Bifurcation analysis of a reaction-diﬀusion-advection predator-prey system with delay

A diﬀusive predator-prey system with time delay and advection is considered. By regarding the conversion delay τ as a main bifurcation parameter, we show that Hopf bifurcation occurs when the parameter τ varies. Then by the improved normal form theory and the center manifold theorem for partial functional diﬀerential equations, an algorithm for determining the direction and the stability of Hopf bifurcation is derived. Finally, some numerical simulations are carried out for illustration of the theoretical results.


Introduction
Reaction-diffusion equations with delays arise from many fields such as biology, chemistry, and physics, and have been investigated extensively recently.For example, predator-prey systems with diffusion and delays are paid more attention by many researchers, see [2,6,8,10,18,20,22,23,24]; There are several articles on the bifurcation theory of delay reaction-diffusion models arising from biology or chemical reactions , see [9,15,16,17,21,25].This paper is concerned with a delay-diffusion-advection predator-prey system which is in the following form: where d, ε, α, a, b, c, τ, γ and β are all positive constants, d, εd are diffusion coefficients, α and εα are the advection transport velocities for the prey and predator respectively, a is the growth rate of prey, b is related to the intraspecific competition of prey, c is the predation coefficient of predator, γ is the transformation rate of prey to predator, β is related to the death rate of predator and τ is the average transformation time of prey to predator.We should point out that the diffusion coefficients (advection transport velocities) are proportionate is a need of mathematical technique.
We would like to mention that the model (1.1) with d = α = 0 has been investigated by several researchers, see [1,11,14] and the references therein.The bifurcation of the model (1.1) with α = 0 subject to with Neumann boundary conditions has been considered by Wu [19], where γ is regarded as a bifurcation parameter and the existence of Hopf bifurcation is obtained.And subject to Dirichlet boundary conditions is studied in [13], where a is regarded as a bifurcation parameter, the existence and the properties of the Hopf bifurcation are obtained.
In 1996, Wu [19] established the Hopf bifurcation theory for a class of abstract semilinear partial functional differential equations.In 2000, Faria [5] extended the normal form method introduced in [4] to partial functional differential equations, and successfully applied to a kind of delayed predatorprey model with diffusion under Neumann conditions, see [6].Based on their fundamental theory, we extend the method presented in Hassard et al. [7] to our model, and derive a algorithm for determining the properties of Hopf bifurcation such as the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions, where the delay τ is regarded as bifurcation parameter.Our main findings are that, a family of inhomogeneous periodic solutions, could bifurcate from the positive constant steady state at the first bifurcation value under certain conditions, in virtue of the effect of diffusion and delay.
The rest of this paper is organized in the following way.In Section 2, we obtain the stability of the positive constant steady state and existence of Hopf bifurcations of system (1.1).In Section 3, stability and direction of bifurcating periodic solutions are discussed by applying the normal form theory and the center manifold theorem of partial functional differential equations.We give an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.Finally, in Section 4, some spatially inhomogeneous periodic solutions are presented to illustrate the theoretical results.

Stability of positive steady state and existence of Hopf bifurcations
Clearly, the system (1.1) has a unique constant positive steady state (β, (a − bβ)/c) when a > bβ.So we make the following assumption always Regarding the delay τ as a main bifurcation parameter, we will consider the stability of equilibrium (β, (a − bβ)/c) and existence of Hopf bifurcations in the following.For convenience, we introduce the transformation û = u − β, v = v − (a − bβ)/c (and we drop the hats of û, v for simplicity of notations).
The linearization of (2.1) around the origin is Define the real-valued Sobolev space and the complexification of X to be We have known that the eigenvalues of the following eigenvalue problem are given by {µ n } n≥1 with and lim n→∞ µ n = ∞.In fact, each µ n is determined by the following equation tan The associated eigenfunction of µ k is where and Then the characteristic equations of (2.2) are given by det(λId That is where Clearly, T k > 0 and D k > 0 for all k ≥ 1, then all the roots of (2.6) have negative real parts when τ = 0. Furthermore, the equilibrium is asymptotically stable.We shall seek critical values of τ such that Eq. (2.6) has a pair of purely imaginary roots.Let ±iω(ω > 0) be the roots of equation (2.6).
Then we have Separating the real and imaginary parts gives (2.7) It follows from (2.7) that we know that the equation (2.8) may have positive roots given by Clearly, (2.9) does not make sense when In other words, (2.6) has no purely imaginary roots for a ≤ bβ follows that, if a > bβ + D1 γβ , there exists a integer k 0 > 1 such that (2.9) makes sense for 1 ≤ k < k 0 , and does not for k ≥ k 0 .Combining with that all the roots of Eq.(2.6) with τ = 0 have negative real parts, and the zero is not a root of Eq.(2.6), we have the following conclusions.
, then all the roots of Eq.(2.6) have negative real parts for all τ ≥ 0; γβ , then there exists a integer k 0 > 1 such that (2.9) makes sense for 1 ≤ k < k 0 , and does not for k ≥ k 0 .
We make the following assumption: γβ , k 0 is the ingeger so that (2.9) makes sense for 1 ≤ k < k 0 , and does not when k ≥ k 0 .
Under the hypothesis (H 2 ) , we define where either So far, we have know that ±iω k is a pair of imaginary roots of Eq.(2.6) with Set λ(τ ) be the root of Eq.(2.6) satisfying Reλ(τ k ) = 0 and Imλ(τ The proof of the lemma can be found in [3]. k } and the corresponding purely imaginary roots ±iω 0 .Then we have the following conclusion on the distribution of the roots of Eq.(2.6).Lemma 2.3.Suppose (H 2 ) is satisfied.Then all the roots of Eq.(2.6) have negative real parts when τ ∈ [0, τ 0 ), the other roots of Eq.(2.6) with τ = τ 0 , except the imaginary roots ±iω 0 , have negative real parts, and Eq.(2.6) has at least a couple of roots with positive real parts.
Applying lemmas 2.1-2.3,we have the following conclusions on the dynamics of the system (1.1).

Stability and direction of bifurcating periodic solutions
In this section, we shall study the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from τ 0 .We introduce the variables changes: Then system (2.1) becomes By the results obtained in the previous section, we know that the system (3.1)undergoes a Hopf bifurcation at the origin when ν = 0.Meanwhile, when ν = 0, the characteristic equation of (3.1) associated with the origin has a pair of simple pure imaginary roots ±iτ 0 ω 0 and all the other roots, except the pure imaginary roots, have negative real parts. Denote and and denote Then in C , the system (3.1) has the form where We also know that ±iτ 0 ω 0 are the simply purely imaginary characteristic values of linear equation of (3.2) at (0, 0), the linear equation is as follows.
The infinitesimal generator A ν is given by , where b k is defined by (2.5).Then {β k } ∞ k=1 form an orthogonal basis for X.For ϕ = (ϕ (1) , ϕ (2) ) T ∈ C , let , b k ⟩, ⟨ϕ and where Then it follows that By the Riesz representation theorem, there exists a matrix whose components are bounded variation In fact, we choose Then (3.6) is satisfied.Hence, (3.4) can be rewritten in the following form: Let A * be the adjoint operator of A, we have ⟨ψ, Aφ⟩ = ⟨A * ψ, φ⟩. (3.9) Similar to the method used to solve equation ( 2.3), we provide the solution for (3.9): where Thus the bilinear form can be rewritten as where (•, •) c is the bilinear form defined on C * × C: where η k (ν, θ), k = 1, 2, • • • are defined as (3.7).Therefore, we get the adjoint operator In the following, for a detailed calculation process, please refer to the appendix.Here we give the main conclusions.The two key values µ 2 and β 2 are calculated as follows.

Numerical simulations
In order to better explain the theoretical results, we give some numerical calculation results.The parameters in system (1.1) are selected as follows.

The effect of advection transport velocities
In this section, we will discuss the impact of convection rate on population size and explore the rules of population change when the rate changes.When the time delay τ is constant, the stable region of the system expands as the convection rate increases.When we fix τ to 3.8, other parameters are still selected from (D).The population size has gradually changed from being influenced by space and time at the beginning to a stable state.Let's take the predator as an example, and use Fig. 4 to illustrate the change.Obviously, dη T k (0, s)q * (−s) bk . (5.1) By calculation, we get where , where I 2 is a 2 × 2 identity matrix.Then the center subspace of the linear equation (3.3) with ν = 0 is given by P CN C, where Let P S C be the stable subspace of the linear equation (3.3) with ν = 0, then C = P CN C ⊕ P S C. From [19], we know that the flow of system (3.2) with ν = 0 in the center manifold is given by the following formula: with h(0, 0, 0) = 0 and Dh(0, 0, 0) = 0. Let z = x 1 − ix 2 and Ψ(0) = (Ψ 1 (0), Ψ 2 (0)) T .We know that q = Φ 1 + iΦ 2 , then (5.2) can be transformed into with W (z, z) = h( z+z 2 , i(z−z) 2 , 0).Combining (5.3) and formula (5.4), we know that z should satisfy ż = iω 0 τ 0 z + g(z, z), (5.5)where (5.7) From (5.2),(5.4)and (5.5), we have 11 (0) ) ) To obtain g 21 , we need to calculate W 20 (θ) and W 11 (θ) (θ ∈ [−1, 0]).We let A U denote the generator of the semigroup generated by the linear system (3.3) with ν = 0. Combining with (5.4) with (5.5), we know that W (z, z) satisfies  (5.9) Since 2iω 0 τ 0 and 0 are not eigenvalues of (3.3), the system (5.9) has unique solutions Therefore, W 20 (θ) and W 11 (θ) can be obtained, the expression of g 21 is also obtained.