DYNAMICS OF A TWO-SPECIES STAGE-STRUCTURED MODEL INCORPORATING STATE-DEPENDENT MATURATION DELAYS

. This paper is devoted to a cooperative model composed of two species with stage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to inﬁnity as the time tends to inﬁnity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the even- tual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.


(Communicated by Shigui Ruan)
Abstract. This paper is devoted to a cooperative model composed of two species with stage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to infinity as the time tends to infinity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the eventual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.
1. Introduction. Time-delay in a natural ecosystem has been widely considered since Hutchison proposed a Logistic model with time-delay (see e.g. [22,29,30,31]). Furthermore, most species go through two or more stages from birth to death, and species at two stages may have different behaviors. Thus stage structure is considered in population according to the natural phenomenon. Gurney, Blythe, and Nisbet [13] proposed a time delay growth model of blowflies. They verified solutions of the time delay model according with the data in blowflies growth experiments by Nicholson (see [26]). This implies that time delay and stage structure are necessary to be introduced into population research such that the model is more realistic.
Due to the influence of circumstances such as resources and interaction, however, the constant time delay is not reasonable any more (see [1,4,10,12,17,18]). Aiello, Freedman, and Wu [3] have already considered a system with a state-dependent delay and two stages, motivated by their knowledge about whale and seal populations. They proposed the following state-dependent delay model v(t) = αu(t) − γv(t) − αe −γτ (v+u) u(t − τ (v + u)), where v(t) and u(t) represent the immature and mature populations densities, respectively; the parameters α and γ represent the birth rate and death rate of immatures, respectively; β represents the mature death and overcrowding rate. The state-dependent time delay τ (v + u) is taken to be an increasing differentiable function of the total population (v + u) so that τ (v + u) ≥ 0 and τ m ≤ τ (v + u) ≤ τ M with τ (0) = τ m and τ (+∞) = τ M . These assumptions imply that the maturation time for the species depends on the total number of them (matures plus immatures) around. The greater the number of individuals present, the longer they will take to mature. This assumption is known to be realistic in the case of Antarctic whale and seal populations [3]. Lowering the number of whales apparently causes the remaining whales to mature more quickly (presumably because there is more food for the remaining whales). Since both immature and mature whales need food, this is the motivation for having the maturation delay depending on the sum v + u of the immature and mature populations. The term αe −γτ (v+u) u(t − τ (v + u)) appearing in both equations of system (1) represents the density of individuals survive to leave the immature and just enter the mature class. Aiello, Freedman, and Wu [3] found that there always exists a positive equilibrium, and obtained criteria for uniqueness as well as local asymptotic stability. In particular, Aiello, Freedman, and Wu [3] obtained bounds for the eventual behavior of u(t) and v(t). Since then, more and more researchers have worked on state-dependent time delay population models; see, for example, [1,4,5,19,35]. Recently, Al-Omari and Gourley [4] derived and studied a stage-dependent population model with state-dependent time delay where the immature birth rate is taken to be a general function of the mature population around and the death rate for the mature is linear. The state-dependent time delay is taken to be an increasing differentiable bounded function of the total population (mature and immature).
In biology, a cooperative system is one of the importance interactions among species and is commonly seen in social animals, including humans (see e.g. [11,23,25]). Lv and Yuan [24] investigated a system consisting of two cooperative mature species by assuming that each immature population density v i depends on the mature population density u i and that the time delay τ is just a function of u i not the sum u i + v i . Namely, Lv and Yuan [24] discussed the stability of the equilibria of the following system with a state-dependent delay      du 1 dt = α 1 e −γ1τ (u1) u 1 (t − τ (u 1 )) − β 1 u 2 1 + µ 1 u 1 u 2 , du 2 dt = α 2 e −γ2τ (u2) u 2 (t − τ (u 2 )) − β 2 u 2 2 + µ 2 u 1 u 2 ,

DYNAMICS OF A TWO-SPECIES STAGE-STRUCTURED MODEL 1395
where u 1 and u 2 represent the densities of the two cooperative mature species, µ 1 and µ 2 are positive interspecific cooperative effects of the classical Lotka-Volterra kind. Similarly to [3] and [4], Lv and Yuan [24] analyzed the positivity and boundedness of solutions of (2). Moreover, by using the comparison principle, Lv and Yuan [24] obtained a stability criterion both from local and global points of view. Motivated by Aiello, Freedman, and Wu [3], Lv and Yuan [24], in this paper we shall investigate the following system of two cooperative species with a delay depending on the total populations where z i = u i + v i , v i and u i (i = 1, 2) are the densities of the cooperative immature and mature species at time t, respectively; α is the birth rate; γ is the death rate of the immature and β is the mature death and overcrowding rate, as in the logistic equation; µ is the interspecific cooperative effect of the classical Lotka-Volterra kind. Throughout this paper, we always assume that α, β, γ, µ are all positive constants. Similarly to [3], the state-dependent time delay τ (z i ) is taken to be an increasing differentiable function of the total population The initial condition for (3) is which is the number of immatures that have survived to time t = 0. Here, τ i is the maturation time of the ith species at t = 0, and the lower limit on the integral is −τ i because anyone of the ith species born before that time will have matured before time t = 0. Since τ i is the maturation time at t = 0, τ i is given by Note that τ i (i = 1, 2) appear on both the left-and right-hand sides of the above equation, so that τ i (i = 1, 2) are determined implicitly. For our model to make sense, i.e., to exclude the possibility of adults becoming immatures except by birth, we need to find conditions ensuring that for each i = 1, 2, t − τ (z i (t)) is an increasing function of t as t increases. Namely, we need In section 2, we will consider the positivity and boundedness of solutions to (3) and shall find that if β > µ then 0 ≤ u i ≤ u + := αe −γτm /(β − µ) and v i ≥ 0 for all 1396 SHANGZHI LI AND SHANGJIANG GUO i = 1, 2. In this case, we have Therefore, if β > µ and τ (z i ) < 4β/(α + µu + ) 2 , then t − τ (z i (t)) is strictly increasing.
Compared with [3] and [24], our theoretical results are more accurate and our method is completely different. In particular, we shall employ degree theory and Lyapunov-Schmidt reduction to investigate the existence and patterns of equilibria. Moreover, we find out the relationship among uniqueness, local asymptotical stability and global asymptotical stability of equilibria. Two auxiliary systems and comparison principles are introduced to prove the global asymptotical stability of the synchronous equilibrium point. By replacing certain variables of u with parameters, we obtain the monotonicity and attractiveness of solutions for the auxiliary systems.
This paper is organized as follows: in section 2, the positivity and boundedness of all solutions of (3) are obtained. In section 3, we first investigate the existence of synchronous equilibria by means of degree theory. Then we employ Lyapunov-Schmidt reduction to discuss the existence of asynchronous equilibria. Section 4 is devoted to the stability of equilibria, especially the synchronous ones. In section 5, we examine the global behavior of solutions and in section 6 we investigate the global asymptotical stability by introducing two auxiliary systems and using the comparison principle of the state-dependent delay equations. Moreover, we illustrate our results with some numerical simulations. Finally, some conclusions and discussions are made in Section 7.
2. Positivity and boundedness. Since the solutions of system (3) represent populations and we also anticipate that limited resources will place a natural restriction to how many individuals can survive, we need address positivity and boundedness of the solution of the system.
Assume that there exist some i ∈ {1, 2} such that u i (t * ) = δ, where t * = inf {t ≥ 0| u i (t) = δ}. It follows from u i (0) = ϕ i (0) ≥ 2δ and the continuity of which contradicts the definition of t * . Thus we complete the proof of this theorem.
Theorem 2.2 implies that for a given pair of positive initial functions ϕ 1 and ϕ 2 , the two mature populations u 1 (t) and u 2 (t) are uniformly bounded away from zero. Now, we shall show that the two mature populations u 1 (t) and u 2 (t) are bounded when β > µ.
Proof. Our proof is split into three cases. We start with the first case where both u 1 (t) and u 2 (t) are eventually monotonic. If both u 1 (t) and u 2 (t) are eventually decreasing then the conclusion of this theorem is obvious. Suppose that both u 1 (t) and u 2 (t) are eventually increasing, i.e.,u i (t) ≥ 0 for all t > T for some T ≥ 0. Then for t > T + τ M , . This means that βu 1 − µu 2 ≤ αe −γτm and βu 2 − µu 1 ≤ αe −γτm for all t > T . Thus, it follows from β > µ, u 1 (t) > 0, and u 2 (t) > 0 that for all t > T and i ∈ {1, 2}, giving us the desired result.
Next, we assume that both u 1 (t) and u 2 (t) are oscillatory. Suppose that there exist two sequences {t n } ∞ n=1 and {s m } ∞ m=1 such thatu 1 (t n ) =u 2 (s m ) = 0, u 1 (t n ) and u 2 (s m ) are local maxima of u 1 and u 2 , respectively, and that u 1 (t) ≤ u 1 (t n ) for all 0 < t < t n , and u 2 (t) ≤ u 2 (s m ) for all 0 < t < s m , m, n ∈ N. Using a similar analysis at t = t n and t = s m , we have For any given t n , we take s n = max{s m |s m ≤ t n }. If s n = t n , then using similar arguments as the first case, we see that u 1 (t) ≤ u + and u 2 (t) ≤ u + for all t < t n .
If s n < t n and u 2 (s n ) ≤ u 2 (t n ), thenu 2 (t n ) > 0 and u 2 (t) ≤ u 2 (t n ) for all t ≤ t n . Otherwise, there is t ∈ (s n , t n ) such thatu 2 (t) = 0, which contradicts the definition of s n . Thus, we have 0 <u 2 (t n ) ≤ αe −γτm u 2 (t n ) − βu 2 2 (t n ) + µu 1 (t n )u 2 (t n ), which means that βu 2 (t n ) − µu 1 (t n ) ≤ αe −γτm . This, together with the first inequality of (4) and a similar argument as the first case, implies that both u 1 (t) and u 2 (t) are bounded above. If s n < t n and u 2 (t n ) ≤ u 2 (s n ), then from the first inequality of (4) it follows that Combining this with the second inequality of (4), we see that both u 1 (t) and u 2 (t) are bounded above. Finally, we consider the case where one of u 1 (t) and u 2 (t) is oscillatory and the other is eventually monotonic. Without loss of generality, assume that u 1 is oscillatory and u 2 is eventually increasing because the other cases can be dealt with analogously. Thus, there exists a sequence {t n } ∞ n=1 such thatu 1 (t n ) = 0, u 1 (t n ) are local maxima of u 1 and that u 1 (t) ≤ u 1 (t n ) for all 0 < t < t n , n ∈ N. It follows that βu 1 (t n ) − µu 2 (t n ) ≤ αe −γτm . For the same sequence {t n } introduced above, it follows from the eventual monotonicity of u 2 that there exists N > 0 such thatu 2 (t n ) ≥ 0 for all n > N , and hence that βu 2 (t n ) − µu 1 (t n ) ≤ αe −γτm for all n > N . Hence, we immediately conclude that both u 1 (t) and u 2 (t) are bounded above. Choosing Θ = max{sup −τ M ≤t≤0 ϕ 1 (t), sup −τ M ≤t≤0 ϕ 2 (t), u + }, we complete the proof of the theorem. Theorem 2.3 implies that the two mature populations u 1 and u 2 are bounded above when the coupling strength µ is small (i.e., β > µ). The following theorem states that u 1 and u 2 are unbounded when the coupling strength µ is large enough (i.e., β < µ).
We shall now prove that v 1 (t) and v 2 (t) are bounded above by a number that depends on the initial conditions when β > µ.
−τi αϕ i (s)e γs ds for i = 1, 2, V is indeed a functional depending only on ϕ 1 and ϕ 2 . Then integrating the equation This completes the proof.
We have proved that u 1 and u 2 remain positive and that u 1 , v 1 , u 2 , and v 2 are all bounded above when β > µ. This leaves the question of whether it is possible to establish the positivity of v 1 and v 2 . As we shall see in the proof of the following theorem, proving positivity of v 1 and v 2 depends on our having a strictly positive lower bound δ and an upper bound Θ for u 1 and u 2 .
Proof. Assume that v i (t) = 0 for some value of t and i. Define t * i = inf{t > 0|v i (t) = 0}. Since v i (0) > 0, then t * i > 0 by continuity. Then integrating the first equation > 0 for all t and i, it seems impossible without placing additional restrictions either on the initial conditions or on the delay τ (z i ). For example, if τ (z i ) ≡ 0, it has been shown in [2] that v i (t) is positive for all t. In Theorem 2.6, we give a set of initial conditions on τ (z i ), while maintaining the essential character of the state-dependent time delay. The following is a corollary of Theorem 2.6, both of them are only sufficient conditions ensuring the positivity of v i (t).
Proof. From the proof of Theorem 2.2, we know that (7) holds for such t * i < ∞. The left-hand side of (7) satisfies by Theorem 2.3 and integration. Similarly, the right-hand side satisfies It follows that f 1 (t * ) ≤ f 2 (t * ). On the other hand, f 1 (t) = δe γt and f 2 (t) = Θe −γtm e γt , then we have We get a contradiction. Therefore, if e −γτm ≤ δ/Θ, no such t * exists. This proves Corollary 1.
3. Existence and patterns of equilibria. The purpose of this section is to investigate the existence and patterns of equilibria It follows from the first and third equations of (8) that for all x, y ∈ R. Thus, system (8) can be reduced to which is obviously . It is clear that system (3) has an equilibrium E 0 (0, 0, 0, 0). If system (3) has a positive equilibrium (v 1 , u 1 , v 2 , u 2 ), i.e., u 1 , u 2 , v 1 , v 2 > 0, then it follows from (9) that and hence that β 2 u 1 u 2 > µ 2 u 1 u 2 , i.e., β > µ. Thus, we obtain the following result.
In what follows, we shall investigate the existence, pattern, and multiplicity of equilibria of system (3) with β > µ. The first of our interest is the existence and multiplicity of synchronous equilibria of the form (v, u, v, u), where v = g(u, u) and Note that for all u ∈ Ω. This implies that the graph of the curve y = f (u) is concave upwards. Thus, we have the following result. Proof. If β ≤ µ then αe −γτ (u+g(u,u)) − (β − µ)u > 0 for all u > 0. This means that system (3) has no nontrivial synchronous equilibria. In what follows, we show that system (3) has at least one nontrivial synchronous equilibrium if β > µ. We first seek for a simpler G: Ω → R such that the integer deg(f, Ω) can be calculated by deg(G, Ω). For this purpose, we have to define a continuous mapping H: This kind of H is called Ω-admissible homotopy. Then by homotopy invariance, we can reach our expectancy. Define H: It turns out that H(t, u) is a Ω-admissible homotopy. Thus, by homotopy invariance, This implies that f has at least one zero point in Ω, and hence that system (3) has at least one synchronous equilibrium Finally, it follows from f (u) > 0 for all u ∈ Ω that f has exactly one or two different zeros in Ω if β > µ. In fact, if f has exactly two different zeros in Ω, then deg(f, Ω) = 0, which contradicts deg(f, Ω) = −1. Therefore, f has exactly one zero in Ω when β > µ. Namely, if β > µ then system (3) has exactly one nontrivial synchronous equilibrium. Theorem 3.2 implies that system (3) has a unique nontrivial synchronous equilibrium (v * , u * , v * , u * ). Note that f (u * ) = 0 and u * < α/(β − µ), then we have It follows from the proof of Theorem 3.2 that f (u * ) < 0 and hence that Remark 2. Following Theorem 4.1 of Aiello, Freedman, and Wu [3], we need one of the following assumptions to ensure that system (3) with β > µ has exactly one synchronous equilibrium . In view of Theorem 3.2, we see that neither of the above three assumptions is necessary.
Next, we consider the existence of boundary equilibria. For an equilibrium (v 1 , u 1 , v 2 , u 2 ) of system (3), if u 1 = 0 then v 1 = 0. This means that the boundary equilibria of system (3) takes the form (v, u, 0, 0) or (0, 0, v, u) with the nonzero vector (u, v) satisfying Using a similar argument as Theorem 3.2, we have the following results.
is a solution to equation (12).
Remark 3. Theorem 3.3 implies that one of two species may exists in the absence of the other. However, we see that in system (3) it is not possible for there to be an equilibrium with one component zero and the others non-zero. This is biologically reasonable since the immature and mature populations depend on each other, and neither of the immature and mature populations can survive at an equilibrium level in the absence of the other.
Finally, we will investigate the existence of the interior equilibria which are asynchronous. From Theorem 3.2 we know that system (3) with β > µ has exactly one synchronous equilibrium (u * , v * , u * , v * ), where (u * , u * ) is a solution to (9). The linearized system of (9) It follows from f (u * ) < 0 that det L < 0 when either β > µ and τ (z * ) = 0 or 2βu * < α + γ and τ (z * ) > 0. Thus, applying the implicit function theorem yields that system (9) has no other solutions in a neighborhood of (u * , u * ). Therefore, what we are interested in is the case where det L = 0.
In view of f (u * ) < 0, we see that det L = 0 if and only if β+µ+e −γτ (z * ) ατ (z * )(γ+ α − 2βu * ) = 0. In what follows, we shall employ Lyapunov-Schmidt method to investigate whether there are other equilibria in a neighborhood of the synchronous one. For more details about the applications of Lyapunov-Schmidt procedure, we refer to [12]. Let η = β+µ+(β−µ)(γ+α−2βu * )u * τ (z * ) and define F : for all u = (u 1 , u 2 ) T ∈ R 2 . We want to study the solution set of the Z 2 -equivariant equation F (u, η) = 0 in a neighborhood of (u * p, 0), where p = (1, 1) T . Denote by L η the Jacobi matrix of F with respect to u evaluated at u * p. Obviously, L η is symmetric and we have the following decompositions: where Y 0 = KerL 0 = span{q} and X 0 = RanL 0 = span{p}, q = (1, −1) T . Let P and I − P denote the projection operators from R 2 onto Y 0 and RanL 0 , respectively. Namely, P u = u − p 2 u · p for all u ∈ R 2 . Thus for each u ∈ R 2 , we have u = u * p + xq + yp with x = 1 2 u · q and y = 1 2 u · p − u * . Thus, F (u, η) = 0 is equivalent to the following system: Thus, the second equation of (14) can be rewritten as Notice that G(0, 0, η) = 0 and G y (0, 0, 0) = (I −P )L 0 p = L 0 p. Applying the implicit function theorem, we obtain a positive constant δ and a continuous differential Substituting y = W (x, η) into the first equation of (14), we have Hence we reduce the original problem to the problem of finding zeros of the map Thus, if ω = 0 then the zeros x of G (·, η) undergo a pitchfork bifurcation near x = 0. Namely, if ω < 0 (respectively, > 0) then there exist a constant δ > 0 and a continuously differentiable mapping x 0 from (0, δ) (respectively, (−δ, 0)) to R such that G (·, η) has three zeros: 0 and ±x 0 (η). Obviously, The zero point x = 0 corresponds to the synchronous equilibrium (v * , u * , v * , u * ) of system (3), while the zeros x 0 (η) and −x 0 (η) correspond to asynchronous equilibria Thus, we obtain the following result.
4. Linearized stability. In this section, we will investigate the local stability of the three types of equilibria. Linearizing an system with state-dependent delay is not completely straightforward because the delay is a function depending on the state variables u i and v i . The local stability of equilibria of state-dependent delay differential equations was studied in [6,16]. It was shown that generically the behaviour of the state-dependent delay except for its value has no effect on the stability of an equilibrium, and that a local linearization is valid by treating the delay function as a constant at the equilibrium point. Hence to study the local stability of an equilibrium E(v 0 1 , u 0 1 , v 0 2 , u 0 2 ) of (3), we linearize (3) at E(v 0 1 , u 0 1 , v 0 2 , u 0 2 ) by treating the two delays τ (u 1 + v 1 ) and τ (u 2 + v 2 ) as τ (v 0 1 + u 0 1 ) and τ (v 0 2 + u 0 2 ), respectively. The resulting linear system is a differential equation with two constant delays: for i . This leads to the following characteristic equation For the extinction equilibrium E 0 (0, 0, 0, 0), (16) reduces to (λ + γ)(λ − αe (γ+λ)τm ) = 0.
Proof. If τ (z * ) = 0, the characteristic equation (16) can be rewritten as Obviously, λ = −γ is an eigenvalue, and some of the others are given by Then we compute the real parts and get and hence Reλ ≤ −(β + µ)u * < 0. The remaining eigenvalues λ are given by Using the same method we see that and hence Reλ ≤ −(β − µ)u * < 0. Thus, we complete the proof.
We now consider the case where τ (z * ) > 0. First we investigate the solutions λ to ∆ + (λ) = 0, i.e., Let λ = a + ib, then separating it into real and imaginary parts, we get where ζ = γu * τ (z * )u * (β − µ)(α + γ − 2βu * ) + 2β . From Theorem 4.1, we know that E * is asymptotically stable if τ (z * ) = 0. Now suppose that τ (z * ) > 0 and seek for the value of ζ such that a = 0, i.e., E * loses its stability. Then (18) becomes Squaring and adding the above two equations yield For such ζ to exist, (19) must have real roots b. After substituting for ζ and rearranging, we see that b is a zero of the following function For this function, we have the following observation.
5. Global behaviors. In this section, we shall discuss the global behavior of solutions of the model (3), and obtain explicit bounds for the eventual behaviors of u i (t) and v i (t), i = 1, 2. Throughout this section, we always assume that β > µ, since from Theorem 2.4 we know that the solutions are unbounded when β < µ. For convenience, let Proof. (i) Without loss of generality, we suppose that there exists t This is a contradiction. And the other cases can be discussed similarly.
(ii) For either i = 1 or i = 2, the result can be obtained with an analogous method in (i), sinceu . This completes the proof.

Remark 4.
This theorem means that if the mature population remains below or above a certain value depending on τ m and τ M for length of time τ M , it will do so from then on.
The following result gives the state bounds on the eventual behaviour of u i (t), independent of admissible initial conditions. Theorem 5.2. Assume that β > µ. Let (v 1 (t), u 1 (t), v 2 (t), u 2 (t)) be a solution of (3). Then where u ± are defined as (21).
Proof. We distinguish three cases to complete the proof of this theorem. The first case is that both u 1 (t) and u 2 (t) are eventually monotonic and bounded. In this case, there exists 0 <ū i < ∞ such that lim t→∞ u i (t) =ū i and lim t→∞ui (t) = 0, i = 1, 2. Hence from (3), taking the limit superior as t → ∞, we havē from which we have u − ≤ū i ≤ u + , i = 1, 2 Next, we consider the case where both u 1 (t) and u 2 (t) are oscillatory. We only show that lim sup t→∞ u i (t) ≤ u + for i = 1, 2, because the other inequalities follow analogously. Define two sequences t n and s m as those times for which u 1 (t) and u 2 (t) achieve their local maxima, respectively, i.e.,u 1 (t n ) = 0,ü 1 (t n ) < 0,u 2 (s m ) = 0,ü 2 (s m ) < 0. Let then lim sup t→∞ u i (t) =ũ i , i = 1, 2. Ifũ i ≤ u + for i = 1, 2, we are done. Hence assume thatũ i > u + (22) is true for at least one of i = 1, 2.
If (22) holds for only i = 1 andũ 2 ≤ u + , we now choose a subsequence of {t n } ∞ n=1 , relabelled as t k such that lim k→∞ u 1 (t k ) =ũ 1 and t k+1 ≥ t k + τ M . We then choose a subsequence of t k , relabelled so that lim k→∞ z i (t k ) =z 1 ,z 1 = lim sup k→∞ z k 1 , ) for this subsequence t k , we choose a subsequence of t k , once again relabelled t k , such that lim k→∞ u 1 (t k − τ (z k 1 )) = u # 1 andȗ 2 = lim sup k→∞ u 2 (t k ). At last, we choose a final subsequence of t k , once again relabelled t k , such that lim k→∞ u 2 (t k ) =ȗ 2 . Obviously, we havȇ u 2 ≤ũ 2 , sinceȗ 2 is just a limit of subsequence of u 2 . Then from (3) and (22), taking the limit as k → ∞, If u # 1 ≤ũ 1 , we get a contradiction. Hence we suppose that u # 1 >ũ 1 . Then we have that, for each k, we can choose a value t p , such thatu 1 (t p ) = 0,ü 1 (t p ) < 0, lim sup p→∞ u 1 (t p ) ≥ u # 1 >ũ 1 , which contradicts the definition ofũ 1 , so u # 1 >ũ 1 cannot be true. If (22) is valid forũ 2 , the same arguments can be done with subsequence s k .
If (22) holds for both i = 1 and 2, we assume thatũ 1 ≥ũ 2 > u + , because the case whereũ 2 ≥ũ 1 > u + can be dealt with analogously. Similarly to (23), we have . Using a similar argument, we see that this is also a contradiction. Therefore, if both u 1 (t) and u 2 (t) are oscillatory then lim sup t→∞ u i (t) ≤ u + .
Finally, we need to consider the case where one of u 1 (t) and u 2 (t) is oscillatory and the other is eventually monotone. Without loss of generality, suppose that u 1 (t) is oscillatory and u 2 (t) is eventually monotone. Using a similar argument as above, we also have (23), since u 2 (t) is monotone and bounded. Thus, lim sup t→∞ u i (t) ≤ u + , i = 1, 2. This completes the proof.
We now use the estimates obtained in Theorem 5.2 to obtain estimates on v 1 and v 2 . We first note that there is a T (ε) > 0 large enough such that for any given ε > 0 whenever t > T . And the equation of v i (t), i = 1, 2 from (3) can be written in the integral form The proofs for Theorems 5.3 and 5.4 are similar to those in [3] and hence are omitted. The condition τ M < 2τ m is required for the lower bound to be positive. 6. Global asymptotical stability. In this section, we shall investigate the global asymptotical stability of the positive synchronous equilibrium when E * (v * , u * , v * , u * ) of (3). For this purpose, we first consider the following system v (t) = αu − γv − aαe −γτ (u+v) , where function τ (·) is the same to that in system (3), c ≥ 0, α, β, γ are positive constants, a ∈ [c − , c + ], b ∈ [c − , c + ], and Note that (26) is a mixed quasi-monotone system (see [27,34]). Firstly, we have the following observations.

Note that
In addition,f a (a, b, c) >0.
In what follows, consider the following system where γ are positive constants, c ≥ 0, and function τ (·) is the same to that of system (3). We have the following result on the existence, uniqueness, and global attractivity of a positive equilibrium point of system (30). Proof. It follows from the proof of Lemma 6.3 that we can obtain the existence and uniqueness of the positive equilibrium point (v(c),û(c)). In what follows, we only need to prove the global attractivity of the positive equilibrium point (v(c),û(c)). Using a similar argument as that of [3], we see that then for all large enough t, we have u (t) < αc + e −γτ (u+v) − βu 2 + cµu.
It follows from Lemma 6.2 that It follows from (31) and Lemma 6.3 that Thus, the positive equilibrium point (v(c),û(c)) attracts all of the positive solutions of system (30). This completes the proof of this lemma.

This process can be continued to construct four sequences {u
Finally, we consider some properties of the functionû(c) for c > 0. Note that u(c) is the unique zero of the function ϑ(c, ·), where ϑ: , where u ± are defined as (21).
Theorem 6.7. The synchronous equilibrium E * (v * , u * , v * , u * ) is globally asymptotically stable if one of the following assumptions is satisfied: ). Furthermore, it follows from Corollary 3 that system (3) has exactly one interior equilibrium, i.e., the synchronous equilibrium E * (v * , u * , v * , u * ), which is locally asymptotically stable. So we only need to prove the global attractivity of E * . For each i = 1, 2, let In view of Theorem 5.2, we obtain u − ≤ u i ≤ u i ≤ u + for i = 1, 2. For each ε ∈ (0, σ), there exists t 0 > τ M such that u i (t) > u − − ε for all t > t 0 , i = 1, 2.
From Figures 1 and 2, we see that all solutions converge to a positive, synchronous, and globally asymptotically stable equilibrium point even though the birth rate and death rate are quite diffferent.
Finally, take α = 2, γ = 0.1, µ = 0.4, and β = 0.365. It follows from Theorem 2.4 that every solution of (3) tends to infinity as t tends to infinity (see Figure 3).   (3) illustrate that every solution of (3) is asymptotically synchronous and tends to infinity as t tends to infinity, where α = 2, γ = 0.1, µ = 0.4, β = 0.365 7. Conclusions and discussions. In this paper, we have investigated a cooperative model composed of two identical species with stage structure and statedependent maturation delays. Despite the low number of units, two-species networks with delay often display the same dynamical behaviors as large networks and, can thus be used as prototypes for us to understand the dynamics of large networks with delayed feedback. Much has been done when the function the delay is constant. When the delay is state-dependent, however, results in the aforementioned work can not be verified as the dynamical systems theory which usually requires the associated semi-flow is continuously differentiable with respect to its initial conditions.
In this paper, we are mainly concerned with the coexistence of the two species. In particular, the existence, patterns, and nonexistence of nonnegative equilibria have been established. Based on our investigation, we may hope to reveal some interesting phenomena of pattern formation in population ecology. The main results of Sections 4 and 5 are the stability analysis of equilibria and the global behaviors of solutions. Moreover, we investigate the global asymptotical stability by introducing two auxiliary systems and using the comparison principle of the state-dependent delay equations. On the one hand, these theoretical results are important for complementing the experimental and numerical observations made in populations population ecological systems, in order to understand the mechanisms underlying the state-dependent delay differential systems dynamics better. On the other hand, the results obtained in this paper suggest that the state-dependent delay plays a very important role on the dynamical behaviors of population system. A properly chosen state-dependent delay can stabilize the system, produce new equilibria, change the stability of the equilibria, and produce much more complex dynamical behaviors. Therefore, state-dependent delay may be used as a simple but efficient switch to control the dynamical behaviors of a system. This paper is only a first step toward networks modeling with state-dependent delays, which can describe more realistic complex networks. There are also some limitations in our model, for example, the two species are identical and the delay function is the same. So future work regarding this topic will include, for example, the dynamics of the following coupled system with state-dependent delay: where z i = u i + v i , α i , β i , γ i , µ i are all positive constants, the state-dependent time delay functions τ i (z i ) are taken to be an increasing differentiable function of the total population z i so that τ i (z i ) ≥ 0, τ i (z i ) ≤ 0, and τ m ≤ τ i (z i ) ≤ τ M with τ i (0) = τ m and τ i (+∞) = τ M , i = 1, 2. The initial condition for (3) is which is the number of immatures that have survived to time t = 0. Here, τ i0 is the maturation time of the ith species at t = 0, and the lower limit on the integral is −τ i0 because anyone of the ith species born before that time will have matured before time t = 0. Since τ i0 is the maturation time at t = 0, τ i0 is given by τ i0 = τ i (u i (0) + v i (0)), i.e., α i ϕ i (s)e γs ds , i = 1, 2.
Note that τ i0 (i = 1, 2) appear on both the left-and right-hand sides of the above equation, so that τ i0 (i = 1, 2) are determined implicitly. Using similar arguments as the proof of Theorems 2.1 and 2.2, we can conclude that for a given pair of positive initial functions ϕ 1 and ϕ 2 on [−τ M , 0], the two mature populations u 1 (t) and u 2 (t) of system (39) are positive and uniformly bounded away from zero. Similar to Theorem 2.3, the two mature populations u 1 (t) and u 2 (t) and the two immature populations v 1 (t) and v 2 (t) are bounded above when β 1 > µ 1 and β 2 > µ 2 .
(v 1 , u 1 , v 2 , u 2 ) is an equilibrium of system (39) if and only if v 1 = g 1 (u 1 , u 2 ), v 2 = g 2 (u 2 , u 1 ), and (u 1 , u 2 ) is a solution to the following system α 1 e −γ1τ1(u1+g1(u1,u2)) − β 1 u 1 + µ 1 u 2 = 0, α 2 e −γ2τ2(u2+g2(u2,u1)) − β 2 u 2 + µ 2 u 1 = 0, where g i : R 2 → R (i = 1, 2) are defined as for all x, y ∈ R. It is clear that system (39) has the origin E 0 (0, 0, 0, 0) as one equilibrium and that has no positive equilibria if β 1 β 2 < µ 1 µ 2 . Thus, it is very interesting to investigate the existence and multiplicity of equilibria of system (39) when β 1 β 2 > µ 1 µ 2 . This problem can be dealt with by means of degree theory and Lyapunov-Schmidt reduction (using similar arguments as that of Section 3), but becomes more complicated. Using similar arguments as that of Sections 5 and 6, we can describe the global behavior of solutions, establish the explicit bounds for the eventual behaviors of the two mature populations and two immature populations, and investigate the global asymptotically stability by using the comparison principle of the state-dependent delay equations. Our another interest is the interaction between the two species maybe not excitatory. For example, the following system may display different dynamical behaviors as (39): du 1 dt = α 1 e −γ1τ1(z1) u 1 (t − τ 1 (z 1 )) − β 1 u 2 1 + µ 1 u 1 u 2 , dv 2 dt = α 2 u 2 − γ 2 v 2 − α 2 e −γ2τ2(z2) u 2 (t − τ 2 (z 2 )), du 2 dt = α 2 e −γ2τ2(z2) u 2 (t − τ 2 (z 2 )) − β 2 u 2 2 − µ 2 u 1 u 2 , where z i = u i + v i , α i , β i , γ i , µ i are all positive constants, the delay functions τ i (z i ) are taken to be the same to that of (39). In (41), we may expect for the existence of bifurcation phenomena and mode interactions. As a result of codimension two mode interaction, the primary branches may undergo secondary bifurcations to branches of mixed-mode solutions. Moreover, the primary branches above may undergo secondary Hopf bifurcations leading to periodic solutions (or quasi-periodic solutions) with trivial spatial isotropy and nontrivial spatiotemporal symmetry. In particular, we may expect that secondary branches of periodic solutions undergo further bifurcations leading to chaotic dynamics. Unfortunately, no systematic method is in the flavor of bifurcation theory for the analysis of the dynamics of state-dependent delay differential equations.