RESEARCH ARTICLE The Periodic Ross-Macdonald ...

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LouZhaoApplAnalFinal

Applicable Analysis Vol. 00, No. 00, Month 200x, 1–21

RESEARCH ARTICLE The Periodic Ross-Macdonald Model with Diffusion and Advection∗ Yijun Lou† and Xiao-Qiang Zhao‡ (Received 00 Month 200x; in final form 00 Month 200x) In this paper, we propose a periodic Ross-Macdonald type model with diffusion and advection to study the possible impact of the mobility of humans and mosquitoes on malaria transmission. We establish the existence of the leftward and rightward spreading speeds and their coincidence with the minimum wave speeds in the left and right directions, respectively. For the model in a bounded domain, we obtain a threshold result on the global attractivity of either zero or the positive periodic solution.

Keywords: Malaria transmission; reaction-diffusion-advection model; spreading speeds; periodic traveling waves; positive periodic solution; global attractivity AMS Subject Classification: 35K57; 37B55; 37N25; 92D30

1.

Introduction

Human malaria is a viral disease which is caused by protozoan parasites of the genus Plasmodium, transmitted from human-to-human by the female Anopheles mosquito. Currently, malaria is still endemic in 109 countries [17]. Since the climate change induces the change of the population dynamics and biting pattern of the mosquito vector, the malaria cases may significantly increase due to climate change [13, 16, 29]. Moreover, modern transport facilitates the movement of human and disease vectors, which may play a role in the global dissemination of malaria [19, 22]. To plan and implement effective control measures, we should understand the spatial-temporal distribution of risk for malaria infections [20]. Mathematical models have long provided important insight into the malaria dynamics and control [1]. The earliest model of malaria transmission is the RossMacdonald model, which captures the essentials of the transmission process. Much has been done based on this classical model (see, e.g., [2, 7, 18] and references therein). The classical Ross-Macdonald model is highly simplified. One omission is the temporal heterogeneity in the distribution of mosquito populations and human biting rate. Transmission and distribution of vector-borne diseases are greatly influenced by environmental and climatic factors. Seasonality and circadian rhythm of mosquito populations, as well as other ecological and behavioral features, are strongly influenced by climatic factors such as temperature, rainfall, humidity, ∗ Supported

in part by the NSERC of Canada and the MITACS of Canada. of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada ([email protected]). ‡ Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada ([email protected]). † Department

ISSN: 0003-6811 print/ISSN 1563-504X online c 200x Taylor & Francis

DOI: 10.1080/0003681YYxxxxxxxx http://www.informaworld.com

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LouZhaoApplAnalFinal Y. Lou and X.-Q. Zhao

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wind, and duration of daylight [6, 16, 20]. Another omission is the spatial movements of reservoirs and vectors. Spatial dispersal of mosquitoes and reservoirs has also contributed to the spread of mosquito-borne diseases [11, 12, 22]. To address these two omissions, we propose a reaction-diffusion-advection malaria model in a periodic environment. As in the classical Ross-Macdonald model, the adult female mosquito and human populations are divided into two epidemiological categories: the susceptible class and infectious class. We assume the total density of human and mosquito population at any point x and time t are H and M (t), respectively. Let the spatial density of infectious humans and vectors be h(t, x) and v(t, x), respectively, then the density of susceptible humans and susceptible mosquitoes are H − h(t, x) and M (t) − v(t, x). Suppose the mortality rate of the humans and mosquitoes are dh and dv (t), respectively. Assume a(t) is the mosquito biting rate, that is, a(t) is the average number of bites per mosquito per unit time at time t. Based on the fact that the total number of bites made by mosquitoes is equal to the total number of bites received by humans, the average number of bites per human receives per unit (t) time at time t is a(t)M . Suppose the transmission probabilities from infectious H vectors to humans and from infectious humans to vectors are denoted by b and c, respectively. Thus, the infection rates per susceptible human and susceptible vector are given by b

a(t)M (t) v(t, x) a(t)b h(t, x) = v(t, x) and a(t)c , respectively. H M (t) H H

To take into account climate effects on mosquito development, we further assume that M (t), dv (t) and a(t) are positive and periodic functions with the same period being ω = 365 days. To describe the random movement of humans and mosquitoes, we use Fick’s law to model the diffusion for simplicity. The diffusion coefficients for humans and mosquitoes are Dh and Dv , respectively. To account for the wind advection to the mosquito dispersal, we use g to denote the constant velocity flux. We will always assume that the advection points to the right. Combining the viral dynamics and dispersal process together, we then have the following mathematical model on malaria dispersal: {

∂h(t,x) ∂t ∂v(t,x) ∂t

2

h(t,x) = a(t)b H−h(t,x) v(t, x) − dh h(t, x) + Dh ∂ ∂x , 2 H ∂ 2 v(t,x) ∂ = a(t)c h(t,x) (M (t) − v(t, x)) − d (t)v(t, x) + D − g ∂x v(t, x). v v ∂x2 H

(1) This paper is devoted to the study of the asymptotic behavior of system (1) in both unbounded and bounded spatial domains. In section 2, we prove the existence of the rightward and leftward spreading speeds c∗+ and c∗− , and their coincidence with the minimal wave speeds for monotone periodic traveling waves in the right and left directions. In section 3, we establish a threshold result on the global dynamics of system (1) in a bounded domain Ω ⊂ Rn . Section 4 presents some numerical simulations to illustrate our analytic results. The paper concludes with a brief discussion in section 5.

2.

Spreading speeds and traveling waves

In this section, we study the spatial dynamics of system (1) in terms of spreading speeds and traveling waves.

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Periodic Ross-Macdonald model with diffusion and advection

2.1.

3

The periodic Ross-Macdonald model

We first study the global dynamics of the following periodic version of RossMacdonald model: { dh(t) H−h(t) v(t) − dh h(t), dt = a(t)b H (2) dv(t) h(t) dt = a(t)c H (M (t) − v(t)) − dv (t)v(t). We can rewrite system (2) as dy = G(t, y) dt

(3)

) ) ( 1 h(t) a(t)b H−y y2 − d h y1 H . Denote D := {(t, y) : with y= , G(t, y) = v(t) (M (t) − a(t)c yH1( )y2 ) − dv (t)y2 H 0 ≤ y ≤ y¯(t), t ≥ 0} with y¯(t) = and L0 = max{H, max M (t)}. Let M (t) t≥0 Dt := {y : 0 ≤ y ≤ y¯(t)}. Clearly, D0 = {(h, v) : 0 ≤ h ≤ H, 0 ≤ v ≤ M (0)}. (

Lemma 2.1: For any (h(0), v(0)) ∈ [0, l] × [0, l] with l ≥ L0 , system (3) has a unique solution (h(t), v(t)) ∈ [0, l] × [0, l] through (h(0), v(0)), ∀t ≥ 0. Furthermore, (h(t), v(t)) ∈ Dt , ∀t ≥ 0, whenever (h(0), v(0)) ∈ D0 . Proof : Since for all y ≥ 0, G(t, y) is continuous and locally Lipschitzian in y in any bounded set, there is a unique solution for system (3) through (h(0), v(0)) ∈ [0, l] × [0, l]. It then follows from [21, Remark 5.2.1] that for any initial value (h(0), v(0)) ∈ [0, l] × [0, l], the unique solution (h(t), v(t)) admits 0 ≤ h(t) ≤ l, 0 ≤ v(t) ≤ l on its maximal interval of existence. Hence, all solutions exist globally. Using a similar argument, we can further get the second statement. Note that (0, 0) is a ω-periodic solution of (3), and the corresponding linearized system for (3) is dz = Dy G(t, 0)z = dt

[

] −dh a(t)b z. c H a(t)M (t) −dv (t) [

Using the notations in [24], we set F (t) = ] [ dh 0 . Then we can rewrite system (4) as 0 dv (t)

0 a(t)b c a(t)M (t) 0 H

(4) ] and V (t) =

dz(t) = (F (t) − V (t))z(t). dt Assume Y(t,s), t ≥ s, is the evolution operator of the linear-periodic system dy = −V (t)y. dt That is, for each s ∈ R, the 2 × 2 matrix Y (t, s) satisfies d Y (t, s) = −V (t)Y (t, s), dt where I is the 2 × 2 identity matrix.

∀t ≥ s,

Y (s, s) = I,

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Let Cω be the Banach space of all ω-periodic functions from R to R2 , equipped with the maximum norm. Suppose φ(s) ∈ Cω is the initial distribution of infectious individuals in this periodic environment, then F (s)φ(s) is the rate of new infections produced by the infected individuals who were introduced at time s, and Y (t, s)F (s) represent the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t for t ≥ s. Hence, ∫

∫

t

ψ(t) =

∞

Y (t, s)F (s)φ(s)ds = −∞

Y (t, t − a)F (t − a)φ(t − a)da

0

gives the distribution of accumulative new infections at time t produced by all those infected individuals φ(s) introduced at previous time. We define the next infection operator L : Cω → Cω by ∫ (Lφ)(t) =

∞

Y (t, t − a)F (t − a)φ(t − a)da,

∀t ∈ R,

φ ∈ CT .

0

Then the basic reproduction ratio is R0 := ρ(L), the spectral radius of L. Let ρ be the principal Floquet multiplier of the linear system (4). The following result comes from [24, Theorem 2.2]. Lemma 2.2:

The following statements are valid:

(i) R0 = 1 if and only if ρ = 1; (ii) R0 > 1 if and only if ρ > 1; (iii) R0 < 1 if and only if ρ < 1. Thus, 0 is asymptotically stable for system (2) if R0 < 1, and unstable if R0 > 1. We further have the following result on the global dynamics of system (2). Lemma 2.3:

The following statements are valid:

(i) If R0 > 1, then system (2) admits a unique positive ω-periodic solution (h∗ (t), v ∗ (t)), and it is globally asymptotically stable for (2) with initial values in D0 \ {0}; (ii) If R0 ≤ 1, then (0, 0) is globally asymptotically stable for system (2) in D0 . Proof : Let yt (y0 ) = y(t, y0 ) be the solution map of system (2) through y0 . Denote ∂yt X(t) = ∂y (y0 ) and A(t) = Dy (G(t, y(t, y0 ))). Then, X(t) = (xij (t))2×2 satisfies 0 X 0 (t) = A(t)X(t),

X(0) = I.

0 i Since ∂G ∂yj ≥ 0, i 6= j, ∀(t, y) ∈ D, then xik (t) ≥ aii (t)xik (t), ∀t ≥ 0 and i, k ∈ {1, 2}. If t0 ≥ 0 and xik (t0 ) > 0, it then follows that xik (t) > 0 for all t ≥ t0 . Since xii (0) = 1, we have xii (t) > 0, ∀t ≥ 0, i = 1, 2. We further prove that xij (tij ) > 0 for some tij ∈ [0, ω], ∀i 6= j, and hence xij (t) > 0, ∀t ≥ ω, i 6= j. Assume, by contradiction, that there is an element xij (t) = 0 for all t ∈ [0, ω] with i, j ∈ {1, 2} and i 6= j. Then

0 = x0ij (t) =

2 ∑

ail (t)xlj (t) = aij (t)xjj (t),

∀t ∈ [0, ω].

l=1

Since xjj (t) > 0, it then follows from the above equality that aij (t) ≡ 0, ∀t ∈ [0, ω].

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5

Note that ( A(t) =

a(t)b − a(t)b H y2 (t, y0 ) − dh H (1 − y1 (t, y0 )) a(t)c a(t)c (M (t) − y (t, y )) − 2 0 H H y1 (t, y0 ) − dv (t)

) .

0) If a12 (t) = 0, ∀t ∈ [0, ω], then y1 (t, y0 ) = H and dy1 (t,y = −dh y1 (t, y0 ). Since dt dy1 (t,y0 ) = 0 while −dh y1 (t, y0 ) = −dh H, we get a contradiction. If a21 (t) ≡ 0, ∀t ∈ dt 0) [0, ω], then y2 (t, y0 ) = M (t) and dy2 (t,y = −dv (t)y2 (t, y0 ), which contradicts the dt ∂yt periodicity of M (t). Thus, we get ∂y0 (y0 ) 0, t ≥ ω. Furthermore, if y2 , y3 ∈ D0 satisfy y2 < y3 , then for all t ≥ ω, we have

∫ yt (y2 ) − yt (y3 ) = 0

1

∂yt (y2 + r(y3 − y2 ))(y3 − y2 )dr 0. ∂y0

Hence, we have yt (y2 ) yt (y3 ), ∀t ≥ ω, and in particular, yω is strongly monotone. It is easy to check that the following two conditions hold for system (3): (B1) G(t, y) ≥ 0 for every (t, y) ∈ D with yi = 0, i = 1, 2; (B2) For each t ≥ 0, y ∈ Dt , G(t, y) is strictly subhomogeneous on y in the sense that G(t, αy) > αG(t, y), ∀y ∈ Dt and y 0, α ∈ (0, 1). Using the same proof as in [28, Theorem 2.3.4], as applied to the Poincaré map associated with system (2) on D0 (see, e.g., [28, Theorem 3.1.2]), we see that two statements are valid. 2.2.

Spatial dynamics

In the rest of this section, we always assume that R0 > 1. According to Lemma 2.3, there exist two periodic solution, (0, 0) and u∗ (t) = (h∗ (t), v ∗ (t)), for the spatially homogeneous system (2). We will consider system (1) with initial conditions 0 ≤ h(0, x) = φ1 (x) ≤ H, 0 ≤ v(0, x) = φ2 (x) ≤ M (0), ∀x ∈ R.

(5)

Let X be the set of all bounded and continuous functions from R to R2 and X+ = {φ ∈ X : φ(x) ≥ 0, ∀x ∈ R}. Clearly, any vector in R2 can be regarded as a function in X. For u = (u1 , u2 ), w = (w1 , w2 ), we write u ≥ v (u v) provided ui (x) ≥ vi (x) (ui (x) > vi (x)), ∀i = 1, 2, x ∈ R, and u > v provided u ≥ v but u 6= v. For any r 0, we define [0, r] := {u ∈ R2 : 0 ≤ u ≤ r} and Xr := {u ∈ X : 0 ≤ u ≤ r}. We equip X with the compact open topology, i.e., um → u in X means that the sequence of um (x) converges to u(x) as m → ∞ uniformly for x in any compact set in R. Define

kukX =

∞ max |u(x)| ∑ |x|≤k k=1

2k

,

∀u ∈ X,

where | · | denotes the usual norm in R2 . Then (X, k · k) is a normed space. Let d(·, ·) be the distance induced by the norm k · k. It follows that the topology in the metric space (Xr , d) is the same as the compact open topology in Xr . Moreover, Xr is a complete metric space. Given any y ∈ R, define the translation operator Ty by Ty [u](x) := u(x − y). Let Q : Xβ → Xβ be a map, where β 0 in R2 . To use the theory of spreading speeds

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6

and traveling waves developed in [8, 9], we need the following assumptions on Q: (A1) (A2) (A3) (A4)

Ty [Q[u]] = Q[Ty [u]], ∀y ∈ R. Q : Xβ → Xβ is continuous with respect to the compact open topology. Q[Xβ ] is precompact in Xβ . Q : Xβ → Xβ is monotone (order-preserving) in the sense that Q[u] ≥ Q[v] whenever u ≥ v in Xβ .

Note that the hypothesis (A1) implies that Q[u] ∈ [0, β] whenever u ∈ [0, β]. Thus, Q is also a map from [0, β] to [0, β]. (A5) Q admits exactly two fixed points 0 and β, and for any positive number ε > 0, there is an α ∈ [0, β] with kαk < ε such that Q[α] α. Let E := {(t, φ) ∈ [0, ∞) × X+ : φ ≤ y¯(t)} be the ) of [0, ∞) × X+ and ( subset H . Assume Y is the set Et := {φ ∈ X+ : (t, φ) ∈ E} = Xy¯(t) , where y¯(t) = M (t) of all bounded and continuous functions from R to R. Let S1 (t) and S2 (t) be the solution semigroups on Y generated by two equations ∂u1 ∂ 2 u1 = Dh , ∂t ∂x2

∂ 2 u2 ∂u2 ∂u2 = Dv −g , 2 ∂t ∂x ∂x

and

respectively. Then ∫

2

− (x−y) 4D t

e √

S1 (t)φ1 (x) = R

∫

h

4πDh t

φ1 (y)dy, t > 0,

(x−gt−y)2 4Dv t

e− √

S2 (t)φ2 (x) = R

4πDv t

φ2 (y)dy, t > 0.

Note that the equations ∂ 2 u1 ∂u1 = Dh − dh u1 , ∂t ∂x2

and

∂ 2 u2 ∂u2 ∂u2 = Dv − dv (t)u2 , −g· 2 ∂t ∂x ∂x

admit evolution operators T1 (t, s) and T2 (t, s) on Y, respectively. Indeed, T1 (t, s) and T2 (t, s) can be defined as follows: T1 (t, s)φ1 = e−dh (t−s) S1 (t − s)φ1 , T2 (t, s)φ2 = e−

Rt s

dv (τ )dτ

S2 (t − s)φ2 .

Define B : E → X by ( B(t, φ)(x) :=

1 (x) a(t)b H−φ φ2 (x) H φ1 (x) a(t)c H (M (t) − φ2 (x))

) , ∀(t, φ) ∈ E, x ∈ R.

(6)

Let u1 (t, x) = h(t, x) and u2 (t, x) = v(t, x). Then (1) becomes  

(

2

∂ Dh ∂x 2 u1 − dh u1 ∂2 ∂ Dv ∂x2 u2 − g · ∂x u2 − dv (t)u2  u(0, x) = φ(x), x ∈ R.

∂t u(t, x) =

) + B(t, u), t > 0,

(7)

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7

Integrating two equations of system (7) together with (5), we have {

∫t u1 (t, ·, φ) = T1 (t, 0)φ1 + 0 T1 (t, s)B1 (s, u(s))ds, ∫t u2 (t, ·, φ) = T2 (t, 0)φ2 + 0 T2 (t, s)B2 (s, u(s))ds.

(8)

It follows that system (7) can be written as an integral equation ∫

t

u(t, ·, φ) = T (t, 0)φ +

T (t, s)B(s, u(s))ds,

(9)

0

whose solutions are called mild solutions to system (7). Definition 2.4: A function u(t, x) is said to be an upper (a lower) solution of (7) if it satisfies ∫ u(t) ≥ (≤)T (t, 0)u(0) +

t

T (t, s)B(s, u(s))ds.

(10)

0

Theorem 2.5 : For any φ ∈ Xy¯(0) , system (7) has a unique mild solution u(t, ·, φ) = (u1 (t, ·, φ), u2 (t, ·, φ)) ∈ Xy¯(t) with u(0, ·, φ) = φ ∈ Xy¯(0) , ∀t ≥ 0, and u(t, x, φ) is a classical solution when t > 0. Moreover, if u(t, x) and u ¯(t, x) are a pair of lower and upper solutions of system (7), respectively, with u(0, ·) ≤ u ¯(0, ·), then u(t, ·) ≤ u ¯(t, ·), ∀t ≥ 0. Proof : We first show that B is a quasi-monotone map from E to X in the sense that lim d(ψ − φ + k[B(t, ψ) − B(t, φ)]; X+ ) = 0,

k→0+

(11)

for all ψ, φ ∈ Xy¯(t) with φ(x) ≤ ψ(x), x ∈ R. In fact, for any ψ, φ ∈ Xy¯(t) with φ(x) ≤ ψ(x), we have B(t, ψ) − B(t, φ) ( ) a(t) b[(H − ψ )ψ − (H − φ )φ ] 1 2 1 2 = a(t) H H c[(M (t) − ψ2 )ψ1 − (M (t) − φ2 )φ1 ] ( ) a(t) b[(H − ψ )ψ − (H − φ )ψ ] 1 2 1 2 ≥ a(t) H H c[(M (t) − ψ2 )ψ1 − (M (t) − φ2 )ψ1 ] ( ) ( ) a(t) b(φ1 − ψ1 )ψ2 − a(t) bM (t)(ψ1 − φ1 ) H H ≥ a(t) ≥ . − a(t) H c(φ2 − ψ2 )ψ1 H cH(ψ2 − φ2 ) Thus, for any k ≥ 0 satisfying

1 k

> max { a(t) H bM (t), a(t)c}, we have 0≤t≤ω

ψ − φ + k[B(t, ψ) − B(t, φ)] ≥ 0, and hence, (11) holds. By [15, Corollary 5] with v + (t, x)=u(t, x, y¯(0)), v − (t, x) = 0, system (7) has a unique solution u(t, ·, φ) on [0, ∞) for any φ ∈ E0 and u(t, ·, φ) ∈ Xy¯(t) , ∀t ≥ 0. It follows from [15, Theorem 1] that u(t, x, φ) is a classical solution if t > 0. Moreover, the comparison principle holds for the lower and upper solutions.

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To study spreading speeds and traveling waves for system (7), we define a family of maps {Qt }t≥0 from Xy¯(0) to Xy¯(t) by Qt (φ)(x) = u(t, x, φ) = (u1 (t, x, φ), u2 (t, x, φ)), ∀t ≥ 0, x ∈ R, where u(t, x, φ) is the mild solution of system (7) with u(0, ·, φ) = φ. We then have the following observation. Lemma 2.6: Xy¯(0) → Xy¯(t) :

The following two statements hold for the solution map Qt :

(1) Qt [Xy¯(0) ] is precompact in Xy¯(t) for all t > 0. (2) {Qt }t≥0 is an ω-periodic semiflow in the sense that (i) Q0 (v) = v, ∀v ∈ Xy¯(0) . (ii) Qt+ω [v] = Qt [Qω (v)], ∀t ≥ 0, v ∈ Xy¯(0) . (iii) Q(t, v) = Qt (v) is continuous in (t, v) ∈ R+ × Xy¯(0) with respect to the compact open topology. Proof : It is easy to see that (2)(i) and (ii) are satisfied for the solution map associated with the periodic system. To prove the remaining parts, we just need to show that T (t, s) is compact whenever t > s ≥ 0, and then use a same argument as in [14, Theorem 8.5.2] to prove that (1) and (2)(iii) hold. In fact, we can write T1 (t, s) and T2 (t, s) explicitly as follows: −dh (t−s)

∫

2

− (x−y) 4D t

e √

T1 (t, s)φ1 (x) = e

R −

T1 (t, s)φ2 (x) = e

Rt s

∫

h

4πDh t

φ1 (y)dy,

(x−gt−y)2 4Dv t

e− √

dv (τ )dτ R

4πDv t

φ2 (y)dy.

For any YM := {φ ∈ Y : 0 ≤ φ(x) ≤ M, ∀x ∈ R} with M > 0, it is easy to see that T1 YM ⊂ YM . Moreover, for any φ ∈ YM and x1 , x2 ∈ R, we have |T1 (t, s)φ(x1 ) − T1 (t, s)φ(x2 )| ∫ ∫ (x1 −y)2 (x −y)2 e−dh (t−s) − 4D − 2 t h = √ | e φ(y)dy − e 4Dh t φ(y)dy| 4πDh t R R ∫ 2 2 −d (t−s) (x −y) (x −y) e h − 1 − 2 ≤ √ |e 4Dh t − e 4Dh t | · |φ(y)|dy 4πDh t R ∫ (x −y)2 (x −y)2 e−dh (t−s) − 1 − 2 ≤ √ ·M |e 4Dh t − e 4Dh t |dy 4πDh t R ∫ 2 (x −x +y)2 e−dh (t−s) − 1 4D2 t − y h = √ ·M |e − e 4Dh t |dy 4πDh t R = g(x1 − x2 ), where g(ξ) =

−dh (t−s) e√ 4πDh t

·M

∫

2

− (ξ+y) 4D t

R |e

h

2

y − 4D

−e

ht

|dy. Clearly, lim g(ξ) = 0. Therefore, ξ→0

T1 YM is a family of equicontinuous functions. It then follows from Arzel` a-Ascoli theorem and a standard diagonal argument that T1 YM is precompact with respect to the compact open topology. Thus, T1 is compact. Similarly, we can prove T2 is compact.

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Lemma 2.7: Qω is subhomogeneous and monotone from Xy¯(0) to Xy¯(0) . Moreover, for any φ ∈ Xy¯(0) with φ > 0, u(t, x, φ) 0 for all t > 0, x ∈ R. Proof : For any φ ∈ Xy¯(0) , let u(t, x, φ) be the solution of system (7) with u(0, x, φ) = φ(x) for x ∈ R. Since B(t, φ) is strictly subhomogeneous in φ, ∀(t, φ) ∈ E, then, for any k ∈ [0, 1], we have ∫

t

ku(t, x, φ) = kT (t, 0)φ + k

T (t, s)B(s, u(s))ds 0

∫

t

= T (t, 0)(kφ) +

T (t, s)[kB(s, u(s))]ds 0

∫ ≤ T (t, 0)(kφ) +

t

T (t, s)[B(s, ku(s))]ds. 0

Hence, ku(t, ·, φ) is a lower solution of system (7) with initial value kφ. By Theorem 2.5, we then have ku(t, x, φ) ≤ u(t, x, kφ) for t ≥ 0, i.e., Qt (kφ) ≥ kQt (φ). Thus, Qt is subhomogeneous. By Theorem 2.5, {Qt }t≥0 is a periodic monotone semiflow from Xy¯(0) to Xy¯(t) . Since for any t > 0, u(t, x, φ) satisfies ∂ 2 u1 H − u1 ∂u1 (t, x, φ) = a(t)b u2 − dh u1 + Dh , ∂t H ∂x2 ∂u2 (t, x, φ) ∂u2 ∂ 2 u2 u1 = a(t)c (M (t) − u2 ) − dv (t)u2 − g + Dv . ∂t H ∂x ∂x2 It then follows from [23, Theorem 5.5.4] that ui (t, x, φ) > 0 for all t > 0, x ∈ R whenever ui (0, ·, φ) = φi > 0. Since φ > 0, we have φ1 > 0 or φ2 > 0. Without loss of generality, we assume that φ1 > 0. Then u1 (t, x, φ) > 0, ∀x ∈ R, ∀t > 0. By contradiction, suppose u2 (t0 , x0 , φ) = 0 for some t0 > 0 and x0 ∈ R. It then follows that u2 (t, x, φ) = 0, ∀t ∈ [0, t0 ], ∀x ∈ R. Since the second equation of system (7) implies 0 = a(t)c

u1 (t, x, φ) M (t), H

∀t ∈ [0, t0 ], ∀x ∈ R,

we get a contradiction. Therefore, for any φ ∈ Xy¯(0) with φ > 0, u(t, x, φ) 0 for all t > 0, x ∈ R. Lemma 2.8: The Poincaré map Qω satisfies all hypotheses (A1)-(A5) with β = u∗ (0) and Qt satisfies (A1) and (A4) for any t > 0. Proof : If u(t, x) is a solution for system (7), then u(t, x + y), ∀y ∈ R, is also a solution, and hence (A1) holds. (A2) and (A3) come from Lemma 2.6. (A4) follows directly from the comparison principle in Theorem 2.5. ˆ ω = Qω |[0,u∗ (0)] 2 . Then Q ˆ ω : D0 → D0 is the Poincaré map generated Let Q R by (2). Note that (2) has a positive ω-periodic solution u∗ (t) which is globally ˆ ω has only two fixed points 0 asymptotically stable in D0 \ {0}. We see that Q ∗ and u (0) in D0 . Thus, by the Dancer-Hess connecting orbit lemma (see, e.g., [28, Section 2.2.1]), it follows that there exists a strictly monotone full orbit {an }∞ −∞ ⊂ ∗ ˆ D0 connecting 0 to u (0) and ai < ai+1 for all i ∈ Z. Since Qω is strongly monotone ˆ ω (ai ) Q ˆ ω (ai+1 ) = ai+2 for any i ∈ Z. from the proof of Lemma 2.3, then ai+1 = Q Therefore ai ai+1 for any i ∈ Z. This implies that (A5) holds for Qω .

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According to [8, Theorem 2.1] and [25, Theorem 6.1], the map Qω admits a rightward spreading speed c∗+ and a leftward spreading speed c∗− . Let r+ (µ) and r− (µ), respectively, be the spectral radiuses of the Poincaré maps associated with the following two linear ordinary differential systems: {

d¯ u1 (t) dt d¯ u2 (t) dt

= a(t)b¯ u2 (t) − dh u ¯1 (t) + Dh µ2 u ¯1 (t), u ¯1 (t) = a(t)cM (t) H − dv (t)¯ u2 (t) + Dv µ2 u ¯2 (t) + gµ¯ u2 (t),

(12)


= a(t)b¯ u2 (t) − dh u ¯1 (t) + Dh µ2 u ¯1 (t), a(t) = H cM (t)¯ u1 (t) − dv (t)¯ u2 (t) + Dv µ2 u ¯2 (t) − gµ¯ u2 (t),

(13)

and {

where µ ≥ 0 is a parameter. It then follows from the Krein-Rutman Theorem (see, e.g., [5, Theorem 7.2]) that r± (µ) > 0. We further have the following computation formulas for c∗± . Proposition 2.9:

c∗± = inf

µ>0

ln r± (µ) . µ

Proof : Let (¯ u1 (t, u ¯0 ), u ¯2 (t, u ¯0 )) be the solution of system (12) satisfying (¯ u1 (0, u ¯0 ), u0 ∈ R2 . It is easy to see that u ¯2 (0, u ¯0 )) =¯ (u1 (t, x), u2 (t, x)) := e−µx (¯ u1 (t, u ¯0 ), u ¯2 (t, u ¯0 )) is a solution of the following linear parabolic system: {

∂u1 (t,x) ∂t ∂u2 (t,x) ∂t

2

1 (t,x) = a(t)bu2 (t, x) − dh u1 (t, x) + Dh ∂ u∂x , 2 a(t) ∂ 2 u2 (t,x) ∂ = H cM (t)u1 (t, x) − dv (t)u1 (t, x) + Dv ∂x2 − g ∂x u2 (t, x).

(14)

Let {Mt }t≥0 be the solution map associated with the system (14). Then we have Bµt (φ) := Mt (φe−µx )(0) = (¯ u1 (t, φ), u ¯2 (t, φ)), ∀φ ∈ R2 , t ≥ 0. Therefore, Bµt is also the solution map of the linear differential equations (12) on R2 . By [26, Lemma 2.1], there exists a positive ω-periodic function w(t) such that v(t) = eλ+ (µ)t w(t) is a solution of (12), where λ+ (µ) = ω1 ln r+ (µ). Thus Bµt (w(0)) = eλ+ (µ)t w(t), and by the ω-periodicity of w(t), it follows that Bµω (w(0)) = eλ+ (µ)ω w(0). This implies that eλ+ (µ)ω is the principal eigenvalue of Bµω with the positive eigenfunction w(0). Define the function Φ+ (µ) :=

λ+ (µ)ω ln r+ (µ) 1 ln(eλ+ (µ)ω ) = = , ∀µ > 0. µ µ µ

(15)

When µ = 0, system (12) reduces to system (4). Since R0 > 1, we have r+ (0) > 1. Hence, condition (C7) in [9] is satisfied. Now we prove that Φ+ (∞) = ∞. Since v(t) = eλ+ (µ)t w(t) is a solution of (12), we have v10 (t) ≥ (Dh µ2 − dh )v1 (t). It then follows that w10 (t) ≥ (Dh µ2 − dh − λ+ (µ)). w1 (t)

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11

Integrating the above inequality from 0 to ω, we obtain ∫

ω

0= 0

w10 (t) dt ≥ (Dh µ2 − dh − λ+ (µ))ω, w1 (t)

and hence Φ+ (µ) = λ+ (µ)ω ≥ Dh µµ−dh , which implies that Φ+ (∞) = ∞. Thus µ Φ+ (µ) attains its minimum at some finite value µ∗ . Since the solution of system (7) u(t, x, φ) is a lower solution of the linear system (14), we have Qt [φ] ≤ Mt [φ], ∀φ ∈ Xu∗ (0) , t ≥ 0. It then follows from [9, Theorem 3.10 (i)] that c∗+ ≤ inf Φ+ (µ). 2

µ>0

Note that the reflection invariance property is assumed in [9] for Mt and Qt in (A1), but this property is not needed in the proof of [9, Theorem 3.10]. ε (µ) be the spectral radius of the Poincar´ For any µ > 0, let r+ e map associated with the following differential system: {


= (1 − ε)a(t)b¯ u2 (t) − dh u ¯1 (t) + Dh µ2 u ¯1 (t), a(t) = (1 − ε) H cM (t)¯ u1 (t) − dv (t)¯ u2 (t) + Dv µ2 u ¯2 (t) + gµ¯ u2 (t).

(16)

Let {Mtε }t≥0 be the solution map associated with {

∂u1 (t,x) ∂t ∂u2 (t,x) ∂t

2

= (1 − ε)a(t)bu2 − dh u1 + Dh ∂∂xu21 , ∂ 2 u2 ∂ = (1 − ε) a(t) H cM (t)u1 − dv (t)u1 + Dv ∂x2 − g ∂x u2 .

(17)

By the continuous dependence of solutions on initial conditions, it follows that for any ε ∈ (0, 1), there is a sufficiently small η ∈ R2 and η 0 such that the solution w(t, η) of the periodic system (3) with w(0, η) = η satisfies w(t, η) ≤ ε · (H, min0≤t≤ω {M (t)})T , ∀t ∈ [0, ω]. Thus, the comparison principle (Theorem 2.5) implies that u(t, x, φ) ≤ w(t, η) ≤ ε · (H, min {M (t)})T , ∀x ∈ R, φ ∈ Xη , t ∈ [0, ω]. 0≤t≤ω

(18)

Since Qt (φ) is an upper solution of linear system (17) for t ∈ [0, ω], φ ∈ Xη , we have Mtε (φ) ≤ Qt (φ), ∀φ ∈ Xη , t ∈ [0, ω].

(19)

In particular, Mωε (φ) ≤ Qω (φ), ∀φ ∈ Xη . Define the function Φε+ (µ) :=

ε (µ) ln r+ , µ

∀µ > 0.

(20)

By a similar analysis and [9, Theorem 3.10 (ii)], we have inf Φε+ (µ) ≤ c∗+ ≤ µ>0

inf Φ+ (µ), ∀ε ∈ (0, 1). Letting ε → 0, we obtain c∗+ = inf Φ+ (µ).

µ>0

µ>0

Let w ˆ1 (t, x) = h(t, −x) and w ˆ2 (t, x) = v(t, −x), we get {

∂w ˆ1 (t,x) ∂t ∂w ˆ2 (t,x) ∂t

2

w ˆ1 = a(t)b H− ˆ2 − dh w ˆ1 + Dh ∂∂xwˆ21 , H w 2 w ˆ1 ∂ = a(t)c H (M (t) − w ˆ2 ) − dv (t)w ˆ2 + Dv ∂∂xwˆ22 + g ∂x w ˆ2 .

(21)

If we denote c∗− as the leftward spreading speed of system (7), then c∗− is the rightward spreading speed of system (21). As argued for (7), we then have c∗− =

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inf

µ>0

ln r− (µ) . µ

Lemma 2.10: c∗+ + c∗− > 0. Proof : If we define r(µ), µ ∈ R, as the spectral radius of the Poincaré map of (12), then r+ (µ) = r(µ), ∀µ ≥ 0, and r− (µ) = r(−µ), ∀µ ≥ 0. It follows from [9, Lemma 3.7] that r(µ) is log convex on R. Thus, for all µ1 , µ2 ∈ R and θ ∈ (0, 1), we have θ ln r(µ1 ) + (1 − θ) ln r(µ2 ) ≥ ln r(θµ + (1 − θ)µ2 ). Let µ1 and µ2 be two positive numbers such that inf

µ>0

inf

µ>0

ln r− (µ) µ

=

ln r− (µ2 ) . µ2

Choosing θ =

µ2 µ1 +µ2 ,

ln r+ (µ) µ

=

ln r+ (µ1 ) µ1

and

we then obtain

ln r+ (µ) ln r− (µ) + inf µ>0 µ>0 µ µ

c∗+ + c∗− = inf =

ln r+ (µ1 ) ln r− (µ2 ) + µ1 µ2

ln r(µ1 ) ln r(−µ2 ) + µ1 µ2 µ1 + µ2 = [θ ln r(µ1 ) + (1 − θ) ln r(−µ2 )] µ1 µ2 µ1 + µ2 µ1 + µ2 ln r(θµ1 − (1 − θ)µ2 ) = ln r(0). ≥ µ1 µ2 µ1 µ2 =

Note that R0 > 1 implies r(0) > 1. Thus, we have c∗+ + c∗− > 0. c

∗ +

The following result shows that ω and ing speeds for system (7), respectively.

∗ −

c ω

are the rightward and leftward spread-

Theorem 2.11 : Let c∗± be defined as in Proposition 2.9. Then the following two statements are valid for system (7): (i) If φ ∈ Xu∗ (0) and φ(x) = 0 outside a bounded interval, then u(x, t, φ) = 0 for all c >

∗ +

c ω

, and

(ii) For any c and c0 satisfying

−c∗− ω

lim

u(x, t, φ) = 0 for all c

t→∞,x≤−ct < −c0 < c

0.

lim

t→∞,x≥ct c∗ > ω− .

lim

t→∞,−c0 t≤x≤ct

Proof : Statement (i) is a consequence of Lemma 2.8, [25, Theorem 6.1] and the proof of [8, Theorem 2.1 (i)]. For the last statement, since Qt is subhomogeneous, by [8, Theorem 2.1], rσ can be chosen to be independent of σ 0. Thus, we can write rσ as r¯. For every φ ∈ Xu∗ (0) with φ > 0, it then follows from Lemma 2.7 that Qt (φ)(x) 0, ∀x ∈ R, t > 0. Fix a t0 = ω > 0, then there is a vector σ 0 such that u(ω, x, φ) ≥ σ for x on an interval of length 2¯ r. Taking Qω (φ) as a new initial date, it then follows from Lemma 2.8, [25, Theorem 6.2] and the proof of [8, Theorem 2.1 (ii)] that statement (ii) is also valid for all φ ∈ Xu∗ (0) with φ > 0. We say that W (t, x − ct) is a rightward periodic traveling wave of the ωperiodic semiflow {Qt }t≥0 if the vector-valued function W (t, z) is ω-periodic in t and Qt [W (0, ·)](x) = W (t, x − ct), and that W (t, x − ct) connects β(t) to 0 if

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13

W (t, −∞) = β(t) and W (t, ∞) = 0. A leftward periodic traveling wave V (t, x + ct) can be defined for the ω-periodic semiflow {Qt }t≥0 in a similar way. The existence and non-existence of periodic traveling wave solutions are consequences of Lemma 2.8 and [8, Theorem 2.2 and 2.3]. Theorem 2.12 : Let c∗± be defined as in Proposition 2.9. Then the following statements are valid: (i) For every c ≥ c∗+ /ω, system (7) has a traveling wave solution U (t, x − ct) connecting u∗ (t) to 0 such that U (t, s) is continuous and nonincreasing in s ∈ R, and for any c < c∗+ /ω, system (7) admits no traveling wave solution U (t, x − ct) connecting u∗ (t) to 0. (ii) For every c ≥ c∗− /ω, system (7) has a traveling wave solution V (t, x + ct) connecting 0 to u∗ (t) such that V (t, s) is continuous and nondecreasing in s ∈ R, and for any c < c∗− /ω, system (7) admits no traveling wave solution V (t, x + ct) connecting 0 to u∗ (t).

3.

Threshold dynamics in a bounded domain

In this section, we consider system (1) on a bounded spatial domain  ∂u1 (t,x) H−u1 ∂ 2 u1   ∂t = a(t)b H u2 − dh u1 + Dh ∂x2 ,  2 ∂u2 (t,x) ∂ = a(t)c uH1 (M (t) − u2 ) − dv (t)u2 + Dv ∂∂xu22 − g ∂x u2 , ∂t  B u = 0 on (0, ∞) × ∂Ω, i = 1, 2, i i   ui (0, x) = φi (x), i = 1, 2,

(22)

where Ω ⊂ RN (N ≥ 1) is a bounded domain with boundary ∂Ω of class C 1+θ i (0 < θ ≤ 1), either Bi ui = ui or Bi ui = ∂u ∂n + αi (x)u for some nonnegative function ∂ αi ∈ C 1+θ (∂Ω, R), ∂n denotes the differentiation in the direction of outward normal n to ∂Ω. N Let N < p < ∞ be fixed and W = Lp (Ω). For each β ∈ ( 12 + 2p , 1), let Xβi be the fractional power space of Lp (Ω) with respect to (−4, Bi ) (see, e.g., [4]). Then Wβ := Xβ1 × Xβ2 is an ordered Banach space with the positive cone Wβ+ consisting of all nonnegative functions in Wβ , and Wβ+ has nonempty interior ¯ with continuous inclusion for m ∈ [0, 2β − int(Wβ+ ). Moreover, Wβ ⊂ C 1+m (Ω) N 1 − p ) (see, e.g., [5]). Let k · kβ be the norm on Wβ . It then follows that there exists a constant Kβ > 0 such that kφk∞ := max |(φ1 (x), φ2 (x))| ≤ Kβ kφkβ , for ¯ x∈Ω

all φ ∈ Wβ . Denote Wy¯(t) := {φ ∈ Wβ : 0 ≤ φ(x) ≤ y¯(t)}, by a similar argument as in the previous section, we can write system (22) as an integral equation with u(0, ·) = φ ∈ Wy¯(0) . It then follows from [15, Corollary 5] that the system (22) has a unique solution u(t, φ) ∈ Wy¯(t) on [0, ∞) with u(0, φ) = φ ∈ Wy¯(0) , and the comparison principle holds for system (22). Define a family of maps {Qt }t≥0 from Wy¯(0) to Wy¯(t) by Qt (φ)(x) = u(t, x, φ), ¯ t ≥ 0. Then {Qt }t≥0 is a monotone ω-periodic semiflow from ∀φ ∈ Wy¯(0) , x ∈ Ω, Wy¯(0) to Wy¯(t) . Moreover, we can show that {Qt }t≥0 is strongly monotone for t ≥ ω by similar arguments as in the proof of [21, Theorem 7.4.1 and Corollary 7.4.2]. Since Qω : Wy¯(0) → Wy¯(0) is compact (see, e.g., [5]), it then follows from [28, Theorem 1.1.3] that the following lemma holds. Lemma 3.1: The Poincaré map Qω admits a global attractor on Wy¯(0) .

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Consider the following linearized system of system (22)  ∂ u˜1 (t,x) 2 u2 − dh u ˜1 + Dh ∂∂xu˜21 ,  ∂t = a(t)b˜ 2 ∂u ˜2 (t,x) a(t) ∂ u1 − dv (t)˜ u2 + Dv ∂∂xu˜22 − g ∂x u ˜2 ,  ∂t = H cM (t)˜ Bi ui = 0 on (0, ∞) × ∂Ω, i = 1, 2.

(23)

Similarly, we can show that the solution u ˜(t, x, φ) exists for all φ ∈ Wβ and the comparison principle holds for (23). Define the Poincaré map of system (23) P1 : Wβ → Wβ by P1 (φ) = u ˜(ω, ·, φ) for all φ ∈ Wβ . Then P1 is compact. Moreover, P1 is strongly positive by the standard parabolic maximum principle (see, e.g., [21, Theorem 7.4.1]). Let r1 = r(P1 ) be the spectral radius of P1 . By the KreinRutman theorem (see, e.g., [5, Theorem 7.2]), it follows that r1 > 0 and P1 has an ˜ ˜ = r1 φ. eigenfunction φ˜ ∈ int(Wβ+ ) corresponding to r1 , that is, P1 (φ) Lemma 3.2: Let λ = − ω1 ln r1 . Then there exists a positive ω-periodic function v˜(t, x) such that e−λt v˜(t, x) is a solution of system (23). ˜ be the solution of system (22) through φ. ˜ Denote v˜(t, x) = Proof : Let u ˜(t, x, φ) λt ˜ ˜ = e u ˜(t, x, φ), then v˜(t, x) 0 for all (t, x) ∈ (0, ∞) × Ω. Substituting u ˜(t, x, φ) −λt e v˜(t, x) into (23), we obtain the following linear periodic system with parameter λ: { 2 ∂˜ v1 (t,x) 1 (t,x) = λ˜ v1 (t, x) + a(t)b˜ v2 (t, x) − dh v˜1 (t, x) + Dh ∂ v˜∂x , 2 ∂t ∂˜ v2 (t,x) a(t) ∂ 2 v˜2 (t,x) ∂ = H cM (t)˜ v1 (t, x) + (λ − dv (t))˜ v2 (t, x) + Dv ∂x2 − g ∂x v˜2 (t, x), ∂t (24) for all (t, x) ∈ (0, ∞) × Ω. Thus, v˜(t, x) is a solution of the ω-periodic system (24) ˜ ˜ = r1 φ˜ and with Bi v˜i = 0 on ∂Ω and v˜(0, x) = φ(x) for all x ∈ Ω. Since P1 (φ) λω λω λω λω ˜ = e P1 (φ)(x) ˜ ˜ ˜ e r1 = 1, we have v˜(ω, x) = e u ˜(ω, x, φ) = e r1 φ(x) = φ(x) = v˜(0, x). Therefore, the existence and uniqueness of solutions of (24) imply that v˜(t, x) = v˜(t + ω, x), ∀t ≥ 0, x ∈ Ω, and hence, v˜(t, x) is an ω-periodic solution of (24) and e−λt v˜(t, x) is a solution of (23). Theorem 3.3 : For any φ ∈ Wy¯(0) , let u(t, x, φ) be the solution of system (22) with u(0, x, φ) = φ(x) for all x ∈ Ω. Then the following two statements are valid: (i) If r1 < 1, then lim ku(t, ·, φ)kβ = 0 for all φ ∈ Wy¯(0) . t→∞

(ii) If r1 > 1, then system (22) admits a unique positive ω-periodic solution u∗ (t, x) and lim ku(t, ·, φ) − u∗ (t, ·)kβ = 0 for all φ ∈ Wy¯(0) \ {0}. t→∞

Proof : In the case where r1 < 1, we have λ = − ω1 ln r1 > 0. Then the following inequalities hold: {

∂u1 (t,x) ∂t ∂u2 (t,x) ∂t

2

1 (t,x) ≤ a(t)bu2 (t, x) − dh u1 (t, x) + Dh ∂ u∂x , 2 ∂ 2 u2 (t,x) ∂ ≤ a(t) cM (t)u (t, x) − d (t)u (t, x) + D − g ∂x u2 (t, x). 1 v 2 v H ∂x2

(25)

Let u ˜(t, x, φ) be the solution of (23) through φ. Then the comparison theorem implies that u(t, x, φ) ≤ u ˜(t, x, φ), ∀t ≥ 0, x ∈ Ω. Since for any φ ∈ Wy¯(0) , we can ˜ Hence, 0 ≤ u(t, x, φ) ≤ u ˜ = ζe−λt v˜(t, x). choose ζ > 0 such that φ ≤ ζ φ. ˜(t, x, ζ φ) ˜ ∞ = 0, we have lim ku(t, ·, φ)k∞ = 0. Next, we show that u(t, ·, ζ φ)k Since lim k˜ t→∞

t→∞

lim ku(t, ·, φ)kβ = 0. Let ω(φ) be the omega-limit set of the orbit {Qnω (φ)}n≥1

t→∞

with respect to the k · kβ norm. It suffices to show that ω(φ) = {0}. For any given ψ ∈ ω(φ), there exists a sequence {ni } such that lim kQni ω (φ) − ψkβ = 0, and ni →∞

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15

hence, lim ku(ni ω, ·, φ)−ψk∞ = 0. Thus, lim ku(ni ω, ·, φ)k∞ = 0 implies ψ = 0. ni →∞

ni →∞

In the case where r1 > 1, we have λ < 0. Let W0 = {φ ∈ Wy¯(0) : φ 6= 0}, ∂W0 = Wy¯(0) \ W0 = {0}. Note that for all φ ∈ W0 , the solution u(t, x, φ) 0 for all t > 0, x ∈ Ω. It follows that Qnω (W0 ) ⊂ W0 for all n > 0. Clearly, Qt (0) = 0 for all t ≥ 0. We now prove the following claim. Claim. Zero is a uniform weak repeller for W0 in the sense that there exists δ0 > 0 such that lim sup kQnω (φ)kβ ≥ δ0 for all φ ∈ W0 . n→∞ We consider the following linear system:  ∂uε (t,x) 2 ε H− ε 1 1 (t,x) dh uε1 (t, x) + Dh ∂ u∂x ,  2  ε∂t = a(t)b H u2 (t, x) −  ε 2 ε ∂u2 (t,x) u1 (t,x) ∂ ε ε 2 (t,x) − g ∂x = a(t)c(M (t) − ε) H − dv (t)u2 (t, x) + Dv ∂ u∂x u2 (t, x), 2 ∂t ε = 0 on (0, ∞) × ∂Ω,  B u i = 1, 2, i   ε i ui (0, x) = φi (x), x ∈ Ω, i = 1, 2. (26) Let uε (t, x, φ) be the solution of system (26) with uε (0, x, φ) = φ(x). Define the Poincaré map of system (26), Pε : W → W, by Pε (φ) = uε (ω, ·, φ). Let rε = r(Pε ) be the spectral radius of Pε . Since r1 = r(P1 ) > 1, there exists a sufficiently small positive number ε1 such that rε > 1 for all ε ∈ [0, ε1 ). Fix an ε ∈ [0, ε1 ). Then there exists some δ > 0 such that kuε (t, ·, φ)k∞ < ε for all t ∈ [0, ω] whenever kφk∞ < δ. Let δ0 = Kδβ . Suppose, by contradiction, that lim sup kQn (φ0 )kβ < δ0 for some n→∞

φ0 ∈ W0 . Then there exist n0 > 0 such that kQnω (φ0 )k∞ ≤ Kβ kQnω (φ0 )kβ < δ for all n ≥ n0 . For any t ≥ n0 ω, we can rewrite t = nω + t0 with n ≥ n0 ω and t ∈ [0, ω]. Therefore, we have kQt (φ0 )k∞ = kQt0 (Qnω (φ0 ))k∞ < ε, ∀t ≥ n0 ω, and u(t, x, φ0 ) satisfies the following system {

2

∂u1 (t,x) ∂t ∂u2 (t,x) ∂t

∂ u1 (t,x) ≥ a(t)b H− , H u2 (t, x) − dh u1 (t, x) + Dh ∂x2 2 u1 (t,x) 2 (t,x) ∂ ≥ a(t)c(M (t) − ε) H − dv (t)u2 (t, x) + Dv ∂ u∂x − g ∂x u2 (t, x), 2 (27) for all t ≥ n0 ω, x ∈ Ω. Let φ˜ε be the positive eigenfunction of Pε associated with rε and λε = − ω1 ln rε < 0. Then by Lemma 3.2, there is a solution uε (t, x,ε ) = e−λε t vε (t, x), with vε (t, x) a periodic positive function. Since u(t, x, φ0 ) 0 for all t > 0, x ∈ Ω, there exists η > 0 such that u(n0 ω, x, φ0 ) ≥ ηε . By (27) and the comparison principle, we have

¯ u(t, x, φ0 ) ≥ ηuε (t − n0 ω, x,ε ) = ηe−λε (t−n0 ω) vε (t, x), ∀t ≥ n0 ω, x ∈ Ω. Since λε < 0, it then follows that u(t, x, φ0 ) is unbounded, a contradiction. This proves the claim. By the claim above, Qω is weakly uniform persistence with respect to (W0 , ∂W0 ). Since Qω admits a global attractor on Wy¯(0) , it follows from [28, Theorem 1.3.3] that Qω is uniformly persistent with respect to (W0 , ∂W0 ) in the sense that there exists δ1 such that lim inf kQnω (φ)kβ ≥ δ1 for all φ ∈ W0 . Note that Qω is n→∞

compact, point dissipative and uniformly persistent. It follows from [28, Theorem 1.3.6] that Qω : W0 → W0 admits a global attractor A0 and has a fixed point φˆ in A0 . Since Qω is strongly monotone semiflow on W0 , we have A0 = Qω (A0 ) 0, and hence φˆ 0. By a similar argument as in the Lemma 2.7, it is easy to see that for each t > 0,

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Qt is strictly subhomogeneous. Then [27, Lemma 1] implies that Qω has at most ˆ one fixed point in W0 . Moreover, it follows from [28, Theorem 2.3.2] that A0 = {φ}. ˆ Thus, φ is globally attractive in W0 for Qω . ˆ be the solution of system (22) with u(0, x, φ) ˆ = φ(x) ˆ Let u(t, x, φ) for all x ∈ Ω. ˆ ˆ Since φ is a fixed point of Qω and is globally attractive in W0 , we see that u(t, x, φ) is an ω-periodic solution of system (22) which attracts all solution of (22) in W0 . ˆ β = 0, ∀φ ∈ W0 . Thus, u∗ (t, x) := u(t, x, φ) ˆ is That is, lim ku(t, ·, φ) − u(t, ·, φ)k t→∞

the desired ω-periodic solution.

4.

Numerical simulations

To numerically illustrate our analytic results, we study the malaria cases in KwaZulu-Natal province in South Africa.

4.1.

Model coefficients and the basic reproduction ratio

1 We choose H = 109 humans/km, dh = 49.1×365 day−1 . As estimated in [10], the mortality rate for adult mosquitoes dv (t) in KwaZulu-Natal Province can be approximated by

4π 2π t) + 0.00082 cos( t) 365 365 8π 10π 6π t) + 0.00118 cos( t) + 0.00018 cos( t) +0.00063 cos( 365 365 365 2π 4π 6π +0.00298 sin( t) + 0.00112 sin( t) − 0.00016 sin( t) 365 365 365 10π 8π t) + 0.00139 sin( t) day−1 . +0.00078 sin( 365 365

dv (t) = 0.1047 + 0.00445 cos(

The biting rate per unit time of mosquitoes in KwaZulu-Natal Province can be fitted by 2π 4π t) − 0.00578 cos( t) 365 365 6π 8π 10π t) − 0.00542 cos( t) − 0.00582 cos( t) +0.00547 cos( 365 365 365 4π 6π 2π −0.04509 sin( t) + 0.00975 sin( t) + 0.00702 sin( t) 365 365 365 10π 8π t) − 0.00664 sin( t) day−1 . −0.00971 sin( 365 365

a(t) = 0.16938 − 0.06043 cos(

Suppose the mosquito density M (t) = 20 × H × a(t) mosquitoes/km, a linear function of the biting rate, such that the average mosquito density is about three times as that of the human density. The proportions of infected bites on humans and mosquitoes that produce an infection are b = 0.011 and c = 0.2. For illustration, we choose Dv = 1.25 × 10−2 km2 /day, Dh = 1 km2 /day and g = 5.0 × 10−2 km/day. Using the method introduced in [24], we can numerically compute the basic reproduction ratio R0 = 7.04. Fig. 1 shows the variation of the basic reproduction ratio R0 as a function of the mosquito density and mosquito biting rate.

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17

1

0.9

0.8

0.7

R0>1 R0=1

0.6

k

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0.5

0.4

R00

ln eλ(µ) µ

= inf

µ>0

λ(µ) µ .

[a]2 bc[M ] ]). H

0 Let µ∗ be the positive root of ( λ(µ) µ ) = 0.

Then the rightward spreading speed for the time-averaged autonomous system is ∗ ) c¯∗+ = λ(µ ¯∗− . Two spreading µ∗ . Similarly, we can obtain the leftward spreading speed c ∗ speeds can be numerically computed as c¯+ = 0.0884 km/day and c¯∗− = 0.0866 km/day.

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The periodic system

4.3.

For the periodic system, using Proposition 2.9, we can numerically get c∗+ = 0.1019 km/day and c∗− = 0.0996 km/day. This implies that the spreading speeds of the time-averaged autonomous system underestimate the real spreading speeds. Fig. 2 shows a plot of the spreading speed c∗+ and c∗− as functions of the advection velocity g. The downstream spreading speed increases with advection velocity, while the upstream spreading speed decreases with advection velocity. Fig. 3 indicates that the rightward spreading speed c∗+ and leftward spreading speed c∗− both increase with human diffusion coefficient Dh . 0.3

0.18

0.16

0.25

Rightward spreading speed 0.14

Rightward spreading speed

*

c (g)

c*(Dh)

0.2 0.12

0.15 0.1

Leftward spreading speed

Leftward spreading speed 0.1

0.05

0.08

0

0.5

1

1.5 2 Advection velocity (g)

2.5

3

0.06 0.5

1

1.5

2

2.5

3

Dh

Figure 2. Leftward and rightward spreading speeds as functions of the advection velocity g.

Figure 3. Leftward and rightward spreading speeds as functions of Dh .

To simulate the spatial spread of malaria, we discretize system (1) by the difference method on a finite spatial interval [−L, L] with the Neumann boundary condition, where L is sufficiently large. Figs. 4 and 5 show numerical plots of the solution through the initial condition  if |x| ≤ 50  80, 3 h(0, x) = 85 (100 − |x|), if 50 ≤ |x| ≤ 100 , v(0, x) = × h(0, x).  2 0, if |x| ≥ 100 The infectious host and vector spread in both directions with a bias towards downstream. 110 100 90 80 Infectious host density

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70 60 50 40 30 20 10 0 −400

−300

−200

−100

0 x

100

200

300

400

Figure 4. The spread of infectious host. The left plot shows the density of infectious host at different times t year, with t = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

To get rightward traveling waves, we choose the initial condition as  220 if x ≤ − 100  3 , 3 11 100 . h(0, x) = v(0, x) = 10 ( 3 − x), if |x| ≤ 100 3  100 0, if x ≥ 3

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40 35 30 Infectious vector density

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25 20 15 10 5 0 −400

−300

−200

−100

0 x

100

200

300

400

Figure 5. The spread of infectious vector. The left plot shows the density of infectious vector at different times t year, with t = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

The evolution of the solution is shown as in Fig. 6.

Figure 6. The rightward periodic traveling waves observed for two components.

To simulate the global dynamics of system (1) on a bounded domain, we choose the following initial condition h(0, x) = v(0, x) = 100 × cos(

πx ), 2L

and the Dirichlet boundary condition h(t, −L) = h(t, L) = v(t, −L) = v(t, L) = 0. The evolution of the solution is shown in Fig. 7 for L = 50. It indicates that in this case the disease persists in the host and vector populations.

Figure 7. The evolution of two components when L = 50.

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Y. Lou and X.-Q. Zhao

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5.


Discussion

In this project, we analyzed a periodic Ross-Macdonald type model with diffusion and advection to account for the movement of host and vector population and the seasonal fluctuation of mosquito dynamics. For the spatially homogeneous system, we determined the basic reproduction ratio R0 , and proved that R0 is a threshold value for the malaria transmission in a periodic environment. This implies that the disease dies out when R0 ≤ 1; while the disease can invade the population when R0 > 1. In order to study the spatial propagation of malaria, we should consider the spatially inhomogeneous system in the case where R0 > 1. In the case of the unbounded domain, the risk of invasion can be described in terms of spreading speeds and periodic traveling waves. In this case, we obtained the existence of the leftward and rightward spreading speeds, and showed that they coincide with the minimal wave speeds for monotone traveling waves in the left and right directions, respectively. To control the disease, we should use physical or chemical strategies to reduce the values of the rightward and leftward spreading speeds, c∗± , as close to zero as possible. As shown in Proposition 2.9, the spreading speeds depend on parameters in the model, which permits the assessment of control strategies. Numerically, we studied how advection increases the rightward spreading speed, whereas decreases the leftward spreading speed. Since human movements are the source of long-distance transmission of malaria, numerical simulations were also performed to investigate the effect of human movements on disease propagation. Both the rightward and leftward spreading speeds increase with diffusion coefficient for humans, which means that global traffic networks may deteriorate malaria situation. In the case of the bounded domain, we established a threshold result on the global attractivity of either zero or a positive periodic solution. Biologically, this result shows that malaria disease stabilizes at a unique positive periodic solution when the zero solution is linearly unstable; while it dies out when the zero solution is linearly stable. Several authors have used reaction-diffusion systems to study the spatial dynamics of vector-borne diseases (see, e.g., [3, 11]). The authors of [11] proposed a reaction-diffusion system to describe the spatial spread of West Nile virus, and simplified their original model as the following one under some reasonable assumptions: ∂IV IR ∂ 2 IV = αV βR (AV − IV ) − dV IV + ε , ∂t NR ∂x2 NR − IR ∂ 2 IR ∂IR = αR βR IV − γR IR + D , ∂t NR ∂x2

(29)

where AV , NR are constant. Further, they proved the existence of traveling waves and calculated the spreading speed for system (29), and also showed that the spreading speed for system (29) is an upper bound for that of the original model, provided that the spreading speed for the latter exists. We should mention that the techniques in our current paper can be employed to study spreading speeds and traveling waves for the time-periodic version of system (29) and other cooperative type vector-borne diseases models with temporal and spatial heterogeneities. Finally, we remark that there are quite a few spaces to improve and generalize our periodic reaction-diffusion-advection malaria model. For example, we can consider the malaria model either in a spatially periodic habitat, or with various “infected” categories of humans and mosquito host to take account of the different developmental stages of the parasite. We leave all these as future investigations.

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LouZhaoApplAnalFinal REFERENCES

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