HOPF BIFURCATION FOR A SPATIALLY AND AGE STRUCTURED POPULATION DYNAMICS MODEL

. This paper is devoted to the study of a spatially and age structured population dynamics model. We study the stability and Hopf bifurcation of the positive equilibrium of the model by using a bifurcation theory in the context of integrated semigroups. This problem is a ﬁrst example for Hopf bifurcation for a spatially and age/size structured population dynamics model. Bifurcation analysis indicates that Hopf bifurcation occurs at a positive age/size depen- dent steady state of the model. The results are conﬁrmed by some numerical simulations.


1.
Introduction. In this article we consider a mathematical population dynamics model to describe the growth of trees. The main characteristic taken into account for the growth of such population are the size of trees and the spatial location. Since the small trees are growing in the shade of the bigger trees, we should take into account the competition for light between big and small trees. Therefore we introduce a competition for light between the small trees and the big trees. The model considered in this article is the following ∂t + ∂u(t, s, x) ∂s = −µu(t, s, x) for s ≥ 0 and x ∈ (0, 1), γ(s)u(t, s, x)ds for x ∈ (0, 1), u(0, ., .) = u 0 ∈ L 1 (0, +∞) , L 1 where µ > 0 is the mortality rate, α > 0 is the birth rate in absence of birth limitations and the birth limitations (competition for food, space or light) are described by h(x) = x exp(−βx), where β > 0. This function is known as Ricker's [24,25] type birth limitation. This type of birth function has been commonly used in the literature. One may observe that the Ricker's type birth limitation was introduced for fishes population in order to describe the cannibalism of adult fishes on lava during the reproduction season.
Here we refer to Ducrot, Magal and Seydi [12] for a mathematical justification of the Ricker function by using a singular perturbation idea. For tree populations the 1736 ZHIHUA LIU, HUI TANG AND PIERRE MAGAL process is similar since the large tree takes most of the light and the young tree can not (or can almost not) grow in the shade of the adult trees.
Here the density of population u(t, s, x) depends on t the time, s the size of individuals (which serves as a clock for the reproduction of trees), and x the spatial location of tree. The density of population here means that s2 s1 x2 x1 u(t, s, x)dxds is the number of trees with a size s ∈ [s 1 , s 2 ] and located in the region [x 1 , x 2 ] at time t.
In practice, the function γ(s) should be understood as a probability to reproduce at size s.
In this article we explore the oscillating properties of the solutions around the positive equilibrium. One observe that similar model has been studied by Ducrot [10]. In [10], the spatial domain is the real line and the existence of oscillating traveling waves has been proved around the positive equilibrium. Here the domain is bounded and we will analyze the oscillation by using a Hopf bifurcation theorem for structured population dynamics models.
In the special case γ(θ) = 1 [τ,+∞) (θ), by setting Therefore the system (1) can also be regarded as a delay differential equation in some special cases. We refer to Wu [32] for more results on this subject.
Recently based on the center manifold theorem proved in Magal and Ruan [19], a Hopf bifurcation theorem has been presented for abstract non densely defined Cauchy problem in Liu Magal and Ruan [17]. These theorems have been successfully applied to study the existence of Hopf bifurcation for some age/size-structured models, see [2,4,5,6,19,21,26,31]. More results can be found in the context of cell population dynamics in Doumic et al. [13], and in the context of structured neuron population in Pakdaman et al. [22]. We refer to Cushing [8,9], Prüss [23], Swart [27], Kostova and Li [16], Bertoni [1] for more results on this subject. Early examples of periodic solutions suspected to appear by Hopf bifurcations in age/sizestructured models are mentioned in the literature (Castillo-Chavez et al. [3], Inaba [14,15], Zhang et al. [33]). In this article we consider a first example for Hopf bifurcation appearing in a spatially and age/size structured population dynamics model. As we will see, due to the spatial structure the bifurcation analysis is more complex than for a model without spatial structure.
The plan of the paper is the following. In section 2 we formulate the model (1) as a non-densely defined Cauchy problem and recall the Hopf bifurcation theorem for the abstract non-densely defined Cauchy problem obtained in [17]. In section 3 we investigate the existence and the uniqueness of the positive equilibrium and consider the linearized system around this positive equilibrium. In section 4, we derive a family of characteristic equations and the main result of this paper, that is, the existence of Hopf bifurcation is obtained by analyzing the spectrum property of the non-densely defined linear operator. In section 5 we present some numerical simulations of the model.

Preliminaries. Consider the Banach space
The space X is endowed with the product norm We consider the linear operator A : .
Define H : X 0 → X by Then by identifying u(t) to v(t) = 0 u(t) , the problem (1) can be considered as the following Cauchy problem The resolvent of A is defined for each λ > −µ by that is the linear operator on X 0 defined as follows Since A is a Hille-Yosida operator, it follows that A 0 generates a strongly continuous semigroup of bounded linear operators {T A0 (t)} t≥0 on X 0 . This semigroup is defined by The global existence, uniqueness and positive of solution of the equation (3) follow from the results of Thieme [28], Magal [18] and Magal and Ruan [20]. Consider the positive cone There exists a unique continuous semiflow {V (t)} t≥0 on X 0+ , such that for each x ∈ X 0+ the map t → V (t)x is the unique mild solution of (3), that is to say that In the following, we recall the Hopf bifurcation theorem obtained in [17] for the following abstract Cauchy problem where A : D(A) ⊂ X → X is a linear operator on a Banach space X, F : R×D(A) → X is a C k map with k ≥ 4, and µ ∈ R is the bifurcation parameter. Set where A 0 is the part of A in X 0 , which is defined by We make the following assumptions on the linear operator A and F.
Assumption 2.2. Let A : D(A) ⊂ X → X be a linear operator on a Banach space (X, . ). We assume that A is a Hille-Yosida operator. That is to say that there exist two constants, Assumption 2.2 implies that A 0 is the infinitesimal generator of a strongly continuous semigroup {T A0 (t)} t≥0 of bounded linear operators on X 0 and A generates a uniquely determined integrated semigroup {S A (t)} t≥0 .
, which is an integrated solution of (5) with the parameter value equals µ(ε) and the initial value equals x ε . So for each t ≥ 0, u ε satisfies Moreover, we have the following properties (i) There exist a neighborhood N of 0 in X 0 and an open interval I in R containing 0, such that for µ ∈ I and any periodic solution u(t) in N with minimal period γ close to 2π ω(0) of (5) for the parameter value µ, there exists ε ∈ (0, ε * ) such that u(t) = u ε (t + θ) (for some θ ∈ [0, γ (ε))), µ(ε) = µ, and γ (ε) = γ.
(ii) The map ε → µ(ε) is a C k−1 function and we have the Taylor expansion where ω(0) is the imaginary part of λ (0) defined in Assumption 2.3.
Remark 1. The Crandall and Rabinowitz's [7] condition applys here. If we only assume that k ≥ 2, and in Assumption 2.3, that is to say that the spectrum of A 0 does not contain a multiple of λ (0) . Then the first part Theorem 2.4 hold (excepted (ii) and (iii)).

Existence of equilibrium and linearized equation.
which is equivalent to the following equations ∂u ∂s + µu = 0, for s ≥ 0, and x ∈ (0, 1) , Hence we can obtain Then or equivalently χ satisfies the non-linear boundary value problem By using some boot strapping we deduce that , R) and the following Lemma holds.
Proof. It is sufficient to consider the case χ = 0. Define Then G is C 1 and By multiplying the first equation of (6) by G χ(x) − ln α β and integrating on (0, 1), we obtain By integrating by parts, we obtain Now by using the boundary values χ (0) = χ (1) = 0, and the fact that αh(x) ≤ x, ∀x ≥ ln α β , we deduce that Thus we obtain From this equality it follows that Indeed, assume that there exists Then and by using (7) we deduce that Thus by the continuity of χ we deduce that which is impossible. By multiplying the first equation of (6) by G ln α β − χ(x) and integrating on (0, 1), we obtain By using the same arguments as above and the fact that αh(x) ≥ x, ∀x ∈ 0, ln α β , we deduce that and the result follows.
Thus we can get that Define Ω := {λ ∈ C : Re(λ) > −µ}, and B α : D(A) ⊂ X → X the linear operator defined by B α := A + DH (v α ) , that is to say that We observe that Ω ⊂ ρ (A) the resolvent set of A, and for each λ ∈ Ω, Lemma 3.5. The linear operator B α : D(A) ⊂ X → X is a Hille-Yosida operator and its part (B α ) 0 in X 0 satisfies Proof. Since DH(v α ) is a bounded linear operator and A is a Hille-Yosida operator, it follows that B α = A + DH(v α ) is a Hille-Yosida operator. From (4), we deduce that ω 0,ess (A 0 ) ≤ ω 0 (A 0 ) ≤ −µ. Moreover, DH(v α ) is compact. By using the perturbation results in Thieme [30] or Ducrot, Liu and Magal [11], we obtain Then it is well known that the spectrum of Consider the system Since (12) can be written as the following The first equation of the above system is rewritten as We have Hence and By using (13), (14) and (15), we have It makes sense to consider the characteristic operator ∆(λ, α) : D(∆(λ, α)) ⊂ L 1 (0, 1) → L 1 (0, 1) From the above discussion, we obtain the following lemma.
(ii) If λ ∈ Ω ∩ ρ (B α ) , we have the following explicit formula for the resolvent Proof. We already proved that To prove the converse inclusion, let Then there exists j 0 ∈ N such that For each j ∈ N, is a simple eigenvalue of d 2 ∆ x associated to the eigenfunction That is to say that where the convergence is understood in norm of operator and Π j is the projector defined by Then λ j is a simple and isolate eigenvalue of B α associated to the eigenfunction ϕ(s) = e −(λj +µ)s cos(jπx) and the projector of the generalized eigenspace of B α is defined by Proof. Since ∂ λ Λ( λ j , α) = 0, by the inverse function theorem there exists at most one root λ Λ(λ, α) = λ j in some neighborhood of B λ j , ε . Therefore for each λ ∈ B λ j , ε \ λ j the
Next we will study the existence of Hopf bifurcation when α is regarded as the bifurcation parameter of the system. By Theorem 4.4 we already knew that the positive equilibrium v α of the system (3) is locally asymptoticlly stable if 1 < α ≤ e.
So we will study the existence of a bifurcation value α > e.
Thus for any fixed Re(λ), we can find at most one Im(λ) > 0 satisfying the characteristic equation in (17). The proof is complete.
In the following, we will consider the existence of the purely imaginary solutions of the characteristic equations in (17). ω µ + π + ωτ = 2kπ (18) and Moreover, for each k ∈ N + , there exists a unique ω k > 0 (which is a function of τ , µ) satisfying equation (18).
Thus we can obtain that at α = α j k , k ∈ N + , j ∈ N the characteristic equations ∆ j (λ, α) = 0, j ∈ N have at most a finite number of pair of purely imaginary eigenvalues. We denote the maximum of these purely imaginary eigenvalues by iω j k . Then

Transversality condition.
In this subsection we will prove the transversality condition for the model.
Now we are in the position to present the main result of this paper. By using the Crandall and Rabinowitz's [7] condition for their Hopf bifurcation theorem (see Remark 1), we obtain the following result. Theorem 4.9. The spatially and age structured population model (1) undergoes a Hopf bifurcation at α = α j k and u α = u α j k given in theorem 3.4, where α j k > 0, k = 1, 2, . . . , j ∈ N are defined in (22). In particular, a non-trivial periodic solution bifurcates from the equilibrium u α = u α j k when α = α j k .
5. An example and numerical simulations. In the following, we provide some numerical simulations to illustrate the Hopf bifurcation for the model (1). We consider system (1) In the figures 1-3 we only modify the parameter α. As predicted by the theory when the parameter α passes from α = 10 (in Figure 1) to α = 20 (in Figure 2) and α = 30 (in Figure 3) we observe a Hopf bifurcation of the positive equilibrium.
In figure 1 we observe the global stability of the positive equilibrium. In figures 2-3 the positive equilibrium is destabilized and the solutions converge to a periodic orbit.