Existence of periodic wave trains for an age-structured model with diffusion

In this paper, we make a mathematical analysis of an age-structured model with diffusion including a generalized Beverton-Holt fertility function. The existence of periodic wave train solutions of the age structure model with diffusion are investigated by using the theory of integrated semigroup and a Hopf bifurcation theorem for second order semi-linear equations. We also carry out numerical simulations to illustrate these results.

1. Introduction. Cannibalism (or predation within species) is an important and universal characteristics of animal species. For fishery models, cannibalism is slightly different from predator-prey interactions among potential competitors, Ricker [23] suggested β(N ) = β 0 e −αN , (1.1) while Beverton and Holt [2] suggested β(N ) = β 0 1 + αN , (1.2) where β 0 is the coefficient involving compensatory density dependent predation. Heavy cannibalization of juveniles by adults of the same species shortly after birth can be explained by the above both functional forms [25]. Different forms come from different hypotheses about the distribution of the length of the juvenile period; in the Ricker equation (1.1) the juvenile period has a fixed length, while in the Beverton-Holt equation (1.2) its length is exponentially distributed, which means individuals make the transition from juvenile to adult stage at a fixed constant rate. To some extend, it is disconcerting that different assumptions about the length distribution of the juvenile period lead to these dramatically different forms. Models as the Beverton-Holt and Ricker equations [25,Chapter 9], or the well-known quadratic family (also known as the discrete logistic model [20,Sect. 2.3]) were investigated by a large number of authors [2,3,23,25,20,24,8]. In this paper, the generalized Beverton-Holt map β(N ) = β0 1+αN n is considred as the boundary condition, where n ≥ 1 measures the abruptness of the onset of density dependence [8]. When n = 1, β(N ) is the well known Beverton-Holt equation (1.2) and is an increasing, saturating, concave function. The generalized form of n > 1 is the hump shape function, which is very consistent with the insect population census data [1].
In population dynamics, age becomes the major way to affect the size and growth of the population. Individuals with different ages have different reproduction, survival abilities and behaviors. In many situation, population dynamical models become considerably intricate when age structure will be considered in individual interactions. In recent years, considering the significance of age structure in populations has become more and more prevalent, the literature on all aspects of the interacting populations with age structure has proliferated [4,5,19,11,27,28,30,31,32,15].
The influence of spatial variables on population dynamics is crucial. When the spatial variables are added to the age structure model, more complicated behavior is possible, including stable patterns, spiral waves and periodic wave trains [9,16,21,29,17,22,26,18,10]. The existence of traveling wave solutions for a class of diffusive predator-prey type systems was investigated by [9] and their results can apply to various kinds of ecological models. [16] considered periodic travelling wave solutions and computed numerically the dispersion relation curves for periodic wave trains of the FitzHugh-Nagumo nerve axon equation. Authors [21] discussed timeand space-periodic solutions of reaction-diffusion systems via bifurcations when the source function parameter, the frequency and the wave vector change. The slow dispersion of nonlinear water waves and periodic wave trains in water of arbitrary depth were studied in [29]. The selection and stability of periodic wave trains behind the invasion fronts were concerned with [17] in predator-prey systems where the prey has an intraspecific nonlocal competition. The bifurcation of periodic wave trains with large wave length in reaction-diffusion systems was studied in [21]. For other literature about periodic wave trains, we refer to the references [9,16,21,29,17,22,26,18,10].
At present, a large number of existing literatures [9,16,21,29,17,22,26,18,10] mainly consider the wave train solutions of the reaction-diffusion equation. In this paper, we explore the periodic wave train solutions of the following age-structured model with diffusion: where µ, β 0 and α are positive parameters; u(t, a, z) denotes the density of population of baleen whale with respect to the age of individuals a in the location z at time t; β(a) is an age-specific fertility function related to individual-age a and satisfies Assumption 1.1. Here ∆ z denotes the Laplace operator for the variable z ∈ R N , for some given integer N ≥ 1.
Recall that a couple (c, U ≡ U (x, a)) is said to be a periodic wave train profile with speed c ∈ R if the function U is periodic with respect to its variable x ∈ R, namely there exists a period T > 0 such that U (T + ·, ·) = U (·, ·) and such that for each direction e ∈ S N −1 the function u(t, a, z) := U (z · e + ct, a), x := z · e + ct, where c is the speed, is an entire solution of (1.3). In other words, the profile (c, U ) is a periodic-in x-solution of the following second order problem (1.4) In order to prove the model (1.3) has periodic wave train solutions, we just need to prove (1.4) has periodic solutions. To achieve the results, we first performed a variable transform on the systems (1.4) and transformed it into a second order abstract semilinear equation in which ε > 0 is a given parameter; A : D(A) ⊂ X → X is a weak Hill-Yosida linear operator acting on a Banach space (X, · ) while G : D(A) → X is a smooth nonlinear operator (see Section 3 below). The second order equation (1.5) can be regarded as the singularly perturbed system of the first order equation From the results in [7], one can know that non-degenerate Hopf bifurcation for (1.6) is persistent for (1.5) when ε is small enough, which means if the system (1.6) has periodic solutions emanating from the positive equilibrium, then the system (1.5) will also have periodic solutions. Therefore, if we can prove (1.4) has periodic solutions, then the age-structured model with diffusion (1.3) will have periodic wave train solutions. The major contribution of this paper is organized as follows. Using the Hopf bifurcation theory to prove the existence of periodic solutions of the age-structured model without diffusion are presented in Section 2. The existence of periodic wave train solutions for the age-structured model with diffusion are derived in Section 3. A detail numerical simulation is included in Section 4, and finally, the paper ends with a conclusion.

2.
The age-structured model without diffusion. Let a → u(t, a) be the density of population of baleen whale with respect to the age of individuals at time t. This means that for each a 1 ≤ a 2 the number of individuals with a size in between a 1 and a 2 is a2 a1 u(t, a)da.
In particular the total number of baleen whale is The baleen whale equation with an age structure can be rewrite as β(a)e −µa da is called the net reproduction rate and gives the number of the newborn that an individual is expected to produce during his reproductive life, where e −µa denotes the survival probability, i.e., the probability for an individual to survive to age a.

2.1.
Rescaling time and age. In this subsection, our destination is to get a smooth dependency of the system (2.1) about τ . We first use the time-scalingt = t τ , age-scalingâ = a τ , and the change of variablesû(t,â) = τ u(τt, τâ) andβ(â) = β(τâ) to normalize τ in (2.1). For abbreviation, after the change of variables we drop the hat notation and obtain the following system where the new function β(a) is given by where τ ≥ 0, β * > 0 and 0 < R < +∞. Next we consider the following Banach space The linear operator A τ : Then by identifying u(t) with u(t, a) and w(t) = 0 u(t) , system (2.1) can be rewritten as the following non-densely defined abstract Cauchy problem The results of global existence and uniqueness of solution of system (2.5) can be obtained by [14] and [13].

Equilibria and linearized equation.
In this subsection, we will obtain the equilibria of system (2.5) and linearized equation of (2.5) around the positive equilibrium.

Existence of equilibria.
Suppose that w(a) = 0 u(a) ∈ X 0 is a steady state of system (2.5). Then Hence, we have the following lemma. In the remainder of our paper we assume that β 0 R − 1 > 0.

Linearized equation.
Here we will obtain the linearized equation of (2.5) around the positive equilibrium w τ . By applying the change of variable y(t) := w(t) − w τ , (2.5) becomes (2.7) Therefore we can write the linearized equation (2.7) around the equilibrium 0 in the form Accordingly system (2.7) can be rewritten as is a linear operator and satisfying H(0) = 0 and DH(0) = 0.

Characteristic equation.
In this subsection, we will get the characteristic equation of (2.5) around the positive equilibrium w τ . Denote Following the results of [12], we derive the following lemma.
Consequently we derive the following proposition.

Existence of Hopf bifurcation.
In this section, we consider the length of the juvenile period τ as a bifurcation parameter and investigate the existence of Hopf bifurcation by applying the Hopf bifurcation theory [12] to the Cauchy problem (2.5). From (2.22), we have where p 0 and q 0 are given by (2.21). Obviously, if β 0 R − 1 > 0, then p 0 + q 0 = 3µ(β0R−1) β0R > 0 and ζ = 0 is not a eigenvalue of (2.23). Note that when β 0 R − 1 > 0, the all roots of (2.23) have negative real parts for τ = 0.
3. The age-structured model with diffusion. In this section we consider the existence of periodic wave train solutions for the age-structured model with diffusion Observing that X 0 = D(A) = X. We also consider the nonlinear map G : X 0 → X defined by Next setting v(x) = 0 U (x, ·) , then system (1.4) can be rewritten as Setting c = 1 ε , t = εx and v(x) =ṽ(εx), then we drop the hat notation and system (3.1) becomes It is obviously that system (3.2) is the singularly perturbed system of the following system , we obtain that system (3.3) is equivalent model (2.1).
From section 2, we know that Hopf bifurcation occurs in model (3.3). By using the corollary 7.8 in [7], we obtain the following result. Additionally, β 0 R = 4 > 3 satisfies the conditions of Assumption 2.1. Calculating it further, we can easily obtain that ω 0 ≈ 0.15 and the first critical value τ 0 ≈ 16.6539.
In Figure 1, we choose the bifurcation parameter τ = 10 < τ 0 and the positive equilibrium u τ =10 (a) = 10.8480e −2a is locally asymptotically stable. Figure 1 (a) demonstrates the solution behaviors of the population. Figure 1 (b) describes the change of the distribution function of the population u(t, a) as the time and age vary. By further continuously increasing τ to 20 > τ 0 , there appears a solution of system (2.1) around the positive equilibrium that has an oscillating behavior and gradually tends to closed orbit. u τ =20 (a) = 10.8480e −0.4a , meanwhile the conclusion of Theorem 2.1 is also numerically demonstrated (see Figure 2). In Figure 2 (a), the solution curves illustrate an oscillation behavior gradually maintaining closed orbit. The variation of u(t, a) as time and age vary at τ = 20 > τ 0 is demonstrated in Figure 2    In Figure 3, we choose the bifurcation parameter τ = 10 < τ 0 and set c = 1000, then the solution of system (1.4) is locally asymptotically stable. In Figure 4, we choose the value of the bifurcation parameter τ = 20 > τ 0 and set the speed c = 1000. It is not difficult to find that oscillatory solution tending closed orbit for system (1.4) is consistent for system (2.1). From these numerical analysis results, we can demonstrate the theoretical findings in Section 3.

5.
Conclusions. In this paper, we introduce an age-structured model with diffusion including a generalized Bevert-Holt fertility function. The existence of the Hopf bifurcation for the age-structured model without diffusion is studied. Bifurcation analysis indicates that there exist some parameter values such that this age-structured model without diffusion has a non-trivial periodic solution which bifurcates from the positive equilibrium. In particular, periodic wave train solutions of the age-structured model with diffusion are considered by employing the methods developed in [7]. Additionally, numerical simulations of model (1.4) and (2.1) are also provided respectively to exemplify the theoretical findings.