ − τ * , we can rewrite Eq. (4.1) aṡ Let B = (iω τ * , −iω τ * , iω τ * , −iω τ * ) and P is the generalized eigenspace associated with B, P * is the adjoint space of P. Then C can be decomposed as C = P ⊕ Q where Q = (φ ∈ C :< ψ, φ >= , for all ψ ∈ P * ). Using the decomposition z t = Φx(t) + y(t), then we have We can decompose Eq. with x ∈ C , y ∈ Q . We will write the Taylor expansion and we have Using the idea of Faria [37], we know that Eq.(4.4) can be written aṡ where f i (x, y, µ) is homogeneous polynomials of degree j about (x, y, µ) with coe cients in C . Then the normal form of Eq. (1.2) on the center manifolḋ where g , g will be calculated in the following part of this section. Fist of all, we get These are the second-order terms of (µ, x) in Eq. (4.6). From Faria and Hal [36][37][38][39], we have the secondorder terms of (µ, x) in the normal form on center manifold as follows: Here, de ne M j to be the operator in V j (C × Kerπ) with the range in the same space by In particular, To compute g (x, , µ), we rst note that from Eq.  In fact, from the e (±iω τ * ) = c∓iω bc , we have Applying the de nition of A Q and π, we obtaiṅ whereḣ denotes the derivative of h(θ) relative to θ. Let Comparing the coe cients of Thus and h ijmn will be calculated in Appendix. From Cases I, II, III, we get So, we can express Eq. (4.1) as If we use double polar coordinates α = ρ cosχ , α = ρ sinχ , and α = ρ cosχ , α = ρ sinχ , then we get Introducing periodic variable parameters ς, and Similarly, we get an equation for z (t). Thus, ignoring the terms o(µ |z| ) + o(|z| ), we get the normal form Let g : C ⊕ C ⊕ R → C ⊕ C be given so that −g(z , z , µ) is the right-hand side of Eq.(4.12), then Eq.(4.12) can be written as ( + ς)ż + g(z, µ) = (4.13) Note that Dz g( , )(z , z ) = i(z , z ) z = (z , z ) ∈ C ⊕ C Also note that g(., µ) : C ⊕ C → C ⊕ C is D × S −equivariant with respect to the following D × S −action on C ⊕ C: According to [28] and [35], the bifurcations of small-amplitude periodic solutions of Eq.(4.13) are completely determined by the signs of three eigenvalues of −i( + ς)z + g(z, µ) = (4.14) and their orbital stability is determined by the signs of three eigenvalues of that are not forced to zero by the group action.
To be more precise, we note that Eq. (4.13) is equivalent to It is known that Eq. (4.16) can be written as for some complex numbers A , A N , B given by By the results of [6,28] and [35], Re(An + b ) > or Re(An + B ) < determines whether the bifurcation of the phase-locked oscillation occurring in the system is supercritical or subcritical. When Re(An + B ) > and Re(B ) < these are orbitally asymptotically stable. In addition, Re( A N + B ) > or Re( A N + B ) < determines whether the bifurcation of mirror-re ecting waves and standing waves are supercritical or subcritical. When Re( A N + B ) > and Re(B ) > these are orbitally asymptotically stable.
Note that Theorem 4.1. Assume b > and de ne (τ * , ω ) as in (2.6), near the critical value τ = τ * . Eight asynchronous periodic solutions of Eq. (1.2) are branched from the trivial solution x = , and the periodic T is close to ( π ω ). These waves are 1. If H < , there exists two supercritical phase-locked oscillation bifurcations: x i (t) = x i− (t ± T ), for i(mod ), t ∈ R, and bifurcated periodic solution exists at τ > τ * , otherwise it's subcritical, and these are orbitally asymptotically stable if and only if H > , H < . 2. If H < , there exists three mirror-re ecting waves: x i (t) = x j (t) ≠ x k (t), for t ∈ R and for some distinct (i, j, k) in ( , , ). Three standing waves: x i (t) = x ij (t + T ),for t ∈ R and for some pair of distinct elements (i, j) in ( , , ), and bifurcated periodic solution exists at τ > τ * , otherwise it's subcritical, and these are orbitally asymptotically stable if and only if H > , H > .

Numerical simulations
In this part, we use Matlab to simulate the research results of Eq.(1.2). As shown in g.1, take b = .
, τ = . , Eq.(1.2) has an asymptotic stable equilibrium point ( . , . , . ). That means the growth of three identical species gradually tends to a balanced state. As shown in g.2, take b = .
, τ = . , Eq.(1.2) occurs the synchronous bifurcating periodic solutions. That means three identical species will change synchronously and periodically over a period of time, eventually reach an equilibrium state. , τ = . .

Conclusion
This paper introduces a typical biological model: Lotka-Volterra, which is mainly used in space ecology, disease transmission and species evolution. In recent years, many mathematicians have made gratifying achievements in the study of this model. However, due to the necessary of considering the evolution problem, delay di erential equation can often describe a real development system more objectively than ordinary di erential equations. In this paper, we introduce time-delay into the Lotka-Volterra model based on biological background, considering the introduction of time-delay into two adjacent species and making the coe cient of in uence between adjacent species is b(b > ). under such a circumstance and based on the theory of Equivariant Hopf bifurcation, we discuss that the Lotka-Volterra ring system composed by three species can produce some singular and interesting bifurcation phenomena.
According to the stability theory of symmetric periodic solutions of Golubitsky [6], many scholars have made classical academic researches. Based on the three dimensional ring neural network model, Wu [28] extended the symmetric local Hopf bifurcation theory to delay di erential equation, and gave a feasible method to solve the case of non-single pairs of purely imaginary eigenvalues. Guo [35] and Fan [39] discussed the bifurcation of the n-dimensional ring neural network model. There is no second-order term in the normal form given by the former, and there is a simple second-order term in the normal form given by the latter. We have studied the Equivariant Hopf bifurcation of the symmetric Lotka-Volterra ring system with delay on the basic of predecessors' theory. When the eigenvalues have multiple pure imaginary roots, the classical Hopf bifurcation theorem of delay di erential equation have lost its validity. Since we introduce the concept of Lie group, we know that the system that we consider now is D -Equivariant. This enables us to use the Hopf bifurcation existence theorem of the delay di erential equation with symmetric structure of Wu [28] to obtain the bifurcation periodic solutions of the system. In this model, the normal form contains such cumbersome second and third order terms, and di erent oscillation periodic solutions are obtained by simpli cation. It is summarized that when the time delay varies and passes through some critical values, eight asynchronous periodic solutions can be derived at the zero equilibrium point of the system in some speci c subdomains. Among them, there are two stable phase-locked periodic solutions, three unstable mirror re ections and three unstable standing waves.
By using the software Mathematica, after easy but long computation, we have the expression of h ( ) and h (− ).

Proof.
From the rst equation of Eq.

Φ(t)Ψ( )e − iω τ * t A dt
By using the software Mathematica, after easy but long computation, we have the expression of h ( ) and h (− ).