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

1 Introduction

In 1990, Aiello and Freedman  developed the following time-delay model of single-species growth with stage structure:

[u.sub.i,t] = [alpha][u.sub.m](t) - [gamma][u.sub.i](t) - [alpha][e.sup.- [gamma][tau]] [u.sub.m](t - [tau]),

[u.sub.m,t] = [alpha][e.sup.-[gamma][tau]] [u.sub.m](t - [tau]) - [beta][u.sup.2.sub.m], (1.1)

where [alpha], [beta], [gamma] and [tau] are positive constants. In this model, [u.sub.i] and [u.sub.m] denote, respectively, the numbers of immature and mature members of the population. The parameter [tau] 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 [alpha][u.sub.m](t) term. And [e.sup.- [gamma][tau]] terms allow for the fact that not all immatures survive to maturity ([gamma] is the death rate of immatures, while [beta] measures deaths of matures).

When the individuals are allowed for moving around, in 2003 Gourley and Kuang  initially studied the diffusive version of the system (1.1):

[mathematical expression not reproducible], (1.2)

where [d.sub.i] > 0 ([d.sub.m] > 0) is the diffusivity of the immature (mature) species. If the immatures are not moving, then we would allow [d.sub.i] [right arrow] 0 in

[mathematical expression not reproducible] (1.3)

to conclude that the corresponding expression is [e.sup.-[gamma][tau]] [alpha][u.sub.m](t - [tau], x). If [d.sub.i] > 0, expression (1.3) total up the individuals of population u born at time t - [tau] in all parts of the domain that are still alive at time t and have just reached maturity and arrived at x.

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.sub.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  studied the following Lotka-Volterra competition model:

[mathematical expression not reproducible]. (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:

[u.sub.,t] = [du.sub.,xx] - [vu.sub.,x] + R(u). (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 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:

[mathematical expression not reproducible]. (1.6)

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:

[u.sub.,t] = [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] + [[alpha].sub.1]u(t - [[tau].sub.1], x) - [[eta].sub.1][u.sup.2] - [p.sub.1]uv,

[v.sub.,t] = [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [[alpha].sub.2]v(t - [[tau].sub.2], x) - [[eta].sub.2][v.sup.2] - [p.sub.2]uv. (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.sub.1] > 0 ([d.sub.2] > 0) is the diffusion coefficient of the adult population u (v). [a.sub.1] > 0 ([a.sub.2] > 0) is the advection speed experienced by the organism. [[alpha].sub.1] ([[alpha].sub.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  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 . This presents the invasion by the strong species of territory previously inhabited only by the weaker. The results in  made significant progress in establishing travelling wave solutions for model (1.4). It is pointed out in  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 . 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 . Some concluding remarks are given in Section 4.

2 Some Preliminaries

2.1 Equilibria and stability

Based on the model (1.7), the corresponding nonspatial system is yield:

[u.sub.,t] = [[alpha].sub.1]u(t - [[tau].sub.1]) - [[eta].sub.1][u.sup.2] - [p.sub.1]uv,

[v.sub.,t] = [[alpha].sub.2]v(t - [[tau].sub.2]) - [[eta].sub.2][v.sup.2] - [p.sub.2]uv (2.1)

with initial conditions

u(t), v(t) > 0 for -[[tau].sub.i] [less than or equal to] t [less than or equal to] 0. (2.2)

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, [infinity]). Positivity of (2.1) is important for both the modeling and the analysis. From the point of biology, positivity implies that the system persists.

System (2.1) has the trivial equilibrium [E.sub.0] = (0,0), the mono-culture equilibria [E.sub.u] = ([u.sup.*], 0) and [E.sub.v] = (0, [v.sup.*]) with

[u.sup.*] = [[alpha].sub.1]/[[eta].sub.1], [v.sup.*] = [[alpha].sub.2]/[[eta].sub.2]

and the coexistence equilibrium [E.sub.+] = ([u.sub.+], [v.sub.+]) with

[u.sub.+] = [[alpha].sub.2][[.sub.1] - [[alpha].sub.1][[eta].sub.2]/[p.sub.1][p.sub.2] - [[eta].sub.1][[eta].sub.2], [v.sub.+] = [[alpha].sub.1][[.sub.2] - [[alpha].sub.2][[eta].sub.1]/[p.sub.1][p.sub.2] - [[eta].sub.1][[eta].sub.2].

Clearly, [E.sub.+] exists if and only if [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2] and [[alpha].sub.1][p.sub.2] < [[alpha].sub.2][[eta].sub.1] or [[alpha].sub.2][p.sub.1] > [[alpha].sub.1][[eta].sub.2] and [[alpha].sub.1][p.sub.2] > [[alpha].sub.2][[eta].sub.1].

We summarize the local stability results for nonspatial system (2.1) in Table 2. The following global stability results can be found in Zhang, Li, Shang  or Al-Omari and Gourley .

Theorem 1. Let [E.sub.u], [E.sub.v] and [E.sub.+] be defined as above, then the following statements are valid:

i. If [[alpha].sub.1][p.sub.2] > [[alpha].sub.2][[eta].sub.1], [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2], then the mono-culture equilibrium [E.sub.u] is globally asymptotically stable;

ii. If [[alpha].sub.1][p.sub.2] < [[alpha].sub.2][[eta].sub.1], [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2], then the mono-culture equilibrium [E.sub.v] is globally asymptotically stable;

iii. If [[alpha].sub.1][p.sub.2] < [[alpha].sub.2][[eta].sub.1], [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2], then the unique coexistence equilibrium [E.sub.+] is globally asymptotically stable.

By the study of the local and global stability of the equilibria of the non- spatial 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.sub.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.sub.u] and [E.sub.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.sub.+] is globally asymptotically stable under certain conditions given in the Theorem 1 part iii.

2.2 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 [tau] be a nonnegative real number. We shall use boldface Roman symbols like u([theta], x) to denote k-vector-valued functions of the two variable [theta] and x, and boldface Greek letters to stand for k-vectors, which may be thought of as constant vector-valued functions. We define u [greater than or equal to] v to mean that [u.sup.i]([theta], x) [greater than or equal to] [v.sup.i]([theta], x) for all i = 1, 2, ..., k, [theta] [member of] [-[tau], 0] and x [member of] H, and u [much greater than] v to mean that [u.sup.i]([theta], x) > [v.sup.i]([theta], x) for all i, [theta] and x. We also define max{u([theta], x), v([theta], x)} to mean the vector-valued function whose ith component at ([theta], x) is max{[u.sup.i]([theta], x), [v.sup.i]([theta], x)}. 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 [-[tau], 0] x H to [R.sup.k], [bar.C] be the set of all bounded continuous functions from [-[tau], 0] to [R.sup.k], and X be the set of bounded continuous functions from H to [R.sup.k]. If r [member of] [bar.C] with r [much greater than] 0, we define the set of continuous functions

[C.sub.r] := {u [member of] C: 0 [less than or equal to] u [less than or equal to] r}.

Moreover, we define the metric function d(*,*) in C by

d([phi], [psi]) = [[infinity].summation over (k=0)] [max.sub.[absolute value of x] [less than or equal to] k, [theta] [member of][-[tau],0]] [absolute value of [phi]([theta], x) - [psi]([theta], x)]/[2.sup.k] [for all][phi], [psi] [member of] C (2.3)

so that (C, d) is a metric space. The convergence of a sequence [[phi].sub.n] to [phi] with respect to this topology is equivalent to the uniform convergence of [[phi].sub.n] to [phi] on bounded subsets of [-[tau], 0] x H.

We study the following discrete-time recursion

[u.sub.n+1] = Q[[u.sub.n]], n = 0, 1, 2, ..., (2.4)

where [u.sub.n]([theta], x) = ([u.sup.1.sub.n]([theta], x), [u.sup.2.sub.n]([theta], x), ..., [u.sup.k.sub.n]([theta], x)), [theta] [member of] [-[tau], 0], and x [member of] H represents the population densities of the populations of k species at time n and point x with time delay [tau]. The operator Q is said to be order--preserving if u [greater than or equal to] v implies that Q[u] [greater than or equal to] Q[v]. A recursion (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.sub.l] = w in the recursion (2.4), then [u.sub.n] = w for all n [greater than or equal to] l. We shall study the evolution of the solution [u.sub.n] of the recursion (2.4) from a [u.sub.0] near an unstable constant equilibrium [theta]. By introducing the new variable [??] = u - [theta] if necessary, we shall assume the unstable equilibrium [theta] from which the system moves away is the origin 0.

We define the translation operators

[T.sub.y] [v]([theta], x) = v([theta], x - y).

A set D [member of] [C.sub.r] is said to be T-invariant if [T.sub.y][D] = D for any y [member of] H.

We shall make the following hypotheses on Q.

Hypotheses 2.1

i. Q = 0, and there is a vector [beta]([theta]) [member of] [bar.C] with [beta]([theta]) [much greater than] 0 such that Q[[beta]] = [beta], and if [u.sub.0] is any vector in [bar.C] with [beta]([theta]) [much greater than] [u.sub.0] [much greater than] 0, then the vector-valued, function [u.sub.n] obtained from the recursion (2.4) converges to [beta]([theta]) uniformly on [-[tau], 0] as n approach infinity.

ii. Q is order-preserving on nonnegative functions.

iii. Q is translation invariant so that Q[[T.sub.y][v]] = [T.sub.y][Q[v]] for all y.

iv. Q is continuous with respect to the topology determined by d(*,*) given in (2.3).

v. One of the following two properties holds:

a. Q[[C.sub.[beta]]] is precompact in [C.sub.[beta]].

b. The set Q[[C.sub.[beta]]](0, *) is precompact in X, and there is a positive number [zeta] [less than or equal to] [tau] such that Q[u]([theta], x) = u([theta] + [zeta], x) for all [theta] [member of] [- [tau], -[zeta]], and the operator

[mathematical expression not reproducible]

has the property that S[D] is precompact in [C.sub.[beta]] for any T-invariant set D [member of] [C.sub.[beta]] with D(0, *) precompact in X.

Remark 1. Hypotheses 2.1 i-ii imply that Q takes [C.sub.[beta]] into itself, and that the equilibrium [beta] attracts all initial functions in [C.sub.[beta]] with uniformly positive components. In biological terms, [beta] is a globally stable coexistence equilibrium. There may also be other equilibria lying between [beta] 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 [v.sub.1]([theta]), [v.sub.2]([theta]) [member of] [bar.C] of (2.4), if [v.sup.i.sub.1]([theta]) [not equal to] [v.sup.i.sub.2]([theta]) for some [theta] [member of] [-[tau], 0] then [v.sup.i.sub.1]([theta]) [not equal to] [v.sup.i.sub.2]([theta]) for all [theta] [member of] [-[tau], 0].

Remark 2. Here we have dropped the hypothesis in  that operator Q is reflect invariant. In  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.

The following lemma can be found in Lui , Weinberger et. al , and Liang and Zhao .

Lemma 1 [Comparison Lemma]. Let R be an order preserving operator. If [u.sub.n] and [v.sub.n] satisfy the inequalities [u.sub.n] [less than or equal to] R[[u.sub.n]] and [v.sub.n] [greater than or equal to] R[[v.sub.n]] for all n, and if [u.sub.0] [less than or equal to] [v.sub.0], then [u.sub.n] [less than or equal to] [v.sub.n] for all n.

We choose a continuous vector-valued function [phi]([theta], x) = ([[phi].sup.1], ..., [[phi].sup.k]) [member of] [C.sub.[beta]] that has the properties

(b1) [[phi].sup.i]([theta], x) is nonincreasing in x for any [theta] [member of] [-[tau], 0] and 1 [less than or equal to] i [less than or equal to] k;

(b2) [[phi].sup.i]([theta], x) = 0 for any [theta] [member of] [-[tau], 0], x [greater than or equal to] 0, and 1 [less than or equal to] i [less than or equal to] k;

(b3) [phi]([theta], -[infinity]) = [alpha]([theta]) for any [theta] [member of] [-[tau], 0], where 0 [much less than] [alpha]([theta]) [much less than] [beta].

For any real number c and u [member of] [C.sub.[beta]]. Define the operator

[R.sub.c][u]([theta], s) = max{[phi]([theta], s), [T.sub.-c][Q[u]]([theta], s)}.

Define a sequence of vector-valued functions [a.sub.n](c; [theta], s) of ([theta], s) [member of] [-[tau], 0] x H by the recursion

[a.sub.n+1](c; [theta], s) = [R.sub.c] [[a.sub.n](c; *]([theta], s), [a.sub.0](c; [theta], s) = [phi]([theta], s). (2.5)

Denote a(c; [theta], [infinity]) as the limit of [a.sub.n](c; [theta], [infinity]). Note that [a.sub.n] [less than or equal to] [a.sub.n+1] [less than or equal to] [beta] for all n, and [a.sub.n](c; [theta], s) is nonincreasing in c and s and continuous in (c, [theta], s).

Define

[c.sup.*] := sup{c : a(c; [theta], [infinity]) [not equal to] 0}. (2.6)

Denote [{[Q.sub.t]}.sup.[infinity].sub.t=0] as the continuous time semiflow on [C.sub.[beta]]. In  we established the following existence result (see [7, Theorem 2.2]).

Theorem 2. Suppose that for any t > 0, [Q.sub.t] satisfies Hypotheses 2.1. Let [c.sup.*] be given by (2.6) where Q is replaced by [Q.sub.1]. Then the following statements are true for [{[Q.sub.t]}.sup.[infinity].sub.t=0].

i. If c [greater than or equal to] [c.sup.*], then there is a nonincreasing travelling wave solution W(c; [theta], x - ct) of speed c with W(c; [theta], [infinity]) = 0 and W(c; [theta], -[infinity]) an equilibrium other than 0.

ii. If there is a nonincreasing travelling wave W(c; [theta], x - ct) with W(c; [theta], [infinity]) = 0 and W(c; [theta], -[infinity]) = [beta], then c [greater than or equal to] [c.sup.*].

We shall employ this abstract result to prove the existence of travelling wave solutions in model (1.7).

3 Existence of Traveling Waves in (1.7)

Suppose that

[[alpha].sub.1][p.sub.2] < [[alpha].sub.2][[eta].sub.1]. (3.1)

By Theorem 1, [E.sub.u] is unstable, [E.sub.+] exists and it is globally attracting if [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2], and [E.sub.+] does not exist and [E.sub.v] is globally attracting if [[alpha].sub.2][p.sub.1] > [[alpha].sub.1][[eta].sub.2]. We shall study the existence of travelling wave solutions connecting [E.sub.u] to [E.sub.+] if [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2], and to [E.sub.v] if [[alpha].sub.2][p.sub.1] > [[alpha].sub.1][[eta].sub.2], see Figure 1.

Let u := [u.sup.*] - u and v := v. We convert the competition system (1.7) into the following cooperative system

[u.sub.,t] = [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] - [[alpha].sub.1]([u.sup.*] - u(t - [[tau].sub.1], x)) + [[eta].sub.1][([u.sup.*] - u).sup.2] + [p.sub.1]([u.sup.*] - u)v, [v.sub.,t] = [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [[alpha].sub.2]v(t - [[tau].sub.2], x) - [[eta].sub.2][v.sup.2] - [p.sub.2]([u.sup.*] - u)v. (3.2)

For this system, we denote

[mathematical expression not reproducible]. (3.3)

Clearly, 0 = (0,0) and [beta] are the only two equilibria in [C.sub.[beta]] if [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2]; there is an extra equilibrium v = ([u.sup.*], 0) in [C.sub.[beta]] if [[alpha].sub.2][p.sub.1] > [[alpha].sub.1][[eta].sub.2]. Hence, we now only to prove the existence of travelling wave solutions connecting 0 to [beta], see Figure 2.

We now state and prove our main result.

Theorem 3. Assume that the condition (3.1) holds. Let [beta] be defined by (3.3), and [c.sup.*] be defined by (2.6) where Q is the time one solution map [Q.sub.1] of (3.2). Then system (3.2) has a nonincreasing traveling wave solution W(x - ct) = (u(x - ct), v(x - ct)) with W(+[infinity]) = 0 and W(-[infinity]) = [beta] if and only if c [greater than or equal to] [c.sup.*].

Remark 3. When the advection terms [a.sub.i] =0, i = 1, 2, model (1.7) reduces to model (1.4). Theorem 3 indicates that the technical assumption made in  is indeed unnecessary and [c.sup.*] is the slowest travelling wave speed.

In what follows we shall prove Theorem 3 by using Theorem 2, that is verifying the abstract Hypotheses 2.1.

Let [tau] = max{[[tau].sub.1], [[tau].sub.2]}. Define [f.sub.i] : C x C [right arrow] X, i = 1, 2, by

[f.sub.1] ([phi], [psi])(x) = -[[alpha].sub.1] ([u.sup.*] - [phi](- [[tau].sub.1], x)) + [[eta].sub.1] [([u.sup.*] - [phi](0, x)).sup.2]

+ [p.sub.1]([u.sup.*] - [phi](0, x))[psi](0, x),

[f.sub.2] ([phi], [psi])(x) = -[[alpha].sub.2][psi] (-[[tau].sub.2], x) - [[eta].sub.2][psi][0, x).sup.2] - [p.sub.2]([u.sup.*] - [phi](0, x))[psi](0, x). (3.4)

Then system (3.2) can be rewritten as

[u.sub.,t] = [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] + [f.sub.1]([u.sub.t], [v.sub.t])(x),

[v.sub.,t] = [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [f.sub.2]([u.sub.t], [v.sub.t])(x), (3.5)

where t > 0, x [member of] R, [u.sub.t], [v.sub.t] [member of] C with [u.sub.t]([theta], x) = u(t + [theta], x) and [v.sub.t]([theta], x) = v(t + [theta], x) for [theta] [member of] [-[tau], 0], x [member of] R.

Definition 1. A function (u(t, x), v(t, x)) : [-[tau], b) x R [right arrow] [R.sup.2], b > 0, with the properties that (u, v) is [C.sup.2] in x [member of] R and [C.sup.1] in t [member of] (0, b) is called a supersolution (subsolution) of (3.2) on [0, b) if for t [member of] [0, b), x [member of] R

[u.sub.,t] [less than or equal to] ([greater than or equal to]) [d.sub.1] [u.sub.,xx] - [a.sub.1][u.sub.,x] + [f.sub.1]([u.sub.t], [v.sub.t])(x),

[v.sub.,t] [less than or equal to] ([greater than or equal to]) [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [f.sub.2]([u.sub.t], [v.sub.t])(x).

Lemma 2. If initial function ([phi], [psi]) [member of] [C.sub.[beta]], then system (3.2) has a unique classical solution (u(t, x; [phi], [psi]), v(t, x; [phi], [psi])) on ([tau], [infinity]) x R, where (u(0, x; [phi], [psi]), v(0, x,; [phi], [psi])) = ([phi], [psi]). If super-solution ([bar.u](t, x), [bar.v](t, x)) and sub-solution ([u.bar](t, x), [v.bar](t, x)) of (3.2) satisfy 0 [less than or equal to] ([u.bar](t, x), [v.bar](t, x)) [less than or equal to] ([bar.u](t, x), [bar.v](t, x)) [less than or equal to] [beta] for t [member of] [-t, 0) and x [member of] R, then 0 [less than or equal to] ([u.bar](t, x), [v.bar](t, x)) [less than or equal to] ([bar.u](t, x), [bar.v](t, x)) [less than or equal to] [beta] for t [greater than or equal to] 0 and x [member of] R.

Proof. Let [{[T.sub.u](t)}.sub.t [greater than or equal to] 0] and [{[T.sub.v](t)}.sub.t [greater than or equal to] 0] be the solution semigroup on X generated by the heat equations

[u.sub.,t] = [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] and [v.sub.,t] = [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x].

Then we can write (3.5) as the following integral equations

u(t, x) = [T.sub.u](t)u(0, *)(x) + [[integral].sup.t.sub.0] [T.sub.u](t - s)[f.sub.1]([u.sub.s], [v.sub.s])(x)ds,

v(t, x) = [T.sub.v](t)v(0, *)(x) + [[integral].sup.t.sub.0] [T.sub.v](t - s)[f.sub.2]([u.sub.s], [v.sub.s])(x)ds, (3.6)

where

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Under the abstract setting in , a mild solution of (3.5) is a solution to its associated integral equation (3.6). One can easily verify that [f.sub.1] and [f.sub.2] are Lipschitz continuous on any bounded subset of C x C. Let Z = BUC(R, [R.sup.2]) be the Banach space of all bounded and uniformly continuous functions from R into [R.sup.2] with the usual supremum norm. Let [Z.sup.+] = {([[phi].sub.1], [[phi].sub.2]) : ([[phi].sub.1], [[phi].sub.2]) [member of] Z, [phi][(x).sub.i] [greater than or equal to] 0, i = 1, 2}. We show that [f.sub.1] and [f.sub.2] are quasi-monotone on C in the sense that

[mathematical expression not reproducible] (3.7)

for all [[phi].sub.j], [[psi].sub.j] [member of] [C.sub.[beta]], j = 1, 2 with ([[phi].sub.2], [[psi].sub.2]) [greater than or equal to] ([[phi].sub.1], [[psi].sub.1]). From the definitions of [f.sub.1] and [f.sub.2] in (3.4) we get

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

It follows that for sufficiently small h > 0,

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Which indicate that (3.7) holds. By Corollary 5 in , we can show the existence and uniqueness of (u(t, x; [phi], [psi]), v(t, x; [phi], [psi])) with

([S.sub.1](t, s), [S.sub.2](t, s)) = ([S.sub.1](t - s), [S.sub.2](t - s)),

([T.sub.1](t, s), [T.sub.2](t, s)) = ([T.sub.1](t - s), [T.sub.2](t - s))

for t [greater than or equal to] s [greater than or equal to] 0,

([B.sub.1](t, [phi], [psi]), [B.sub.2](t, [phi], [psi])) = ([f.sub.1]([phi], [psi]), [f.sub.2]([phi], [psi])),

and [v.sup.+] = [beta], [v.sup.-] = 0. Moreover, by the semigroup theory given in the proof of Theorem 1 in , it follows that (u(t, x; [phi]), v(t, x; [psi])) is a classical solution for t > [tau].

Let [PSI]([theta], x) = ([bar.u]([theta], x), [bar.v]([theta], x)), [HI]([theta], x) = ([u.bar]([theta], x), [v.bar]([theta], x)), for [theta] [member of] [-[tau], 0], x [member of] R. Then 0 [less than or equal to] [PHI] [less than or equal to] [PSI] [less than or equal to] [beta] with [PHI] [less than or equal to] [PSI] in [C.sub.[beta]]. Again by Corollary 5 in , we have

0 [less than or equal to] (u(t, x, [PHI]), v(t, x, [PHI])) [less than or equal to] (u(t, x, [PSI]), v(t, x, [PSI])) [less than or equal to] [beta]. (3.8)

for t [greater than or equal to] 0, x [member of] R.

Let [v.sup.+] = [beta] and [v.sup.-] = ([u.bar](t, x), [v.bar](t, x)), [v.sup.+] = ([bar.u](t, x), [bar.v](t, x)) and [v.sup.-] = 0, respectively, we obtain

([u.bar](t, x), [v.bar](t, x)) [less than or equal to] (u(t, x, [PHI]), v(t, x, [PHI])) [less than or equal to] [beta], t [greater than or equal to] 0, x [member of] R, (3.9)

0 [less than or equal to] (u(t, x, [PSI]), v(t, x, [PSI])) [less than or equal to] ([bar.u](t, x), [bar.v](t, x)), t [greater than or equal to] 0, x [member of] R. (3.10)

It follows from (3.8)-(3.10) that ([bar.u](t, x), [bar.v](t, x)) [greater than or equal to] ([u.bar](t, x), [v.bar](t, x)) for all t [greater than or equal to] 0, x [member of] R. This completes the proof.

Lemma 2 together with the Theorem 1 shows that the time t solution map [Q.sub.t] of system (3.2) with t > 0 exists, and it satisfies Hypotheses 2.1 i- ii. Hypotheses 2.1 iii is satisfied by [Q.sub.t] since system (3.2) is autonomous.

Lemma 3. For any t > 0, [Q.sub.t] satisfies Hypothesis 2.1 iv with [beta] given by (3.3).

Proof. Let [PHI] := ([phi], [psi]), [[PHI].sub.1] := ([[phi].sub.1], [[psi].sub.1]), [[PHI].sub.2] := ([[phi].sub.2], [[psi].sub.2]), [member of] [C.sub.[beta]]. For any [epsilon] > 0 and [t.sub.0] > 0, we define

[mathematical expression not reproducible];

where [DELTA] := [[alpha].sub.1] + [[alpha].sub.2] + 4[[eta].sub.1][u.sup.*] + 2[[eta].sub.2][v.sup.*] + ([p.sub.1] + [p.sub.2])(2[u.sup.*] + [v.sup.*]). Without loss of generality, we assume K [greater than or equal to] [sup.sub.[theta] [member of] [-[tau], 0], x [member of] R] H([theta], x). Then, there exists ([t.sup.*], [x.sup.*]) [member of] [0, [t.sub.0]] x R such that H([theta], x) [less than or equal to] H([t.sup.*], [x.sup.*]) + [[epsilon].sub.0] for (t, [theta], x) [member of] [0, [t.sub.0]] x [-[tau], 0] x R. We choose [mathematical expression not reproducible] and M = M([epsilon], [t.sub.0]) > 0 such that for any t [member of] [0, [t.sub.0]]

[mathematical expression not reproducible].

For [mathematical expression not reproducible]

[mathematical expression not reproducible]

By a similar argument, we have

[mathematical expression not reproducible].

We thus get

[mathematical expression not reproducible].

By Gronwall's inequality we get

[mathematical expression not reproducible].

We thus have obtained that for any [epsilon] > 0, and compact subset S [subset] [-[tau], 0] x R, there exist [sigma] > 0 and a compact set [[OMEGA].sub.M] ([x.sup.*]) such that S [subset] [[OMEGA].sub.M] ([x.sup.*]) and

[mathematical expression not reproducible].

This shows that [Q.sub.t] is continuous in [PHI] with respect to the compact open topology uniformly for t [member of] [0, [t.sub.0]]. Note that the metric space ([C.sub.[beta]], d) is complete. By the triangle inequality and the continuity of [Q.sub.t] in t from Lemma 2, it then follows that [Q.sub.t]([phi]) is continuous in (t; [PHI]) with respect to the compact open topology. Since (3.6) is an autonomous system, [{[Q.sub.t]}.sub.t [greater than or equal to] 0] is a semiflow on [C.sub.[beta]]. This completes the proof of this Lemma.

Lemma 4. For any t > 0, [Q.sub.t] satisfies Hypothesis 2.1 v with [beta] given by (3.3).

Proof. Let T(t) = ([T.sub.u](t), [T.sub.v](t)), where [T.sub.u](t) is the solution map of [partial derivative]u/[partial derivative]t = [d.sub.u] [[partial derivative].sup.2]u/[partial derivative][x.sup.2] - [a.sub.1] [partial derivative]u/[partial derivative]x and [T.sub.v](t) is the solution map of [partial derivative]v/[partial derivative]t = [d.sub.v] [[partial derivative].sup.2]v/[partial derivative][x.sup.2] - [a.sub.2] [partial derivative]v/[partial derivative]x. It follows that [{[T.sub.u](t)}.sub.t [greater than or equal to] 0] and [{[T.sub.v](t)}.sub.t [greater than or equal to] 0] are linear semigroup on X and [T.sub.u](t) and [T.sub.v](t) are compact for each t > 0. Let [Q.sub.t] = ([Q.sup.u.sub.t], [Q.sup.v.sub.t]). Given [t.sub.0] > [tau], then

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

By the properties of [T.sub.u](t) and [T.sub.v](t) and the boundness of [f.sub.i], we see that [Q.sup.u.sub.t] and [Q.sup.v.sub.t] are compact for each t > [tau]. Thus [Q.sub.t] satisfies Hypotheses 2.1 v.a. when t > [tau].

Given [t.sub.0] [member of] (0, [tau]], we now show that [mathematical expression not reproducible] satisfies Hypotheses 2.1 v.b. with [mathematical expression not reproducible]. To prove S[D] is precompact, it suffices to show that for any given compact interval I [subset] R, u(t, x; [phi], [psi]) and v(t, x; [phi], [psi]) are equicontinuous in (t, x) [member of] [0, [t.sub.0]] x I for all ([phi], [psi]) [member of] D. We shall only show that u(t, x; [phi], [psi]) is equicontinuous in (t, x) [member of] [0, [t.sub.0]] x I for all ([phi], [psi]) [member of] D, since the other statement can be proved similarly. By the continuity of integral, we have that for any [epsilon] > 0, there exists [[delta].sub.0] > 0 such that for any t [member of] (0, [[delta].sub.0]],

[absolute value of [[integral].sup.t.sub.0] [T.sub.u](t - s)[f.sub.1]([u.sub.s], [v.sub.s])(x)ds] < [epsilon]/12.

On the other hand, since D(0, *) is precompact in X, for the above interval I, there exists [[delta].sub.1] > 0 such that [absolute value of [phi](0, [x.sub.1]) - [phi](0, [x.sub.2])] < [epsilon]/6 and [absolute value of [psi](0, [x.sub.1]) - [psi](0, [x.sub.2])] < [epsilon]/6 for all ([phi], [psi]) [member of] D when [x.sub.1], [x.sub.2] [member of] I with [absolute value of [x.sub.1] - [x.sub.2]] < [[delta].sub.1]. Thus for any [t.sub.1], [t.sub.2] [member of] [0, [[delta].sub.0]], [x.sub.1], [x.sub.2] [member of] I with [absolute value of [x.sub.1] - [x.sub.2]] < [[delta].sub.1], we have

[absolute value of u([t.sub.1], [x.sub.1]; [phi], [psi]) - u([t.sub.2], [x.sub.2]; [phi], [psi])] [less than or equal to] [absolute value of u([t.sub.1], [x.sub.1]; [phi], [psi]) - [phi](0, [x.sub.1])]

+ [absolute value of u([t.sub.2], [x.sub.2]; [phi], [psi]) - [phi](0, [x.sub.2])] + [absolute value of [phi](0, [x.sub.1]) - [phi](0, [x.sub.2])].

We claim that [absolute value of u([t.sub.1], [x.sub.1]; [phi], [psi]) - [phi](0, [x.sub.1])] < [epsilon]/6 under appropriate choice of [[delta].sub.0]. In fact, if we choose M > 0 and [[delta].sub.0] properly so that

[mathematical expression not reproducible].

We then obtain that

[mathematical expression not reproducible].

Similarly, we can show that [parallel]u([t.sub.2], [phi], [psi])[parallel] < [epsilon]/6 under appropriate choice of [[delta].sub.0]. Hence we obtain that [absolute value of u([t.sub.1], [x.sub.1]; [phi], [psi]) - u([t.sub.2], [x.sub.2]; [phi], [psi])] < [epsilon]/2. Since [Q.sup.u.sub.t] is compact when t > [tau], it follows that u(t, x; [phi], [phi]) is equicontinuous in (t, x) [member of] [[[delta].sub.0], [t.sub.0]] x I for all ([phi], [psi]) [member of] D. That is, for the above [epsilon] and I, there exists [[delta].sub.2] > 0 so that [absolute value of u([t.sub.1], [x.sub.1]; [phi], [psi]) - u([t.sub.2], [x.sub.2]; [phi], [psi])] < [epsilon]/2 for all ([phi], [psi]) [member of] D if [t.sub.1], [t.sub.2] [member of] [[[delta].sub.0], [t.sub.0]] and [x.sub.1], [x.sub.2] [member of] I with [absolute value of [t.sub.1] - [t.sub.2]] + [absolute value of [x.sub.1] - [x.sub.2]] < [[delta].sub.2]. Let [delta] := min{[[delta].sub.0], [[delta].sub.1], [[delta].sub.2]}, then we have for any [epsilon] > 0 and I [member of] R, there exists [delta] > 0 such that [absolute value of u([t.sub.1], [x.sub.1]; [phi], [psi]) - u([t.sub.2], [x.sub.2]; [phi], [psi])] < [epsilon] for all ([phi], [psi]) [member of] D if [t.sub.1], [t.sub.2] [member of] [0, [t.sub.0]] and [x.sub.1], [x.sub.2] [member of] I with [absolute value of [t.sub.1] - [t.sub.2]] + [absolute value of [x.sub.1] - [x.sub.2]] < [delta]. This completes the proof of this Lemma.

Lemma 5. Let [c.sup.*] be defined by (2.6) where Q is replaced by the time one map [Q.sub.1] of (3.2), then 0 < [c.sup.*] < [infinity].

Proof. We write the time one solution operator [Q.sub.1] of (3.2) as [Q.sub.1] = ([Q.sub.u1], [Q.sub.v1]). Let [[??].sub.v1] be the time one solution map of

[v.sub.,t] = [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [[alpha].sub.2]v(t - [[tau].sub.2], x) - [[eta].sub.2][v.sup.2] - [p.sub.2][u.sup.*]v. (3.11)

The reaction term in (3.11) is [f.sub.2](0, [v.sub.t])(x) where [f.sub.2] given in (3.4). Since [f.sub.2]([u.sub.t], [v.sub.t])(x) [greater than or equal to] [f.sub.2](0, [v.sub.t])(x), [Q.sub.v1](u, v) [greater than or equal to] [[??].sub.v1] (v) for (u, v) [member of] [C.sub.[beta]]. The definition of [c.sup.*] and Lemma 1 show that

[c.sup.*] [greater than or equal to] [c.sup.*]v, (3.12)

where [c.sup.*.sub.v] is given by (2.6) where Q is [[??].sub.v1].

The equation (3.11) has two constant equilibria v = 0 and v = ([[alpha].sub.2][[eta].sub.1] - [[alpha].sub.1][p.sub.2])/([[eta].sub.1][[eta].sub.2]) > 0 due to (3.1). One can verify that under the condition (3.1), 0 is unstable and ([[alpha].sub.2][[eta].sub.1] - [[alpha].sub.1][p.sub.2])/([[eta].sub.1][[eta].sub.2]) is stable. Since

[v.sub.,t] [less than or equal to] [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [[alpha].sub.2]v(t - [[tau].sub.2], x) - [p.sub.2][u.sup.*]v,

the time t solution map of (3.11) is dominated by that of the linearized model. One can then use Theorem 3.10 in Liang and Zhao  and the argument given on the pages 2758-2759 of Fang, Wei and Zhao  to find that

[mathematical expression not reproducible],

where [lambda]([mu]) is the principal eigenvalue of the characteristic equation

[mathematical expression not reproducible]. (3.13)

By the equation (3.13) and the condition (3.1) one can easily show that [lambda]([mu]) > 0 for [mu] [greater than or equal to] 0. On the other hand, [lambda]([mu]) [greater than or equal to] [[eta].sub.1][d.sub.v][[mu].sup.2] for large positive [mu] so that [lambda]([mu])/[mu] [right arrow] [infinity] as [mu] [right arrow] [infinity]. It follows that the infimum of [lambda]([mu])/[mu] occurs at a finite number and thus [c.sup.*.sub.v] > 0. It follows from (3.12) that [c.sup.*] > 0.

We next show that [c.sup.*] < [infinity]. From (3.2) we obtain

[u.sub.,t] [less than or equal to] [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] + [[alpha].sub.1]u(t - [[tau].sub.1], x) + [p.sub.1][u.sup.*]v,

[v.sub.,t] [less than or equal to] [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [[alpha].sub.2]v(t - [[tau].sub.2], x). (3.14)

We shall show that there exist [[lambda].sub.0] > 0 and [c.sub.0] > 0 such that

[mathematical expression not reproducible] (3.15)

satisfies

[u.sub.,t] [greater than or equal to] [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] + [[alpha].sub.1]u(t - [[tau].sub.1], x) + [p.sub.1][u.sup.*]v,

[v.sub.,t] [greater than or equal to] [d.sub.2][v.sub.,xx] - [a.sub.2][v.sub.,x] + [[alpha].sub.2]v(t - [[tau].sub.2], x). (3.16)

Substituting (u, v) given by (3.15) into (3.16), we find that (3.16) is satisfied if and only if

f([c.sub.0], [[lambda].sub.0]) > 0, g([c.sub.0], [[lambda].sub.0]) > 0,

where

[mathematical expression not reproducible].

It is easily checked that for any fixed [lambda] > 0

[mathematical expression not reproducible].

Which indicate that there exist [[lambda].sub.0] > 0 and 0 < [c.sub.0] < [infinity] such that f([c.sub.0], [[lambda].sub.0]) > 0 and g([c.sub.0], [[lambda].sub.0]) > 0. Since [mathematical expression not reproducible] satisfies (3.16) and solutions of (3.2) satisfy (3.14), [mathematical expression not reproducible] is an upper solution for (3.2). Let [L.sub.1] be the time one solution map of the linear equation system corresponding to (3.16). We have that for (u, v) [member of] [C.sub.[beta]], if [mathematical expression not reproducible] with [theta] [member of] [-[tau], 0] then

[mathematical expression not reproducible]. (3.17)

One can then use (2.5) to define the sequence [a.sub.n]([c.sub.0]; [theta], x) with Q replaced by [Q.sub.1] and ([phi]([theta], x), [psi]([theta], x)) satisfying [mathematical expression not reproducible] for [theta] [member of] [-[tau], 0]. Induction and (3.17) show that [mathematical expression not reproducible] for all n and thus the limit function a([c.sub.0]; [theta], x) satisfies [mathematical expression not reproducible]. It follows immediately that

[mathematical expression not reproducible].

The definition of [c.sup.*] shows that [c.sub.0] [greater than or equal to] [c.sup.*]. This completes the proof of this lemma.

Proof of Theorem 3. By Theorem 2, we only need to show that for any c [greater than or equal to] [c.sup.*], there is no nonincreasing travelling wave solution in (3.2) that connects 0 = (0,0) with v = ([u.sup.*], 0). Assume to the contrary that (u(x - ct), v(x - ct)) is a nonincreasing travelling wave solution in (3.2) with c [greater than or equal to] [c.sup.*] connecting 0 = (0, 0) with v = ([u.sup.*], 0). Such a travelling wave solution is equivalent to a travelling wave solution for the following scalar equation:

[u.sub.,t] = [d.sub.1][u.sub.,xx] - [a.sub.1][u.sub.,x] - [[alpha].sub.1]([u.sup.*] - u(t - [[tau].sub.1], x)) + [[eta].sub.1][([u.sup.*] - u).sup.2].

We therefore have [??](x - ct) = [u.sup.*] - u(x - ct) is a nondecreasing traveling wave solution of

[mathematical expression not reproducible] (3.18)

with [??](-[infinity]) = 0 and [??](+[infinity]) = [u.sup.*]. By Lemma 5, [c.sup.*] > 0, so that c > 0. Therefore, [??](x - ct) is a nondecreasing traveling wave solution of (3.18) with the speed -c < 0 connecting the unstable equilibrium 0 with the stable equilibrium [u.sup.*].

On the other hand

[mathematical expression not reproducible],

which indicates that the time t solution map of (3.18) is dominated by that of the linearized model. By Theorem 3.10 in Liang and Zhao  and the argument given on the pages 2758-2759 of Fang, Wei and Zhao  we obtain that the minimal traveling wave speed for (3.18) is given by [inf.sub.[mu]>0] [lambda]([mu])/[mu] where [lambda]([mu]) is the principle eigenvalue of the characteristic equation

[mathematical expression not reproducible].

Since [lambda](0) > 0 and [lambda]([mu]) [greater than or equal to] [d.sub.u][[mu].sup.2] for [mu] > 0, [inf.sub.[mu]>0] [lambda]([mu])/[mu] > 0. We obtain a contradiction and the proof of this Theorem is complete.

4 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  and Zhang et al. . There is no advection terms in their models, which can provide a resolution to the drift paradox in stream ecology (see ). "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  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  is improved.

In 2012, Li  showed that for the general partial cooperative reaction- diffusion system

[u.sub.,t] = D[u.sub.,xx] - E[u.sub.,x] + f (u(t, x)) (4.1)

[bar.c] can be characterized as the slowest speed of a class of travelling wave solutions by verifying the linear determinacy conditions under appropriate assumptions, where

[mathematical expression not reproducible] (4.2)

and [[gamma].sub.1](u) is the principal eigenvalue of the first irreducible block of the moment generating matrix 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 [[gamma].sub.1](u) of [C.sub.[mu]] and the corresponding eigenvector [xi]([mu]) cannot be expressed explicitly. Thus the ideas used in  cannot be applied in our system.

In order to verify the linear determinacy conditions, one might first write the [xi]([mu]) as a function of the principle eigenvalue [[gamma].sub.1]([mu]). Then use the fact that the wave speed equation [phi]([mu]) = [[gamma].sub.1]([mu])/[mu] is a convex function and thus infimum in Eq. (4.2) exists, one might provide an estimation of [[gamma].sub.1]([mu]) and therefore show that the linear determinacy conditions are satisfied under appropriate assumptions. Liang and Zhao  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.

http://dx.doi.org/10.3846/13926292.2015.1020455

Acknowledgements

The authors are grateful to the anonymous reviewers for his/her thoughtful comments and constructive suggestions on improving our original manuscript. This work is supported by Doctor Foundation of Northwest A&F University (201104054460), NNSFC (39970112, 30470268, 11371294, and 11401474), and Research Fund for the Doctoral Program of Higher Education (20130204110004).

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Liang Zhang (a,b) and Huiyan Zhao (b)

(a) Institute of Applied Mathematics, College of science, Northwest A&F University No. 22 Xinong Road, Yangling, 712100 Shaanxi, China

(b) State Key Laboratory of Crop Stress Biology in Arid Areas, School of Plant Protection, Northwest A&F University No. 22 Xinong Road, Yangling, 712100 Shaanxi, China

E-mail: zhanglsd@126.com

E-mail(corresp.): zhaohy@nwsuaf.edu.cn

Received May 11, 2014; revised January 15, 2015; published online March 15, 2015

Caption: Figure 1. The spatial transition from [E.sub.u] to [E.sub.v] if [[alpha].sub.2][p.sub.1] > [[alpha].sub.1][[eta].sub.2], and to [E.sub.+] if [[alpha].sub.2][p.sub.1] < [[alpha].sub.1][[eta].sub.2] in competition system (1.7).

Caption: Figure 2. The spatial transition from 0 to 13 in cooperative system (3.2).
```Table 1. Parameter description for model (1.6).

Parameter                  Description of parameter

[U.sub.i](t, x)            population density of immature members U at
time t and point x
[U.sub.m] (t, x)           population density of mature members U at
time t and point x
[V.sub.i] (t, x)           population density of immature members V at
time t and point x
[V.sub.m] (t, x)           population density of mature members V at
time t and point x
[mathematical expression   diffusion coefficient of mature population
not reproducible]          U
[mathematical expression   diffusion coefficient of mature population
not reproducible]          V
[mathematical expression   advection speed of mature population U
not reproducible]          experienced by the organism
[mathematical expression   advection speed of mature population V
not reproducible]          experienced by the organism
[[alpha].sub.1]            birth rate of population U
[[alpha].sub.2]            birth rate of population V
[[gamma].sub.1]            death rate of immature population U
[[gamma].sub.2]            death rate of immature population V
[[beta].sub.1]             death rate of mature population U
[[beta].sub.2]             death rate of mature population V
[[tau].sub.1]              time delay from birth to maturity of
population U
[[tau].sub.2]              time delay from birth to maturity of
population V
[c.sub.1]                  competitive effect of the species V has on
the mature of species U
[c.sub.2]                  competitive effect of the species U has on
the mature of species V

Table 2. Summary of existence and local stability criteria of
equilibria.

[E.sub.0]      always exists
[E.sub.u]      always exists
[E.sub.v]      always exists
[E.sub.+]      [[alpha].sub.2][p.sub.1] <
[[alpha].sub.1][[eta].sub.2], [[alpha].sub.1][p.sub.2]
< [[alpha].sub.2][[eta].sub.1] or [[alpha].sub.2]
[p.sub.1] > [[alpha].sub.1][[eta].sub.2],
[[alpha].sub.1][p.sub.2] > [[alpha].sub.2][[eta].sub.1]

[E.sub.0]      unstable
[E.sub.u]      [[alpha].sub.1][p.sub.2] > [[alpha].sub.2][[eta].sub.1]
[E.sub.v]      [[alpha].sub.2][p.sub.1] > [[alpha].sub.1][[eta].sub.2]
[E.sub.+]      [[alpha].sub.1][p.sub.2] < [[alpha].sub.2][[eta].sub.1]
and [[alpha].sub.2][p.sub.1] < [[alpha].sub.1]
[[eta].sub.2]
```

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