Traveling Wave Solutions in a Stage-Structured Delayed Reaction-Diﬀusion Model with Advection

. We investigate a stage-structured delayed reaction-diﬀusion model with advection that describes competition between two mature species in water ﬂow. Time delays are incorporated to measure the time lengths from birth to maturity of the populations. We show there exists a ﬁnite positive number c ∗ that can be characterized as the slowest spreading speed of traveling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium. The model and mathematical result in [J.F.M. Al-Omari, S.A. Gourley, Stability and travelling fronts in Lotka–Volterra competition models with stage structure, SIAM J. Appl. Math. 63 (2003) 2063–2086] are generalized. We use H to denote the habitat where the species grow, interact and migrate. H is either the real line (the continuous habitat) or the subset of the real line which consists of all integral multiples of positive mesh size h (a discrete habitat). Let τ be a nonnegative real number. We shall use boldface Roman symbols like u ( θ, x ) to denote k -vector-valued functions of the two variable θ and x , and boldface Greek letters to stand for k -vectors, which may be thought of as constant vector-valued functions. We deﬁne u ≥ v to mean that u i ( θ, x ) ≥ v i ( θ, x ) for all i = 1 , 2 , . . . , k , θ ∈ [ − τ, 0] and x ∈ H , and u (cid:29) v to mean that u i ( θ, x ) > v i ( θ, x ) for all i , θ and x . We also deﬁne max { u ( θ, x ) , v ( θ, x ) } to mean the vector-valued function whose i th component at ( θ, x ) is max { u i ( θ, x ) , v i ( θ, x ) } . We use the notation 0 for the constant vector all of whose components are 0.


Introduction
In 1990, Aiello and Freedman [1] developed the following time-delay model of single-species growth with stage structure: where α, β, γ and τ are positive constants. In this model, u i and u m denote, respectively, the numbers of immature and mature members of the population.
The parameter τ in model measures the time from birth to maturity. The rate at which individuals are born is taken to be proportional to the number of matures at that time; this is the αu m (t) term. And e −γτ terms allow for the fact that not all immatures survive to maturity (γ is the death rate of immatures, while β measures deaths of matures). When the individuals are allowed for moving around, in 2003 Gourley and Kuang [5] initially studied the diffusive version of the system (1.1): It is well known that competition both within and between species is an important topic in ecology, especially community ecology. Observe that system (1.2) is not a fully coupled system in that the second equation, for the mature population u m , can be solved independently of the first. Consideration of this second equation alone is an interesting and non-trivial mathematical problem in its own way. Therefore Al-Omari and Gourley [2] studied the following Lotka-Volterra competition model: x)U 2 (t, x), U 2,t = d 2 U 2,xx +α 2 e −γ2τ2 U 2 (t − τ 2 , x) − β 2 U 2 2 (t, x) − c 2 U 1 (t, x)U 2 (t, x). (1.4) In model (1.4), it is assumed that competition occurs only between the adults, since many species strongly protect their young, and the immature population are not moving.
In the real world, there are many cases that species live in the media where the diffusion moves to a certain direction. Typical examples include aquatic organisms in stream, and marine organisms with larval dispersal influenced by ocean currents. When one direction in the random walk is favored, an equation with a first order derivative is the result: (1.5) The first order derivative term on the right hand side of the equation (1.5) is called an advection term, and the equation (1.5) is called a reaction-diffusion equation with advection. When the mature individual movement is described as a combination of advection corresponding to the one-dimensional medium with a undirectional Table 1. Parameter description for model (1.6).

Parameter
Description of parameter U i (t, x) population density of immature members U at time t and point x Um(t, x) population density of mature members U at time t and point x V i (t, x) population density of immature members V at time t and point x Vm(t, x) population density of mature members V at time t and point x d Um diffusion coefficient of mature population U d Vm diffusion coefficient of mature population V a Um advection speed of mature population U experienced by the organism a Vm advection speed of mature population V experienced by the organism α 1 birth rate of population U α 2 birth rate of population V γ 1 death rate of immature population U γ 2 death rate of immature population V β 1 death rate of mature population U β 2 death rate of mature population V τ 1 time delay from birth to maturity of population U τ 2 time delay from birth to maturity of population V c 1 competitive effect of the species V has on the mature of species U c 2 competitive effect of the species U has on the mature of species V flow as experienced by the organisms and diffusion corresponding to the heterogeneous stream flow and individual swimming, we are concerned with the following delayed competition reaction-diffusion model with stage structure and advection: Here, we assume the immature individuals always stay on the stationary until they grow to maturity during a period of time, only the mature individuals drift in the water flow, and competition occurs only between the adults. A descriptions of all parameters can be found in Table 1 and all c-values are positive. In the model (1.6) it is assumed that competition effects are of the classical Lotka-Volterra kind, while death of the matures for each population is modelled by quadratic terms, as in the logistic equation.
Since the second and the fourth equations in system (1.6) are uncoupled from the first and the third equations, it is sufficient to consider a simplified model with the first and third equations merged: (1.7) Here u(t, x) and v(t, x) represent densities of adult members of two species u and v at time t and point x, respectively. d 1 > 0 (d 2 > 0) is the diffusion coefficient of the adult population u (v). a 1 > 0 (a 2 > 0) is the advection speed experienced by the organism. α 1 (α 2 ) combining two factors: the per capita birth rate and the survival rate of immature for the population u (v) during the immature stage. Mathematical modelling has been central to the development of general invasion theory (e.g., [12,13,14,15,16]). System in the form of reaction-diffusion equations and integro-difference equations are commonly used to describe biological invasion process. In ecology, the existence of travelling wave solutions connecting two equilibria means that the unstable equilibrium is taken over by the stable one in space as time increases. Studies on existence of traveling waves in such system have received considerable attention, and many noteworthy findings have come out of this field, e.g., [2,3,5,6,7,8,9,10,17,18,19,20,21,22].
Al-Omari and Gourley [2] studied the existence of traveling wave solutions connecting two mono-culture equilibria in model (1.4) under a technical assumption by using the theory of monotone iteration scheme developed by Wu and Zou [20]. This presents the invasion by the strong species of territory previously inhabited only by the weaker. The results in [2] made significant progress in establishing travelling wave solutions for model (1.4). It is pointed out in [2] that the technical assumption for the existence of traveling wave solutions may not be essential. In this paper, we attempt to provide the sharpest result regarding the existence of travelling wave solutions connecting an unstable mono-culture equilibrium with a nontrivial equilibrium in model (1.7) using our abstract result for general delayed recursion model established in [7]. We show that there exists a finite positive number c * that can be characterized as the slowest speed of travelling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium.
The rest of this paper is organized as follows: In Section 2, we present some preliminaries including equilibria and stability for system (1.7), and abstract results for delayed recursion. In Section 3, we prove the existence of traveling wave solutions using our abstract result in [7]. Some concluding remarks are given in Section 4.

Equilibria and stability
Based on the model (1.7), the corresponding nonspatial system is yield: It is straightforward to show that the solutions of system (2.1), subject to (2.2) above, satisfy u(t), v(t) > 0 on (0, ∞). Positivity of (2.1) is important for both the modeling and the analysis. From the point of biology, positivity implies that the system persists.
By the study of the local and global stability of the equilibria of the nonspatial system (2.1), we conclude that: 1. At least one of two interactive species with stage structure can persist in a stream due to the fact that the trivial equilibrium E 0 is always unstable.
2. One species out-competes the other one. In other words, one of them will die out due to the competition for the limited source in the long run.
3. However, under the conditions that the two mono-culture equilibria E u and E v are both unstable, the two species can coexist and approach a stable population density in the long term. This is the explanation of the fact that the unique coexistence equilibrium E + is globally asymptotically stable under certain conditions given in the Theorem 1 part iii .

Abstract result for delayed recursion
We use H to denote the habitat where the species grow, interact and migrate.
H is either the real line (the continuous habitat) or the subset of the real line which consists of all integral multiples of positive mesh size h (a discrete habitat). Let τ be a nonnegative real number. We shall use boldface Roman symbols like u(θ, x) to denote k-vector-valued functions of the two variable θ and x, and boldface Greek letters to stand for k-vectors, which may be thought of as constant vector-valued functions. We define u ≥ v to mean We use the notation 0 for the constant vector all of whose components are 0.
Let C be the set of all bounded continuous functions from [−τ, 0] × H to R k , C be the set of all bounded continuous functions from [−τ, 0] to R k , and X be the set of bounded continuous functions from H to R k . If r ∈C with r 0, we define the set of continuous functions Moreover, we define the metric function d(·,·) in C by so that (C, d) is a metric space. The convergence of a sequence φ n to φ with respect to this topology is equivalent to the uniform convergence of φ n to φ on bounded subsets of [−τ, 0] × H.
We study the following discrete-time recursion , and x ∈ H represents the population densities of the populations of k species at time n and point x with time delay τ . The operator Q is said to be order − preserving (2.4) in which Q has this property is said to be cooperative. A function is said to be an equilibrium of Q if Q[w] = w, so that if u l = w in the recursion (2.4), then u n = w for all n ≥ l. We shall study the evolution of the solution u n of the recursion (2.4) from a u 0 near an unstable constant equilibrium θ. By introducing the new variableû = u − θ if necessary, we shall assume the unstable equilibrium θ from which the system moves away is the origin 0.
We define the translation operators We shall make the following hypotheses on Q.
ii. Q is order-preserving on nonnegative functions.
iv. Q is continuous with respect to the topology determined by d(·,·) given in (2.3).
v. One of the following two properties holds: , and the operator Remark 1. Hypotheses 2.1 i-ii imply that Q takes C β into itself, and that the equilibrium β attracts all initial functions in C β with uniformly positive components. In biological terms, β is a globally stable coexistence equilibrium. There may also be other equilibria lying between β and the extinction equilibrium 0, in each of which at least one of the species is extinct. Throughout this paper, we shall assume that the recursion (2.4) has a finite number of equilibria and that the equilibria of (2.4) are completely separate in the sense that for any two equilibria ν 1 (θ),

Remark 2.
Here we have dropped the hypothesis in [7] that operator Q is reflect invariant. In [7] the reflection invariance was assumed, but it was not used in the proof of Theorem 2.1 and Theorem 2.2. Consequently, Theorem 2.1 and Theorem 2.2 are still valid without the reflection assumption.

Lemma 1 [Comparison Lemma].
Let R be an order preserving operator. If u n and v n satisfy the inequalities u n ≤ R[u n ] and v n ≥ R[v n ] for all n, and if u 0 ≤ v 0 , then u n ≤ v n for all n.
Denote a(c; θ, ∞) as the limit of a n (c; θ, ∞). Note that a n ≤ a n+1 ≤ β for all n, and a n (c; θ, s) is nonincreasing in c and s and continuous in (c, θ, s). Define Denote {Q t } ∞ t=0 as the continuous time semiflow on C β . In [7] we established the following existence result (see [7,Theorem 2.2]). We shall employ this abstract result to prove the existence of travelling wave solutions in model (1.7).

Existence of Traveling Waves in (1.7)
Suppose that (3.1) By Theorem 1, E u is unstable, E + exists and it is globally attracting if α 2 p 1 < α 1 η 2 , and E + does not exist and E v is globally attracting if α 2 p 1 > α 1 η 2 . We shall study the existence of travelling wave solutions connecting E u to E + if α 2 p 1 < α 1 η 2 , and to E v if α 2 p 1 > α 1 η 2 , see Figure 1. Let u := u * − u and v := v. We convert the competition system (1.7) into the following cooperative system (3.2) Figure 1. The spatial transition from Eu to Ev if α 2 p 1 > α 1 η 2 , and to E + if α 2 p 1 < α 1 η 2 in competition system (1.7).
For this system, we denote Clearly, 0 = (0, 0) and β are the only two equilibria in C β if α 2 p 1 < α 1 η 2 ; there is an extra equilibrium ν = (u * , 0) in C β if α 2 p 1 > α 1 η 2 . Hence, we now only to prove the existence of travelling wave solutions connecting 0 to β, see Figure 2. We now state and prove our main result. In what follows we shall prove Theorem 3 by using Theorem 2, that is verifying the abstract Hypotheses 2.1.
Proof. Let {T u (t)} t≥0 and {T v (t)} t≥0 be the solution semigroup on X generated by the heat equations Then we can write (3.5) as the following integral equations where Under the abstract setting in [11], a mild solution of (3.5) is a solution to its associated integral equation (3.6). One can easily verify that f 1 and f 2 are Lipschitz continuous on any bounded subset of C × C. Let Z = BUC (R, R 2 ) be the Banach space of all bounded and uniformly continuous functions from R into R 2 with the usual supremum norm. Let We show that f 1 and f 2 are quasi-monotone on C in the sense that for all φ j , ψ j ∈ C β , j = 1, 2 with (φ 2 , ψ 2 ) ≥ (φ 1 , ψ 1 ). From the definitions of f 1 and f 2 in (3.4) we get It follows that for sufficiently small h > 0, Which indicate that (3.7) holds. By Corollary 5 in [11], we can show the existence and uniqueness of (u(t, x; φ, ψ), v(t, x; φ, ψ)) with and v + = β, v − = 0. Moreover, by the semigroup theory given in the proof of Theorem 1 in [11], it follows that (u(t, x; φ), v(t, x; ψ)) is a classical solution for t > τ .
x)) and v − = 0, respectively, we obtain This completes the proof.
By the properties of T u (t) and T v (t) and the boundness of f i , we see that Q u t and Q v t are compact for each t > τ . Thus Q t satisfies Hypotheses 2.1 v.a. when t > τ .
Lemma 5. Let c * be defined by (2.6) where Q is replaced by the time one map Q 1 of (3.2), then 0 < c * < ∞.
Proof. We write the time one solution operator Q 1 of (3.2) as Q 1 = (Q u1 , Q v1 ). LetQ v1 be the time one solution map of (3.11) The reaction term in (3.11) is f 2 (0, v t )(x) where f 2 given in (3.4). Since The definition of c * and Lemma 1 show that where c * v is given by (2.6) where Q isQ v1 .

Concluding Remarks
In this paper we investigated a system of delayed reaction-diffusion equations which modelled growth, spread and competition of two species with stage structure in water flow. This model is an extension of the time-delayed population system with stage structure studied by Al-Omari and Gourley [2] and Zhang et al. [22]. There is no advection terms in their models, which can provide a resolution to the drift paradox in stream ecology (see [4]). "Drift paradox" is one key issue for theory in stream ecology and states that extinction is inevitable in a closed population subjected only to downstream drift. By using the abstract theorem in [7] we show that there exists a finite positive number c * that can be characterized as the slowest speed of travelling waves connecting two mono-culture equilibria or a mono-culture with the coexistence equilibrium. The existence result of travelling waves in [2] is improved.
In 2012, Li [6] showed that for the general partial cooperative reactiondiffusion system u ,t = Du ,xx − Eu ,x + f u(t, x) and γ 1 (µ) is the principal eigenvalue of the first irreducible block of the moment generating matrix C µ of the linearized system of (4.1). However, in our system (1.7) the delay is considered in the vector function f (u(t, x)). Consequently the principal eigenvalue γ 1 (µ) of C µ and the corresponding eigenvector ξ(µ) cannot be expressed explicitly. Thus the ideas used in [6] cannot be applied in our system. In order to verify the linear determinacy conditions, one might first write the ξ(µ) as a function of the principle eigenvalue γ 1 (µ). Then use the fact that the wave speed equation φ(µ) = γ 1 (µ)/µ is a convex function and thus infimum in Eq. (4.2) exists, one might provide an estimation of γ 1 (µ) and therefore show that the linear determinacy conditions are satisfied under appropriate assumptions. Liang and Zhao [8] developed the analytical theory on the spreading speeds for delayed cooperative system, which can be applied to our cooperative system (3.2). We shall study the issue on spreading speeds for model (1.7) in the following work.