MULTIPLE SPATIOTEMPORAL COEXISTENCE STATES AND TURING-HOPF BIFURCATION IN A LOTKA-VOLTERRA COMPETITION SYSTEM WITH NONLOCAL DELAYS

. We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspeciﬁc and interspeciﬁc competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and suﬃcient condition that Turing instability occurs in the weak competition case, and ﬁnd that the strength of nonlocal intraspeciﬁc competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the diﬀerence compared with system without nonlocal terms in calculating coeﬃcients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspeciﬁc competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.

In population dynamics, since species take time to move and it can affects the density of species in the neighborhood of their current positions, the effect interacting the past history (delay) and different locations (nonlocality) should be included in modeling process [3,11,14]. Thus it would better incorporate the spatial weighted functions into competition terms with delays (spatiotemporal average or nonlocal delays) for the population models, see more examples [3,6,14,15,19] for single species models and [8,12,13,17] for two species models.
The kernels are chosen as spatially homogeneous incorporated with discrete delays in system (1), which can reflect that the species do not simply depend on its density at the current positions and time but on all positions in region (0, π) and previous time τ . Both local and nonlocal intraspecific and interspecific competitions are considered and the effects of their strength are also presented to system (1).
For the two-species Lotka-Volterra competition model, the global stability of (non-)constant steady states [5,8,9,13,18,24,25,26] or Hopf bifurcations induced by time delays at the (non-)constant steady states [17,31,34,39] have been investigated by many researchers. In recent years, Guo and Yan [17] investigated a two-species Lotka-Volterra competition system with nonlocal delays under homogeneous Dirichlet boundary condition and the stability of spatially inhomogeneous steady-state solutions and the existence of Hopf bifurcation have been derived, which presents interesting spatiotemporal pattern formations. In [8,26], besides the global dynamics for two constant semi-trivial and the coexistence equilibrium, the existence of spatial patterns (or Turing instability) have been rigorously proved for a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under homogeneous Neumann boundary condition. Naturally, the effects of nonlocality in competitions to the Lotka-Volterra type systems are still not completely clear and are deserved to study further. Thus we investigate the spatiotemporal pattern formations to system (1) and it shows that nonlocal competitions can lead to the occurrence of Turing-Hopf bifurcation.
As far as we know, although the existence of spatiotemporal phenomena has been studied to much extent through the Turing-Hopf bifurcation in many population dynamical models [1,4,7,21,27,32,33], there are few or no results in Turing-Hopf bifurcations for the competition system. Thus investigating what cause the occurrence of the Turing-Hopf bifurcation and how the spatiotemporal pattern can be formed is meaningful and worth exploring. Furthermore, the nonlocal terms can bring about difference in calculating normal forms for the codimension-two Turing-Hopf bifurcation.
In this paper, we firstly analyze the distribution of all eigenvalues for the linearized system of (1) evaluated at the coexistence equilibrium, and prove that system (1) can undergo Hopf bifurcations, Turing bifurcations and Turing-Hopf bifurcations when the nonlocal intraspecific competitions are suitably large. Moreover, the necessary and sufficient condition that the coexistence equilibrium becomes Turing instability are derived, and it can be expressed by explicit curves in d 2 − d 1 parameter plane. One can easily see the eigenvalue distribution for the characteristic equations associated with the linearized system in different regions of d 2 − d 1 parameter plane. Secondly, since center manifold theory and normal form method are efficient to study the codimension-two Turing-Hopf bifurcation, we apply the method and have found that the explicit formulas of normal forms in [20] cannot be directly used because of the existence of nonlocal terms. Thus we recalculate the terms and derive the explicit formulas of third order normal forms restricted on the center manifold at the codimension-two Turing-Hopf singularity, which shows the difference from the system without effects of nonlocal terms. Moreover, the explicit formulas only depend on parameters of original system and the calculating method can be applied in other similar systems with nonlocal terms. And via analyzing the normal form truncated to order 3 (amplitude equations), the existence of interesting and complex spatiotemporal patterns, like the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved in the weak competition case, which is new phenomenon to show variety of coexistence states in the Lotka-Volterra competition system in the literature.
The rest of the paper is organized as follows. In Sect. 2, we give bifurcation analyses by analyzing distribution of all roots to the characteristic equations. In Sect. 3, by applying the results in Sect. 2, the explicit formulas of third order normal forms are derived and the existence of complex spatiotemporal patterns are proved. In Sect. 4, we summarize the results briefly and give some conjectures for the future work.
2. Bifurcation analysis. The well-posedness of solution for system (1) can be easily checked, see [8,36,37]. Following definitions in [8,26], denote m ij = a ij + b ij the combined strength of local and nonlocal competitions of species u j to u i , i, j = 1, 2 and m 11 m 22 > m 12 m 21 (m 11 m 22 < m 12 m 21 ) the weak competition case (strong competition case).
Evidently, g(λ, τ ) is an analytic function in λ and continuous in τ . By (5), it is clear to see that the degree of P 1 (λ) and P 2 (λ) with respect to λ is lower than 2. Then the following result holds. Proof. See Theorem 2.4 in [10] or Theorem 1.1 in [2] and it can also be proved by the results in [29] since zeros in the right half complex plane of D(λ, τ, 0) = 0 are uniformly bounded.
Then we apply the geometric stability switch criterion established in [2] to investigate the distribution of all roots of D(λ, τ, 0) = 0.
Let λ = iw, w > 0 be a root of D(λ, τ, 0) = 0. From the characteristic equation (5), separating real and imaginary parts, we have From (12), w > 0 must satisfies the following equations (14) holds if w belongs to the set where From (16) and (13), F (w) is an even function and a polynomial function with respect to w, and I w is a finite set. Thus, if I w is not empty, we can take an element w * ∈ I w and substitute it to (14) to derive where θ(τ ) := w * τ and θ(τ ) ∈ [0, 2π]. Then we have From (14) and Lemma 2.1 of [2], θ(τ ) = 0, 2π and Define the map S k : [0, +∞) → R: By (18) and (20), S k (τ ), k ∈ N 0 , is continuous and differentiable with respect to τ . From the analysis above, we know there must exist τ k (τ ) such that characteristic equation D(λ, τ k (τ ), 0) = 0 have purely imaginary roots if I w is not empty. Then we give the following result: and By (7), (16) and direct calculations, F (w) is an eight degree polynomial with respect to w and it can be rewrite as the following form: where the highest degree of G(w) with respect to w is no more than 6 and is the sum of all terms of F (w) independent of w. Thus we have G(0) = 0 and From (22), the sign of C is the same as the one of ( By direct calculations, we have From assumption (H) and (21), we obtain that C < 0, which implies that By (27), (28) and the continuity of F (w), there must exist w > 0 such that F (w) = 0 holds. This completes the proof.

Remark 1.
From assumption (H), we can easily verify that (21) could not be satisfied if nonlocal intraspecific competitions are weak enough, which is consistent with the results in [8] that nonlocal intraspecific competitions play an important role in resulting in the complex pattern formations to the system (1).
To show that there is no contradiction between (21), (22) and assumption (H), we take parameter values as Table 1 shows and plot the graph of F against w, see Figure 1.
Since the explicit expression of F (w) is complex, we make the assumption to simplify the analysis of roots of D(λ, τ, 0). Then we give the following results: and λ ± (τ w ) = ±iw is a pair of simple conjugate pure imaginary roots of D(λ, τ, 0) which crosses the imaginary axis from left to right if δ(τ w ) > 0 and crosses the imaginary axis from right to left if δ(τ w ) < 0, where Moreover, all roots of D(λ, τ, 0) have negative real parts if τ < τ w * = min{τ 0 (τ w ) : Proof. The first part follows from Theorem 2.1 in [2]. Since τ k (τ w ), k ∈ N 0 , is strictly increasing with respect to k, we have We can find the critical value τ w * by the finiteness of I w . Thus the rest part holds true from Theorem 2.1.
For convenience, in the rest of the paper, we assume that a 12 a 21 − a 11 a 22 > 0 holds and denote curves in the d 2 − d 1 plane to describe the distribution of all roots of D(λ, n 2 ) when d 1 and d 2 vary in the first quadrant of and with n ∈ N.
. Evidently, if a 12 a 21 − a 11 a 22 > 0 holds, the definition of (33) is valid since d n 1 (d 2 ), d 2,n > 0, and (34) and (35) is feasible since d 2,n is decreasing with respect to n. And from Remark 3.3 in [8] or assumption (H), a 12 a 21 −a 11 a 22 > 0 holds if the nonlocal intraspecific competitions are suitably strong, which implies the strength of nonlocal intraspecific competitions play an important role in result in pattern formations of system (1).
We also define that the curve L n S n is the combination of L n and S n for n ∈ N and a two function sequence { d 1 Proof. For the case that d 2 ∈ [d 2,n+1 , d 2,n ), we have For the case that d 2 ∈ (0, d 2,n+1 ), we only need to show d n . Note that f (n 2 ) = 0 if and only if d 1 = d n 1 (d 2 ). And the point P n := (d 2 , d n 1 (d 2 )) is on L n in the d 2 − d 1 plane for fixed d 2 ∈ (0, d 2,n+1 ). Firstly, we claim that P n ∈ L n+1 . By contradiction, we know d n , which contradicts that f (n 2 ) = 0 when d 1 = d n 1 (d 2 ) since f (n 2 ) is strictly increasing with respect to n 2 .
Secondly, by (40) and the monotonicity of d 2,n with respect to n, we have and lim d2→d2,n+1 d n+1 n+1 ). Thus by the continuity of curve L n , we ). This completes the proof.  Table 1 shows. The left figure represents the enlarged figure inside the pink rectangular part of the right figure and the red hollow circles are intersection points of L n and S n . The solid black circles means that the remaining curves L n and S n are omitted here.
Remark 3. Lemma 2.5 give an completed geometry description on sequence of curves {L n S n } n∈N and {L n } n∈N in the d 2 − d 1 plane. We take system parameter values as Table 1 shows and draw these curves, see Figure 2.
Thus we can derive the distribution all roots of D(λ, n 2 ) = 0, n ∈ N as follows Theorem 2.6. Assume that a 12 a 21 − a 11 a 22 > 0 holds.
Following the definitions in Theorem 2.6, we derive the necessary and sufficient conditions for the occurrence of Turing instability at the coexistence equilibrium E * . Theorem 2.7. Assume that assumption (H) and a 12 a 21 −a 11 a 22 > 0 hold. If delay τ < τ w * , the coexistence equilibrium E * is locally asymptotic stable when (d 2 , d 1 ) is in region Λ 0 . Moreover, the coexistence equilibrium E * will become unstable if (d 2 , d 1 ) in region Λ 0 goes through L 1 .
Proof. From Lemma 2.3 and Theorem 2.6, the proof is completed.
Remark 4. By Theorem 2.7, L 1 is called the first Turing bifurcation curve and it is sufficiently smooth but not piecewise smooth, which is different from the one derived in [21].
By combining the results of Lemmas 2.3 and 2.6, Theorem 2.4 and following the definitions above, we have the following results.

3.
Turing-Hopf normal forms and spatiotemporal dynamics. In this section, we assume that assumptions and conditions in Theorem 2.8 are satisfied, and calculate the normal form of (1, 0)-mode Turing-Hopf bifurcation at the coexistence equilibrium for system (1). Then we take a group of system parameters to show spatiotemporal dynamics of system (1).
We have to emphasize that the explicit formulas for the case k 1 = 0 and k 2 = 0 (see Propsition 4.3 in [20]) cannot be directly used since the effects of nonlocal terms in system (44). It is well known that −∆ on domain (0, π) subject to homogeneous Neumann boundary condition has simple eigenvalues µ n = n 2 with corresponding normalized eigenfunctions β n for n ∈ N 0 , where and < ·, · > the standard inner product in Hilbert space X, i, j ∈ N 0 . From (47), we have where with φ i,j the j-th element of φ i , i, j = 1, 2.
Furthermore, we obtain that where , with h j,k q the k-th element of h j q , k = 1, 2. Here, h j q =< h q , β j > and h q is defined by (6.32)-(6.34) in [20].
Then for notational simplicity to calculate the coefficients of normal forms, let and (71) has same definitions if φ i is substituted by its conjugate, i = 1, 2.
From [20], normal forms restricted on center manifold up to third-order for system (44) at Turing-Hopf singularity are Following the definitions of (71), the coefficients in (72) can be calculated out by the following explicit expressions.
Proof. By Theorem 3.1 and calculations of case 3 in [20], the results quickly follows.
Remark 6. Although formulations in Lemma 3.1 are similar to the one of Proposition 4.3 in [20], recalculating the coefficients in (72) is necessary since it can be affected by nonlocal terms in (44) in reduction process. And for other cases (not the case k 2 = 0, k 1 = 0), explicit formulas in Lemma 3.1 may also be different. Despite all this, we provide the method to calculate explicit third-order normal forms, which can also be applied to the system with nonlocal terms similar to (1).

3.2.
Spatiotemporal dynamics. In this part, based on the explicit formulas in subsection 3.1, we take concrete system parameters to show the complex spatiotemporal dynamical behaviors of system (44) near (1, 0)−mode Turing-Hopf singularity point.
All constant equilibria of the (76) are E 0 = (0, 0), 2 and E ± 3 of (76) correspond to the coexistence equilibrium, a spatially homogeneous periodic orbit, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic orbits of original system (44) respectively.
From direct calculations and [16], the unfolding of (76) is case II and all pitch-fork bifurcation curves are H 0 :α 2 = 0, where H 0 and L 1 are also the Hopf bifurcation and Turing bifurcation curves for the original reaction-diffusion systems respectively.
Then we give the bifurcation sets and phase portraits in Figure 3. For convenience, recording all fixed system parameters in Table 2. Therefore, we have Proposition 1. Assume that system parameters are fixed as Table 2 shows. If (τ, d 1 ) are chosen in the neighborhood of (τ w * , d 1 1 (d 2 )), then system (44) exhibits the following complex dynamics 1. When (τ, d 1 ) ∈ D 1 , the coexistence equilibrium (u * 1 , u * 2 ) is locally asymptotic stable; 2. When (τ, d 1 ) ∈ D 2 , a stable spatially homogeneous periodic orbit bifurcating from (u * 1 , u * 2 ); 3. When (τ, d 1 ) ∈ D 3 , a spatially homogeneous periodic orbit remains stable and a pair of unstable spatially inhomogeneous steady states occurs; 4. When (τ, d 1 ) ∈ D 4 , a pair of spatially inhomogeneous periodic orbits bifurcating from a pair of spatially homogeneous periodic orbits is stable, which indicates that D 4 is a bistable region; 5. When (τ, d 1 ) ∈ D 5 , a pair of spatially inhomogeneous periodic orbits remains stable and a spatially homogeneous periodic orbit disappears, which implies that D 5 is a bistable region; 6. When (τ, d 1 ) ∈ D 6 , a pair of spatially inhomogeneous periodic orbits disappears and a pair of spatial inhomogeneous steady states becomes locally asymptotic stable. Thus D 6 is a bistable region.      4. Conclusions and discussions. When combining the results in [8], we know that the strength of nonlocal intraspecific competitions play an important role in result in complex spatiotemporal pattern formations to the Lotka-Volterra competition system with nonlocal delays in the weak competition case, but the exclusion principle (see [13]) is still being preserved if the strength of nonlocal intraspecific competitions is sufficiently weak regardless of the delay effects or nonlocal kernels satisfied with assumptions on the spatiotemporal kernels in [8]. Moreover, the complex spatiotemporal patterns induced by Turing-Hopf bifurcations are proved in mathematical way and verified by numerical results, which is very interesting and can be a new guide for investigating on dynamics in competition systems.
Furthermore, we believe the coexistence equilibrium E * is globally asymptotic stable if delay τ < τ w * and (d 2 , d 1 ) ∈ Λ 0 (see Figure 2) despite lack of rigorously proofs here. If the delay τ in each nonlocal term can be chosen different or the nonlocal competitions vary in different spatial positions for system (1), more complex spatiotemporal patterns, like spatially inhomogeneous quasi-periodic orbits may be formed.