ASYMPTOTIC SPREADING FOR A TIME-PERIODIC PREDATOR-PREY SYSTEM

. This paper is concerned with asymptotic spreading for a time-periodic predator-prey system where both species synchronously invade a new habitat. Under two diﬀerent conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of as- ymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspeciﬁc nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative eﬀect on the spreading of the prey while the prey may play a positive role on the spreading of the predator.


1.
Introduction. To formulate the invasion process when the initial habitat size of the invader is finite in population dynamics [25,30], the corresponding asymptotic spreading is an important topic. In particular, asymptotic speed of spreading (for short, spreading speed) is a useful index, which was first defined by Aronson and Weinberger [1] and has been widely studied for monotone semiflows, see Aronson and Weinberger [2], Fang and Zhao [13], Liang et al. [18], Liang and Zhao [19], Lui [24], Weinberger [39,40], Weinberger et al. [41] and a survey paper by Zhao [46]. For convenience, we first give the following definition. Definition 1.1. Assume that u(x, t) is a nonnegative function for x ∈ R, t > 0. Then c * is called the spreading speed of u(x, t) if (a) lim t→∞ sup |x|>(c * + )t u(x, t) = 0 for any given > 0, u(x, t) > 0 for any given ∈ (0, c * ).
Clearly, the definition of spreading speed states that if an observer were to move to the right or left at a fixed speed greater (less) than c * , then the local population density u(x, t) would eventually look like 0 (greater than 0) [41], and it also describes the speed at which the geographic range of the new population expands in population dynamics. In the case of monotone semiflows and cooperative systems, the works cited above include many important results with applications. When a scalar equation is not monotone due to time delay or discrete, we may refer to [12,16,17,23,42,44] and references cited therein. In spatial ecology, the spatial propagation dynamics of predator-prey system has attracted considerable attention [6,9,10,25,27]. But when the asymptotic spreading in predator-prey systems is involved, the above results do not work, and there are a few results describing the interspecific actions between the prey and the predator. In particular, Lin [20] and Pan [28] studied the asymptotic spreading of the following predator-prey system in which x ∈ R, t > 0 and all the parameters are positive. They estimated the invasion speeds of u, v when both u, v are invaders. Ducrot [8] investigated the spatial propagation for an SIR model which has the same monotonicity as the predator-prey system. Recently, Pan [29] also estimated the asymptotic spreading of a predatorprey system having negative intrinsic growth rate of the predator, which obtained a spreading speed of the predator that equals to the minimal wave speed in Lin [21]. Moreover, Wang [35,36] and Wang et al. [37] studied the spreading phenomena of (1) with free boundaries and obtained a spreading-vanishing dichotomy.
In this paper, we consider the following time-periodic predator-prey system [22,38] where u(x, t), v(x, t) denote the densities for the prey and the predator at time t and in location x, respectively. Furthermore, d 1 , d 2 > 0 are the diffusive rates for the prey and the predator, respectively. The functions r i (t), a i (t) and b i (t), i = 1, 2 satisfy the following assumptions: (A) r i (t), a i (t) and b i (t) ∈ C θ (R, R), i = 1, 2 are T -periodic functions for t ∈ R, some θ ∈ (0, 1) and some T > 0. In addition, The purpose of this paper is to investigate the asymptotic spreading that both species synchronously invade a new habitat. So, we shall estimate the spreading speeds of u, v formulated by the corresponding initial value problem of (2), assuming that the initial values have nonempty compact supports. On the one hand, (2) has a monotone condition similar to that of (1), so we can apply the idea in [20,28] to study (2). On the other hand, the parameters in (2) depend on t, so there are some significant differences between the study of (1) and (2). For example, we can not obtain the convergence result by dominated converge theorem because of time periodicity. Therefore, we need some techniques different from [20,28] to show the long time behavior of (2).
In this paper, we obtain the bounds of spreading speed of the predator and the prey in two cases by constructing suitable upper and lower solutions and auxiliary equations, applying the generalized eigenvalue and further combining the theory of asymptotic spreading of scalar equations with comparison principle. To study the long time behavior of solutions, we prove that the solution of (2) converges to the positive T -periodic solution by the idea in [4,5]. These results show that either the predator or the prey has a spreading speed that is determined by the linearized while the spreading speed of the other species also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator, which is similar to the phenomena in Fagan and Bishop [11], Owen and Lewis [27].
2. Preliminaries. We first introduce some notations. In this paper, we use the standard partial ordering in R 2 . Let , v(x)) : R → R 2 is bounded and uniformly continuous .
For any T -periodic continuous function f (t), we denote We now give some properties of the corresponding kinetic systems. Let U (t) be a positive periodic solution of By (A), U (t) is globally asymptotically stable with respect to R + [7]. Similarly, we can write V (t) as the unique positive periodic solution of Moreover, admits a trivial solution (0, 0) and two nonnegative semi-trivial periodic solutions (U (t), 0) and (0, then system (3) is uniformly persistent and has a positive T -periodic solution (u * (t), v * (t)) (see [32,33]). We say that (u provided that the initial value of (3) satisfies u(0) > 0, v(0) > 0. To our knowledge, some sufficient conditions on the asymptotic stability of (u * (t), v * (t)) have been established. For example, Teng and Chen [34, Theorem 1] implies that (u * (t), v * (t)) is asymptotically stable if (4) holds and there exist positive constants l 1 , l 2 such that There are also some other sufficient conditions on the asymptotic stability and uniqueness of positive periodic solution of (3) and we shall not focus on precise conditions in this paper.
In order to obtain the spreading speeds of predator-prey system described by (2), we consider the following initial value problem in which u(x), v(x) admit nonempty compact supports and K = (U (0), V (0)).
Obviously, (5) can be analysed by the classical theory of reaction-diffusion systems [14,43]. Following Fife and Tang [14, Definition 4 and Remark 2], Zhao and Ruan [45], we introduce an admissible pair of irregular upper and lower solutions of (5) as follows, which will be called upper and lower solutions for the sake of convenience.
(v) the initial value functions of (5) satisfy are said to be a pair of upper and lower solutions of (5), respectively.
Assume that (u, v), (u, v) are a pair of upper and lower solutions of Lemma 2.2 is clear by the classical theory of reaction-diffusion systems (see Smoller [31] and Ye et al. [43]) and the proof is omitted here.
(b) Assume that z(x,t) is a solution of (6) and z 0 (x) admits a nonempty compact support. Then Consider the following initial value problem where the function f satisfies (1) f is of class C δ,δ/2 in (x, t), locally in z for a given δ ∈ (0, 1), (2) f is locally Lipschitz continuous in z, and of class Applying generalized eigenvalue, Berestycki et al. [3] proved the following result.
Lemma 2.4. Let z(x, t) be the solution of (7) with z(x) > 0. Assume that there exists r(t) : 3. Main results. In this section, we shall study system (5) in two cases. The first case is that v is stronger than u in the sense of d 2 r 2 > d 1 r 1 , and the second one is that u is stronger than v in the sense of d 1 r 1 − b 1 V > d 2 r 2 + a 2 U . Now, we define some constants where U (t), V (t), V (t) are defined in Section 2.
Assume that (4) holds. Let U (t) be the unique positive T -periodic solution of

XINJIAN WANG AND GUO LIN
3.1. Case I: d 2 r 2 > d 1 r 1 . Throughout this subsection, we assume that then Under the assumption, we shall estimate the spreading speeds of the predator and the prey, which is motivated by Lin [20].

Remark 2.
In view of Lemma 2.3 and Theorem 3.1, we can conclude that (1) under the assumption (8), the spreading speed of the predator does not change by introducing the prey, (2) if (8) holds, then the spreading speed of the prey may be slowed by introducing the predator.
This completes the proof. Proof. Define continuous functions where t > 0 is large enough such that u(x) ≤ u(x, 0), ϕ 1 (t) is defined by (10). Similar to the proof of Lemma 3.2, we can easily verify that (u, v), (u, v) are a pair of upper and lower solutions of system (5).
Notice that Then Lemma 2.2 implies the desirable limit. This completes the proof.

Lemma 3.4. For any given
Proof. Clearly, v(x, t) is an upper solution of the initial value problem By the comparison principle and Lemma 2.3, we obtain that This completes the proof. Proof. It suffices to consider the case c 2 − > c 1 + c 3 . Let δ > 0 be small enough such that (c 3 + /4)

Define a positive constant
then the proof of Lemma 3.4 implies that for any given δ ∈ (0, 1), there exists a T 1 > 0 such that (a) the solution w(x, t) of (13) satisfies We define constants where w(x, t) satisfies (13), ϕ 1 (t) is given by (10) and t 2 > 0 is large such that u(x, T 1 ) ≥ u(x, T 1 ) for all x ∈ R, here t 2 is admissible by Lemmas 3. (1) If u(x, t) = U (t), then the proof is the same as that of Lemma 3.2.
Due to the positivity of v(x, t) and the item (a), we obtain that for any t > T 1 , =0.

XINJIAN WANG AND GUO LIN
Lemma 3.6. For any given ∈ (0, c 4 ), we have lim inf Proof. Recalling the upper and lower solutions of (5) in Lemma 3.2, u(x, t) satisfies By the comparison principle and Lemma 2.3, we have This completes the proof.
To prove the convergence of (u(x, t), v(x, t)), we first prove the following result by the idea in [4,5].
Then we obtain This completes the proof.

3.2.
Case II: We shall confirm that the spreading speed of the prey is 2 √ d 1 r 1 and give the lower bounds of the spreading speed of the predator. Here we refer to Pan [28]. Let V (t) be the unique positive T -periodic solution of Then we define two positive constants c 6 := 2 d 2 (r 2 + a 2 U ) m , c 7 := 2 d 2 r 2 + a 2 U .
Remark 3. In view of Lemma 2.3 and Theorem 3.9, we can conclude that (1) from the item (i), the spreading speed of the prey is exactly c 1 = 2 √ d 1 r 1 , which illustrates that the interspecific action does not change the spreading speed of the prey when (21) holds, (2) the item (ii) gives a lower bounds c 6 of the spreading speed for the predator if the intrinsic growth rate r 2 (t) has a small amplitude, which implies that the prey may accelerate the asymptotic spreading of the predator.
Subsequently, we shall prove several lemmas to verify our main results. Proof. Due to the proof of Lemma 3.
By Lemma 2.3 and the comparison principle, we have This completes the proof.
Remark 1 and the comparison principle imply that for any ε 2 ∈ (0, U m ), if |x| ≤ (c 1 − /2)t 1 − r 2 with some positive constant r 2 > 0, then there is T 3 > 0 such that Furthermore, for any given > 0, we can find a constant T 4 > 0 such that Therefore, we have u(x, t) ≥ U (t) − ε 2 for any t > max T 2 + T 3 , T 4 and |x| < Since ε 2 is arbitrary, the desirable result holds. This completes the proof.
By the arbitrariness of ε 3 , we obtain the desirable result. This completes the proof.
The proof of (iii) in Theorem 3.9 is similar to that of Lemma 3.8, so we omit it here. From what we have done, the proof of Theorem 3.9 is complete. Remark 4. We have proved that c 6 is a lower bounds of spreading speed of the predator if r 2 (t) has a small amplitude. Inspired by the result in autonomous sense [28,Theorem 4.1], we conjecture that the lower bounds of spreading speed of the predator is not less than c 7 .