BIFURCATION ANALYSIS OF A DIFFUSIVE PLANT-WRACK MODEL WITH TIDE EFFECT ON THE WRACK

. This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the sta- bility and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results. (cid:101) P ) is a function describing the positive eﬀect of plant biomass on its own growth, s is the speciﬁc rate of plant senescence, I ( (cid:101) P, (cid:102) W ) is a function describing the inhibiting eﬀect of wrack on plant growth as a function of plant and the wrack biomass, b is the decay rate of wrack, and d 1 and d 2 are diﬀusion constants describing lateral movement of plants and wrack. Here, ∆ = ∂ 2 ∂x 21 + · · · ∂ 2 ∂x 2 N is the usual Laplacian operator in N


1.
Introduction. More recently, many ecologists have paid more and more attention to the experimental investigation of regular spatial patterning in Carex stricta. Carex stricta, the tussock sedge, is a species with widespread distribution in freshwater marshes of North America. Spatial dispersals of vegetation (through tillers) and wrack (resulting from dead plant leaves dropping to the soil surface and movement by the tides) are modeled using a diffusion approximation. The model, which describes the interaction of the plant and wrack, is as follows [20]: where Ω ⊂ R N is a bounded domain, P is the plant biomass, W is the wrack biomass, F ( P ) is a function describing the positive effect of plant biomass on its own growth, s is the specific rate of plant senescence, I( P , W ) is a function describing the inhibiting effect of wrack on plant growth as a function of plant and the wrack biomass, b is the decay rate of wrack, and d 1 and d 2 are diffusion constants describing lateral movement of plants and wrack. Here, ∆ = ∂ 2 ∂x 2 1 + · · · ∂ 2 ∂x 2 N is the usual Laplacian operator in N -dimension space.

JUN ZHOU
In this paper, we focus on the following specific type with an indirect facilitation of growth by the root mound by lowering of inhibition by the wrack, which leads to F ( P ) = 1, I( P , W ) = a P W K P + K , where K P +K is added to the inhibition term, lowering inhibition as P increases, K is the level of plant biomass where inhibition is lowered by half, and a is an inhibition coefficient [20]. As a result, model (1) with homogeneous Neumann boundary condition has the following from: where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, ν is the unit outward normal on ∂Ω, and the homogeneous Neumann boundary conditions indicate that the system is self-contained with zero population flux across the boundary ∂Ω. The constant a, b, s, K, d 1 , d 2 are assumed to be positive. P 0 (x) and W 0 (x) are nonnegative nontrivial continuous functions. For the sake of simplicity, we let P = P , W = b W , a = a/b, then problem (2) becomes x ∈ ∂Ω, t > 0, P (x, 0) = P 0 (x) = P 0 (x), W (x, 0) = W 0 (x) = b W 0 (x), x ∈ Ω. (3) In order to provide guidelines on the dynamics of the full reaction-diffusion system, it is important to consider the steady states corresponding to (3), which satisfies the following elliptic system: It is, naturally, the dynamics in the biologically meaningful region {(P, W ) : P, W ≥ 0} are of interest. Furthermore, we want to find the positive steady state of the non-spatial model, (P * , W * ), which is corresponding to the coexistence of plant and wrack. By direct calculation, we find that if s ≥ 1, problem (4) admits no positive constant solution. On the other hand, if 0 < s < 1, problem (4) admits a unique positive constant solution (P * , W * ), where Here, Spatial, temporal and spatiotemporal patterns could occur in the reaction-diffusion model (3) via three possible mechanisms: Turing instability, Hopf bifurcation, and positive non-constant steady states. There are a great deal of research have been devoted to the study of spatial, temporal and spatiotemporal patterns in chemical and biology contexts (see [1,3,9,10,11,13,19,27,32,37,53,58,59] for Brusselator model; [4,6,14,23,24,30,44,54,55,57] for Gray-Scott model; [7,17,18,25,26,50,52] for Lengyel-Epstein model; [31,48,56] for Oregonator model, [12,16,34,43,45,46,49] for Schnakenberg model, [5,8,21,28,29,33,39] for Sel'klov model).
The goal of this article is to show that the diffusive plant-wrack model (2) exhibits various spatial, temporal and spatiotemporal patterns via the aforementioned three mechanisms. The organization of the remaining part of this paper is as follows. In Section 2, we investigate the asymptotic behavior of the positive equilibrium (P * , W * ) and occurrence of Hopf bifurcation of the local system of (3). In section 3, we firstly consider the asymptotic behavior and Turing instability of the positive equilibrium (P * , W * ) for the reaction-diffusion system (3), then we study the existence of Hopf bifurcation. In Section 4, we consider the existence and nonexistence of nonconstant positive solutions for problem (4) by bifurcation theory, energy method and Implicit Function Theorem. We end our study with numerical simulations in Section 5. Throughout this paper, N is the set of natural numbers and N 0 = N ∪ {0}. The eigenvalues of the operator −∆ with homogeneous Neumann boundary condition in Ω are denoted by 0 = µ 0 < µ 1 ≤ µ 2 ≤ · · · ≤ µ n ≤ · · · , and the eigenfunction corresponding to µ n is φ n (x).
2. Analysis of the local system. In this section, we mainly consider the following local system corresponding to problem (3): The dynamical behavior of the solutions near the positive constant equilibrium (P * , W * ) can be studied by computing the eigenvalues of the Jacobin matrix L 0 (b) of the system (8), namely, The characteristic equation of L 0 (b) is A series of calculations shows that 1. If s + K ≥ 1 or s + K < 1 and then Theorem 2.1. Assume (5) holds. Let b 0 be the constant defined as (15). Then the positive equilibrium (P * , W * ) of the local system (8) given as (6) is locally asymptotically stable if (i): s + K ≥ 1; or (ii): s + K < 1 and (13) holds; or (iii): (14) holds and b > b 0 .
While the positive equilibrium (P * , W * ) is unstable with respect to (8) if (14) holds and b < b 0 . System (8) undergoes a Hopf bifurcation at (P * , W * ) as b passes through b 0 .
Proof. (i), (ii), (iii) have been proved in the previous paragraphs. We only focus on the Hopf bifurcation occurring at (P * , W * ) by using b as the bifurcation parameter. According to Poincaré-Andronov-Hopf Bifurcation Theorem [47, Theorem 3.1.3], system (8) has a small amplitude non-constant periodic solution bifurcating from (P * , W * ) when b crosses through b 0 if the transversal condition is satisfied.
be the roots of (10). Then (12)). This shows that the transversal condition holds, and thus (8) undergoes a Hopf bifurcation at (P * , W * ) as b passes through b 0 .
To illustrate the above result. We give an numerical example.
Example 2.2. Consider problem (8) with s = K = 0.25 and a = 25.6 such that (5) and (14) hold. Then we get the following system and the positive constant equilibrium is (P * , W * ) = (0.15, 0.0375), the constant 3. Analysis of the PDE model (3). In this section, we mainly consider the model (3), and the studies include the parameter regions for the stability and instability of the unique positive constant equilibrium (P * , W * ), the occurrence Turing instability and the existence of time periodic orbits.
3.1. Stability analysis. The local stability of (u * , v * ) with respect to (3) is determined by the following eigenvalue problem which is got by linearizing the system (4) about the positive constant equilibrium (P * , W * ) where L 0 (b) is defined as (9). Denote For each n ∈ N 0 , we define a 2 × 2 matrix The following statements hold true by using Fourier decomposition: 1. If µ is an eigenvalue of (17), then there exists n ∈ N 0 such that µ is an eigenvalue of L n (b). 2. The constant equilibrium (P * , W * ) is locally asymptotically stable with respect to (3) if and only if for every n ∈ N 0 , all eigenvalues of L n (b) have negative real part. 3. The constant equilibrium (u * , v * ) is unstable with respect to (3) if there exists an n ∈ N 0 such that L n (b) has at least one eigenvalue with positive real part. The characteristic equation of L n (b) is where Here, T (b) and D(b) are defined as (11) and (12) respectively. Then (P * , W * ) is locally asymptotically stable if T n (b) < 0 and D n (b) > 0 for all n ∈ N 0 , and (P * , W * ) is unstable if there exists n ∈ N 0 such that T n (b) > 0 or D n (b) < 0. Since D(b) > 0 and µ n ≥ 0 for all n ∈ N 0 , a sufficient condition to ensure T n (b) < 0 and D n (b) > 0 is T (b) ≤ −b, which is equivalent to asK ≤ M 2 + 4(1 − s)K, i.e., s + K ≥ 1 or s + K < 1 and (13) holds (see the analysis in Section 2).
In the following we consider the case that (14) holds, which implies b 0 , defined as (15), is positive. Then we have T (b) = b 0 − b, and then it follows from (12) that where We define and Then H is the Hopf bifurcation curve and S is the steady state bifurcation curve. Furthermore, the sets H and S are graphs of functions defined as follows (ii): Let µ * 1 := Then µ = µ * 1 is the unique critical value of b S (µ), the function b S (µ) is strictly increasing for µ ∈ (0, µ * 1 ), and it is strictly decreasing for µ > µ * (vi): Let . Now we can give a stability result regarding the constant equilibrium (P * , W * ) by the analysis above. To this end, we define Theorem 3.2. Assume (5) holds. Then the constant equilibrium (P * , W * ) of the system (3) given as (6) is locally asymptotically stable if (i): s + K ≥ 1; or (ii): s + K < 1 and (13) holds; or (iii): (14) holds and b > max{b 0 , b}.
Next we consider the occurrence of Turing instability, which means the constant equilibrium (P * , W * ) is stable with respect to the ODE model (8) (5) and (14) hold. Then Turing instability happens if where D 2 , µ L , µ R , b 0 and b are positive constants defined as (38), (39), (15) and (40) respectively, b S (µ) is the function given in (31).

Hopf bifurcation.
In this part, we study the existence of periodic solutions of (3) by analyze the Hopf bifurcation from the constant equilibrium (P * , W * ) under the assumption (5) and (14) since there is no change of stability for other cases. We assume that all eigenvalues µ i are simple, and denote the corresponding eigenfunction by φ i (x), i ∈ N 0 . Note that this assumption always holds when N = 1 for Ω = (0, π), as for i ∈ N 0 , µ i = i 2 / 2 and φ i (x) = cos(ix/ ), where is a positive constant. We use b as the main bifurcation parameter. To identify possible Hopf bifurcation value b H , we recall the following necessary and sufficient condition from [15,50,51].
(HS) There exists i ∈ N 0 such that where T i (b) and D i (b) are given in (23) and (24) respectively, and for the unique pair of complex eigenvalues A(b) ± iB(b) near the imaginary axis, where the function b H (µ) is given in (30). Fig. 2). Finally, we consider the conditions in (42). Let the eigenvalues close to the pure imaginary one Then all conditions in (HS) are satisfied if i ∈ {0, · · · , n 0 }. Now by using the Hopf bifurcation theorem in [51], we have Theorem 3.4. Assume (5) and (14) hold. Let Ω be a smooth domain so that all eigenvalues µ i , i ∈ N 0 , are simple. Then there exists a n 0 ∈ N 0 such that µ n0 < µ H ≤ µ n0+1 , and b i,H , defined as (43), is a Hopf bifurcation value for i ∈ {0, · · · , n 0 }, where µ H is given in (35). At each b i,H , the system (3) undergoes a Hopf bifurcation, and the bifurcation periodic orbits near (b, P, (24) is the corresponding time frequency, φ i (x) is the corresponding spatial eigenfunction, and (a i , b i ) is the corresponding eigenvector, i.e., where L(b) is given in (18). Moreover, 1. The bifurcation periodic orbit from b 0,H = b 0 are spatially homogeneous, where b 0 is given in (15); 2. The bifurcation periodic orbit from b i,H , i ∈ {1, · · · , n 0 }, are spatially nonhomogeneous.
Next we calculate the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits bifurcating from b = b 0 . We use the normal form method and center manifold theorem in [15] to study it. Let L * (b) be the conjugate operator of L(b) defined as (18) i.e., with domain where b 0 is given in (15) and , and χ be the constants given in (25). It holds denotes the inner product in L 2 (Ω) × L 2 (Ω). According to [15], we decompose X = X C ⊕ X S with For any (P, W ) ∈ X, there exists z = q * , (P, W ) T ∈ C and ω = (ω 1 , ω 2 ) ∈ X S such that Thus, Then system (3) in (z, ω) coordinates become where A direct calculation shows that H(z, z, ω) = (0, 0) T . Let It follows [15, Appendix A] that the system (45) possesses a center manifold, then we can write ω in the form Thus we have For later uses, we denote with all the partial derivatives evaluated at the point (P, W ) = (P * , W * ). Therefore, the model (3) restricted to the center manifold in z, z coordinates is given by According to [15], we have Based on the above analysis, we give our results in the following theorem.

4.
Analysis of the PDE model (4). In this section, we study the model (4) by analyzing the existence and nonexistence of nonconstant positive solutions. We obtain existence/nonexistence results for by using energy estimates, Implicit Function Theorem and bifurcation methods.

4.1.
A priori estimates. Firstly, we give some estimates for the positive solutions of (4), which will be used later. The following lemma is given in [22].
If (1 − s)K ≤ ρ(P (x 1 )), then we get from (57) that P (x 1 ) ≥ 1−s as . On the other hand, if (1 − s)K > ρ(P (x 1 )), it follows from (57) that In all, without the assumptions in Theorem 49, there exists a positive constant C 1 depending on s ∈ (0, 1) and a, K ∈ (0, ∞) such that the positive solution (P, W ) of problem (4) satisfies max Now we introduce a Harnack inequality derived in [36]. Upon (58) and above lemma, we can discard the assumptions in Theorem 4.2 and get the following results.
Proof. By the proof of Theorem 4.2, we get Let's rewrite the first equation of (4) as ∆P (x) + c(x)P (x) = 0 with Furthermore, Then it follows from Lemma 4.5 and Remark 4.4 that there exists a positive constant C, depending only on a, s, d and Ω such that which combining with (60) implies

4.2.
Nonexistence of positive nonconstant steady state solutions.
Theorem 4.7. Let assumptions in Theorem 4.2 hold. Let A and B be the two positive constants given in (51) and (50) respectively. If µ 1 is large enough such that then (4) admits no positive nonconstant solutions.
Before giving the proof, we firstly make some remarks on above theorem. Proof of Theorem 4.7. In the proof we denote |Ω| −1 Ω ξ(x)dx by ξ for ξ ∈ L 1 (Ω). Let (P, W ) be a positive solution of (4), then it is obvious that Ω (P − P )dx = Multiplying the first equation of (4) by P − P , by (49), we obtain Similarly, multiplying the first equation of (4) by W − W , by (49), we obtain Thus, thanks to the well-known Poincaré's inequality, we find from (63) that If W = W on Ω, the second equation of (4) shows P = W /s, so P and W are both constants. Next we assume that W ≡ W . Then (64) leads to which together with (64), infers By virtue of (62), (65) and Poincaré's inequality, we get which combining with (61) implies P ≡ P , and then it follows from the second equation of (4) that W ≡ sP .
Next we will discard the assumptions in Theorem 4.7 and study the nonexistence of positive nonconstant solutions of (4) as d 1 → ∞ or d 2 → ∞. To this end, we firstly introduce the following lemma. Lemma 4.9. Let a, b, K > 0, 0 < s < 1 be fixed and (P * , W * ) be the unique constant equilibrium defined as (6), then the following statements hold.
Proof. (i) Without losing generality, we assume d 1,i ≥ 1 for i = 1, 2, · · · . By Theorem 4.6, there exists a positive constant θ depending only on a, s, K and Ω such that By Sobolev embedding theory and standard regularity theory of elliptic equations, there exists a subsequence of (P i , W i ), relabeled as itself, and (P, W ) ∈ C 2 (Ω) × C 2 (Ω) such that (P i , W i ) → (P, W ) in C 2 (Ω) × C 2 (Ω) as i → ∞. Furthermore, (P, W ) satisfies the following relations From the first and the third relations in (66), we know that and the unique solution of (67) is W = sc. Then it follows the forth relation of (66) that c > 0 satisfies i.e., c = P * , where P * is given in (6), which in turn implies P = P * and W = W * . Then (i) holds. The proof of (ii) is similar to the proof of (i).
(iii) Similar arguments as above imply that there exists a subsequence of (P i , W i ), relabeled as itself, and (P, W ) ∈ C 2 (Ω) × C 2 (Ω) such that (P i , W i ) → (P, W ) in C 2 (Ω) × C 2 (Ω) as i → ∞. Furthermore, (P, W ) satisfies the following relations Then P ≡ c > 0 and W ≡ c > 0, where c and c are constants. By the fifth relation of (69), we get c = sc, and then c satisfies (68). So as above we get P = P * and W = W * .
Based on Lemma 4.9, we can obtain the following result by using Implicit Function Theorem. Proof. (i) We write P as P = U + ξ with ξ = |Ω| −1 Ω P dx such that Ω U dx = 0. Then we observe that finding solutions of (4) is equivalent to solving the following problem where σ = 1/d 1 . Clearly, (U, W, ξ) = (0, W * , P * ) is a solution of (70). From above analysis, to verify our assertion, we only need to prove there exists a positive constant σ 0 which depends only on a, b, s, K, d 2 and Ω such that (U, W, ξ) = (0, W * , P * ) is the unique solution of (70) when σ < σ 0 . For this, we define Then (70) is equivalent to solving F (σ, U, W, ξ) = 0. Moreover, similar to the proof of Lemma 4.9, (70) admits a unique solution (U, W, ξ) = (0, W * , P * ) when σ = 0. By simple computations, we have In order to use Implicit Function Theorem, we need to prove Φ is invertible, that is Φ one-to one and onto. It is easy to see that Φ is a surjection. So we only need to prove the homogeneous equation Φ(y, z, τ ) = 0 has unique solution y = z = τ = 0.
Note the relationship between b and d 2 in the second equation (4), we get d 2 → ∞ is equivalent to b → 0. Then we get the following corollary from (ii) of Lemma 4.9 and (ii) of Theorem 4.10.

4.3.
Existence of positive nonconstant steady state solutions. In this part, we analyze model (4) by bifurcation theory with b as the bifurcation parameter. As in Section 3, we assume (5) and (14) hold, and all eigenvalues µ i are simple, and denote the corresponding eigenfunction by φ i (x), i ∈ N 0 . We identify state bifurcation value b S of (4), which satisfies the following conditions [51].
(SS) There exists i ∈ N 0 such that where D i (b) and T i (b) are given in (24) and (23) respectively. Since D 0 (b) = χb > 0, where χ is defined as (25), we only consider i ∈ N . In the following, we determine b-values satisfying (SS). We notice that D i (b) = 0 is equivalent to b = b S (µ i ), where b S (µ) is defined as (31). Hence we make the following additional assumption on the spectral set {µ n } n∈N0 .
(SP) There exist p ∈ N such that µ p < µ * 3 ≤ µ p+1 and µ H = µ i for i = 1, · · · , p, where µ * 3 and µ H are given in (33) and (35) respectively. In the following, for p satisfy (SP), we denote The points b i,S defined above are potential steady state bifurcation points. In follows from Lemma 3.1 that for each i = 1, · · · , p, (SS) is not satisfied if (SQ) holds, and we shall not consider bifurcations at such a point. On the other hand, it is also possible that (SR) b i,S = b j,H for some i, j ∈ {1, · · · , p} and i = j, where b j,H is a Hopf bifurcation value defined as (43).
However, from an argument in [51], for N = 1 and Ω = ( π), there are only countably many , such that (SQ) or (SP) occurs. One also can show that (SQ) or (SP) does not occur for generic domains in R N (see [42]).
Summarizing the above discussion, we obtain the main result of this part on bifurcation of steady state solutions.
Theorem 4.12. Assume (5) and (14) hold. Let Ω be a bounded smooth domain so that all eigenvalues µ i , i ∈ N 0 , are simple, and satisfy (SP). Then for any i ∈ {1, · · · , p}, there exists a unique b i,S defined as (75) such that D i (b i,S ) = 0, D i (b i,S ) = 0 and T i (b i,S ) = 0. If in addition, we assume that b i,S = b j,S , b i,S = b j,H f or any j ∈ {1, · · · , p} and i = j, where b j,H is defined as (43). Then the following conclusions hold.
where is a small positive constant and for some smooth functions b i , ψ 1,i and ψ 2,i such that b i (0) = b i,S and ψ 1,i (0) = ψ 2,i (0) = 0, and (l i , m i ) satisfies here L is the operator defined in (18). (ii): Γ i is contained a global branch Σ i of positive nontrivial solution of problem (4) and 1. Σ i connects another bifurcation point (b j,S , P * , W * ) for some j ∈ {1, · · · , p} and j = i; or 2. the projection of Σ i on to b-axis contains the interval (b i,S , ∞), and then for b ∈ (b i,S , ∞) \ (∪ p k=1 b k,S ), problem (4) admits at least one positive nonconstant solution.
Proof. The condition (SS) has been proved in the previous paragraphs, and the bifurcation of solutions to (4) occur at b = b i,S . Note that we assume (SQ) and (SR) hold, so b = b i,S ia always a bifurcating from simple eigenvalue point, then by using the general bifurcation theorem in [51], we know the conclusion (i) holds. Moreover, similar to the proof of Theorem 4.6, there exists a positive constant θ independent of b i (τ ) such that From the global bifurcation in [35] and (77), Γ i is contained in a global branch Σ i of positive solutions. Furthermore, Σ i must satisfy (1): Σ i connects to another bifurcation point (b j,S , P * , W * ) for some j ∈ {1, · · · , p} and j = i; or Assume (1) does not happen, then (2) occurs. By (77), we know the projection of Σ i on to b-axis is not compact. Furthermore, by Corollary 4.11, we know that the projection of Σ i on to b-axis can not extend to −∞, and so the projection of Σ i on to b-axis contains the interval (b i,S , ∞). The conclusion (ii) holds.

Numerical simulations.
To visualize the cascade of Turing instability, Hopf bifurcation and steady state bifurcation described in Theorems 3.3, 3.4 and 4.12, we consider two numerical examples.