A diffusive logistic problem with a free boundary in time-periodic environment: favorable habitat or unfavorable habitat

We study the diffusive logistic equation with a free boundary in timeperiodic environment. To understand the effect of the dispersal rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$, and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d$, $h_0$, $\mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha, \gamma, h(t), T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha, \gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha, \gamma, h_0, T)>0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the dispersal rate is slow or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small, while survive successfully if the initial value is big. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.

In the special case that α, γ and β are independent of time t, problem (1.1) was studied more recently in [5]. They showed that, if the diffusion is slow or the occupying habitat is large, the species can establish itself in the favorable habitat, while the species will die out if the initial value of the species is small in an unfavorable habitat. However, big initial number of the species is benefit for the species to survive.
We remark that there are some related recent research about diffusive logistic problem with a free boundary in the homogeneous or heterogeneous environment. In particular, Du and Lin [8] are the first ones to study the spreading-vanishing dichotomy of species in the homogeneous environment of dimension one, which has been extended in [9] to the situation of higher dimensional space in a radially symmetric case. In [29], some of the results of [8] were extended to the case that the solution satisfies a Dirichlet boundary condition at x = 0 and a free boundary at x = h(t), covering monostable and bistable nonlinearities. Other theoretical advances can also be seen in [12,14,13,25,33,36,4]. Moreover, some Lotka-Vottera competitive type problems with a free boundary were introduced in [21,22,31,10]. Other studies of Lotka-Vottera prey-predator problem with a free boundary can be found in [3,32,34,35].
But the model in [11] do not exactly describe the survival of the species in real environment, for example, some part of the habitat has been polluted or destroyed. To describe the feature of environment, as in [17,19], we say that r is a favorable site if the birth rate α(t, r) is greater than the local death rate γ(t, r) over the time interval [0, T ], that is, T 0 (α(t, r) − γ(t, r))dt > 0. An unfavorable site is defined by reversing the inequality. Define the favorable set and unfavorable set in B R (a ball with radius R) as respectively. The habitat B R is characterized as favorable (resp. unfavorable) if the average of the birth rate 1 |[0,T ]×BR| T 0 BR α(t, r)dxdt is greater than (resp. less than) the average of the death rate The aim of this paper is to study the dynamics of problem (1.1) in the time-periodic case, a situation that more closely reflects the periodic variation of the natural environment, such as daily or seasonal changes. Further, we will consider our problem both in the favorable habitat and unfavorable habitat. To best of our knowledge, the present paper seems to be the first attempt to consider the unfavorable habitat with time-periodic environment in the moving domain problem. It should be pointed out here that the arguments developed in the previous works [11] do not work in the situation of unfavorable habitat with time-periodic environment, since now α(t, r) − γ(t, r) admitted to change sign in (t, r) ∈ [0, T ] × [0, ∞). Instead of h 0 which is only used in [11], here we choose the parameters d, h 0 , u 0 and µ as the varying parameters to study problem (1.1) both in the favorable habitat and an unfavorable habitat. We derive some sufficient conditions to ensure that spreading and vanishing occur, which yield the spreading-vanishing dichotomy, and sharp criteria governing spreading and vanishing both in the favorable habitat and an unfavorable habitat. Furthermore, we demonstrate that the species will spreading in a favorable habitat if the dispersal rate is slow or the initial occupying habitat radius is large. In an unfavorable habitat, the species will vanish if the initial density of the species is low. However, the species can also spread in an unfavorable habitat, if the initial number of the species is large. Therefore, we derive that the diffusion, the initial value and the original habitat play a significant role in determining the spreading and vanishing to problem (1.1). Finally, we also extend the asymptotic spreading speed of a free boundary in favorable time-periodic environment, when spreading of the species happens, to unfavorable time-periodic environment.
The rest of our paper is arranged as follows. In section 2, we exhibit some fundamental results, including the global existence and uniqueness of the solution to problem (1.1) and the comparison principle in the moving domain. An eigenvalue problem is given in section 3. In section 4, a spreading-vanishing dichotomy is proved. Section 5 is devoted to show the sharp criteria governing spreading and vanishing. In section 6, we investigate the asymptotic spreading speed of the free boundary when spreading occurs.

Preliminaries
In this section, we first give the existence and uniqueness of a global solution to problem (1.1), then a comparison principle which can be used to estimate both u(t, r) and the free boundary r = h(t) is given.
Therefore, the function f (t, r, u) satisfies conditions in Theorem 3.1 in [11], then we can prove the local existence and uniqueness result by the contraction mapping theorem, and use suitable estimates to show that the solution is defined for all t > 0. The proof is a simple modification of Theorem 4.1 in [9] and Theorem 3.1 in [11], so we omit it. ✷ For later applications, we need a comparison principle as follows.

An eigenvalue problem
In this section, we mainly study an eigenvalue problem and analyze the property of its principle eigenvalue. These results play an important role in later sections.
In what follows, we present some further properties of λ 1 = λ 1 (d, α − γ, R, T ). We now discuss the dependence of λ 1 on d for fixed R.
(ii) From the condition (1.3) and the monotonicity of It is also easy to check that It is also easy to see that This completes the proof. ✷ The above theorem implies the following result.
If we replace R in (3.1) by h(t), then it follows from the strict increasing monotony of h(t) and

The Spreading-vanishing dichotomy
In this section, we prove the spreading-vanishing dichotomy. Though our approach here mainly follows the lines of [11], considerable changes in the proofs are needed, since the situation here is more general and difficult.
In order to study the long time behavior of the spreading species, we firstly consider the fixed boundary problem where R is a positive constant, and u 0 (r) satisfies (1.2). The related T −periodic problem is Proof. the proof of (i) and (ii) can be see in [23,16]. Taking p = 2 in Theorem 3.1 in [24], we can easily get (iii). ✷ Next we consider the T −periodic parabolic logistic problem on the entire space Proof. Taking p = 2 in Theorem 1.3 in [24], we can easily get (i) and (ii). ✷ It follows from Theorem 2.1 that h(t) is monotonic increasing and therefore there exists h ∞ ∈ (0, +∞] such that lim t→+∞ h(t) = h ∞ . The spreading-vanishing dichotomy is a consequence of the following two lemmas.
Proof. Firstly, we use the contradiction argument to prove lim t→+∞ u(t, We claim that the unique positive solution Thus we have By the parabolic regularity (see [15,20,18]), we have, up to a subsequence if necessary, u k →û as k → ∞ withû satisfyinĝ We note thatû By the strong maximum principle, Furthermore, using the Hopf lemma to the equation ofû at the On the other hand, since h ∞ < ∞ and the bound of h ′ (t) is independent of t (the proof is a simple modification of that for Theorem 4.1 in [11]). We claim that h ′ (t) → 0 as t → ∞. In fact, support that there exists a sequence {t n } such that t n → ∞ and h ′ (t n ) → ε 1 as n → ∞ for some ε 1 > 0. We can find ε 2 > 0 small enough such that h ′ (t) ≥ ε1 2 for all t ∈ (t n − ε 2 , t n + ε 2 ) for all n. Then we obtain We argue by contradiction. By the continuity of Then by the comparison principle (see Remark where u * (t, r) is the unique positive T −periodic solution of the problem Hence, lim inf n→∞ u(t+nT, r) ≥ u * (t, r) > 0 in (0, h(τ )). This contradicts to lim t→+∞ u(t, ·) C( Proof. Since (H) holds, we have lim inf r→+∞ (α(t, r) − γ(t, r)) = η * (t) > 0, and then there exists a large positive constant r * such that α(t, r) − γ(t, r) > 0 and λ 1 (d, α − γ, R, T ) < 0 for any R > r * (by Corollary 3.2). Thus, we can choose an increasing sequence of positive numbers R m with R 1 > r * and R m → +∞ as m → +∞, such that λ 1 (d, α − γ, R m , T ) < 0 for all m ≥ 1. And we note that there exists a t * > 0 such that r * = h(t * ).
Let W Rm (t, r) be the unique positive T −periodic solution to (4.2) with replacing R by R m .
It follows from proposition 4.2 that W Rm (t, r) converge to U (t, r) as R m → +∞. Since h ∞ = ∞, we can find T m > 0 such that h(t) ≥ R m for all t ≥ T m . We consider the following problem (4.7) Next, using a squeezing argument similar in spirit to [7], we prove that We first consider the following T-periodic boundary blow-up parabolic problems It follows from Lemma 3.1 in [24] that (4.9) has a unique positive T −periodic solution V Rm (t + t * , r + r * ) V * Rm (t, r). Moreover, for any integer K ≥ max [0,T ]×BR m α(t+t * ,r+r * )−γ(t+t * ,r+r * ) β(t+t * ,r+r * ) , we consider the problem (4.10) Since V = 0 and V = K are the sub-supersolutions to (4.10), then it has one positive solution. By the same analysis as the uniqueness of positive T −periodic solution to (4.2), the uniqueness of positive T −periodic solution follows, denote it by V * Rm,K (t, r). By the comparison principle, it is obvious to check that V * Rm,K (t, r) < V * Rm,K+1 (t, r). Thus, lim K→+∞ V * Rm,K (t, r) exists and is a positive T −periodic solution to (4.9). Since (4.9) has a unique positive T − periodic solution, then lim K→+∞ V * Rm,K (t, r) = V * Rm (t, r). Furthermore, letV Rm (t, r) =V (t+t * , r +r * ) be the positive solution to the following parabolic problem where k m is a positive constant and satisfies By the comparison principle, u(t + t * , r + r * ) ≤V Rm (t, r) for t > 0 and 0 ≤ r ≤ R m , which implies that lim sup n→∞ u(t + t * + nT, r + r * ) ≤ V * Rm,K (t, r) uniformly for (t, r) ∈ [0, T ] × [0, R m ]. Therefore, from Proposition 4.2, we get lim sup n→∞ u(t + nT, r) ≤ U (t, r) locally uniformly for (t, r) ∈ [0, T ] × [0, ∞). Clearly, (4.5) is a consequence of (4.7) and (4.

Sharp criteria for spreading and vanishing
In this section, we aim to use parameters d, h 0 , µ and u 0 (r) to derive sharp criteria for species spreading and vanishing established in Theorem 4.1.

Now we set
Then by a direct calculation, we obtain By the comparison principle, we have According to Lemma 4.1, we see that h ∞ = ∞. ✷ Assume that there is a favorable site in B h0 and the diffusion is slow (0 < d ≤ d * ), by applying The above result implies that, if the average birth rate of a species is greater than the average death rate, the invasive species with slow diffusion or large habitat occupation will survive in the new environment. In a biological view, the species will survive easily in a favorable habitat.
In the following, with the parameters h 0 and d satisfying 0 < h 0 < h * and d > d * , respectively, which implies that λ 1 (d, α − γ, h 0 , T ) > 0, and u 0 fixed, let us discuss the effect of the coefficient µ on the spreading and vanishing when µ is sufficiently large. Proof. The idea of this proof comes from Lemma 3.6 in [25]. Note that from (4.4) there exists a positive constant M such that 0 < u(t, r) ≤ M for t ∈ (0, ∞) and 0 ≤ r < h(t), where u(t, r) is the unique positive solution to problem (1.1). Thus, there exists a positive constantL such that for all t ∈ (0, ∞) and r ∈ [0, h(t)]. We next consider the auxiliary free boundary problem W r (t, 0) = 0, W (t, δ(t)) = 0, t > 0, δ ′ (t) + µW r (t, δ(t)) = 0, t > 0, Arguing as in proving the existence and uniqueness of the solution to problem (1.1), one will easily see that (5.3) also admits a unique global solution (W, δ) and δ ′ (t) > 0 for t > 0. To stress the dependence of the solutions on the parameter µ, in the sequel, we always write (u µ , h µ ) and (W µ , δ µ ) instead of (u, h) and (W, δ). By the comparison principle, we have u µ (t, r) ≥ W µ (t, r), h µ (t) ≥ δ µ (t) for any t ≥ 0, r ∈ [0, δ µ (t)]. (5.4) In what follows, we are in a position to prove that for all large µ, for any positive constant T * > 1.
To this end, we first choose a smooth function δ(t) with δ(0) = h0 T * , δ ′ (t) > 0 and δ(T * ) = T * h * . We then consider the following initial-boundary problem Here, for the smooth initial value W 0 (r), we require The standard theory for parabolic equations ensures that (5.6) has a unique positive solution W , and W r (t, δ(t)) < 0 for all t ∈ [0, T * ] due to the well-known Hopf boundary lemma. According to our choice of δ(t) and W 0 (r), there is a constant µ 0 > 0 such that for all µ ≥ µ 0 , δ ′ (t) ≤ −µW r (t, δ(t)), for all 0 ≤ t ≤ T * .
Theorem 5.2. There exists µ * > 0, depending on u 0 , h 0 and d, such that h ∞ = ∞ if µ > µ * , and Proof. Recall Lemma 5.4 and 5.5, the proof is similar to that of Theorem 2.10 in [9], so we omit the details. ✷

Sharp criteria
In this subsection, we will establish the sharp criteria by selecting d, h 0 , µ and u 0 as varying factors to distinguish the spreading and vanishing for the invasive species.
If h 0 is fixed, the spreading or vanishing of an invasive species depends on the diffusion rate d, the initial number u 0 (r) of the species and the expanding ability µ. Now we give the sharp criteria for species spreading and vanishing. The result directly follows from Theorems 5.1 − 5.2, and Lemmas 5.2 − 5.3. (ii.2) there exists µ * > 0, depending on u 0 , h 0 and d, such that spreading happens if µ > µ * , and vanishing occurs if µ ≤ µ * ; (iii) moreover, µ * = 0 if h 0 ≥ h * . Theorem 5.4 tells us an unfavorable habitat is bad for the species with small number at the beginning, the endangered species in an unfavorable habitat will become extinct in the future. However, even the habitat is unfavorable, if the occupying area B h0 is beyond a critical size, namely h 0 ≥ h * , then regardless of the initial population size u 0 (r), spreading always happens (see (i) in Theorem 5.4). And if h 0 < h * , spreading is also possible for big initial population size u 0 (r).
If d and h 0 are fixed, the initial number u 0 (r) governs the spreading and vanishing of the invasive species.
Proof. The result follows from Theorem 3.2, Lemmas 5.1 − 5.3. The proof is similar to Theorem 5.7 in [5] with obvious modification. So we omit it.
Theorems 5.1−5.5 imply that slow diffusion, large occupying habitat and big initial population number are benefit for the species to survive in the new environment.

Spreading speed
In this section, we always suppose (H) holds. In the spreading case, we will give the asymptotic spreading speed of the free boundary x = h(t).
Proof. The proof is a simple modification of that for Theorem 4.4 in [11]. So we only briefly describe the main steps.
Step respectively.
This step can be obtained by the sub-supersolution argument, which is similar to the step 1 in Theorem 4.4 in [11] with the help of Proposition 4.1 and 4.2.
Direct calculations show that (u, h) and (ū,h) are sub-supersolutions to (1.1), respectively. Therefore, we have Since ε > 0 can be arbitrarily small, this implies our desired result. ✷ The result below follows trivially from Theorem 6.1.