## Abstract

This paper is devoted to the investigation of the asymptotic behaviour for a reaction–diffusion model with a quiescent stage. We first establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone travelling waves. Then we obtain a threshold result on the global attractivity of either zero or positive steady state in the case where the spatial domain is bounded.

## 1. Introduction

Reaction–diffusion systems are used widely to study the dynamics of biological invasions, the spread of a disease or the effects of dispersal in general, e.g. Cantrell & Cosner (2003), Murray (2003) and Gourley & Wu (2006). A classical reaction–diffusion equation is(1.1)where *D* is the diffusion coefficient. This equation assumes that individuals are always mobile. Since individuals in a population might vary with respect to motility, Cook (Murray 2003) first investigated the effect of the individual variability on the speed of travelling waves in a reaction–diffusion system with mobile and non-mobile stages. Several authors have since used this idea and have derived different forms of the models (Lewis & Schemitz 1996; Lutscher *et al*. 2005). Lewis & Schemitz (1996) considered the following single population model with mobile and non-mobile stages(1.2)where *u*_{1} and *u*_{2} are the densities of the mobile and non-mobile subpopulations, *γ*_{1} and *γ*_{2} are the switching rates, respectively, and *μ* is the mortality rate. The sedentary subpopulation reproduces with the intrinsic growth rate *s* and is subject to a finite carrying capacity *N*. All of the parameters in this model are positive constants. Lewis & Schemitz (1996) determined the minimal wave speed for monotone travelling waves under the assumption that the emigration rate *γ*_{2} is less than the intrinsic growth rate *s* for the sedentary class.

The spreading speed for model (1.2) has been studied by Hadeler & Lewis (2002), where they employed the theory presented by Weinberger *et al*. (2002). However, the existence of monotone travelling waves for this system cannot be proved by the result in Li *et al*. (2005) since the solution maps associated with (1.2) are not compact due to the absence of diffusion in one equation. Recently, this problem has been solved by Wang & Zhao (2006). In their paper, using the theory developed in Thieme & Zhao (2003) for nonlinear integral equations, they proved that for system (1.2), the spreading speed is indeed the minimal wave speed for monotone travelling waves.

Hadeler & Lewis (2002) have also presented and discussed briefly the following model:(1.3)where *f* is the reproduction function. The model (1.3) describes a population where the individuals alternate between mobile and non-mobile states, and only the mobile reproduce. Such behaviour is typical for invertebrates living in small ponds in arid climates, which dry up and reappear subject to rainfall (Hadeler & Lewis 2002). However, Hadeler & Lewis (2002) did not provide further mathematical analysis.

The purpose of this paper is to study the asymptotic behaviour of system (1.3) in both unbounded and bounded spatial domains. Although system (1.3) can be rewritten as a scalar integral equation, the reduced equation does not satisfy all assumptions in Thieme & Zhao (2003). It then follows that the theory presented in Thieme & Zhao (2003) on spreading speeds and travelling waves, which worked for the model (1.2) in unbounded spatial domains, does not apply to (1.3). Therefore, we use the monotone dynamical systems approach to study the asymptotic behaviour of system (1.3).

This paper is organized as follows. In §2, we study system (1.3) on the spatial domain . By appealing to the theory recently developed in Liang & Zhao (2007) for monotone semiflows, we establish the existence of the spreading speed and show that it coincides with the minimal wave speed for monotone travelling waves. In §3, we use the theory of monotone and subhomogeneous dynamical systems to investigate the global dynamics of system (1.3) in a bounded domain *Ω*⊂^{n}. A brief discussion section completes the paper.

## 2. Spreading speed and travelling waves

In this section, we first define the ordered spaces of functions and then introduce the assumptions (A1)–(A5) on monotone semiflows from Liang & Zhao (2007). Lastly, we prove that the solution semiflow of system (1.3) satisfies all these assumptions, so that we can use the results in Liang & Zhao (2007) to prove the existence of the spreading speed and travelling wave solutions for system (1.3).

Let be the set of all bounded and continuous functions from to ^{2}. Clearly, any vector in ^{2} can be regarded as a function in . For *u*=(*u*_{1},*u*_{2}),*v*=(*v*_{1},*v*_{2})∈, we write *u*≥*v*(*u*≫*v*) provided , and *u*>*v* provided *u*≥*v* but *u*≠*v*. For *r*≫0 in ^{2}, we define [0,*r*]≔{*u*∈^{2} : 0≤*u*≤*r*} and _{r}≔{*u*∈ : 0≤*u*≤*r*}. It is easy to see that _{+}≔{*u*∈ : *u*≥0} is a positive cone of .

We equip with the compact open topology, i.e. *u*^{m}→*u* in means that the sequence of *u*^{m}(*x*) converges to *u*(*x*) as *m*→∞ uniformly for *x* in any compact set. Definewhere |.| denotes the usual norm in ^{2}. Then (, ‖.‖) is a normed space. Let *d*(., .) be the distance induced by the norm ‖.‖. It follows that the topology in the metric space (,*d*) is the same as the compact open topology in . Moreover, (_{r},*d*) is a complete metric space.

Define the reflection operator by [*u*](*x*)=*u*(−*x*), and the translation operator *T*_{h} by *T*_{h}[*u*](*x*)=*u*(*x*−*h*) for any given *h*∈. Let *r*≫0 in ^{2} and *Q* be a map from _{r} to _{r}. In order to use the theory in Liang *et al*. (2006) and Liang & Zhao (2007), we need the following hypotheses on *Q*:

*Q*:_{r}→_{r}is continuous with respect to the compact open topology.*Q*:_{r}→_{r}is monotone (order preserving) in the sense that whenever .

Note that the hypothesis (A1) implies that *Q*[*u*]∈[0,*r*] whenever *u*∈[0,*r*]. Thus, *Q* is also a map from [0,*r*] to [0,*r*].

*Q*: [0,*r*]→[0,*r*] admits exactly two fixed points 0 and*r*, and for any*ϵ*>0, there is*ξ*∈[0,*r*] with ‖*ξ*‖<*ϵ*such that*Q*[*ξ*]≫*ξ*.

Given a function *u*∈_{r} and a bounded interval *I*=[*a*, *b*]⊂, we define a function *u*_{I}∈*C*(*I*, ^{2}) by *u*_{I}(*x*)=*u*(*x*). Moreover, for any subset of _{r}, we define _{I}={*u*_{I}∈*C*(*I*, ^{2}): *u*∈}. In order to obtain the existence of the travelling wave with the wave speed *c*≥*c*^{*}, we need the following additional assumption:

For any number

*δ*>0, there exists*l*=*l*(*δ*)∈[0,1] such that for any_{r}and any interval*I*=[*a*,*b*] of the length*δ*, we have*α*(*Q*[]_{I})≤*lα*(_{I}), where*α*is the Kuratowski measure of non-compactness on the Banach space*C*(*I*,^{2}).

In this section, we consider the model system (1.3) on the spatial domain . We assume that function *f*∈*C*^{1}(_{+}, ) satisfies

*f*(0)=0,*f*′(0)>0, (d/d*v*)(*f*(*v*)/*v*)<0 for*v*>0.There exists

*H*>0 such that*f*(*v*)≤0 for all*v*≥*H*.

We first study the reaction system associated with (1.3)(2.1)By assumption (C1), we see that *f* is strictly subhomogeneous in the sense that *f*(*ρw*_{1})>*ρf*(*w*_{1}), ∀*w*_{1}>0, *ρ*∈(0,1). It is easy to see that system (2.1) is cooperative and irreducible (Smith (1995) for the definition). Thus, the solution maps associated with system (2.1) are strictly subhomogeneous (e.g. Zhao 2003) and strongly monotone (e.g. Smith 1995). Note thatLet *r*(*DF*(0)) be the spectral radius of *DF*(0). We then have

By the assumptions (C1) and (C2), there exists such that and . Define . Then is an equilibrium of equation (2.1). The assumption (C2) also implies that with is positively invariant for any . By the continuous-time version of Zhao (2003, lemmas 2.3.4 and 2.2.1), we have the following result:

*Assume that* (C1) *and* (C2) *hold*. *Then* *is globally asymptotically stable for* (2.1) *in* ^{2}\{0}.

We consider system (1.3) with initial conditions(2.2)Let *Γ*(*t*,*x*) be the Green function associated with the parabolic equation ∂_{t}*v*=*D*Δ*v*. Note that equation ∂_{t}*u*_{1}=*D*Δ*u*_{1}−*γ*_{1}*u*_{1} and ∂_{t}*u*_{2}=−*γ*_{2}*u*_{2} generate linear semigroups *T*_{1}(*t*) and *T*_{2}(*t*), respectively. *T*_{1}(*t*) and *T*_{2}(*t*) are defined as follows:Integrating two equations of system (1.3) together with (2.2), we have(2.3)

It follows that system (1.3) can be written as the following integral equation(2.4)

A function is said to be a lower solution of (1.3) ifA function is said to be an upper solution of (1.3) if

*Let* (C1) *and* (C2) *hold*. *For any* , *system* (1.3) *has a unique mild solution u*(*t*,*x*,*ϕ*)=(*u*_{1}(*t*,*x*,*ϕ*),*u*_{2}(*t*,*x*,*ϕ*)) *with u*(0,., *ϕ*)=*ϕ and* . *Moreover, if* *and* *are a pair of lower and upper solutions of* (1.3), *respectively, with* , *then* .

We first show that *B* is quasi-monotone on in the sense thatfor all with . Indeed, it is easy to see that there is a constant *ρ*>0 such thatand hence for any *h*>0 satisfying *hρ*<1,By Martin & Smith (1990, Corallary 5), (1.3) has a unique mild solution *u*(*t*, ., *ϕ*) on [0,∞) for each , and the comparison principle holds for the lower and upper solutions. ▪

Define a family of operators {*Q*_{t}}_{t≥0} on by(2.5)Note that for any , we have(2.6)By Martin (1976, theorem 8.5.2), it follows that *Q*_{t}(*ϕ*) is continuous at (*t*_{0}, *ϕ*_{0}) with respect to the compact open topology. Thus, {*Q*_{t}}_{t≥0} is a semiflow on (e.g. Liang & Zhao (2007) for the definition of semiflows).

{*Q*_{t}}_{t≥0} *is a subhomogeneous and strongly monotone semiflow on* .

Since *f* is strictly subhomogeneous, we can see that for any *ρ*∈[0,1] and , *ρu*(*t*,*ϕ*) is a lower solution of (1.3) with initial value *ρϕ*. By lemma 2.2, we then have *ρu*(*t*,*ϕ*)≤*u*(*t*,*ρϕ*) for *t*≥0, i.e. *ρQ*_{t}(*ϕ*)≤*Q*_{t}(*ρϕ*). Thus, *Q*_{t} is subhomogeneous.

By lemma 2.2, {*Q*_{t}}_{t≥0} is a monotone semiflow on . Next, we show that for each *t*>0, *Q*_{t} is strongly monotone in the sense that *Q*_{t}(*ϕ*)≪*Q*_{t}(*ψ*) whenever *ϕ*<*ψ* in . Given *ϕ*<*ψ* in , let *U*(*t*,*x*)=*u*(*t*,*x*, *ψ*)−*u*(*t*,*x*,*ϕ*). Then *U*(*t*,*x*)≥0, ∀*t*≥0 and . Note that the first and second components *U*_{1}(*t*,*x*) and *U*_{2}(*t*,*x*) satisfy(2.7)(2.8)(2.9)

In the case where , the strict positivity theorem (e.g. Volpert *et al*. (1994), theorem 5.5.4) and inequality (2.8) imply that *U*_{1}(*t*,*x*)>0, ∀*t*>0, ∀*x*∈. It follows from equation (2.9) that *U*_{2}(*t*,*x*)>0, ∀*t*>0, ∀*x*∈.

In the case where , by equation (2.9), we haveThus, (2.7) implies thatwhere *T*_{3}(*t*) is the linear semigroup generated by ∂_{t}*U*_{1}=*D*Δ*U*_{1}−(*ρ*+*γ*_{1})*U*_{1}, i.e.Hence, by Volpert *et al*. (1994, theorem 1.4.5), we get *U*_{1}(*t*,*x*)>0, ∀*t*>0, ∀*x*∈. It follows from equation (2.9) that *U*_{2}(*t*,*x*)>0, ∀*t*>0, ∀*x*∈.

Therefore, *u*(*t*,*x*, *ψ*)≫*u*(*t*,*x*,*ϕ*), ∀*t*>0, *x*∈, which implies that is a strongly monotone semiflow. ▪

*For each t*>0, *the map Q*_{t} *satisfies* (*A*1)–(*A*5) *with r*=*u*^{*}.

It is easy to see that assumptions (A1)–(A3) hold for *Q*_{t}. Let be the restriction of *Q*_{t} to [0,*u*^{*}]. Then is the solution semiflow generated by the ordinary differential system (2.1). By lemma 2.1, *u*^{*} is a global asymptotic stable equilibrium of in ^{2}\{0}. Note that is a strongly monotone semiflow on [0,*u*^{*}]. By the Dancer–Hess connecting orbit lemma (e.g. Zhao 2003), it follows that for each *t*>0, the map admits a strongly monotone full orbit connecting 0 and *u*^{*}, and hence, *Q*_{t} satisfies (A4).

Define a linear operator *L*(*t*)*ϕ*=(0,*T*_{2}(*t*)*ϕ*_{2}), ∀*ϕ*∈ and a nonlinear mapIt is easy to see that . Sincewe have . By the expression of (2.3) and the compactness of *T*_{1}(*t*), it then follows that is compact for each *t*>0. Thus, for any number *r*>0, any interval *I* of the length *r*, and any , we havewhich implies that (A5) is satisfied. ▪

According to Liang & Zhao (2007, theorems 2.11 and 2.15), the map admits a spreading speed *c*^{*}. In order to compute *c*^{*}, we consider the linear differential equation(2.10)where *μ*≥0 is a parameter. Let be the solution of (2.10) satisfying .

It is easy to see that is the solution of the linear differential equation with diffusion(2.11)Let {*M*_{t}}_{t≥0} be the solution semiflow associated with system (2.11). Note that *Q*_{t}(*ϕ*) is a lower solution of linear system (2.11) for *t*∈[0,∞). It then follows thatNote that the fundamental solution matrix of (2.10) is e^{A(μ)t} withDefine asTherefore, is the solution map of the linear differential equations (2.10) on ^{2}, and its principal eigenvalue is e^{λ(μ)t}, where *λ*(*μ*) is the spectral radius of the matrix *A*(*μ*), andSince *λ*(0)>0, the map *M*_{t} satisfies assumptions (B1)–(B7) in Liang & Zhao (2007) for each *t*>0.

Letting *t*=1, we see that e^{λ(μ)} is the principal eigenvalue of . Define the function(2.12)Since and , *Φ*(μ) assumes its minimum at some finite value *μ*^{*}. It then follows from Liang & Zhao (2007, theorem 3.10) that .

For any 0<*ϵ*<1, there is *δ*>0 such that *f*(*v*)≥(1−*ϵ*)*f*′(0)*v*, ∀0≤*v*≤*δ*. By the continuous dependence of solutions on initial conditions, it follows that there is a sufficient small *η*>0 such that the solution of (2.1) with satisfies , where . Thus, the comparison theorem (lemma 2.2) implies thatThen for all *t*∈[0,1] and *x*∈, *u*(*t*,*x*,*ϕ*) with satisfies(2.13)Let be the solution semiflow associated with(2.14)Since *Q*_{t}(*ϕ*) is an upper solution of linear system (2.14) for *t*∈[0,1] and , it then follows thatIn particular, . As we did for {*M*_{t}}_{t≥0}, similar analysis can be made for . By Liang and Zhao (2007, theorem 3.10), we haveLetting *ϵ*→0, we obtain . Setting *Φ*′(*μ*)=0, we then have the following equation(2.15)Thus, *c*^{*}=*Φ*(*μ*^{*}), where *μ*^{*} is the positive root of (2.15) at which *Φ*(*μ*) takes its minimum value.

The following result shows that the above-defined *c*^{*} is the spreading speed for solutions of (1.3) with initial functions having compact supports.

*Assume that* (C1) *and* (C2) *hold, and let* . *Then the following statements are valid*:

*For any c*>*c*^{*},*if**with*0≤*ϕ*≪*u*^{*}*and ϕ*(*x*)=0 for*x outside a bounded interval, then*.*For any c*∈(0,*c*^{*}),*if**and*,*then*.

Conclusion (1) is a straightforward consequence of the first part of Liang & Zhao (2007, theorem 2.17). Let *c*<*c*^{*} be given. Since *Q*_{t} is subhomogeneous, then *r*_{σ} can be chosen to be independent of *σ*≫0. Thus, we can write *r*_{σ} as . If and *ϕ*(*x*)≫0 for *x* on an interval *J* of length , then there exists a vector ** σ**≫0 such that

*ϕ*(

*x*)≫

**, ∀**

*σ**x*∈

*J*and hence, the second part of Liang & Zhao (2007, theorem 2.17) implies that . For any with , it follows from the strong monotonicity of

*Q*

_{t}that

*u*(

*t*,

*x*,

*ϕ*)≫0, ∀

*x*∈,

*t*>0. Fix a

*t*

_{0}>0. Then

*u*(

*t*

_{0},

*x*,

*ϕ*)≫0, ∀

*x*∈. By taking

*u*(

*t*

_{0},

*x*,

*ϕ*) as a new initial value, we have the conclusion (2). ▪

The existence and non-existence of travelling wave solutions are straightforward consequences of Liang & Zhao (2007, theorems 4.3 and 4.4) and Liang *et al*. (2006, Remark 2.3).

*Assume that* (C1) *and* (C2) *hold, and let* . *Then the following statements are valid*:

## 3. Dynamics in a bounded domain

In this section, we consider system (1.3) on the bounded spatial domain(3.1)where *Ω*⊂^{n}(*n*≥1) is a bounded domain with boundary ∂*Ω* of class *C*^{1+θ}(0<*θ*≤1), the boundary condition is either *Bu*=*u* (Dirichlet boundary condition) or *Bu*=(∂*u*/∂*ν*)+*g*(*x*)*u* (Robin type boundary condition) for some non-negative function *g*∈*C*^{1+θ}(∂*Ω*, ), ∂*u*/∂*ν* denotes the differentiation in the direction of outward normal *ν* to ∂*Ω*.

Let =*L*^{p}(*Ω*), ∀*n*<*p*<∞, and for *β*∈((1/2)+(*n*/2*p*),1), let _{β} be the fractional power space of with respect to −Δ and the boundary condition *Bu*=0 (e.g. Henry 1981). Then _{β} is an ordered Banach space with the order cone consisting of all non-negative functions in _{β}, and has non-empty interior . Moreover, with continuous inclusion for *m*∈[0,2*β*−1−(*n*/*p*)) (e.g. Hess 1991). Let *E*=_{β}×_{β} and . Then (*E*, *P*) is an ordered Banach space. Denote the norm on *E* by ‖.‖_{β}. Thus, there exists a constant *k*_{β}>0 such that .

By the similar analysis as in §2, we can write equation (3.1) as an integral equation (2.4) with *u*(0)=*ϕ*∈*P*. For any , let . By Martin & Smith (1990, Proposition 3 and Remark 2.4), equation (3.1) has a unique solution *u*(*t*,*ϕ*) with *u*(0,*ϕ*)=*ϕ* on [0,∞) for each *ϕ*∈*P*_{N}, and the comparison theorem holds for (3.1). In addition, *P*_{N} is positively invariant for (3.1).

Define a family of operators {*Q*_{t}}_{t≥0} on *P* by(3.2)By similar arguments as in the proof of lemma 2.3, it then follows that {*Q*_{t}}_{t≥0} is a strongly monotone semiflow on *P*.

*Let* (C1) *and* (C2) *hold*. *Then the solution semiflow* {*Q*_{t}}_{t≥0} *admits a connected global attractor on P*.

By a similar argument as in the proof of lemma 2.4, we can prove that for each *t*>0, *Q*_{t} is an *α*-contraction on *P* with contracting function being .

Next, we show that the solution semiflow {*Q*_{t}}_{t≥0} is point dissipative, i.e. there exists a positive number *R* such thatFor any given *ϕ*∈*P*, let . Denote the solution of (2.1) with initial value *w*_{0} as *w*(*t*,*w*_{0}). By lemma 2.1, we have . By the comparison theorem, we have . It follows that , and hence, there is *t*_{0}>0 such that ‖*u*(*t*, ., *ϕ*‖_{∞}<2*u*^{*}, ∀*t*≥*t*_{0}. By the definition of ‖.‖_{0} on _{0}=, we have ‖*u*‖_{0}≤*k*‖*u*‖_{∞} for some positive number *k*. It follows that , and hence By Hess (1991, lemma 19.3), where *β*<*γ*<1, and *c* depends on *γ*, *β* and *ku*^{*}. Hence,

Since *P*_{N} is positively invariant for all , the orbits of bounded sets are bounded. By the continuous-time version of Zhao (2003, theorem 1.1.2), {*Q*_{t}}_{t≥0} admits a connected global attractor on *P*, which attracts any bounded set in *P*. ▪

Note that (0,0) is an equilibrium of system (3.1). Linearizing system (3.1) at (0,0), we have(3.3)Substituting *u*_{i}(*t*,*x*)=e^{λt}*ϕ*_{i}(*x*), *i*=1, 2, we obtain the associated eigenvalue problem(3.4)

By the proof of Smith (1995, theorem 7.6.1) and a generalized Krein–Rutman Theorem (e.g. Jiang *et al*. (2004), lemma 2.2), equation (3.4) has a principal eigenvalue, denoted by *λ*^{*}, with an associated eigenvector .

According to Smith (1995, theorem 7.6.1), the eigenvalue problem(3.5)has a principal eigenvalue with an eigenfunction . Moreover, we have the following observation.

*The following statements are valid*:

*and λ*^{*}*has the same sign as*.*in the case where Bu*=∂*u*/∂*ν and**in the case where Bu*=*u and Ω*=(0,*L*).

Let *λ*^{*} be the principal eigenvalue of (3.4) with eigenfunction . Then , and hence *λ*^{*}+*γ*_{2}>0. It follows that(3.6)Since in _{β}, we must have , and hence *λ*^{*} is a real zero of the quadratic equation(3.7)It remains to prove that *λ*^{*} is the maximum zero of (3.7). Let *λ* be a given zero of (3.7). Since *P*(−*γ*_{2})=−*γ*_{1}*γ*_{2}<0, we have *λ*≠−*γ*_{2}, i.e. *λ*+γ_{2}≠0. It is easy to see that . Since *λ*+γ_{2}≠0, we have . Note that satisfies (3.5) with . Set . It then follows that *λ* is an eigenvalue of (3.4) with eigenfunction . Thus, any zero of (3.7) is an eigenvalue of (3.4). Since *λ*^{*} is the principal eigenvalue of (3.4), it follows that *λ*^{*} is the maximum zero of (3.7), and henceWith the above expression of *λ*^{*}, we can easily see that *λ*^{*}>0 if , *λ*^{*}=0 if and *λ*^{*}<0 if . This implies that *λ*^{*} has the same sign as .

In the case where *Bu*=∂*u*/∂*ν*, it is easy to verify that *λ*=*f*′(0)−*γ*_{1} is an eigenvalue of (3.5) with the eigenfunction . Since only the principal eigenvalue admits strongly positive eigenfunction, we have . In the case where *Bu*=*u* and *Ω*=(0,*L*), we see that *λ*=−(*π*^{2}*D*)/(*L*^{2})+*f*′(0)−*γ*_{1} is an eigenvalue of (3.5) with the eigenfunction . Since in _{β}, it follows that . ▪

Now we are ready to prove the following threshold result on the global dynamics of (3.1).

*Assume that* (C1) *and* (C2) *hold. Let u*(*t*, ., *ϕ*) *be the solution of* (3.1) with *u*(0,., *ϕ*)=*ϕ*∈*P*.

If

*λ*^{*}<0,*then**for every ϕ*∈*P*.*If λ*^{*}>0,*then*(3.1)*admits a unique positive steady state ϕ*^{*}*and**for every ϕ*∈*P*\{0}.

(1) In the case of *λ*^{*}<0, Smith (1995, theorem 7.6.2) implies that , where *v*(*t*, ., *ϕ*) is the unique solution of (3.3) with *v*(0,., *ϕ*)=*ϕ*. Note that the solution *u*(*t*, ., *ϕ*) of (3.1) satisfies(3.8)By the comparison theorem, we have *u*(*t*,*x*,*ϕ*)≤*v*(*t*,*x*,*ϕ*), ∀*t*≥0, *x*∈*Ω*, and hence, . Next we show that . Let *ω*(*ϕ*) be the omega limit set of the orbit {*u*(*t*, ., *ϕ*) : *t*≥0} with respect to the norm ‖.‖_{β}. It suffices to show that *ω*(*ϕ*)={0}. For any *ψ*∈*ω*(*ϕ*), there exists a sequence *t*_{n}→∞ such that , and hence, . Thus, implies that *ψ*=0. It follows that .

(2) In the case of *λ*^{*}>0, let , . Clearly, . We further have the following claim.

Zero is a uniform weak repeller in the sense that there exists *δ*_{0}>0 such that .

Indeed, let *λ*_{ϵ} be the principal eigenvalue of(3.9)with a positive eigenfunction *ϕ*_{ϵ}. Since , we can fix a sufficiently small number *ϵ*>0 such that *λ*_{ϵ}>0. Choose *δ*_{ϵ}>0 such that *f*(*u*_{1})>(*f*′(0)−*ϵ*)*u*_{1} for all *u*∈(0,*δ*_{ϵ}). Let *δ*_{0}=*δ*_{ϵ}/*k*_{β}. Suppose, by contradiction, there exists *ϕ*_{0}∈*P*_{0} such that , and hence, there exists *t*_{0}>0 such that . It is easy to see that *u*(*t*,*x*,*ϕ*_{0}) satisfies(3.10)Note that is a solution of(3.11)Since *u*(*t*_{0},.,*ϕ*_{0})≫0 in *E*, i.e. *u*(*t*_{0},.,*ϕ*_{0})∈int(*P*), it follows that there exists a sufficiently small *a*>0 such that *u*(*t*_{0},*x*,*ϕ*_{0})≥*aϕ*_{ϵ}(*x*), ∀*x*∈*Ω*. By the comparison theorem, we have ∀*t*≥*t*_{0}, *x*∈*Ω*. Since *λ*_{ϵ}>0, it follows that *u*(*t*,*x*,*ϕ*_{0}) is unbounded, a contradiction.

By the continuous-time version of Zhao (2003, theorem 1.3.3), {*Q*_{t}}_{t≥0} is uniformly persistent with respect to *P*_{0} in the sense that there exists *δ*_{1}>0 such thatBy the continuous-time version of Magal & Zhao (2005, theorem 3.7), the semiflow *Q*_{t}: *P*_{0}→*P*_{0}, *t*≥0, admits a global attractor *A*_{0}. Thus, Magal & Zhao (2005, theorem 4.7) implies that {*Q*_{t}}_{t≥0} has an equilibrium *ϕ*^{*}∈*A*_{0}. Since {*Q*_{t}}_{t≥0} is a strongly monotone semiflow on *P*, we have *A*_{0}=*Q*_{t}(*A*_{0})⊂int(*P*) for any *t*>0, and hence *ϕ*^{*}≫0.

It is easy to see that for each *t*>0, *Q*_{t} is strictly subhomogeneous. Then, Zhao (1996, lemma 1) implies that for each *t*>0, the map *Q*_{t} has at most one fixed point, and hence the semiflow {*Q*_{t}}_{t≥0} has at most one eqilibrium. Thus, *A*_{0} only contains one equilibrium *ϕ*^{*}. By the continuous-time version of Zhao (2003, theorem 2.3.2), it follows that *A*_{0}={*ϕ*^{*}}, i.e. *ϕ*^{*} is globally attractive in *P*_{0}. ▪

## 4. Discussion

In this paper, we consider a reaction–diffusion model with mobile and stationary compartments. We obtain the existence of the spreading speed and a formula how to compute it, and prove that it coincides with the minimal wave speed for monotone travelling waves. It turns out that the invasion rate of the population can be determined by the linearization of the model system at the trivial solution. This result is unsurprising from the mathematical and ecological perspective, but the proof is challenging owing to missing compactness in the model. We should mention that the method we used to prove the existence of the spreading speed and monotone travelling waves in this paper also works for model (1.2). We further study the global dynamics of the model in the bounded domain and obtain the threshold condition on the global attractivity of either zero or positive steady state. Biologically, this result shows that the population dies out when the zero solution is linearly stable, while the population stablizes at a unique positive steady state when the zero solution is linearly unstable.

Note that in the case where *Bu*=*u* and *Ω*=(0,*L*), we can discuss the critical domain size for the persistence of the population. It has been shown in theorem 3.2 that the global dynamics of the model system is determined by the sign of *λ*^{*}, which is the principal eigenvalue of (3.4). In this case, we see from lemma 3.1 that , and hence, *λ*^{*} has the same sign as −(*π*^{2}*D*)/(*L*^{2})+*f*′(0), which implies that the sign of *λ*^{*} does not depend on the switching rates but only on *f*′(0), diffusion coefficient and domain size.

A straightforward computation shows that

By theorem 3.2, it follows that there exists a critical domain size *L*^{*} such that the population stablizes at a positive steady state when the domain size is larger than *L*^{*} and the population becomes extinct when the domain size is smaller than *L*^{*}.

Although model (1.3) gives us valuable insights into the spatial dynamics of the population, it is more realistic to assume that the parameters and the reaction function in this model are time dependent in view of the fluctuating environment. As a first step, it is worthy to consider the periodic version of model (1.3). We leave it as a future work.

## Acknowledgements

We are grateful to two anonymous referees for their careful reading and valuable comments, which led to an important improvement of our original manuscript. Supported in part by the NSERC of Canada and the MITACS of Canada.

## Footnotes

- Received August 28, 2006.
- Accepted December 14, 2006.

- © 2007 The Royal Society