Dynamics of a Time-periodic and Delayed Reaction–diffusion Model with a Quiescent Stage

In this paper, we study a time-periodic and delayed reaction–diffusion system with quiescent stage in both unbounded and bounded habitat domains. In unbounded habitat domain R, we first prove the existence of the asymptotic spreading speed and then show that it coincides with the minimal wave speed for monotone periodic trav-eling waves. In a bounded habitat domain Ω ⊂ R N (N ≥ 1), we obtain the threshold result on the global attractivity of either the zero solution or the unique positive time-periodic solution of the system.


Introduction
In population ecology, dormancy or quiescence plays an important role in the growing process of some species such as reptiles and insects, which is an attractive biological phenomenon.A typical example is the growth of invertebrates living in small ponds in semi-arid region.Since the varying of growing environment subject to the disappear and reappear of rainfall, the individuals can be grouped into two parts: mobile sub-populations and non-mobile subpopulations.It means that the individuals switch between mobile and non-mobile states, while only the mobile sub-populations can reproduce.
It is well known that mathematical models have become basic tools in studying the evolution of population.Recently, considerable attentions have been paid to investigate population models with a quiescent stage or dormancy from the mathematical view (see e.g., [1,4,5,20,22]).Precisely speaking, a reaction-diffusion equation coupled with a quiescent stage can be used to describe aforementioned biological phenomena.Hadeler and Lewis [5] proposed the following basic model: ∂ ∂t u(t, x) = d∆u(t, x) + f (u(t, x)) − γu(t, x) + βv(t, x), ∂ ∂t v(t, x) = γu(t, x) − βv(t, x), (1.1) Corresponding author.Email: wsm@lzcc.edu.cnwhere u(t, x) and v(t, x) are the densities of mobile and stationary sub-populations at time t and location x, respectively; f is the recruitment function and only depends on the density of mobile sub-populations; d is the diffusion rate of the mobile, γ and β are the switch rates between two states.Based on the mathematical analysis of (1.1), the authors provided some appropriate biological interpretation for their results.Zhang and Zhao [22] further investigated the asymptotic behavior of system (1.1) in both unbounded and bounded spatial domains.In the case where the habitat domain is R, they established the existence of the asymptotic spreading speed which coincides with the minimal wave speed for monotone traveling waves.In the case where the habitat domain is bounded, they obtained a threshold result on the global attractivity of either zero or positive steady state.In addition, Zhang and Li [23] studied the monotonicity and uniqueness of traveling waves of (1.1).
As mentioned in [4], to study the effect of a quiescent phase, it is meaningful to incorporate the time delays, which can be caused by many factors such as hatching period or maturation period.Motivated by this, Wu and Zhao [20] studied the following time-delayed reactiondiffusion model with quiescent stage: ∂ ∂t u(t, x) = d∆u(t, x) + f (u(t, x), u(t − τ, x) − γu(t, x) + βv(t, x), ∂ ∂t v(t, x) = γu(t, x) − βv(t, x), (1.2) where f (u(t, x), u(t − τ, x)) is the reproduction function, τ is a nonnegative constant.They established the existence of the minimal wave speed and further studied the asymptotic behavior, monotonicity and uniqueness of the traveling wave fronts.We mentioned that the analysis for (1.2) on bounded spatial domain remains open.
On the other hand, the effect from varying environment (e.g., the seasonal fluctuations and periodic availability of nutrient supplies) should not be ignored in reality.Therefore, it is more reasonable to assume that the reproduction rate and the two switching rates are time heterogeneous, especially, time periodic.More recently, Wang [19] considered a time-periodic version of (1.1): ∂ ∂t u(t, x) = d∆u(t, x) + f (t, u(t, x)) − γ(t)u(t, x) + β(t)v(t, x), ∂ ∂t v(t, x) = γ(t)u(t, x) − β(t)v(t, x), (1.3) where f (t, •) = f (t + ω, •), γ(t) = γ(t + ω), β(t) = β(t + ω), ∀t > 0, ω is a positive constant.For (1.3), the author [19] proved the existence of the spreading speed and showed that it coincides with the minimal wave speed of monotone periodic traveling waves.In the case where the spatial domain is bounded, a threshold result on the global attractivity of either zero or positive periodic solution was established.
(H2) There exists a constant L > 0 such that The purpose of this paper is to investigate the asymptotic behavior of system (1.4).We first apply the results on monotone semiflow in [12][13][14] to obtain the spreading speed in a weak sense.Due to the zero diffusion arising from the quiescent stage, the system (1.4) has a weak regularity, which leads to a difficulty in obtaining the existence of traveling waves.To overcome this problem, we adopt the ideas involving the minimal wave speeds for monotone and "point-α-contraction" systems with monostable structure developed in [3].
The organization of this paper is as follows.Section 2 is devoted to obtain the existence of spreading speed and to show that the spreading speed exactly coincides with the minimal wave speed for monotone periodic traveling waves.In Section 3, we study the global dynamics of system (1.4) in a bounded domain Ω ⊂ R N .In Section 4, we give the appendix on spreading speeds and periodic traveling waves for monotonic systems, which is used in Sections 2 and 3.The frameworks, concepts and results presented by this section are adapted from [3,12,13].

Dynamics in unbounded domain
In this section, we consider the system (1.3) on an unbounded spatial domain Ω = R : In the following, we are mainly concerned with the spreading speed and traveling wave solutions for (2.1).In the first subsection, we present some fundamental results, including the global dynamics of the spatially homogeneous system associated with (2.1), the existence of solutions to (2.1), a comparison principle and the properties of the periodic semiflow.In the second subsection, by appealing the abstract results established in [12,13], we study the spreading speeds for (2.1).The third subsection is devoted to the existence of periodic traveling wave solutions for (2.1) by applying the results established in [3].It needs to be noticed that due to the lack of compactness of the semiflow of (2.1), the abstract results on traveling waves in [12,13] cannot be directly applied.

Preliminaries
).Let X be the set of all bounded and continuous functions from R to R and X + = {ϕ ∈ X : Clearly, X + is a positive cone of X.We equip X with the compact open topology (i.e., ϕ m → ϕ in X means that the sequence of ϕ m (x) converges to ϕ(x) as m → ∞ uniformly for x in any compact set on R) induced by the following norm is an ordered Banach space.For convenience, we identify an element ϕ ∈ C as a function from + , we let N denote the constant function with vector value N in C, C. Firstly, we consider the following spatial-independent system associated with (2.1), Note that (0, 0) is a solution of (2.2).Linearizing (2.2) at the zero solution, we get Due to the periodicity of f , γ, β, and assumptions (H1), we see that for any φ = (φ 1 , φ 2 ) ∈ C+ , (2.3) has a unique solution Ū(t, φ) = ( ū(t, φ), v(t, φ)) on [0, ∞) with Ū(s, φ) = φ ∈ C+ .Hence, we can define a solution semiflow {Ψ t } t≥0 for (2.3) by Define the Poincaré map P : C+ → C+ by P(φ) = ( ūω (φ), vω (φ)) for all φ ∈ C+ , and let r = r( P) be the spectral radius of P. By arguments similar to [21, Proposition 2.1], we show the following results.Proposition 2.1.r = r( P) is positive and is an eigenvalue of P with a positive eigenfunction φ * .
Proof.Since f is strictly sub-homogeneous, Note that the solutions of system (2.
Then S is monotone, and S n is strongly monotone for nω ≥ 2τ.Moreover, it is easy to conclude from the sub-homogeneity of f that S is sub-homogeneous.
By the continuity and differentiability of solutions with respect to initial values, it follows that S is differentiable at zero, and DS(0) = P. Furthermore, since ∂ 2 f (t, u, w) > 0, [7, Theorem 3.6.1]and [18,Theorem 5.3.2]imply that (DS(0)) n is compact and strongly positive for all nω ≥ 2τ.Consider S n 0 , n 0 ω ≥ 2τ.Then, S n 0 is strongly monotone, and (DS(0)) n 0 is compact and strongly positive.
(i) If r ≤ 1, then zero is a globally asymptotically stable fixed point of S n 0 with respect to V h .
(ii) If r ≥ 1, then S n 0 has a unique positive fixed point φ * in V h , and φ * is globally asymptotically stable with respect to V h \ {0}.
Consider the following time-periodic reaction-diffusion equation By virtue of [9, Chapter II], it follows that (2.4) admits an evolution operator where w(t, x, ϕ) is the solution of (2.4).Moreover, for any 0 ≤ s < t, T 1 (t, s) is a compact and positive operator on X, and Integrating two equations of (2.1), we have where A function Û is said to be an upper solution of (2.1) if Thus, we can choose h sufficiently small such that By [16, Corallary 5], (2.1) admits a unique mild solution U(t, •, φ) on [0, +∞) for each φ ∈ C K , and the comparison principle holds for the lower and upper solutions.This completes the proof.
Define a family of operators {Q t } t≥0 on C K by where (u Lemma 2.4.For each t > 0, Q t is strictly subhomogeneous in the sense that Q t (θφ) > θQ t (φ) for any fixed θ ∈ (0, 1).
In the case where φ 2 (x) ≡ 0, by the strong positivity of T 2 (t, s), t > s ≥ 0, we have v(t, x) > 0 for t > 0 and x ∈ R. Since u(t, x, φ) satisfies by an argument similar to (2.6), we can prove that u(t, x, φ) > 0 for all t > 0 and x ∈ R. Therefore, for any φ ∈ C K with φ ≡ 0, we have U(t, x, φ) > 0 for all t > τ and x ∈ R.This completes the proof.

Spreading speed
In what follows, the theory for spreading speeds for monotone autonomous semiflows and periodic semiflows in the monostable case developed in [12,13] will be used to study the spatial dynamics of (2.1).For the sake of convenience, the abstract results in [12,13] can be found in the Appendix. Define . Throughout this subsection, we further assume that (H4) r > 1.
Let Qt be the restriction of Q t to CK .It is easy to see that Qt : CK → CK is an ω-periodic semiflow generated by (2.2) with initial date V0 = φ ∈ CK .Moreover, Qt is strictly monotone for any t > τ and strongly monotone for any t ≥ 2τ on CK .By (H4) and Theorem 2.2, we can conclude from Dancer-Hess connecting orbit lemma (see, e.g., [24]) that Qω admits a strongly monotone full orbit connecting 0 to Therefore, it follows from Theorem 4.2 and Remark 4.3 in the Appendix that Q ω has an asymptotic speed of spread c * ω .This completes the proof.
It then follows from comparison principle that Q t (φ) ≤ M t (φ), ∀φ ∈ C V * 0 .Consequently, we can conclude from Theorem 4.5 (1) in the Appendix that c * ω ≤ inf ρ>0 Φ(ρ).By virtue of (H1), there exists a positive number ς such that ς It is not difficult to see that for any ∈ (0, 1), there is and hence, Let r (ρ) be the spectral radius of the Poincaré map associated with the following linear periodic cooperative and irreducible system (2.10) Let M t be the solution map associated with the linear periodic system (2.12) and hence, the comparison principle implies that Q t (φ) ≥ M t (φ), ∀φ ∈ C ξ , t ∈ [0, ω].By an argument for M t similar to that for M t , from Theorem 4.5 (2) Furthermore, in view of the strong positivity of Q t for all t > 2τ (see Lemma 2.5), it follows that for a fixed t 0 > 2τ, Q t 0 (φ) 0. Taking Q t 0 (φ) as an initial value for U(t, x, φ) and by the above analysis, we complete the proof of part (ii).

Periodic traveling waves
In this subsection, we show that c * = c * ω ω is the minimal wave speed.Due to the lack of compactness of system (2.1), we apply the abstract results on traveling waves in [3], which are presented in the Appendix.To do it, we introduce a new space.Let M be the space consisting of all monotone functions from R to C. For any φ, ψ ∈ M, we write w ≥ z if w(x) ≥ z(x) for x ∈ R and w > z if w ≥ z but w = z.Equip M with the compact open topology.Similar to C r , we can define M r = {φ ∈ M : φ ∈ Cr }.Giving a subset A ⊆ M and p ∈ R, we define A(p) := {W(p) : W ∈ A}.
Applying Proposition 4.12 in the Appendix, we have the following results, which assert that the spreading speed c * ω established in Proposition 2.7 coincides with the minimal wave speed of traveling waves for {Q n ω } n≥0 on M V * 0 .Proposition 2.9.Assume that (H1)-(H4) hold.Let c * ω is the spreading speed established in Proposition 2.7.Then the following statements are valid.
(i) For any c ≥ c * ω , there is a traveling wave W(x − cn) connecting V * 0 and 0.
(ii) For any c < c * ω , there is no traveling wave connecting V * 0 and 0.
Proof.By Proposition 4.12, it suffices to verify that Similar to the previous subsection, we know that Q ω satisfies (B1), (B2), (B4) and (B5).In the remainder of proof, we only need to show that (B3) is valid for Q ω on M V * 0 .In order to realize this, we define (see, e.g., [6,14]) Obviously, Q t = L t + S t for t > 0. As introduced in [6, Theorem 4.1.11]and [14, Section 3], for any bounded set W in M V * 0 , we have α (L t [W ](0)) ≤ e −νt α (W (0)) for some positive number ν.Since the derivatives (∂ t u(t, 0, φ), ∂ t v(t, 0, φ)) are uniformly bounded for t > 0 and φ ∈ W, the set S t [W ](•, 0) is compact.Hence, we get which further implies that Q ω satisfies (B3).Thus, it follows from Proposition 4.12 in the Appendix that c * ω is the minimal wave speed for traveling waves of {Q n ω } connecting V * (ii) For any c < c * , there is no ω-periodic traveling wave connecting V * (t) and 0.
Proof.Motivated by the proof of [12, Theorem 2.3], we define P t = T −ct Q t , where {Q t } t≥0 is the ω-periodic semiflow on M V * 0 generated by (2.1).Thus, {P t } t≥0 is an ω-periodic semiflow on M V * 0 .Therefore, by Proposition 2.9 and arguments similar to those in [12, Theorem 2.2 and 2.3], we complete the proof of the conclusions.

Dynamics in a bounded domain
In this section, we consider system (1.4) in a bounded spatial domain: where Ω ⊂ R N (N ≥ 1) is bounded domain with boundary ∂Ω of class C 2+θ (0 ≤ θ ≤ 1).The boundary operator is either Bu = u (Dirichlet boundary condition) or Bu = ∂u ∂ν + α(x)u (Robin type boundary condition) for some non-negative function α ∈ C 1+θ (∂Ω, R), and ∂u ∂ν is the differentiation in the direction of outward normal ν to ∂Ω.
Let X = L p (Ω) and p ∈ (N, +∞) be fixed.For any ρ ∈ 1 2 + N 2p , 1 , let X ρ be the fractional power space of X with respect to −∆ and the boundary condition Bu = 0 (see, e.g., [8]).Then X ρ is an ordered Banach space with the order cone X + ρ consisting of all non-negative functions in X ρ , and X + ρ has non-empty interior Int X + ρ .Moreover, Then (X , X + ) is an ordered Banach space with the order cone X + consisting of all non-negative functions in X , and X + has non-empty interior Int(X + ).Denote • ρ as the norm on X ρ .Then there exists a constant l ρ > 0 such that φ For convenience, we will identify an Note that the differential operator ∆ generates an analytic semigroup T0 (t) on L p (Ω) and standard parabolic maximum principle (see, e.g., [18,Corollary 7.2.3])implies that the semigroup T0 (t) : X ρ → X ρ is strongly positive in the sense that T0 (t) X + ρ \ {0} ⊆ Int X + ρ for all t > 0. By similar analysis to that in section 2, system (3.1) can be written as an integral equation (2.5) with U 0 (•, •, φ) ∈ E + .For any φ ∈ E K , it follows from [16,Corollary 5] that (3.1) admits a unique mild solution U(t, •, φ) with U 0 (•, •, φ) = φ on [0, ∞).Moreover, U(t, x, φ) is a classic solution when t > τ and the comparison theorem holds for system (3.1).
Define a family of operator {Q t } t≥0 on E + by where (u(t, x, φ), v(t, x, φ)) is a solution of (3.1) with (u(s, x), v(0, x)) = (φ 1 (s, x), φ 2 (x)) for s ∈ [−τ, 0] and x ∈ Ω. Similarly as in section 2, it is easy to see that {Q t } t≥0 is a monotone periodic semiflow on E + , and Q t is strictly subhomogeneous for each t > 0. When t > τ, U(t, x, φ) > 0, ∀x ∈ Ω, ∀φ ∈ E + with φ ≡ 0, and hence, Q t is strongly positive for t > 2τ.Let Recall the definition of a global attractor (see, e.g., [24,Chapter 1]).Let (G, ρ) be a metric space with metric ρ, is a continuous map.A bounded set A is said to attract a bounded set B in G if lim n→∞ sup x∈B d( n (x), A) = 0.A global attractor for : G → G is an attractor that attracts every point in G. Theorem 3.1.Assume that (H1)-(H4) hold.Then Q n 0 ω admits a connected global attractor on E + .
Proof.Firstly, we prove that {Q t } t≥0 is point dissipative on E + .It suffices to prove that there exists a positive number L such that for any φ ∈ E + , lim t→∞ U(t, •, φ) ρ ≤ L.
Proof.Since P = Ũn 0 ω , P 0 = Ũω , we have P = P n 0 0 .By the properties of spectral radius of linear operators, it follows that r(P) = (r(P 0 )) n 0 , i.e., r = r n 0 0 .As mentioned in [11,Theorem 3.3], the qualitative solutions of (3.1) and (3.2) do not change whether we consider them as n 0 ωperiodic systems or ω-periodic systems.Thus the conditions in Theorem 3.3 can be replaced by r < 1 and r > 1, respectively.In what follows, we consider (3.1) and (3.2) as n 0 ω-periodic systems and prove the conclusions (1) and ( 2) under the conditions of r < 1 and r > 1, respectively.
Note that Q n 0 ω is α-contracting, point dissipative and uniformly persistent.It follows from [24,Theorem 1.3.6]that Q n 0 ω : M 0 → M 0 admits a global attractor A 0 and has a fixed point φ in A 0 .Since Q n 0 ω is strictly subhomogeneous, then Q n 0 ω has at most one fixed point according to [25, Lemma 1].Thus, Q n 0 ω has a unique fixed point φ ∈ M 0 .According to the strong monotonicity of Q n 0 ω , we have φ ∈ Int(E + ).Due to the strong monotonicity and strict sub-homogeneity of Q n 0 ω , it follows from [24, Theorem 2.3.2] that A 0 = { φ}.Consequently, φ is globally attractive in M 0 for Q n 0 ω .

Appendix
In this Appendix, we present some results of [3,12,13] about spreading speeds and traveling waves for monotone evolution systems. is a compact and strongly positive linear operator on C.
It then follows from that B µ has a principal eigenvalue λ(µ) with a strongly positive eigenfunction.The following condition is needed for the estimate of the spreading speed c * .
We say that M has compact support provided there is some ρ such that for any α ∈ C, M[α](θ, x) depends only on the value of ęÁ in   (C6') For any µ ≥ 0, B µ is a positive operator, and there exist n 0 and l ∈ [0, 1) such that is a strongly positive linear operator on C and α(B n 0 µ (A)) ≤ lα(A) for any bounded subset A of C. Definition 4.7.Let ω > 0 and r ∈ C with r 0 be given.A family of mappings {Q t } t≥0 is said to be an ω-periodic semiflow on C r provided Q t has the following properties.
The mapping Q ω is called the Poincaré (or periodic) map associated with this periodic semiflow.
In the following, we collect the abstract results on traveling waves in [3] (see also e.g., [2]).Let M be the space consisting of all monotone functions from R to C. For any φ, ψ ∈ M, we write w ≥ z if w(x) ≥ z(x) for x ∈ R and w > z if w ≥ z but w = z.Equip M with the compact open topology.Similar to C r , we can define M r = {φ ∈ M : φ ∈ Cr }.Giving a subset A ⊆ M and p ∈ R, we define A(p) := {W(p) : W ∈ A}.In the following, we make some assumptions for a given operator Q : M β → M β (see [2,3]).In the case where the system admits no advection, the upper and lower bounds of rightward spreading speeds in [3,Theorem 3.8] are same, hence we have the following observation.Proposition 4.12.Assume that Q satisfies (B1)-(B5).Let c + be the spreading speed of Q, then the following statements are valid: (1) for any c > c + , there exists a continuous traveling wave W(x − cn) connecting β to 0; (2) for any c < c + , there is no traveling wave connecting β to 0.
respect to the compact open topology, by arguments similar to those in[15, Theorem 8.5.2], we know that Q t (φ) is continuous at (t 0 , φ 0 ) with respect to the compact open topology.According to the definition of ω-periodic semiflow (see Definition 4.7 in the Appendix), it follows that {Q t } t≥0 is an ω-periodic semiflow on C K .

( 2 . 11 )
With the aim of the comparison principle, there is ξ

Remark 4 . 6 .
Theorem 4.5 is still valid if we replace (C6) with the following assumption.
is an ordered Banach space.We equip C with the compact open topology and define the norm on •) is cooperative on C+ .Clearly, the system (2.3) is irreducible.Moreover, for any K > L, K = K, max t∈[0,ω] Due to the strict subhomogeneity of f (t, •, •), we have the following results about the global dynamics of (2.2).
as n → ∞.Since S is continuous and the sequence of S n (ς φ * ) is monotone, φ * is a fixed point of S, which implies that ( u(t, φ * ), v(t, φ * )) is a ω-periodic solution.The proof is complete.