INVASION ENTIRE SOLUTIONS IN A TIME PERIODIC LOTKA-VOLTERRA COMPETITION SYSTEM WITH DIFFUSION

. This paper is concerned with invasion entire solutions of a monostable time periodic Lotka-Volterra competition-diﬀusion system. We ﬁrst give the asymptotic behaviors of time periodic traveling wave solutions at inﬁnity by a dynamical approach coupled with the two-sided Laplace transform. Accord- ing to these asymptotic behaviors, we then obtain some key estimates which are crucial for the construction of an appropriate pair of sub-super solutions. Finally, using the sub-super solutions method and comparison principle, we establish the existence of invasion entire solutions which behave as two periodic traveling fronts with diﬀerent speeds propagating from both sides of x -axis. In other words, we formulate a new invasion way of the superior species to the inferior one in a time periodic environment.

1. Introduction. In this paper, we consider the following time periodic Lotka-Volterra competition-diffusion system where u = u(t, x) and v = v(t, x) denote the densities of two competing species at time t ∈ R + and x ∈ R, d ∈ (0, 1] denotes the relatively diffusive coefficient of the two species, r i (t), a i (t) and b i (t) are T-periodic continuous functions, a i (·) and b i (·) are positive in [0,T], and r i := 1 T T 0 r i (t)dt > 0, where i = 1, 2. Systems like (1.1) arise in interactive populations which live in a fluctuating environment, for instance, physical environmental conditions such as temperature and humidity and the availability of food, water and other resources usually vary in time with seasonal or daily variations [45]. Time periodic traveling waves of (1.1) are solutions with the form and where c ∈ R is the wave speed, z = x − ct is the co-moving frame coordinate, and (u ± (t), v ± (t)) are periodic solutions of the corresponding kinetic system (1.2) Traveling wave solutions of system (1.1) with autonomous nonlinearities have been extensively studied. In particular, we can refer to Hosono [17] and Kan-on [20] for the monostable case, Conley and Gardner [8], Gardner [11] and Kan [19] for the bistable case, Tang and Fife [36] and Vuuren [37] for the coexistence case. At the same time, during the past decades, there have been many works on the space/time periodic traveling waves of scalar reaction-diffusion equations. For instance, one can see Alikakos et al. [1], Bates and Chen [4] and Shen [33] on time periodic traveling waves of the local, nonlocal and lattice equations, respectively, Berestycki and Hamel [5] and Hamel [14] on space periodic traveling waves, and Nadin [30,31] and Nolen et al. [32] on the space-time periodic traveling waves.
It is well known that traveling wave solutions are special examples of the socalled entire solutions defined for all time and whole space. As we all know, it is of great significance in studying the entire solution since it is essential for a full understanding of the transient dynamics and structures of the global attractor. In addition, entire solutions can be used to describe the dynamics of two solutions that have distinct histories in the configuration, though their asymptotic profiles as t → +∞ coincide. The study of new types of entire solutions can be traced back to the works of Hamel and Nadirashvili [15,16] and Yagisita [41], see also [10,7,12,28,22,39] for equations with and without delays, and [21,35] with nonlocal dispersal. Note that all these works mainly concentrate on entire solutions of scalar space-time homogeneous equations. Recently, some researchers paid attention to the study of entire solutions for space/time periodic equations (see [34,6,23,25]). With regard to some systems, Morita and Tachibana [29] first established the existence of entire solutions for a homogeneous Lotka-Volterra competition-diffusion system while Li et al [24] lately considered the corresponding nonlocal dispersal system. The basic idea in establishing such entire solutions is to use traveling fronts propagating from both sides of the x-axis to construct sub-super solutions, and then obtain the existence of entire solutions by comparison principle. A similar result was established by Guo and Wu [13] for the discrete system. One can also see Zhang et al. [42] for a nonlocal dispersal epidemic system. However, to the best of our knowledge, the issue on constructing new types of entire solutions other than traveling waves for time periodic reaction-diffusion systems is still open, which is the motivation of our present work. More precisely, we deal with the time periodic system (1.1) focusing on the following monostable case a 2 (t) a 1 (t) r 1 ≥ r 2 > 0, (1.3) which implies that (1.2) has only three nonnegative T-periodic solutions (0, 0), (p(t), 0) and (0, q(t)), with (p(t), 0) globally stable and (0, q(t)) unstable in the positive quadrant R 2 + = {(u, v)| u ≥ 0, v ≥ 0}, where p(t) and q(t) are given by For system (1.1), the time periodic traveling wave solution (X(t, z), Y (t, z)) connecting (0, q(t)) and (p(t), 0) actually satisfies In the past few years, there were a few works devoted to the study of this issue. In particular, Zhao and Ruan [43] established the existence, uniqueness and stability of time periodic traveling waves under the monostable assumption (1.3). In 2014, the authors extended the results to a class of more general time-periodic advectionreaction-diffusion systems in [44]. In addition, Bao and Wang [3] obtained the existence and stability of time periodic traveling waves for the bistable case. Very recently, Bao et al. [2] further studied the existence, non-existence and asymptotic stability of bistable time-periodic traveling curved fronts in two-dimensional spatial space.
In our present paper, we shall consider the invasion entire solutions of system (1.1), that is, an entire solution (u(t, x), v(t, x)) satisfying (1.1) as well as the following conditions For future reference, we denote a vector by u = (u 1 , . . . , u n ), where u i stands for the ith component of u. Let I, Γ ⊂ R be two (possibly unbounded) intervals and M ⊂ R n . Denote by C(I × Γ, M ) the space of continuous functions u : such that all partial derivatives of u are uniformly bounded. Throughout the paper, we always assume that , then (1.1) becomes (omitting * for simplicity) It is easy to see that is a periodic traveling wave solution of (1.5) connecting (0, 0) and (1,1), that is ) is a periodic traveling wave solution of (1.5) with speed c, then (P (t, z),Q(t, z)) : Under assumptions (A1)-(A3), for any c ≤ c * , (1.5) admits a time periodic traveling wave solution (P, Q) ∈ C 1,2 b (R×R, R 2 ) with (P z , Q z ) > (0, 0) and (P, Q) ≤ (1, 1) for all (t, z) ∈ R × R, where is the maximal wave speed (see [43]). In particular, the authors in [43] obtained the exact exponential decay rate of solutions of (1.6) as z → −∞ in establishing the uniqueness of the periodic traveling wave solution. Next we will consider the cooperative system (1.5) to obtain invasion entire solutions of system (1.1). In order to employ the basic idea developed in [29,13,27] to establish the existence of such entire solutions, we essentially need some estimates which are concerned with the asymptotic behavior of the periodic traveling wave solution. One of the main difficulties arises in obtaining the exact exponential decay rate of the periodic traveling wave as it tends to its limiting state. In the autonomous case, the asymptotic behavior is usually obtained by investigating the linearized equations at the equilibrium points (see e.g. [38,18]), which can not be applied to system (1.1) since the presence of time dependent nonlinearities. Inspired by [43], we employ the two-sided Laplace transform method to obtain the exact exponential decay rate, which is essentially based on some a priori exponential decay estimates of the periodic traveling wave tails as z → +∞. In particular, unlike the a priori exponential estimates as z → −∞ characterized by the principle eigenvalue of the linear periodic eigenvalue problem associated with the linearized system at the unstable limiting state (see [43,Lemma 3.3]), the exponential estimates as z → +∞ can only be characterized by a small perturbation of the corresponding principle eigenvalue (see 'λ ± c, ' in Lemma 2.2). Fortunately, this small perturbation can be declined small enough such that it imposes no influence on the Laurent development of the resolvent near the isolated principle eigenvalue.
The rest of this paper is organized as follows. In Section 2, we study the exact exponential decay rate of a periodic traveling wave solution of (1.5) as it approaches its stable limiting state. We then establish some key and useful estimates in Section 3. In Section 4, we establish the existence and qualitative properties of entire solutions by a comparing argument.
2. Asymptotic behavior of periodic traveling wave fronts. In this section we shall study the asymptotic behavior of time periodic traveling waves of (1.5). Denote For completeness, we first state the following asymptotic behavior of solutions of system (1.6) as it approaches its unstable limiting state (see [43,Theorem 3.8]). Proposition 1. Assume (A1)-(A3) hold, and (P (t, z), Q(t, z)) ∈ C 1,2 b (R × R, R 2 ) and c solve (1.6). Then where k 1 > 0 is a constant, l = 0 if c < c * and l = 1 if c = c * .
In order to characterize the asymptotic behavior of time periodic traveling waves as z → +∞, we now list a useful lemma of the Harnack inequalities for cooperative parabolic system, which was given in [9] (see also [43,3]).

Let
then we have where For any c ≤ c * = −2 √ κ, denote To be specific and convenient, we give an additional assumption on the periodic coefficients.
, that is, assumption (A4) is compatible with (A3). It should be emphasized here that (A4) is a technique assumption that ensures λ − c > λ c for any c ≤ c * , which is essential in obtaining the exact exponential decay rate of u in our present work. Indeed, (A3) yields that κ 0 ≤ κ < κ 1 , then a direct computation shows that Thus, Remark 2. We also remark here that when d = 1, condition (A4) can be deleted since λ − c > λ c holds certainly under condition (A3). Noting that g(t, 0, 0) = h(t, 0, 0) = g(t, 1, 1) = h(t, 1, 1) = 0, system (2.2) can be written as , Ω = (z − 1, z + 1) with z ∈ R, τ = 0, and θ = T. Since U (·, z) and V (·, z) are periodic of t, and (U, V ) are both positive and bounded, Lemma 2.1 then implies that there is some N > 0 such that We now state an essential lemma for system (2.2) on the exponential decay estimates of the periodic traveling wave tails as z → +∞ by using the method similar to [2].
Remark 3. The definitions of λ ± c, indicate that there exist ε ± = ε ± ( ) > 0 such that for any c ≤ c * and 0 < < min{1, κ0 C + }, there holds λ ± c, = λ − c ± ε ± with ε ± ( ) → 0 + as → 0 + . Actually, we know from the proof of Lemma 2.2 that the exponential decay as z → +∞ can only be estimated by the perturbation λ ± c, rather than λ − c , since there is no such exponential type sub-super solutions that equipped with λ − c as the exponential decay rate.
be a solution of (2.2). Then there exist C 1 > 0 and C 2 > 0 such that where λ + c, are defined as in Lemma 2.2.
Proof. The proof is similar to [43,Proposition 3.4], using the interior parabolic estimates and Lemma 2.1, so we omit the details here.
We next establish the exact exponential decay rate of the solution of (2.2) as z → +∞. Specifically, regarding variable z as the evolution variable, we employ the Laplace transform method and spectral theory for this purpose. In the following of the current section, we denote (u(t, z), v(t, z)) := (U (t, z), V (t, z)) for convenience of writing.
It is easy to see that A is closed and densely defined in D(A) = H 1 T × L 2 T . Now let w = v z , then the v-equation of (2.2) can be written as a first order system .
Step I. We prove that there exists k 2 > 0 such that

Now let
Then ξ(t, z) satisfies R(t, z) − κ 1 ξ + ξ zz + cξ z − ξ t = 0 for any z ≥ 0, where We know from Step I that sup Thus there exist positive constants M and K M such that In view of Lemma 2.2, we have sup Since ω ± (t, z) is T − periodic in t, it is sufficient to show that ω + (t, z) ≥ 0 for (t, z) ∈ (0, 2T ) × [M, +∞), while the similar argument holds for ω − (t, z) ≤ 0. Assume to the contrary that inf ω + (t, z) < 0, and hence ω + The argument for u z (t, z) is similar and we only give a brief sketch here. Let for (t, z) ∈ R × R + , then The same argument as above implies that sup t∈R ξ(t, z) = o(e λ − c z ) as z → +∞, and then lim z→+∞ u z (t, z) Now we complete all the proof.
The following is a direct result of Theorem 2.4.
) be a solution of (1.6). Then for some constant k 2 > 0.
3. Key estimates. In this section, we give some crucial estimates which are helpful for the construction of sub-super solutions. Throughout this section, we always assume that (A1)-(A4) hold. In view of Proposition 1 and Corollary 1, the following lemma holds obviously.
We now give some key estimates in the following two lemmas.
2) be solutions of (1.6). Assume that c i ≤ c * and p 2 ≤ p 1 ≤ 0. Denote Then there exist positive constants α 2 and K 2 such that Proof. We divide x ∈ R into four intervals.
Proof. The proof is similar to that of [14, Lemma 3.1] and we omit the details here.
To construct a supersolution of (1.5), we first introduce an auxiliary coupled system of ordinary differential equations where c 2 ≤ c 1 ≤ c * , α and K are positive constants. Solving the equations explicitly, we obtain . Then p i (t) is monotone increasing, and by virtue of p 2 (t) − p 1 (t) = c 1 − c 2 ≥ 0, we have p 2 (t) ≤ p 1 (t) ≤ 0 for all t ≤ 0. Let Then and there is a constant C 0 > 0 such that Now we can construct a supersolution of (1.5) as follows.
We now state our main result as follows.
The assertions (ii)-(vi) in Theorem 4.4 are straightforward consequences of (4.8). Therefore, we only prove the assertion (i).
Remark 5. For the autonomous Lotka-Volterra competition system with random (local) and nonlocal dispersal, Morita and Tachibana [29] and Li et al. [24] established the existence of invasion entire solutions, respectively. Notice that in their papers, the following condition is needed, which may be technical: (C): There exists a positive number η 0 such that φ(z) 1−ϕ(z) ≥ η 0 for z ≤ 0, where (φ(z), ψ(z)) is the invasion traveling wave solution. In fact, according to Remark 4, when the time periodic system (1.1) degenerates into the homogeneous case, the condition (C) holds obviously under our assumptions (A1)-(A3). We point out that the following supersolution U (t, x) := P1(t, x + p1(t)) + P2(t, −x + p2(t)) − P1(t, x + p1(t))P2(t, −x + p2(t)), V (t, x) := Q1(t, x + p1(t)) + Q2(t, −x + p2(t)) − Q1(t, x + p1(t))Q2(t, −x + p2(t)), which has been used in [29,24], is also applicable to our problem. In this sense, we generalize the result about entire solutions from autonomous case to periodic case. Remark 6. By the relation between systems (1.5) and (1.1), we get that (1.1) admits an entire solution (u(t, x), v(t, x)) := (p(t)U θ1,θ2 (t, x), q(t)(1 − V θ1,θ2 (t, x))). According to Theorem 4.4 (ii) and (iii), we have lim t→−∞ {|u(t, x)| + |v(t, x)) − q(t)|} = 0 locally in x ∈ R, lim t→+∞ {|u(t, x) − p(t)| + |v(t, x)|} = 0 uniformly in x ∈ R, which indicates that the entire solution (u, v) exhibits the extinction of the inferior species v by the superior one u invading from both sides of x-axis. In fact, this kind of entire solution describes a different type of biological invasion from one presented by traveling waves in a time periodic environment. On the other hand, we point out in particular that the continuous dependence of such an entire solution on parameters such as wave speeds and the shifted variables is important but still open. For some related works on this issue, one can see Hamel and Nadirashvili [15] for a local dispersal KPP equation, Wang et al. [40] for a delayed lattice differential equation, and Li et al. [23] for a nonlocal dispersal periodic monostable equation. We will leave such problems about our system (1.1) for a future study.