Abstract
This paper is devoted to a nonlocal dispersal logistic model with seasonal succession and free boundaries, where the free boundaries represent the expanding front and the seasonal succession accounts for the effect of two different seasons. Technically, this free boundary problem is much more difficult than the case without seasonal succession since the coefficients are all time periodic and piecewise continuous. We prove the existence and uniqueness of global solution, and then examine the long-time dynamical behaviour and the criteria that completely determine when spreading and vanishing can happen, revealing some significant differences from the model without seasonal succession. Moreover, we use a “thin-tail" condition on the kernel function to estimate the asymptotic speeds of (g, h) and the asymptotic spreading speed of u, which is achieved by solving the associated semi-wave problem.
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1 Introduction
In most of spatial population models, the movement of the considered species is described by random diffusion, which induces a reaction-diffusion equation. Recently, the nonlocal diffusion operators are introduced to model the spatial distribution of the species, which can capture local as well as long-distance dispersal. In this paper, we consider a nonlocal dispersal logistic model with seasonal succession, and aim to reveal some similarities and fundamental differences between it and the associated local diffusion version. The model considered here has two key features, one of which is that the species experiences two different seasons: the bad season and the good season; the other of which is that the underlying spatial domain is changing as time varying. This is a nonlocal dispersal system with free boundaries, which encounters considerable technical difficulties for its analysis.
Time-varying environmental conditions are important for the growth and survival of species. Seasonal forces in nature are a common cause of environmental change, affecting not only the growth of species but also the composition of communities [1, 13]. The growth of species is actually driven by both external and internal dynamics. For instance, in temperate lakes, phytoplankton and zooplankton grow during the warmer months and may die or lie dormant during the winter. This phenomenon is termed as seasonal succession. Ignoring the spatial evolution of the involved species, the effects of seasonal succession on the dynamics of population can be analysed by ODE models, see [15, 19] and references therein. There are also some investigations on it by the numerical method, see, e.g. [14, 21].
In reality, individual organisms are distributed in space and typically interact with the physical environment and other organisms in their spatial neighborhood. In [20], Peng and Zhao considered the diffusive logistic system with a free boundary and seasonal succession, which reads as follows:
where u(t, x) denotes the population density of a new or invasive species at time t and location x over a one dimensional habitat [0, h(t)], and h(t) accounts for the moving boundary to be determined. Here \(\omega ,0<\rho <1,\delta ,d,a,b,\mu \) and \(h_0\) are positive constants, and the initial function \(u_0\) satisfies
The parameters \(\omega \) and \(1-\rho \) represent the period of seasonal succession and the duration of the good season, respectively. Here and in what follows, unless specified otherwise, we always take \(i=0,1,2,\cdots \).
In (1.1), it is assumed that the species u undergoes two different seasons: the bad season and the good season. In the bad season: \(i\omega<t< (i+\rho )\omega \), for instance, from winter to spring, the species can not get enough food to feed themselves and its density are declining exponentially. During the bad season, the spreading front \(h(t)=h(i\omega )\) means biologically that the species does not migrate and stays in a hibernating state in the habitat \([0,h(i\omega )]\). In the good season (for instance, from summer to autumn): \((i+\rho )\omega <t\le (i+1)\omega \), as in (1.1), the spatiotemporal evolution of the species are determined by the classical logistic equation and the expanding rate of the range boundary is supposed to satisfy \(h'(t)=-\mu u_x(t,h(t))\), which is to say, it is proportional to the gradient of the population density of the new species at the range movement. One can refer to [3, 9] for detail mathematical derivation about such free boundary condition and for many other population models with the free boundary condition (see [7, 22,23,24]).
By a rigorous analysis, Peng and Zhao [20] have showed that (1.1) has a unique solution defined for all \(t\ge 0\), and if \(a(1-\rho )-\rho \delta \le 0\), the population always vanishes eventually, namely,
When \(a(1-\rho )-\rho \delta >0\), a spreading-vanishing dichotomy holds:
Either (1.3) holds or the population spreads successfully, i.e.
Peng and Zhao [20] also obtained a criteria for vanishing and spreading: there is a critical length \(h_*=\frac{\pi }{2}\sqrt{\frac{d(1-\rho )}{a(1-\rho )-\delta \rho }}\) such that if \(h_0\ge h_*\), then spreading always happens, while if \(h_0<h_*\), then there exists a \(\mu ^*>0\) depending on \(u_0\) such that spreading happens if and only if \(\mu >\mu ^*\). In particular, when spreading occurs, a estimate of asymptotic spreading speeds for (1.1) is obtained in [20]. Look at the Cauchy problem
with initial function having compact supports. It was shown in [20] that the spreading speed \(c^*\) for system (1.4) is given by
and the spreading speed \(c_*\) determined by the free boundary problem (1.1) is always smaller than \(c^*\).
We notice that in (1.1), the term \(du_{xx}\) is used to describe the spatial movement of the species, which is known as ‘local diffusion’. Although the standard Laplacian operator can describe the movement of individuals under a Brownian process, it does not account for the long-range motion of some individuals very well. Recently, various integral operators have been used to model the nonlocal diffusion phenomena. A widely used nonlocal diffusion operator has the form
which can capture the factors of ‘long-range dispersal’ as well as ‘short-range dispersal’ by setting the kernel function J satisfying that \(J:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous nonnegative even function and \(\int _{{\mathbb {R}}}J(x)\mathrm dx=1\).
In the recent paper [4], Cao et al. proposed a nonlocal dispersal Fisher-KPP model with the free boundaries, and extended many basic of [9] to the nonlocal model. The spreading speed of such nonlocal Fisher-KPP equation with free boundaries has also been studied by Du et al. [8]. The results of [8] showed a threshold condition on the kernel function such that spreading grows linearly in time exactly when this condition holds; when the kernel function violates this condition, accelerating spreading happens. See [10,11,12] for more investigations on nonlocal dispersal model with free boundaries.
The following nonlocal Cauchy problem
and its corresponding traveling wave were studied by Jin and Zhao [16], where a(t) and \(F(t,\cdot )\) are both \(\omega \)-periodic in t. Clearly, by choosing suitable form for a(t) and \(F(t,\cdot )\), (1.5) can be reduced to a nonlocal version of (1.4). However, the nonlocal version of (1.1) has not been investigated so far. In this paper, motivated by [4], we consider a nonlocal problem of (1.1) with double free boundaries, where the local diffusion term \(u_{xx}\) is replaced by a nonlocal operator of the form
Notice that the above nonlocal dispersal operator can be rewritten as
by extending u to be zero for \(x\in {\mathbb {R}}\setminus [g(t),h(t)]\).
Our nonlocal version of (1.1) with double free boundaries has the following form:
where \(\omega ,0<\rho <1,\delta ,d,a,b,\mu \) and \(h_0\) are positive constants. The initial function \(u_0(x)\) satisfies
The kernel function \(J:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is assumed to satisfy
- \(\mathbf{(J):}\):
-
\(J\in C({\mathbb {R}})\cap L^{\infty }({\mathbb {R}})\) is nonnegative, symmetric, \(J(0)>0\) and \(\int _{{\mathbb {R}}}J(x)\mathrm dx=1\).
In (1.7), the free boundaries \(g(t)=g(i\omega )\) and \(h(t)=h(i\omega )\) for \(i\omega \le t\le (i+\rho )\omega \) mean that the species does not migrate and stays in a hibernating state in the habitat \([g(i\omega ),h(i\omega )]\) during the bad season. The free boundary conditions
mean that during the good season, the expanding rate of the range [g(t), h(t)] is proportional to the outward flux of the population across the boundary of the range (see [4] for some further interpretation).
Our first aim in current paper is to study the long-time behavior of system (1.7). Our main results are stated as follows.
Theorem 1.1
(Existence and uniqueness) Assume that \(\mathbf{(J)}\) holds and the initial function \(u_0\) satisfies (1.8). Then problem (1.7) admits a unique positive solution (u, g, h) defined for all \(t\ge 0\).
Theorem 1.2
(Spreading-vanishing dichotomy) Assume that \(\mathbf{(J)}\) holds and the initial function \(u_0\) satisfies (1.8). Let (u, g, h) be the unique solution of (1.7) and denote
Then one of the following alternatives must hold true:
-
(i)
Spreading: \(h_{\infty }=-g_{\infty }=+\infty \) and
$$\begin{aligned} \lim _{n\rightarrow \infty }u(t+n\omega ,x)=z^*(t)\ \ \mathrm {in}\ C_{\mathrm {loc}}([0,\omega ]\times {\mathbb {R}}), \end{aligned}$$where \(z^*(t)\) is the unique positive \(\omega \)-periodic solution of the following equation
$$\begin{aligned} {\left\{ \begin{array}{ll} z_t=-\delta z, &{} i\omega< t\le (i+\rho )\omega , \\ z_t=z(a-bz), &{} (i+\rho )\omega <t\le (i+1)\omega , \\ z(t+\omega )=z(t), &{} t\ge 0; \end{array}\right. } \end{aligned}$$(1.9) -
(ii)
Vanishing: \(h_{\infty }-g_{\infty }<+\infty \) and
$$\begin{aligned} \lim _{t\rightarrow +\infty }u(t,x)=0\ \ \mathrm {in}\ \ C([g(t),h(t)]). \end{aligned}$$
Theorem 1.3
(Spreading-vanishing criteria) Assume that \(\mathbf{(J)}\) holds and the initial function \(u_0\) satisfies (1.8). Let (u, g, h) be the unique solution of (1.7). Then the following statements are true:
-
(i)
If \((1-\rho )a-\rho \delta \le 0\), then vanishing always occurs;
-
(ii)
If \(0<(1-\rho )a-\rho \delta \le (1-\rho )d\), then there exists a unique \(\ell ^*>0\) such that
-
(ii.a)
for \(h_0\ge \ell ^*/2\), spreading always occurs;
-
(ii.b)
for \(h_0\in (0,\ell ^*/2)\), there is a unique \(\mu ^*>0\) depending on \(u_0\) so that spreading occurs only when \(\mu >\mu ^*\);
-
(ii.a)
-
(iii)
If \((1-\rho )a-\rho \delta >(1-\rho )d\), then spreading always occurs.
Remark 1.4
In Theorem 1.3, the threshold value \(\ell ^*\) is determined by a related eigenvalue problem, independent of the initial value \(u_0\) (see Proposition 3.1 and Remark 3.2).
The second aim in this paper is to characterize the spreading speed of (1.7) when spreading happens. To this end, we need to consider the existence of semi-wave solutions to (1.7). These are solution pairs \((k,\Psi )\in C([\rho \omega ,\omega ])\times \big [C^{1,0}((0,\rho \omega ]\times (-\infty ,0])\cap C^{1,1}((\rho \omega ,\omega ]\times (-\infty ,0]) \big ]\) which satisfy the following two equations:
and
where \(i\in {\mathbb {Z}}\) and \(z^*(t)\) is the same as in Theorem 1.2 (i). If \((k,\Psi )\) solves (1.10), then we call \(\Psi \) a \(\omega \)-periodic semi-wave. However, only when the semi-wave solution pairs \((k,\Psi )\) satisfy (1.11), they are useful to estimate the spreading speed of the free boundary condition at the moving front. In [20], Peng and Zhao has studied the semi-waves solutions for a time-periodic local diffusion equation. Recently, Du et al. [8] considered the existence of semi-waves solutions of a general nonlocal dispersal problem on the half line, whose coefficients are constants. Du and Ni [12] provided a threshold condition on the existence of semi-waves to a general cooperative system with nonlocal diffusion and free boundaries. It should be mentioned that the method used for local diffusion system and that for nonlocal dispersal equation are distinct since the former is a parabolic equation and the latter is an integral differential equation. We note that the coefficients in (1.10) are \(\omega \)-periodic and piecewise continuous in t due to the presence of seasonal succession, so the methods developed in [8, 12] can not applied directly here. Then we will use some techniques related to time-periodic nonlocal dispersal system to deal with problem (1.10).
To describe the results for such time-periodic semi-wave, we introduce a vital condition on the kernel function J, which is
- \(\mathbf{(J1):}\):
-
There exists \({\hat{\eta }}>0\) such that
$$\begin{aligned} \int _{-\infty }^{+\infty }J(x)e^{\eta x}\mathrm dx<+\infty ,\ \ \forall \eta \in [0,{\hat{\eta }}), \end{aligned}$$which is often called a “thin tail" condition.
Theorem 1.5
Assume that \(\mathbf{(J)}\) holds and \((1-\rho )a-\rho \delta >0\). Then the following statements are true:
-
(i)
If \(\mathbf{(J1)}\) holds, then (1.10)- (1.11) admit a unique solution pair \((k,\Psi )=(k_0,\Psi ^{k_0})\) with \(k_0(t)>0\) in \((\rho \omega ,\omega )\), \(\Psi ^{k_0}(t,\xi )\) nonincreasing with respect to \(\xi \) and \(\omega \)-periodic with respect to t;
-
(ii)
If
$$\begin{aligned} \displaystyle \int _{-\infty }^0\int _0^{+\infty }J(x-y)\mathrm dy\mathrm dx<+\infty ,\ \ \mathrm {i.e.},\ \ \displaystyle \int _{-\infty }^0\int _{-\infty }^xJ(y)\mathrm dy\mathrm dx<+\infty \end{aligned}$$does not hold, then (1.10)-(1.11) has no nontrivial solution pair.
The proof of Theorem 1.5 highly relies on the solution of a nonlocal problem (P) (see Sect. 4). With the semi-wave theory established above, we can construct appropriate upper- and lower-solutions to estimate the spreading fronts g(t), h(t) as \(t\rightarrow \infty \), and obtain the following theorem.
Theorem 1.6
(Spreading speed) Assume that the conditions in Theorem 1.2 are satisfied, and the spreading occurs for (1.7). If \(\mathbf{(J1)}\) is satisfied, then
where \(k_0\) is given by Theorem 1.5.
To simplify the notation, we define
We then have the result on the long-time behavior of the solution to (1.7) inside the region \(\{ x:-c_*(\mu )t\le x\le c_*(\mu )t \}\), which seems to be an improvement of Theorem 1.2 (i) when the spreading occurs.
Theorem 1.7
Assume that the conditions in Theorem 1.6 are satisfied, and the spreading occurs for (1.7). Then for each small \(\epsilon >0\),
where \(z^*(t)\) is the unique positive \(\omega \)-periodic solution to (1.9).
Although this paper focuses on the nonlocal logistic model (1.7) with seasonal succession, many techniques applied here can also apply to other nonlocal dispersal model with free boundaries in time-periodic environment. For example, the techniques of establishing the existence and uniqueness of semi-wave solution in Sect. 3 are useful for a general time-periodic semi-wave equation. If we replace the term \(a-bu\) in (1.7) with a continuous and strictly decreasing function f(u) on \([0,+\infty )\) satisfying \(f(K)\le 0\) for some \(K>0\), then the conclusions showed in Theorems 1.1-1.7 still hold with \(a=f(0)\).
The rest part of this paper is organized as follows. In Sect. 2, we first prove the global existence and uniqueness of solution of (1.7). Sect.3 is devoted to the long-time dynamical behavior of system (1.1). In Sect. 4, we prove Theorem 1.5 on the semi-wave solution, which paves the ground for Theorem 1.6. Then by making use of the semi-wave solution established in Sect.4, Sect. 5 completes the proof of Theorem 1.6 on finite spreading speed. The proof of Theorem 1.7 is also included in Sect.5.
2 Well-posedness
In this section, we show the existence and uniqueness of the global solution of (1.7). At first, we introduce some notations for convenience of discussion. Given any constant \(h_0>0\) and any integer \(m>0\), we set
For \(g\in {\mathbb {G}}_{h_0,m\omega },h\in {\mathbb {H}}_{h_0,m\omega }\) and \(u_0\) satisfying (1.8), denote
Theorem 2.1
Suppose that \(u_0\) satisfies (1.8) and \(\mathbf{(J)}\) holds. Then the problem (1.7) admits a unique positive solution (u(t, x), g(t), h(t)) defined for all \(t\ge 0\). Moreover, \(g\in {\mathbb {G}}_{h_0,m\omega },h\in {\mathbb {H}}_{h_0,m\omega }\) and \(u\in {\mathbb {X}}_{m\omega }\) for all \(m\in {\mathbb {N}}\).
Proof
At first, we set
for \(t\in [0,\rho \omega ]\). Then \({{\hat{u}}}\in C(\Delta _{0,1}^{{{\hat{g}}},{{\hat{h}}}})\) and \(({{\hat{u}}},{{\hat{g}}},{{\hat{h}}})\) satisfies
Consider the following problem
Then we can apply the result in [4, Theorem 2.1] to conclude that (2.1) has a unique solution \(({\bar{u}},{\bar{g}},{\bar{h}})\), where
and
From [4, Lemma 2.3], we also obtain that
Define
We have that \(g\in {\mathbb {G}}_{h_0,\omega },h\in {\mathbb {H}}_{h_0,\omega }\) and \(u\in {\mathbb {X}}_\omega \).
Based on the above obtained function u, g, h, we let
for \(\omega \le t\le (1+\rho )\omega \). Then \(u_1\in C(\Delta _{1,1}^{g_1,h_1})\) and \((u_1,g_1,h_1)\) satisfies
Likewise, the problem
has a unique solution \(({\bar{u}}_1,{\bar{g}}_1,{\bar{h}}_1)\), in which
and
Define
Then it holds that \(g\in {\mathbb {G}}_{h_0,2\omega },h\in {\mathbb {H}}_{h_0,2\omega }\) and \(u\in {\mathbb {X}}_{2\omega }\).
By repeating this procedure, we therefore obtain the existence and uniqueness of the solution (u, g, h) of (1.7). \(\square \)
3 Spreading-vanishing Dichotomy and Criteria
3.1 Some Preparatory Results
3.1.1 Time Periodic Positive Solutions Over Fixed Domain
Here we investigate the nonlocal dispersal equation:
and its associated time periodic positive solution problem:
where \(i=0,1,2,\cdots \).
To this end, we look at the time independent eigenvalue equation
It is well known (see, e.g., [2, 5, 6]) that (3.3) admits a principal eigenvalue \(\sigma _{(l_1,l_2)}\), which satisfies \(\sigma _{(l_1,l_2)}<d-a\). Moreover, there holds that
Proposition 3.1
[4, Proposition 3.4] Assume that \(\mathbf{(J)}\) holds and \(-\infty<l_1<l_2<+\infty \). Then the following hold true:
-
(1)
\(\sigma _{(l_1,l_2)}\) is strictly decreasing and continuous in \(\ell :=l_2-l_1\);
-
(2)
\(\lim _{l_2-l_1\rightarrow +\infty }\sigma _{(l_1,l_2)}=-a\);
-
(3)
\(\lim _{l_2-l_1\rightarrow 0^+}\sigma _{(l_1,l_2)}=d-a\).
Remark 3.2
We can see from Proposition 3.1 that, if \(0<(1-\rho )a-\rho \delta <(1-\rho )d\), then there is a threshold value \(\ell ^*>0\) such that \(\lambda _{(l_1,l_2)}:=(1-\rho )\sigma _{(l_1,l_2)}+\rho \delta <0\) if and only if \(\ell >\ell ^*\).
Proposition 3.3
[18] Assume that \(\mathbf{(J)}\) holds and \(-\infty<l_1<l_2<+\infty \). Then for any given initial value \(u_0(x)\in C([l_1,l_2])\) and \(u_0\ge ,\not \equiv 0\), (3.1) has a unique solution \(u(t,\cdot ;u_0)\) defined for all \(t>0\). Moreover, the following statements are true:
-
(1)
If \(\lambda _{(l_1,l_2)}<0\), then (3.2) admits a unique \(\omega \)-periodic positive solution \(u_{(l_1,l_2)}^*(t,x)\), and \(\lim \limits _{n\rightarrow \infty }u(t+n\omega ,x;u_0)=u_{(l_1,l_2)}^*(t,x)\) in \(C([0,\omega ]\times [l_1,l_2])\);
-
(2)
If \(\lambda _{(l_1,l_2)}\ge 0\), then 0 is the unique nonnegative solution of (3.2), and \(\lim \limits _{t\rightarrow \infty }u(t,x;u_0)=0\) uniformly for \(x\in [l_1,l_2]\).
Proposition 3.4
[18] Assume that \(\mathbf{(J)}\) holds. If \((1-\rho )a-\rho \delta >0\), then there exists \({\hat{\ell }}>0\) such that \(\lambda _{(l_1,l_2)}<0\) for every interval \((l_1,l_2)\) with \(l_2-l_1>{\hat{\ell }}\) and hence (3.2) admits a unique positive \(\omega \)-periodic solution \(u_{(l_1,l_2)}^*(t,x)\). Moreover,
3.1.2 Comparison Principle
In the following, we provide two maximum principle, which are analogue to [4, Lemma 2.2] and [4, Lemma 3.3], respectively.
Proposition 3.5
Let m be a positive integer. Assume that \(\mathbf{(J)}\) holds, \(g,h\in C([0,m\omega ])\) satisfy \(-g(0)=h(0)=h_0>0\), \(g(t)\le g(i\omega ),h(t)\ge h(i\omega )\) for \(t\in [i\omega ,(i+\rho )\omega ]\), and both \(-g(t)\) and h(t) are nondecreasing for \(t\in ((i+\rho )\omega ,(i+1)\omega ],(i=0,1,\cdots ,m-1)\). Suppose that \(u,u_t\in C({\overline{D}}_{m\omega }), c\in L^{\infty }(D_{m\omega })\) and
Then \(u(t,x)\ge 0\) for \((t,x)\in {\overline{D}}_{m\omega }\). Moreover, if \(u(0,x)>0\) in \((-h_0,h_0)\), then \(u(t,x)>0\) for \((t,x)\in (0,m\omega ]\times (g(t),h(t))\).
Proposition 3.6
Let m be a positive integer, \(h_0\) be the positive constants. Assume that \(\mathbf{(J)}\) holds. Suppose that \(u,u_t\in C([0,m\omega ]\times [-h_0,h_0]), c\in L^{\infty }([0,m\omega ]\times [-h_0,h_0])\) and
where \(i=0,1,\cdots ,m-1\). Then \(u(t,x)\ge 0\) for \((t,x)\in [0,m\omega ]\times [-h_0,h_0]\). Moreover, if \(u(0,x)>0\) in \([-h_0,h_0]\), then \(u(t,x)>0\) for \((t,x)\in (0,m\omega ]\times [-h_0,h_0]\).
Then we have the following comparison principle for system (1.7), which comes from [4, Theorem 3.1].
Proposition 3.7
(Comparison principle) Let m be a positive integer. Assume that \(\mathbf{(J)}\) holds, \(u_0\) satisfies (1.8), \({\overline{h}},{\overline{g}}\in C([0,m\omega ])\), and \({\overline{u}}\in C\Big ({\overline{D}}_{m\omega }^{{\overline{g}},{\overline{h}}}\Big )\) satisfies
where \(i=0,1,\cdots ,m-1\). Then the unique positive solution of system (1.7) satisfies
Remark 3.8
We call the triplet \(({\overline{u}},{\overline{g}},{\overline{h}})\) in Proposition 3.7 is an upper-solution of (1.7). Similarly, a lower-solution of (1.7) can be defined if all reversed inequalities in (3.6) and (3.7) are satisfied.
3.2 Proof of Theorems 1.2 and 1.3
Throughout this subsection, we assume that \(\mathbf{(J)}\) holds, the initial value \(u_0(x)\) satisfies (1.8) and (u, g, h) is the unique solution of (1.7). Since h(t) and \(-g(t)\) are nondecreasing on \([0,+\infty )\), we let
Lemma 3.9
If \(h_{\infty }-g_{\infty }<+\infty \), then \(\lim _{t\rightarrow +\infty }\Vert u\Vert _{C([g(t),h(t)])}=0\) and \(\lambda _{(g_{\infty },h_{\infty })}\ge 0\), where \(\lambda _{(\cdot ,\cdot )}\) is defined as in Remark 3.2.
Proof
We first show that
By way of contradiction, we suppose that \(\lambda _{(g_{\infty },h_{\infty })}<0\). In view of \(J(0)>0\) and Proposition 3.1, it follows that for sufficiently small \(\epsilon \in (0,h_0)\), \(J(x)>0\) for \(x\in [-4\epsilon ,4\epsilon ]\) and there exists a large integer \(m>0\) such that \(\lambda _{(g(m\omega ),h(m\omega ))}<0\) with \(|g(m\omega )-g_{\infty }|<\epsilon \) and \(|h(m\omega )-h_{\infty }|<\epsilon \). Let w(t, x) be the solution of the following initial value problem
where \(i\ge m\). By Proposition 3.6 and a simple comparison argument, there holds that
Since \(\lambda _{(g(m\omega ),h(m\omega ))}<0\), by Proposition 3.3, the equation (3.2) with \([l_1,l_2]\) replaced by \([g(m\omega ),h(m\omega )]\) has a unique \(\omega \)-periodic positive solution \(u_{(g(m\omega ),h(m\omega ))}^*(t,x)\), and \(\lim \limits _{n\rightarrow \infty }w(t+n\omega ,x)=u_{(g(m\omega ),h(m\omega ))}^*(t,x)\) in \(C([0,\omega ]\times [g(m\omega ),h(m\omega )])\). Then there exists large integer \(N\ge m>0\) such that for all \(n\ge N\),
Denote
Now for \(n\ge N\) and \(t\in [0,\omega ]\), we have \([h(t+n\omega )-2\epsilon ,h(t+n\omega )-\epsilon ]\in [g(m\omega ),h(m\omega )]\) and then
which leads to that
This means \(h_{\infty }=+\infty \), contradicting with the assumption that \(h_{\infty }-g_{\infty }<+\infty \). Hence, \(\lambda _{(g_{\infty },h_{\infty })}\ge 0\).
Denote by \({\overline{u}}(t,x)\) the unique solution of the problem
in which
By Proposition 3.5, it holds that \(u(t,x)\le {{\tilde{u}}}(t,x)\) for \(t>0\) and \(x\in [g(t),h(t)]\). In view of \(\lambda _{(g_{\infty },h_{\infty })}\ge 0\), we see from Proposition 3.3 (ii) that \(\lim _{t\rightarrow +\infty }{{\tilde{u}}}(t,x)=0\) uniformly for \(x\in [g_{\infty },h_{\infty }]\). Consequently, \(\lim _{t\rightarrow +\infty }u(t,x)=0\) uniformly for \(x\in [g(t),h(t)]\). The proof is completed. \(\square \)
Lemma 3.10
If \((1-\rho )a-\rho \delta \le 0\), then \(h_{\infty }-g_{\infty }<+\infty \) and vanishing occurs.
Proof
From Lemma 3.9, we only need to prove \(h_{\infty }-g_{\infty }<+\infty \). Note that \(\int _{{\mathbb {R}}}J(x)\mathrm dx=1\) and J is symmetric. Then for \(t\in ((i+\rho )\omega ,(i+1)\omega )(i=0,1,2,\cdots )\), it holds that
Recall the definition of the piecewise function \({\bar{a}}(t)\) in (4.2). By (3.8), (1.7) and \((1-\rho )a-\rho \delta \le 0\), it can be inferred that for \(t\in ((i+\rho )\omega ,(i+1)\omega ]\),
Notice that \([(1-\rho )a-\rho \delta ]t-\int _0^t{\bar{a}}(s)\mathrm ds\) is \(\omega \)-periodic and decreasing in \(t\in ((i+\rho )\omega ,(i+1)\omega ]\). We then have
Thus, when \(t\in [\rho \omega ,\omega ],n\ge 0\),
Since \(h(t+n\omega )-g(t+n\omega )=h(n\omega )-g(n\omega )\) for \(t\in [0,\rho \omega ],n\ge 0\), we must have \(h_{\infty }-g_{\infty }<+\infty \). \(\square \)
Lemma 3.11
If \(\lambda _{(-h_0,h_0)}>0\) and \(\Vert u_0\Vert _{C([-h_0,h_0])}\) is sufficiently small, then \(\lim \limits _{t\rightarrow +\infty }\Vert u\Vert _{C([g(t),h(t)])}=0\).
Proof
In view of Lemma 3.9, it suffices to show \(h_{\infty }-g_{\infty }<+\infty \). Recall the definition of \(\lambda _{(\cdot ,\cdot )}\) in Remark 3.2. Due to \(\lambda _{(-h_0,h_0)}>0\) and the continuity of \(\lambda _{(-h_0,h_0)}\) from Proposition 3.1, we have that \(\lambda _{(-h_1,h_1)}>0\) for some \(h_1=h_0+\varepsilon \) with \(0<\epsilon \ll 1\). Denote by \(\phi _1(x)\) the positive eigenfunction corresponding to the principal eigenvalue \(\sigma _1:=\sigma _{(-h_1,h_1)}\) of (3.3) with \((l_1,l_2)=(-h_1,h_1)\). Let
and define
where
with \(\sigma (t)=\delta \) for \(t\in (i\omega ,(i+\rho )\omega ]\), and \(\sigma (t)=\sigma _1\) for \(t\in ((i+\rho )\omega ,(i+1)\omega ]\). Obviously, \({\overline{h}}(t),{\overline{g}}(t)\) are continuous functions and \({\overline{h}}(t)\in [h_0,h_1)\) for \(t\in [0,+\infty )\).
Since \(\lambda _{(-h_1,h_1)}>0\), by (3.3), there holds that
Seeing that \([{\overline{g}}(t),{\overline{h}}(t)]\subset (-h_1,h_1)\) and restricting \(t\in ((i+\rho )\omega ,(i+1)\omega )\), we obtain that
Similarly,
Since \(-{\overline{g}}(t),{\overline{h}}(t)\in [h_0,h_1)\) for \(t\in [0,+\infty )\), we have
When we choose
there should be
It now follows from Proposition 3.7 that \([g(t),h(t)]\subset [{\overline{g}}(t),{\overline{h}}(t)]\) and hence \(h_{\infty }-g_{\infty }\le {\overline{h}}_{\infty }-{\overline{g}}_{\infty }=2h_1<+\infty \). Clearly, \(\lim _{t\rightarrow +\infty }\Vert u\Vert _{C([g(t),h(t)])}=0\) due to the construction of \({\overline{u}}\) and the fact that \(u(t,x)\le {\overline{u}}(t,x)\) for \(t\ge 0,g(t)\le x\le h(t)\). This completes the proof. \(\square \)
Remark 3.12
It can be seen in the proof of Lemma 3.11 that \(M\rightarrow +\infty \) when \(\mu \rightarrow 0\). Then for any given admissible initial value \(u_0\), there exists \(\mu _*>0\) (relying on the given initial value) such that (3.9) holds for \(\mu \in (0,\mu _*)\). This means that vanishing occurs for (1.7) if \(\mu \le \mu _*\).
Lemma 3.13
\(h_{\infty }<+\infty \) if and only if \(-g_{\infty }<+\infty \).
Proof
By way of contradiction, without loss of generality, we assume that \(h_{\infty }=+\infty \) and \(g_{\infty }>-\infty \). We first show that \((1-\rho )a-\rho \delta >0\) and there is \(T>0\) such that
In fact, if \((1-\rho )a-\rho \delta \le 0\), then \(h_{\infty }-g_{\infty }<+\infty \) by Lemma 3.10, a contradiction. Hence, \((1-\rho )a-\rho \delta >0\). This combined with Proposition 3.1 (1), (2) and \(h_{\infty }=+\infty \) to conclude that \(\lambda _{(g(t),h(t))}<0\) for sufficiently large t, which prove (3.10).
Now, following the proof of Lemma 3.9, we can use (3.10) and \(g_{\infty }>-\infty \) to seek a constant \(\nu >0\) and a positive integer N with \(N\omega >T\) such that
which contradicts with \(g_{\infty }>-\infty \). This completes the proof. \(\square \)
Lemma 3.14
If \(\lambda _{(g(s_0),h(s_0))}<0\) for some \(s_0\ge 0\), then \(g_{\infty }=-\infty ,h_{\infty }=+\infty \) and
where \(z^*(t)\) is the unique positive \(\omega \)-periodic solution to (1.9), and \(\lambda _{(\cdot ,\cdot )}\) is defined as in Remark 3.2.
Proof
Firstly, we show that \(h_{\infty }-g_{\infty }=+\infty \). In fact, since \(g(s_0),h(s_0)\subset [g(t),h(t)]\) for \(t\ge s_0\), it follows from Proposition 3.1 and the definition of \(\lambda _{(\cdot ,\cdot )}\) that
If \(h_{\infty }-g_{\infty }<+\infty \), then we can follow the argument as in the proof of Lemma 3.9 to obtain a contradiction. This is to say \(h_{\infty }-g_{\infty }=+\infty \), and thus \(g_{\infty }=-\infty ,h_{\infty }=+\infty \) by Lemma 3.13. This also implies \((1-\rho )a-\rho \delta >0\) by Lemma 3.10.
In the following, we prove (3.11). Fix a integer \(s>0\) with \(s\omega >s_0\) and denote by \({\underline{u}}(t,x)\) the solution of the initial value problem
By Proposition 3.6 and a comparison argument, we have
Since \(\lambda _{(g(s\omega ),h(s\omega ))}<0\), from Proposition 3.3, the equation (3.2) with \([l_1,l_2]=[g(s\omega ),h(s\omega )]\) admits a unique positive \(\omega \)-periodic solution, denoted by \({\underline{u}}_{(g(s\omega ),h(s\omega ))}^*(t,x)\) and
In view of (3.13), (3.14) and Proposition 3.3 (1), it holds that
Since \(g(s\omega ),h(s\omega )\rightarrow (-\infty ,+\infty )\) as \(s\rightarrow \infty \), by Proposition 3.4 and taking \(s\rightarrow \infty \), we obtain
On the other hand, it is easy to verify that the ODE model
has a unique positive solution defined for all \(t>0\) provided a positive initial value. Set \({\overline{u}}(t)\) be the unique solution of (3.15) with the initial value \({\overline{u}}(0)\ge \max _{x\in [-h_0,h_0]}u_0(x)\). Since
we can derive from Proposition 3.5 that
Recall that \(\lim _{n\rightarrow \infty }{\overline{u}}(t+n\omega )=z^*(t)\) in \(C([0,\omega ])\) when \((1-\rho )a-\rho \delta >0\) (see [15, Theorem 2.1]). Therefore, (3.11) holds true. This completes the proof. \(\square \)
Proof of Theorem 1.2
When \(h_{\infty }-g_{\infty }=+\infty \), we have \((1-\rho )a-\rho \delta >0\) from Lemma 3.10. It then follows from Proposition 3.1 and the definition of \(\lambda _{(\cdot ,\cdot )}\) that \(\lambda _{(g(s_0),h(s_0))}<0\) for large \(s_0>0\). Hence, spreading occurs by Lemma 3.14.
When \(h_{\infty }-g_{\infty }<+\infty \), \(\lim _{t\rightarrow +\infty }\Vert u\Vert _{C([g(t),h(t)])}=0\) from Lemma 3.9. This is, vanishing occurs. \(\square \)
Proof of Theorem 1.3
1. Part (i) follows from Lemma 3.10 directly.
2. Consider the case \(0<(1-\rho )a-\rho \delta \le (1-\rho )d\). When \(h_0\ge \ell ^*/2\) and vanishing occurs, we know that \([g_{\infty },h_{\infty }]\) is a finite interval, whose length is strictly longer than \(2h_0\ge \ell ^*\). Hence by Remark 3.2, \(\lambda _{(g_{\infty },h_{\infty })}<0\), which is a contradiction with the conclusion in Lemma 3.9. This proves part (ii.a).
In the following, we consider (ii.b). If \(h_0\in (0,\ell ^*/2)\), then by Remark 3.2, \(\lambda _{(-h_0,h_0)}>0\). This together with Remark 3.12 means that there exists \({\underline{\mu }}>0\) such that for any given admissible initial value \(u_0\), vanishing occurs for all \(\mu \in (0,{\underline{\mu }}]\).
To emphasize the dependence on \(\mu \), denote by \((u_{\mu },g_{\mu },h_{\mu })\) the unique solution of (1.7). Next, we prove that there is \({\overline{\mu }}>0\) such that spreading occurs as \(\mu \ge {\overline{\mu }}\). For this purpose, we first show that there exists \(\mu _0>0\) such that
In the proof of Lemma 3.14, we see that
where \({\overline{u}}(t)\) is the solution of (3.15) with the initial value \({\overline{u}}(0)\ge \max _{x\in [-h_0,h_0]}u_0(x)\). Thus, there is a large \(C^*>0\), independent of \(\mu >0\), so that
In addition, for such \(C^*\), we can find a constant \(\delta ^*>\delta \) satisfying
Consider the auxiliary free boundary problem
Applying the conclusion of [4, Theorem 1.1] and following the argument as the proof of Theorem 2.1, one can easily derive that (3.17) admits a unique solution (v, p, q) defined for all \(t\ge 0\). To stress the dependence of the solution on \(\mu \), we always write \((v_{\mu },p_{\mu },q_{\mu })\). It now follows from Proposition 3.7 that
In what follows, we want to prove \(q_{\mu }(\omega )-p_{\mu }(\omega )>\ell ^*\) for large \(\mu \). To this end, we first choose two smooth functions l(t) and r(t), in which \(-l(t),r(t)\) are decreasing on \([0,\rho \omega ]\) while increasing on \([\rho \omega ,\omega ]\) with \(-l(0)=r(0)=h_0/2\), \(l'(0)>0,r'(0)<0\), \(l(\rho \omega )<0<r(\rho \omega )\) and \(-l(\omega )=r(\omega )=\ell ^*\). Clearly, \(l(t)\ge l(0)=-h_0/2,r(t)\le r(0)=h_0/2\) for \(t\in [0,\rho \omega ]\). Then we look at the following initial value problem
where the initial value \({\underline{u}}_0\in C([-h_0/2,h_0/2])\) satisfies
By applying [4, Lemma 2.3] and a step by step method, we see that (3.18) has a unique positive solution \({\underline{w}}(t,x)\) defined for all \(t\in [0,\omega ]\). Recall that \(J(0)>0\), which means that there is \(\epsilon _0\) and \(\gamma _0>0\) such that \(J(x)>\gamma _0\) for \(|x|<\epsilon _0\). Based on our choice of l(t), r(t) and \({\underline{u}}_0\), there must be a constant \(\mu _0>0\) such that for all \(\mu \ge \mu _0\),
and similarly, there holds
On the other hand, we can establish a comparison principle for equation (3.17) similar to Proposition 3.7. Thus, it can be inferred from (3.17)-(3.21) that
This also means that \(q_{\mu }(\omega )-p_{\mu }(\omega )\ge r(\omega )-l(\omega )=2\ell ^*>\ell ^*\) and consequently, (3.16) holds true.
Combining (3.16) with Remark 3.2, we have \(\lambda _{(g_{\mu _0}(\omega ),h_{\mu _0}(\omega ))}<0\), and then by Lemma 3.14, spreading occurs if \(\mu =\mu _0\). Notice that the functions \(u_{\mu },-g_{\mu }\) and \(h_{\mu }\) are all nondecreasing in \(\mu >0\). This together with the above arguments implies that vanishing occurs for all small \(\mu >0\) and spreading occurs for all large \(\mu >0\). Define
Then we can apply a similar discussion to the proof of [4, Theorem 3.14] to conclude that vanishing occurs for \(0<\mu \le \mu _*\) and spreading occurs for \(\mu >\mu _*\).
3. For the case \((1-\rho )(a-d)-\rho \delta >0\), we can see from Proposition 3.1 that \(\lambda _{(-h_0,h_0)}<0\). This together with Lemma 3.14 allows us to conclude that spreading always occurs. This completes the proof. \(\square \)
4 Semi-Waves of (1.10)
This section is devoted to proving the existence and uniqueness of a semi-wave \(\Psi (t,\xi )\) of (1.10), and completing the proof of Theorem 1.5, which is useful to construct some suitable upper- and lower-solutions to study the asymptotic profiles of spreading solutions of (1.7). Firstly, we prove the existence of solutions to the following nonlocal problem
i.e., for any constant speed c in certain range, there is a positive \(\omega \)-periodic solution \(\Phi =\Phi ^c(t,x)\in C^{1,1}((i\omega ,(i+\rho )\omega ]\times (-\infty ,0))\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times (-\infty ,0))\) of (P). Then we use the solution \(\Phi \) of (P) as a lower-solution to show the existence of solution \(\Psi \) to (1.10). This allows us to prove that for any given \(\mu >0\), there exists a unique nonnegative continuous function \(k=k(t)\) such that
4.1 Some Preparatory Results
To fully understand the solution to (P), we now recall some results for the following nonlocal dispersal problem on \({\mathbb {R}}\):
where
and J satisfies the assumptions \(\mathbf{(J)}\) and \(\mathbf{(J1)}\). It is easy to verify that (H1) and (H2) in [16] hold. Then following the argument as in [16, Theorem 2.1], we obtain that for any nonnegative, bounded and continuous initial value \(u_0\), (4.2) has a unique nonnegative and bounded classical solution \(U(t,x;u_0)\) defined for all \(t\ge 0\).
An \(\omega \)-periodic traveling wave solution of (4.2) takes the following form:
where \(c\in (0,+\infty )\) denotes the wave speed. Substituting (4.3) into (4.2) yields that
for \(t\ge 0,x\in {\mathbb {R}}\). For the existence and nonexistence of periodic traveling waves, we can see from [16, Theorems 4.1 and 4.5] the following results.
Lemma 4.1
Assume that \(\mathbf{(J)},\mathbf{(J1)}\) are satisfied and \((1-\rho )a-\rho \delta >0\). Let
Then the problem (4.2) has a \(\omega \)-periodic traveling wave \(\phi (t,x-ct)\) connecting \(z^*(t)\) and 0, in which \(\phi (t,\xi )\) is nonincreasing in \(\xi \), if and only if \(c\ge c^*\). Moreover, for each \(c\ge c^*\), \(\phi (t,x-ct)\) is also a classical solution of (4.2).Footnote 1
In the following, we provide two strong maximum principle for later discussion, in which the proof of the second maximum principle is similar to that of the first one.
Lemma 4.2
Suppose that \(v\in C({\mathbb {R}}\times {\mathbb {R}})\cap C^{1,1}((i\omega ,(i+\rho )\omega ]\times ({\mathbb {R}}\setminus \{0\}))\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times ({\mathbb {R}}\setminus \{0\}))\) is \(\omega \)-periodic in t, nonincreasing in \(\xi \) and satisfies
where the constants \(d,\delta >0\) and \(c\ge 0\), and \(k,f\in L_{\mathrm {loc}}^{\infty }([0,\omega ]\times {\mathbb {R}})\). If \(v(t,\xi )\ge ,\not \equiv 0\) for \(t\in [0,\omega ],\xi <0\), then \(v(t,\xi )>0\) for all \(t\in (\rho \omega ,\omega ],\xi <0\). Furthermore,
-
(i)
if \(c=0\), then \(v(t,\xi )>0\) for all \(t\in [0,\omega ],\xi <0\);
-
(ii)
if \(c>0\) and \(v(\omega ,\xi )>0\) on \([0,+\infty )\), then \(v(t,\xi )>0\) for all \(t\in [0,\omega ],\xi <0\).
Proof
On the contrary, assume that there exists \((t_0,\xi _0)\in [0,\omega ]\times (-\infty ,0)\) such that \(v(t_0,\xi _0)=0\). There are three cases to be considered.
Case 1. \(t_0\in (\rho \omega ,\omega )\). Since \(v(t_0,\xi _0)=0\) and \(v(t,\xi )\not \equiv 0\), we can find a sequence \(\{\xi _n\}_{n=1}^{\infty }\) with \(\xi _n\rightarrow \xi _0\) such that \(v(t_0,\xi _n)>0\). Notice that \(v_t(t_0,\xi _0)=v_{\xi }(t_0,\xi _0)=0\). It can be inferred from the inequality satisfied by v that
which means that \(v(t_0,\xi )=0\) for all \(\xi \) near \(\xi _0\) in view of \(J(0)>0\). This is a contradiction with \(v(t_0,\xi _n)>0\). Hence, \(v(t,\xi )>0\) in \((\rho \omega ,\omega )\times (-\infty ,0)\).
Case 2. \(t_0=\omega \), namely, \(v(\omega ,\xi _0)=0\). This together with the conclusion in Case 1 means that \(v_t(\omega ,\xi _0)\le 0\). Notice that \(v_{\xi }(\omega ,\xi _0)=0\). Thus, we can derive from the inequality satisfied by v that
By a similar argument as in Case 1, a contradiction can be obtained.
Case 3. \(t_0\in [0,\rho \omega ]\). When \(t\in (0,\rho \omega ],\xi <0\), we have
which is equivalent to
Then \(\big (e^{\delta t}v(t,x-ct)\big )_t\ge 0\), which implies that \(e^{\delta t_0}v(t_0,x-ct_0)\ge e^{\delta t}v(t,x-ct)\) for \(t\in [0,t_0]\) and \(x-ct<0\). Since \(v(t_0,\xi _0)=0\), there holds that \(v(0,\xi _0+ct_0)=0\). It follows from the periodicity of v that \(v(\omega ,\xi _0+ct_0)=v(0,\xi _0+ct_0)=0\). For this situation, we can induce a contradiction under the assumption in part (i) and part (ii). This completes the proof. \(\square \)
Lemma 4.3
Suppose that \(v\in C({\mathbb {R}}\times {\mathbb {R}})\cap C^{1,1}((i\omega ,(i+\rho )\omega ]\times {\mathbb {R}})\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times {\mathbb {R}})\) is \(\omega \)-periodic in t, nonincreasing in \(\xi \) and satisfies
where the constants \(d,\delta >0\) and \(c\ge 0\), and \(k,f\in L_{\mathrm {loc}}^{\infty }([0,\omega ]\times {\mathbb {R}})\). If \(v(t,\xi )\ge ,\not \equiv 0\) for \(t\in [0,\omega ],\xi \in {\mathbb {R}}\), then \(v(t,\xi )>0\) for all \(t\in [0,\omega ],\xi \in {\mathbb {R}}\).
4.2 A Perturbed Semi-wave Problem
In this subsection, we study an auxiliary problem
with the periodic constraints \(\Phi (t,\xi )\equiv \Phi (t+\omega ,\xi )\), where \(\sigma \in (0,\min _{t\in [0,\omega ]}z^*(t))\) and \(c\in (0,c^*)\). We aims to prove that (4.6) admits a solution \(\Phi _{\sigma }\), which converges to a \(\omega \)-periodic solution of (P) as \(\sigma \rightarrow 0\).
Definition 4.4
A function \({\overline{\Phi }}\in C^{1,1}((i\omega ,(i+\rho )\omega ]\times (-\infty ,0))\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times (-\infty ,0))\cap C({\mathbb {R}}\times {\mathbb {R}})\) is called a upper-solution of (4.6) if
\({\underline{\Phi }}\) is called a lower-solution of (4.6) if the above inequalities are reversed.
Lemma 4.5
Assume that \(\mathbf{(J)},\mathbf{(J1)}\) are satisfied and \((1-\rho )a-\rho \delta >0\). Then for any \(\sigma \in (0,\min _{t\in [0,\omega ]}z^*(t))\) and \(c\in (0,c^*)\), (4.6) admits a solution \(\Phi _{\sigma }(t,\xi )\) which is \(\omega \)-periodic in t and strictly decreasing in \(\xi \). Moreover, \(\lim _{\xi \rightarrow -\infty }\Phi (t,\xi )=z^*(t)\) uniformly in \(t\in [0,\omega ]\).
Proof
Let \(\phi (t,\xi )\) be a \(\omega \)-periodic traveling wave of (4.2) with minimal speed \(c^*\), which is nonincreasing in \(\xi \) and connects \(z^*(t)\) and 0. For any \(\epsilon >0\), define \(v(t,\xi ):=\phi (t,\xi -\epsilon )-\phi (t,\xi )\). Then \(v(t,\xi )\ge ,\not \equiv 0\) in \([0,\omega ]\times {\mathbb {R}}\). By Lemma 4.3, we can derive that \(v(t,\xi )>0\) in \([0,\omega ]\times {\mathbb {R}}\), namely, \(\phi (t,\xi )\) is strictly decreasing in \(\xi \) for any fixed \(t\in [0,\omega ]\). For any fixed \({\bar{t}}\in [0,\omega ]\), there should be a unique \({\bar{\xi }}<0\) such that
From the implicit function theorem, there is neighborhood \(O({\bar{t}},{\bar{\xi }})\subset [0,\omega ]\times {\mathbb {R}}\) of \(({\bar{t}},{\bar{\xi }})\), in which the equation \(\phi (t,\xi )=\sigma \) uniquely determine a continuous function \({\tilde{\xi }}={\tilde{\xi }}(t)\) defined on \(({\bar{t}}-\gamma ,{\bar{t}}+\gamma )\cap [0,\omega ]\) (\(\gamma >0\) is sufficiently small), such that for \(t\in ({\bar{t}}-\gamma ,{\bar{t}}+\gamma )\cap [0,\omega ]\), \((t,{\tilde{\xi }}(t))\in O({\bar{t}},{\bar{\xi }})\), \({\bar{\xi }}={\tilde{\xi }}({\bar{t}})\) and \(\phi (t,{\tilde{\xi }}(t))\equiv \sigma \). By the arbitrariness of \({\bar{t}}\), we obtain a function \({\tilde{\xi }}(t)\in C([0,\omega ])\) satisfying \({\tilde{\xi }}(t)<0\) on \([0,\omega ]\) and \(\phi (t,{\tilde{\xi }}(t))\equiv \sigma \). By the translation \({\tilde{\phi }}(t,\xi )=\phi (t,\xi +{\tilde{\xi }}(t))\), for the simplicity of notation, we may suppose that
Due to the periodicity of \(\phi \) in t, we can extend the domain of \(\phi (t,\xi )\) from \([0,+\infty )\times {\mathbb {R}}\) to \({\mathbb {R}}^2\). In the following, we regard \(\phi \) as a \(\omega \)-periodic function defined on \({\mathbb {R}}^2\) satisfying that \(\phi (t,-\infty )=z^*(t)\) and \(\phi (t,+\infty )=0\) uniformly in \(t\in [0,\omega ]\). If it is necessary to stress the dependence of \(\phi (t,\xi )\) on \(\sigma \), we write \(\phi (t,\xi )=\phi ^{\sigma }(t,\xi )\). Define
We next show that for any \(c\in (0,c^*)\), \({\underline{\Phi }}(t,x)\) is a lower-solution of (4.6). Clearly, \({\underline{\Phi }}(t,\xi )=\sigma \) for \(t\in {\mathbb {R}}, \xi \ge 0\). For \(t\in {\mathbb {R}}, \xi <0\), we have
Likewise, we see that \({\overline{\Phi }}(t,\xi )\equiv z^*(t)\) is a upper-solution of (4.6). By the construction of \({\overline{\Phi }}\) and \({\underline{\Phi }}\), we have that \({\overline{\Phi }}(t,\xi )\ge {\underline{\Phi }}(t,\xi )\) for \((t,\xi )\in {\mathbb {R}}^2\). We now define a set
and an operator \({\mathcal {H}}:\Sigma \rightarrow C({\mathbb {R}}\times {\mathbb {R}})\) by
where \(\beta =d+\delta +2b\sup _{t\in [0,\omega ]}z^*(t)\). Then we can rewrite (4.6) as the following equation
It is easy to check that equation (4.7) is equivalent to the following equation:
Hence we easily see that the existence of solutions of (4.6) is reduced to that of the fixed point of the operator \({\mathcal {F}}:\Sigma \rightarrow C([0,+\infty )\times {\mathbb {R}})\) defined as
In the following, we completes our proof by three claims. \(\square \)
Claim 1. \({\mathcal {F}}\) is a nondecreasing operator and maps \(\Sigma \) to into \(\Sigma \). In addition, if \(\Phi (t,\xi )\in \Sigma \) is nonincreasing with respect to \(\xi \), then \({\mathcal {F}}(\Phi )(t,\xi )\) is nonincreasing in \(\xi \in (-\infty ,0)\).
For any \(\Phi _1,\Phi _2\in \Sigma \) with \(\Phi _1(t,\xi )\le \Phi _2(t,\xi )\) on \([0,\omega ]\times {\mathbb {R}}\), there holds that for \(t\in [0,\omega ], -\infty<\xi <0\),
It then follows from the definition of \({\mathcal {F}}\) in (4.9) that \({\mathcal {F}}(\Phi _1)(t,\xi )\le {\mathcal {F}}(\Phi _2)(t,\xi )\) on \([0,\omega ]\times {\mathbb {R}}\), which implies that
for all \(\Phi \in \Sigma \). Moreover, we have that for any \(t\in [0,\omega ],\xi <0\),
Similarly, \({\mathcal {F}}({\overline{\Phi }})(t,\xi )\le {\overline{\Phi }}(t,\xi )\) for \(t\in [0,\omega ],\xi <0\). Meanwhile, \({\underline{\Phi }}(t,x)=\sigma ={\mathcal {F}}({\underline{\Phi }})(t,\xi )\) and \({\mathcal {F}}({\overline{\Phi }})(t,\xi )=\sigma <{\overline{\Phi }}(t,\xi )\) for \(t\in [0,\omega ],\xi \ge 0\). If \({\hat{\Phi \in }} \Sigma \), then we see that for \(t\in {\mathbb {R}}\) and \(\xi <0\),
Therefore, \({\mathcal {F}}(\Sigma )\subset \Sigma \). If \(\Phi \in \Sigma \) is nonincreasing in \(\xi \in {\mathbb {R}}\), then for any \(t\in [0,\omega ]\) and \(\xi _1<\xi _2<0\), it holds that
Thus, we obtain
This claim is proved.
Claim 2. \({\mathcal {F}}\) has a fixed point in \(\Sigma \).
Let
Using the monotonicity of \({\mathcal {F}}\) on \(\Phi \), it can be derived that
for \(n=1,2,\cdots ,(t,\xi )\in [0,\omega ]\times {\mathbb {R}}\). We define
Obviously, \(\Phi _{\sigma }(t,\xi )=\sigma \) for \(t\in [0,\omega ]\) and \(\xi \ge 0\). Since for \(t\in [0,\omega ],\xi <0\),
where \(C_0:=\beta +2d+a+b\sup _{t\in [0,\omega ],\xi <0}{\overline{\Phi }}(t,\xi )\), by applying the Lebesgue’s dominated convergence theorem, we obtain
which means that \(\Phi _{\sigma }\) is a fixed point of \({\mathcal {F}}\).
Claim 3. \(\Phi _{\sigma }(t,\xi )\) is \(\omega \)-periodic in t and strictly decreasing in \(\xi \in (-\infty ,0)\). Moreover, \(\lim _{\xi \rightarrow -\infty }\Phi (t,\xi )=z^*(t)\) uniformly in \(t\in [0,\omega ]\).
Notice that \({\overline{\Phi }}\) and \({\underline{\Phi }}\) are periodic in t and \({\underline{\Phi }}\) is nonincreasing in \(\xi <0\). This combined with Claim 1 allows us to conclude that \(\Phi ^n\in \Sigma \) and \(\Phi ^n\) is nonincreasing in \(\xi <0\). Thus, \(\Phi _{\sigma }(t,\xi )\) is \(\omega \)-periodic in t and nonincreasing in \(\xi \in (-\infty ,0)\). Since \(\phi ^{\sigma }(t,\xi )={\underline{\Phi }}(t,\xi )\le \Phi _{\sigma }(t,\xi )\le {\overline{\Phi }}(t,\xi )=z^*(t)\) for \(t\in [0,\omega ],\xi <0\), then \(\lim _{\xi \rightarrow -\infty }\Phi _{\sigma }(t,\xi )=z^*(t)\) point by point. Due to the monotonicity of \(\Phi _{\sigma }(t,\xi )\) with respect to \(\xi \), by Dini’ Theorem, we have \(\lim _{\xi \rightarrow -\infty }\Phi _{\sigma }(t,\xi )=z^*(t)\) uniformly in \(t\in [0,\omega ]\). If \(\epsilon >0\), then \(v(t,\xi )=\Phi _{\sigma }(t,\xi -\epsilon )-\Phi _{\sigma }(t,\xi )\ge 0\). Since \(\Phi _{\sigma }(t,-\infty )=z^*(t)\) and \(\Phi _{\sigma }(t,0)=\sigma <\min _{t\in [0,\omega ]}z^*(t)\), there must exist \((t_0,\xi _0)\in [0,\omega ]\times (-\infty ,0)\) such that \(\partial _{\xi }\Phi _{\sigma }(t_0,\xi _0)<0\), which implies that \(v(t_0,\xi _0)=\Phi _{\sigma }(t_0,\xi _0-\epsilon )-\Phi _{\sigma }(t_0,\xi _0)>0\). It follows that \(v(t,\xi )\not \equiv 0\). By applying Lemma 4.2 to v, we see that \(v(t,\xi )>0\) for \(t\in [0,\omega ],\xi <0\), which gives the strict monotonicity of \(\Phi _{\sigma }(t,\xi )\) in \(\xi \in (-\infty ,0)\). \(\square \)
Lemma 4.6
Let \(\Phi _{\sigma }^c\) be the solution of (4.6) obtained through the iteration process in Lemma 4.5, with \(c\in (0,c^*)\) and \(\sigma \in (0,\min _{t\in [0,\omega ]}z^*(t))\). Then
Proof
1. For fixed \(c\in (0,c^*)\), we will adopt the definition of \({\mathcal {F}}\) and \(\Phi ^n\) in Lemma 4.5. To this end, we need to distinguish them between \(\sigma _1\) and \(\sigma _2\) by writing \({\mathcal {F}}^{(i)}\) and \(\Phi ^{i,n}\) for \(\sigma _i(i=1,2)\). It then holds that
Clearly, the function \({\underline{\Phi }}\) defined in Lemma 4.5 is nondecreasing in \(\sigma \). Thus, \({\mathcal {F}}(\Phi )(t,x)\) is nondecreasing with respect to \(\Phi \) and \(\sigma \), respectively, which implies that
if \(\Phi ^{1,n}(t,x)\le \Phi ^{2,n}(t,x)\) on \([0,\omega ]\times {\mathbb {R}}\). From the monotonicity of \({\underline{\Phi }}\) in \(\sigma \), we have \(\Phi ^{1,n}(t,x)\le \Phi ^{2,n}(t,x)\). The above argument together with the induction process yields that \(\Phi ^{1,n}(t,x)\le \Phi ^{2,n}(t,x)\) for all \(n=0,1,2,\cdots \). Hence, \(\Phi _{\sigma _1}^c(t,\xi )\le \Phi _{\sigma _2}^c(t,\xi )\), as desired.
2. For fixed \(\sigma \in (0,\min _{t\in [0,\omega ]}z^*(t))\), we will use the notations \({\mathcal {F}}_{(i)}\) and \(\Phi _i^n\) for \({\mathcal {F}}\) and \(\Phi ^n\) when \(c=c_i(i=1,2)\). Then we see from Lemma 4.5 that
In view of (4.8), if \(c_1<c_2\), then
which means that
Notice that \(\Phi _{\sigma }^{c_1}(t,\xi )\ge {\underline{\Phi }}(t,\xi )=\Phi _2^0(t,\xi )\). Since \({\mathcal {F}}(\Phi )(t,x)\) is nondecreasing in \(\Phi \), we have
By induction, there must be \(\Phi _{\sigma }^{c_1}(t,\xi )\ge \Phi _2^n(t,\xi )\) for all \(n=0,1,2,\cdots \). Therefore, \(\Phi _{\sigma }^{c_1}(t,\xi )\ge \Phi _{\sigma }^{c_2}(t,\xi )\). \(\square \)
4.3 Existence of Periodic Solution to (P)
In this part, we discuss the existence of \(\omega \)-periodic solution to (P).
Theorem 4.7
Assume that \(\mathbf{(J)},\mathbf{(J1)}\) hold and \((1-\rho )a-\rho \delta >0\). Then for each \(c\in (0,c^*)\), problem (P) admits a \(\omega \)-periodic solution \(\Phi ^c\), which is nonincreasing in \(\xi \in (-\infty ,0]\) for fixed \(c\in (0,c^*)\), and is nonincreasing in \(c\in (0,c^*)\) for fixed \(\xi <0\).
Proof
We finish the proof of this theorem by several claims.
Claim 1. For each \(c\in (0,c^*)\), problem (P) admits a \(\omega \)-periodic solution \(\Phi ^c\), which satisfies \(0\le \Phi ^c(t,\xi )\le z^*(t)\) in \([0,\omega ]\times (-\infty ,0)\) and \(\Phi ^c(t,\xi )\not \equiv 0\). Moreover, \(\Phi ^c(t,\xi )\) is nonincreasing in \(\xi \in (-\infty ,0)\) for fixed \(t\in [0,\omega ]\).
Denote \(e_*=\frac{1}{2}\min _{t\in [0,\omega ]}z^*(t)\). Let \(\sigma _n\) be a decreasing sequence in \((0,e_*)\), which satisfies \(\sigma _n\rightarrow 0\) as \(n\rightarrow \infty \). Note that \(\Phi _{\sigma _n}\) satisfies that, for \(t\in [0,\omega ],\xi <0\),
and for \(t\in [0,\omega ],\xi \ge 0\), \(\Phi _{\sigma _n}(t,\xi )=\sigma _n\). It then follows that for \(t\in [0,\omega ],\xi <0\),
Evidently, \(\Phi _{\sigma _n}\) and by the equation subsequently \(\partial _t\Phi _{\sigma _n}, \partial _{\xi }\Phi _{\sigma _n}\) are uniformly bounded for \(n\in {\mathbb {N}}\). By applying the Arzela-Ascoli theorem and a standard argument involving a diagonal process of choosing subsequences, we observe that there is a subsequence of \(\{ \Phi _{\sigma _n} \}_{n=1}^{\infty }\), still denoted by \(\{ \Phi _{\sigma _n} \}_{n=1}^{\infty }\), and a function \(\Phi ^c\in C({\mathbb {R}}\times {\mathbb {R}})\) such that \(\lim _{n\rightarrow \infty }\Phi _{\sigma _n}(t,\xi )=\Phi ^c(t,\xi )\) in \(C_{\mathrm {loc}}([0,\omega ]\times {\mathbb {R}})\). Moreover, \(\Phi ^c(t,\xi )\) is \(\omega \)-periodic in t and nonincreasing with respect to \(\xi \) with \(\Phi ^c(t,\xi )=0\) for \(t\in [0,\omega ],\xi \ge 0\).
For any fixed \((t,\xi )\in [0,\omega ]\times (-\infty ,0)\), by the dominated convergence theorem, we can deduce from (4.11) that, by taking \(n\rightarrow \infty \), there holds that
which gives that
Due the monotonicity of \(\Phi _{\sigma _n}\) with respect to \(\sigma _n\) from Lemma 4.6, we see that \(0\le \Phi ^c(t,\xi )\le \Phi _{\sigma _n}(t,\xi )\le z^*(t)\). Notice that \(\Phi ^c(t,\xi )\) is defined in \({\mathbb {R}}^2\). Since \(\Phi ^c(t,\xi )\) is \(\omega \)-periodic in t, \(\Phi ^c(t,\xi )\) also solves (P).
In the following, we show that \(\Phi ^c\not \equiv 0\). For fixed \(t=0\), there should be a unique \({\tilde{\xi }}_n<0\) such that
since \(\Phi _{\sigma _n}\) is strictly decreasing in \(\xi \in (-\infty ,0)\), \(\Phi _{\sigma _n}(t,-\infty )=z^*(t)\) and \(\Phi _{\sigma _n}(t,0)=\sigma _n\in (0,e_*)\). From Lemma 4.6, we have that \({\tilde{\xi }}_m\le {\tilde{\xi }}_n\) for \(m>n\). Now, we need to consider two possible cases:
-
Case 1. \({\tilde{\xi }}_n\rightarrow -\infty \) as \(n\rightarrow \infty \);
-
Case 2. \({\tilde{\xi }}_n\rightarrow \xi _0\) as \(n\rightarrow \infty \) for some \(\xi _0\in (-\infty ,0)\).
For Case 1, we set
Since \({\tilde{\Phi }}_n\) and by the equation subsequently \(\partial _t{\tilde{\Phi }}_n,\partial _{\xi }{\tilde{\Phi }}_n\) are uniformly bounded, we can find \({\tilde{\Phi }}_{\infty }(t,\xi )\in C({\mathbb {R}}\times {\mathbb {R}})\) and a subsequence of \({\tilde{\Phi }}_n\), still denoted by itself, such that \(\lim _{n\rightarrow \infty }{\tilde{\Phi }}_n(t,\xi )={\tilde{\Phi }}_{\infty }(t,\xi )\) in \(C_{\mathrm {loc}}([0,\omega ]\times {\mathbb {R}})\). Here, \({\tilde{\Phi }}_{\infty }(t,\xi )\) is \(\omega \)-periodic in t and nonincreasing in \(\xi \), and \({\tilde{\Phi }}_{\infty }(0,0)=e^*\). By (4.11), we see that
By the dominated convergence theorem, for fixed \((t,\xi )\in [0,\omega ]\times {\mathbb {R}}\), as \(n\rightarrow \infty \), the above equation gives that
By differentiating this equation, one easily have
which is a contradiction with the fact that \(c^*\) is the minimal speed. Hence, Case 1 can not happen.
The above argument implies that Case 2 must happen. Suppose that \(\Phi ^c(t,\xi )\equiv 0\). Then for any bounded closed set \([0,\omega ]\times [l_1,l_2]\subset [0,\omega ]\times (-\infty ,0)\) and any small \(\epsilon >0\), there exists large \(N>0\) independent of \((t,\xi )\), such that
which contradicts with \(\Phi _{\sigma _n}(0,{\tilde{\xi }}_n)= e_*\) for any \(n\in {\mathbb {N}}\). Hence, \(\Phi ^c\not \equiv 0\).
Claim 2. \(\Phi ^c>0\) in \((\rho \omega ,\omega ]\times (-\infty ,0)\). Furthermore, \(\Phi ^c(t,\xi )\rightarrow z^*(t)\) uniformly for \(t\in [0,\omega ]\) as \(\xi \rightarrow -\infty \).
Since \(\Phi ^c\not \equiv 0\), we have \(\Phi ^c>0\) in \((\rho \omega ,\omega ]\times (-\infty ,0)\) by the boundedness of \(\Phi ^c\) and Lemma 4.2. Since \(\Phi ^c(t,\xi )\) is nonincreasing in \(\xi \in (-\infty ,0]\), by the uniform boundedness of \(\Phi ^c\), for each \(t\in [0,\omega ]\), there is \(0\le z_*(t)\le z^*(t)\) such that \(\Phi ^c(t,\xi )\) convergence to \(z_*(t)\) point by point as \(\xi \rightarrow -\infty \). Clearly, \(\partial _t\Phi ^c\) and \(\partial _{\xi }\Phi ^c\) are uniformly bounded by the equation (4.13) and the uniform boundedness of \(\Phi ^c\). Again, applying the Arzela-Ascoli theorem, we see that
for some \(\Phi _{\infty }(t)\in C([0,\omega ])\). In view the uniqueness of the limit, \(\Phi _{\infty }(t)\equiv z_*(t)\) for \(t\in [0,\omega ]\). Thus, \(\lim _{\xi \rightarrow -\infty }\Phi ^c(t,\xi )=z_*(t)\) in \(C([0,\omega ])\). We claim that
uniformly in \([0,\omega ]\). Since \(\Phi ^c(t,x)\) is nonincreasing in \(\xi \), there holds that
uniformly in \([0,\omega ]\) as \(\xi \rightarrow -\infty \). On the other hand,
which verifies (4.14).
Now, we show that \(z_*(t)\equiv z^*(t)\). Owing to \(\Phi ^c=0\) in \([0,\omega ]\times [0,+\infty )\), we rewrite (4.12) as
Letting \(\xi \rightarrow -\infty \), by the dominated convergence theorem, the above equation yields that
which implies that
If \(\Phi ^c>0\) in \((\rho \omega ,\omega ]\times (-\infty ,0)\), then \(z_*(t)>0\) in \((\rho \omega ,\omega ]\), which together with the uniqueness of the positive \(\omega \)-periodic solution of (1.9) implies that \(z_*(t)\equiv z^*(t)\). Therefore, \(\Phi ^c(t,\xi )\rightarrow z^*(t)\) uniformly for \(t\in [0,\omega ]\) as \(\xi \rightarrow -\infty \).
Claim 3. \(\Phi ^c(t,\xi )\) is nonincreasing in \(c\in (0,c^*)\) for fixed \(\xi <0\).
From Lemma 4.6 and the proof of Claim 1, we see that for any \(0<c_1<c_2<c^*\), \(v(t,\xi ):=\Phi ^{c_1}(t,\xi )-\Phi ^{c_2}(t,\xi )\ge 0\) in \([0,\omega ]\times (-\infty ,0)\) and the monotonicity of \(\Phi ^c\) with respect to \(c\in (0,c^*)\) is obtained. The proof of this theorem is finished. \(\square \)
4.4 Existence, Uniqueness and Monotonicity of Semi-wave
The main purpose of this subsection is to prove the existence and uniqueness of \(\omega \)-periodic semi-wave of (1.10) provided a nonnegative continuous function k(t). In (1.10), k(t) is required to belong to \(C([(i+\rho )\omega ,i\omega ])\). We extend it to be zero on \((i\omega ,(i+\rho )\omega )\) so that \(k\in C([(i+\rho )\omega ,i\omega ])\cap L^{\infty }(i\omega ,(i+1)\omega )\). Then \(k(t+\omega )=k(t)\) a.e. for \(t\in {\mathbb {R}}\). We use E to denote the set of all such extended functions. In one period of \((0,\omega )\), k is equipped with the supremum norm.
Definition 4.8
A function \({\overline{\Psi }}\in C^{1,1}((i\omega ,(i+\rho )\omega ]\times (-\infty ,0))\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times (-\infty ,0))\cap C({\mathbb {R}}\times {\mathbb {R}})\) is called a upper-solution of (1.10) if
where \(i\in {\mathbb {Z}}\). \({\underline{\Psi }}\) is called a lower-solution of (1.10) if the above inequalities are reversed.
Theorem 4.9
Assume that \(\mathbf{(J)},\mathbf{(J1)}\) hold and \((1-\rho )a-\rho \delta >0\). Then for any given nonnegative function \(k(t)\in E\) with \(0\le k(t)< c^*\) in \(t\in [\rho \omega ,\omega ]\), the equation (1.10) admits a unique \(\omega \)-periodic semi-wave solution \(\Psi ^k\). Moreover, \(\Psi ^k(t,\xi )\) is strictly decreasing in \(\xi \in (-\infty ,0]\) for fixed \(k\in E\), and \(\Psi ^{k_1}(t,\xi )>\Psi ^k(t,\xi )\) provided that \(k_1\le ,\not \equiv k\) for fixed \(\xi <0\), where \(k_1\) is a nonnegative function in E.
Proof
Since \(0\le k(t)< c^*\), we can choose a constant
such that \(0\le k(t)<{{\hat{c}}}<c^*\). Define \({\underline{\Psi }}(t,\xi )\equiv \Phi ^{{{\hat{c}}}}(t,\xi )\), where \(\Phi ^c\big ( c\in (0,c^*) \big )\) is the \(\omega \)-periodic solution of (P) obtained in Theorem 4.7. Clearly, \({\underline{\Psi }}(t,\xi )=0\) on \([0,\omega ]\times [0,+\infty )\). We see that for \(t\in (i\omega ,(i+\rho )\omega ],\xi <0\),
while for \(t\in ((i+\rho )\omega ,(i+1)\omega ],\xi <0\),
which means that \({\underline{\Psi }}(t,\xi )\) is a lower-solution of (1.10). Similarly, we can verify that \({\overline{\Psi }}(t,\xi )\equiv z^*(t)\) is a upper-solution of (1.10). From Theorem 4.7, we have that \({\overline{\Psi }}(t,\xi )\ge {\underline{\Psi }}(t,\xi )\) for \(t\ge 0,\xi <0\).
By a similar manner as in the proof of Lemma 4.5, we define a set
and an operator \({\mathcal {G}}:\Lambda \rightarrow C({\mathbb {R}}\times {\mathbb {R}})\) by
where \(\beta =d+\delta +2b\sup _{t\in [0,\omega ]}z^*(t)\). Then the equation (1.10) can be rewritten as
Define the operator \({\mathcal {A}}:\Lambda \rightarrow C({\mathbb {R}}\times {\mathbb {R}})\) by
It is easy to verify that the existence of solution of (1.10) is reduced to finding a fixed point of the operator \({\mathcal {A}}\). Following the argument as in the proof of Lemma 4.5, we obtain that the operator \({\mathcal {A}}\) admits a fixed point \(\Psi ^k(t,\xi )\) in \(\Lambda \), which is \(\omega \)-periodic in t and strictly decreasing in \(\xi \in (-\infty ,0]\). Furthermore, \(\Psi ^k(t,-\infty )=z^*(t)\) uniformly for \(t\in [0,\omega ]\). Thus, we have proved the existence of \(\omega \)-periodic solution to (1.10).
Claim 1. (1.10) has at most one positive \(\omega \)-periodic solution.
Suppose that (1.10) has two positive \(\omega \)-periodic solutions \(\Psi _1\) and \(\Psi _2\) satisfying \(\Psi _i(t,-\infty )=z^*(t)\) and \(\Psi _i(t,0)=0,i=1,2\). Similar to [25, Theorem 1.3], for any given \(\varepsilon >0\), we define
where \(q(t,\xi )\) is a positive and bounded function satisfying
and will be determined later. Clearly, \(M_{\varepsilon }\ne \emptyset \). Set \(m_{\varepsilon }:=\inf M_{\varepsilon }\ge 1\). It is easy to see that for any \(0<\varepsilon _1<\varepsilon _2\), \(m_{\varepsilon _1}\Psi _1(t,\xi )\ge \Psi _2(t,\xi )-\varepsilon _1q(t,\xi )>\Psi _2(t,\xi )-\varepsilon _2q(t,\xi )\). Thus, \(m_{\varepsilon _1}\ge m_{\varepsilon _2}\), namely, \(m_{\varepsilon }\) is nonincreasing with respect to \(\varepsilon \). Define
In the following, we will prove \(m_{\varepsilon }=1\) for all \(\varepsilon >0\), and then \(m^*=1\).
To the contrary, suppose that there is a \(\varepsilon _0>0\) such that \(m_{\varepsilon _0}>1\), which combined with the monotonicity of \(m_{\varepsilon }\) in \(\varepsilon \) means that \(m_{\varepsilon }\ge m_{\varepsilon _0}>1\) for \(0<\varepsilon \le \varepsilon _0\), and thus \(m^*>1\). For any given \(\varepsilon \in (0,\varepsilon _0]\), we denote
Then \(w_{\varepsilon }(t,\xi )\ge 0\) on \([0,\omega ]\times (-\infty ,0)\). Since \(w_{\varepsilon }(t,0)=\varepsilon q(t,0)>0\) and
from the definition of \(m_{\varepsilon }\), we can find \((t_{\varepsilon },\xi _{\varepsilon })\) such that
Otherwise, there exists \(\varepsilon >0\) such that \(w_{\varepsilon }(t,\xi )>0\) for any \((t,\xi )\in [0,\omega ]\times (-\infty ,0]\). Since \(m_{\varepsilon }>1\), for sufficiently small \(\theta >0\), there should be \(m_{\varepsilon }-\theta >1\) and \(\frac{w_{\varepsilon }(t,\xi )}{\Psi _1(t,\xi )}\ge \theta \), and hence \((m_{\varepsilon }-\theta )\Psi _1(t,\xi )\ge \Psi _2(t,\xi )-\varepsilon q(t,\xi )\) in \([0,\omega ]\times (-\infty ,0]\), which contradicts with the definition of \(m_{\varepsilon }\). Now, there are three possible cases for \(\xi _{\varepsilon }\):
-
Case (i): \(\xi _{\varepsilon _n}\rightarrow -\infty \) along some sequence \(\varepsilon _n\rightarrow 0^+\);
-
Case (ii): \(\xi _{\varepsilon _n}\rightarrow \xi ^*\in (-\infty ,0)\) along some sequence \(\varepsilon _n\rightarrow 0^+\);
-
Case (iii): \(\xi _{\varepsilon _n}\rightarrow 0\) along some sequence \(\varepsilon _n\rightarrow 0^+\).
For case (i), there is a sequence \(\{\varepsilon _n\}_{n=1}^{\infty }\subset (0,\varepsilon _0]\) with \(\lim _{n\rightarrow \infty }\varepsilon _n\rightarrow 0\) such that \(\lim _{n\rightarrow \infty }t_{\varepsilon _n}=t^*\in [0,\omega ]\) and \(\lim _{n\rightarrow \infty }\xi _{\varepsilon _n}=-\infty \). Notice that there exists \(\theta _{\varepsilon }>0\) such that when \((t,\xi )\in (0,\rho \omega ]\times (\xi _{\varepsilon }-\theta _{\varepsilon },\xi _{\varepsilon }+\theta _{\varepsilon })\),
while when \((t,\xi )\in (\rho \omega ,\omega ]\times (\xi _{\varepsilon }-\theta _{\varepsilon },\xi _{\varepsilon }+\theta _{\varepsilon })\),
provided that
In fact, we can construct \(q(t,\xi )\) such that (4.20) holds for \(t\in [0,\omega ]\) and \(\xi <-L\) with some sufficiently large \(L>0\). By \(\lim _{\xi \rightarrow -\infty }\Psi _i(t,\xi )=z^*(t),i=1,2\), we have
Then there is a sufficiently large \(L>0\) such that
where
in which \({\bar{a}}\) and \({\bar{b}}\) are defined in (4.2). One can easily verify that \(\int _0^{\omega }\nu (t)\mathrm dt=0\), since
We now choose \(q\in C^{1,1}((i\omega ,(i+\rho )\omega ]\times {\mathbb {R}})\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times {\mathbb {R}})\) satisfying that \(q(t,\xi )\) is \(\omega \)-periodic in t, \(q(t,\xi )=e^{\int _0^t\nu (s)\mathrm ds}\) for \(\xi <-L\), \(q(t,\xi )=0\) for \(\xi \ge 0\) and q is nonincreasing in \(\xi \in [-L,0]\) for fixed t. Thus, a direct calculation gives that
This verification means that (4.20) holds for \(\xi _{\varepsilon }<-L\), and hence (4.18) and (4.19) holds for \(\xi _{\varepsilon }<-L\). Since \(w_{\varepsilon }\ge 0\) on \([0,\omega ]\times (-\infty ,0)\), by (4.17), we obtain a contradiction from Lemma 4.2 (i) as \(\varepsilon _n\rightarrow 0^+\). Hence, case (i) can not happen.
In case (ii), there is a sequence \(\{\varepsilon _n\}_{n=1}^{\infty }\subset (0,\varepsilon _0]\) with \(\lim _{n\rightarrow \infty }\varepsilon _n\rightarrow 0\) such that \(\lim _{n\rightarrow \infty }(t_{\varepsilon _n},\xi _{\varepsilon _n})=(t^*,\xi ^*)\in [0,\omega ]\times (-\infty ,0)\). It then follows that
Let \(w(t,\xi )=m^*\Psi _1(t,\xi )-\Psi _2(t,\xi )\) for \(t\in [0,\omega ],\xi <0\). Since \(w(t,\xi )=\lim _{n\rightarrow \infty }w_{\varepsilon _n}(t,\xi )\), it holds that \(w(t,\xi )\ge 0\) for \(t\in [0,\omega ],\xi <0\) and \(w(t^*,\xi ^*)=0\). Due to \(m^*>1\), we see that \(w(t,0)=0\) and \(w(t,-\infty )=(m^*-1)z^*(t)>0\), which implies that \(w(t,\xi )\not \equiv 0\). Additionally, \(w(t,\xi )\) satisfies
By applying Lemma 4.2 to \(w(t,\xi )\), we have \(w(t,\xi )>0\) for all \(t\in [0,\omega ],\xi <0\), which is a contradiction with \(w(t^*,\xi ^*)=0\).
For the case (iii), we can also derive a contradiction at \(\xi ^*=0\) by a similar argument as in case (ii). Since these three possible cases can not happen, it holds that \(m^*=1\), which leads to that \(\Psi _1\ge \Psi _2\). Similarly, we can prove \(\Psi _2\ge \Psi _1\). Consequently, \(\Psi _1\equiv \Psi _2\) and the uniqueness of the positive \(\omega \)-periodic solution of (1.10) (if exists) follows immediately.
Claim 2. \(\Psi ^{k_1}(t,\xi )>\Psi ^k(t,\xi )\) provided that \(k_1\le ,\not \equiv k\) for fixed \(\xi <0\), where \(k_1\in E\) is nonnegative.
Assume that \(\Psi ^k\) is the unique positive solution of (1.10) and \(k_1\in E\) satisfies \(0\le k_1\le ,\not \equiv k\). Let \(\Psi ^{k_1}\) be the unique positive solution of (1.10) by replacing k as \(k_1\). Following the argument as in the proof of Lemma 4.6, we have \(\Psi ^{k_1}(t,\xi )\ge \Psi ^k(t,\xi )\). Meanwhile, we have \(\Psi ^{k_1}_{\xi }(t,\xi )<0\) for \((t,\xi )\in [0,\omega ]\times (-\infty ,0)\). Then \(\Psi ^{k_1}\) satisfies that
By applying Lemma 4.2 to \(v(t,\xi )=\Psi ^{k_1}(t,\xi )-\Psi ^k(t,\xi )\), it follows that either \(v(t,\xi )>0\) in \([0,\omega ]\times (-\infty ,0)\) or \(v(t,\xi )\equiv 0\). Once \(v(t,\xi )\equiv 0\), the above inequality for \(\Psi ^{k_1}\) becomes an equality, which is impossible since \(\Psi ^{k_1}_{\xi }(t,\xi )<0\) and \(k_1\le ,\not \equiv k\). Therefore, \(v(t,\xi )>0\) in \([0,\omega ]\times (-\infty ,0)\) and the monotonicity of \(\Psi ^k(t,\xi )\) with respect to k is obtained. This completes the proof of this theorem. \(\square \)
In the end of this subsection, we prove the following theorem describing the asymptotic profile of \(\Psi ^k\).
Theorem 4.10
Assume that \(\mathbf{(J)},\mathbf{(J1)}\) hold and \((1-\rho )a-\rho \delta >0\). Then the unique \(\omega \)-periodic solution \(\Psi ^k(t,\xi )\) of (1.10) satisfies
where \(k_n\in E\) with \(0\le k_n< c^*\) on \([\rho \omega ,\omega ]\), and \(k_n\rightarrow c^*\) in \(L^{\infty }((0,\omega ))\) as \(n\rightarrow \infty \).
Proof
Let \(\{k_n\}_{n=1}^{\infty }\subset E\) with \(0\le k_n(t)< c^*\) be an arbitrary increasing function sequence satisfying that \(\lim _{n\rightarrow \infty }k_n(t)=c^*\) in \(L^{\infty }(0,\omega )\). Denote \(\Psi _n(t,\xi ):=\Psi ^{k_n}(t,\xi )\). Clearly, \(\Psi _n(t,\xi )\) is uniformly bounded and \(\partial _t\Psi _n(t,\xi )\) and \(\partial _{\xi }\Psi _n(t,\xi )\) are also uniformly bounded from the equation satisfied by \(\Psi _n\). Thus, there is a subsequence of \(\Psi _n\), still denoted by itself, such that \(\lim _{n\rightarrow \infty }\Psi _n(t,\xi )=\Psi (t,\xi )\) in \(C_{\mathrm {loc}}([0,\omega ]\times (-\infty ,0])\). By a similar manner as in Claim 1 in the proof of Theorem 4.7, one can check that \(\Psi \) satisfies
Clearly, \(0\le \Psi (t,\xi )<\Psi _n(t,\xi )\) in \([0,\omega ]\times (-\infty ,0)\). Fix \(\theta \in (0,1)\) and define \({\tilde{\Psi }}(t,\xi ):=\theta \Psi (t,\xi )\). Then
Consider an auxiliary traveling wave problem
Let \(\phi (t,\xi )\) be a \(\omega \)-periodic traveling wave of (4.2) with minimal speed \(c^*\). Similar to the Definition 4.8 or Definition 4.4, we can check that \(z^*(t)\) and \(\phi (t,\xi )\) are a pair of upper-lower solutions of equation (4.22). Then as in the proof of Lemma 4.5 or Theorem 4.9, we see that the equation (4.22) admits a nontrivial solution, denoted by \(\psi (t,\xi )\), which is \(\omega \)-periodic in \(t\in {\mathbb {R}}\) and strictly decreasing in \(\xi \in {\mathbb {R}}\). Moreover, \(\psi (t,-\infty )=z^*(t)\) and \(\psi (t,+\infty )=0\) uniformly in \([0,\omega ]\). Obviously, \(\psi (t,\xi -\eta )\ge \psi (t,-\eta )\) for any \(\eta >0\) in \((t,\xi )\in [0,\omega ]\times (-\infty ,0]\). Since \(\psi (t,-\infty )=z^*(t)\) and \({\tilde{\Psi }}(t,\xi )\le \theta z^*(t)< z^*(t)\), there exists a large \(\eta _0>0\) independent of t, such that for all \(\eta \ge \eta _0\),
Hence the notation
is well-defined. If \(\eta _*>-\infty \), then \(w^{\eta _*}(t,\xi )\ge 0\) in \([0,\omega ]\times (-\infty ,0]\). This together with the definition of \(\eta _*\) and the fact that \(w^{\eta _*}(t,-\infty )=(1-\theta )z^*(t)>0\) and \(w^{\eta _*}(t,0)=\psi (t,-\eta _*)>0\), implies that there exists \((t_*,\xi _*)\in [0,\omega ]\times (-\infty ,0)\) such that
Combining the equations satisfied by \({\tilde{\Psi }}\) and \(\psi \), we have
By applying Lemma 4.2 to \(w^{\eta _*}\), we immediately obtain that \(w^{\eta _*}(t,\xi )>0\) for \(t\in [0,\omega ],\xi <0\), which contradicts with \(w^{\eta _*}(t_*,\xi _*)=0\). This contradiction means that \(\eta _*=-\infty \). Thus, \(\psi (t,\xi -\eta )\ge {\tilde{\Psi }}(t,\xi )\) for all \(\eta \in {\mathbb {R}}\). Noticing the limit \(\psi (t,+\infty )=0\) and letting \(\eta \rightarrow -\infty \), we have that \({\tilde{\Psi }}(t,\xi )\le 0\), which leads to that \(\Psi (t,\xi )\equiv 0\). Since \(k_n\in E\) is an arbitrary increasing function sequence converging to \(c^*\) in \(L^{\infty }(0,\omega )\), we can conclude that (4.21) holds. \(\square \)
4.5 Proof of Theorem 1.5
In the previous subsection, we have proved that when \(\mathbf{(J)},\mathbf{(J1)}\) hold and \((1-\rho )a-\rho \delta >0\), (1.10) admits a unique semi-wave \(\Psi ^k\), which is \(\omega \)-periodic in t and strictly decreasing in k. Now we prove Theorem 1.5.
Proof of Theorem 1.5
(i) Define a function \({\mathcal {B}}:[0,c^*]\times [\rho \omega ,\omega ]\rightarrow {\mathbb {R}}\) by
where \(\Psi ^k\) is the unique \(\omega \)-periodic semi-wave solution of (1.10) for \(k\in [0,c^*)\), while \(\Psi ^k\equiv 0\) for \(k=c^*\). Denote
A direct calculation gives that
where \(\eta \) satisfies \(\mathbf{(J1)}\), which leads to that
Thus, the function \({\mathcal {B}}\) is bounded.
Obviously, \({\mathcal {B}}(k,t)\) is strictly increasing in k for fixed \(t\in [0,\omega ]\) due to the monotonicity of \(\Psi ^k\) in k. Since \(\Psi ^k\) is the unique semi-wave solution of (1.10), one may induce a argument similar to that used to show the convergence of \(\Psi _n(t,\xi )\) in Theorem 4.10 to conclude that \(\Psi ^k(t,\xi )\) is continuous in k uniformly for \((t,\xi )\) in any bounded subset of \([0,\omega ]\times (-\infty ,0]\). Consequently, \({\mathcal {B}}(k,t)\) is continuous with respect to \((k,t)\in [0,c^*]\times [\rho \omega ,\omega ]\) by the fact that \({\mathcal {B}}\) is continuous for each independent variable k and t, respectively, and is strictly increasing in k for fixed \(t\in [0,\omega ]\).
Notice that \({\mathcal {B}}(0,t)<0\) and \({\mathcal {B}}(c^*,t)>0\) for \(t\in [\rho \omega ,\omega ]\). Then for any fixed \({\bar{t}}\in [\rho \omega ,\omega ]\), there must be a unique \({\bar{k}}\in (0,c^*)\) such that \({\mathcal {B}}({\bar{k}},{\bar{t}})=0\) in view of the strict monotonicity of \({\mathcal {B}}\) in k. By the arbitrariness of \({\bar{t}}\), we obtain a function \(k_0(t)\) defined on \([\rho \omega ,\omega ]\) satisfying that \({\mathcal {B}}(k_0(t),t)=0\) and \(0<k_0(t)<c^*\) for any \(t\in [\rho \omega ,\omega ]\).
In the remainder, we will prove that \(k_0(t)\) is continuous in \(t\in [\rho \omega ,\omega ]\). For any fixed \(t_0\in (\rho \omega ,\omega )\), given any sufficiently small \(\varepsilon >0\) such that
we see that \({\mathcal {B}}(k_0(t_0)-\varepsilon ,t_0)<0\) and \({\mathcal {B}}(k_0(t_0)+\varepsilon ,t_0)>0\). By the continuity of \({\mathcal {B}}\) in (k, t), there exists a neighborhood \((t_0-\tau ,t_0+\tau )\subset (\rho \omega ,\omega )\) of \(t_0\) such that
Thus, there is a unique k such that \({\mathcal {B}}(k,t)=0\), i.e., \(k=k_0(t)\), \(|k-k_0(t_0)|<\varepsilon \). This implies that \(|k_0(t)-k_0(t_0)|<\varepsilon \) for \(|t-t_0|<\tau \), namely, \(k_0(t)\) is continuous in \(t=t_0\). By the arbitrariness of \(t_0\), we obtain that \(k_0(t)\) is continuous in \((\rho \omega ,\omega )\). Similarly, we can show that \(k_0(t)\) is right-continuous in \(t=\rho \omega \) and left-continuous in \(t=\omega \). Moreover, \(k_0(t)\) can be extended to be zero in \((0,\rho \omega )\) so that \(k_0(t)\in E\).
(ii) Suppose that \(\mathbf{(J)}\) holds, \((1-\rho )a-\rho \delta >0\) and (1.10)-(1.11) admits a solution pair \((k_0,\Psi ^{k_0})\), which is \(\omega \)-periodic in t and strictly decreasing in \(\xi \). In view of the continuity of \(k_0(t)\) on \([\rho \omega ,\omega ]\), we see that \(k_0(t)\) is bounded. Note that
since \(\Psi ^{k_0}(t,\xi )\) is nonincreasing in \(\xi \). Recall the definition of \({{\tilde{a}}}(\xi )\) in part (i). Then
Since \({{\tilde{a}}}(\xi )\) is continuous, obviously
Therefore, Theorem 1.5 (ii) is proved. \(\square \)
5 Spreading Speed
Assume that \(\mathbf{(J)}\) and \(\mathbf{(J1)}\) are satisfied. Then there exists a unique pair \((k_0,\Phi ^{k_0})\) solving (1.10)-(1.11). Recall the notation E in Sect. 4.4. Since \(k_0(t)>0\) on \([\rho \omega ,\omega ]\), we can extend it to be zero in \((0,\rho \omega )\) such that \(k_0\in E\). Let (u, g, h) be the unique solution of (1.7) and suppose that spreading happens. In the this subsection, we aims to show
which is the conclusion of Theorem 1.6.
Lemma 5.1
Under the above assumptions, there holds that
Proof
By the proof of Lemma 3.14, we have that
where \({\overline{u}}(t)\) is the unique solution of (3.15) with the initial value \({\overline{u}}(0)\ge \max _{x\in [-h_0,h_0]}u_0(x)\). Recall that \(\lim _{t\rightarrow +\infty }({\overline{u}}(t)-z^*(t))\), where \(z^*(t)\) is the unique \(\omega \)-periodic positive solution of (1.9). As a result, for any given small \(\varepsilon >0\), there is a integer \(m^*>0\) large so that
Recall from Theorem 1.5 that there exists a unique function \(k_0(t)\in E\). To complete the proof, by the fact that \(\Psi ^{k_0}(t,-\infty )=z^*(t)\) uniformly in \([0,\omega ]\), we define
where \(L>0\) is large enough such that \({\overline{h}}(0)=L>h(m^*\omega )\) and
It is easy to check that
When \(t\in ((i+\rho )\omega ,(i+1)\omega ]\), \(k_0(t)\) is continuous and \(k_0(t)>0\). Then
and
for \(t\in ((i+\rho )\omega ,(i+1)\omega ]\) and \(x<{\overline{h}}(t)\).
Now we are ready to show that
In view of (5.2) and \({\overline{h}}(0)>h(m^*\omega )\), we have that the above inequalities hold for small \(t>0\). Once (5.3) does not hold for all \(t>0\), there exists a first time moment \(t^*>0\) such that (5.3) holds for \(t\in (0,t^*)\), and
-
(i)
\(h(t^*+m^*\omega )={\overline{h}}(t^*)\), or
-
(ii)
\(h(t^*+m^*\omega )<{\overline{h}}(t^*)\) and \(u(t^*+m^*\omega ,x^*)={\overline{u}}(t^*,x^*)\) for some \(x^*\in [g(t^*+m^*\omega ),h(t^*+m^*\omega )]\).
Consider case (i). If \(t^*\in (i_0\omega ,(i_0+\rho )\omega ]\) for some \(i_0\ge 0\), then \(h(i_0\omega +m^*\omega )=h(t^*+m^*\omega )={\overline{h}}(t^*)={\overline{h}}(i_0\omega )\), which contradicts with the definition of \(t^*\). Thus, \(t^*\in ((i_0+\rho )\omega ,(i_0+1)\omega ]\) for a integer \(i_0\ge 0\), and necessarily \(h'(t^*+m^*\omega )\ge {\overline{h}}'(t^*)\). On the other hand,
where we have used the fact that
Then a contradiction occurs for case (i).
For case (ii), since \({\overline{u}}(t,x)>0\) for \(x\in \{ g(t+m^*\omega ),h(t+m^*\omega ) \}\) in \(t\in (0,t^*]\), and \({\overline{u}}(0,x)>u(m^*\omega ,x)\) for \(x\in [g(m^*\omega ),h(m^*\omega )]\), one can apply Propositions 3.5 and 3.7 to obtain that
which is also a contradiction with the assumption for case (ii). Consequently, (5.3) holds, and
Letting \(\varepsilon \rightarrow 0\), we thereby obtain (5.1). \(\square \)
Lemma 5.2
Under the assumptions of Lemma 5.1, there holds that
Proof
For any given small \(\epsilon >0\), we define
and
where \(L>0\) is a large constant to be determined. Evidently, for \(t\ge 0\),
as \(L\rightarrow +\infty \). In the following, we will show that for large L,
and
Firstly, a direct calculation yields that for \(t\in ((i+\rho )\omega ,(i+1)\omega ]\),
By \(\mathbf{(J1)}\), we see that there is a large \(L_1>0\) such that for \(L\ge L_1\),
where \({{\tilde{k}}}=\min _{t\in [\rho \omega ,\omega ]}k_0(t)\). In addition,
provided that \(L\ge L_2\) for some large enough \(L_2>L_1\). Thus, for \(L\ge L_2\) and \(t\in ((i+\rho )\omega ,(i+1)\omega ]\),
This proves (5.5).
Next we prove (5.6). By \(\Psi ^{k_0}(t,-\infty )=z^*(t)\), there is \(M>0\) large enough so that
This implies that
Set
Then there holds that
For the simplicity of notations, we will write \(\Psi =\Psi ^{k_0}\). It then follows that for \(t\in (i\omega ,(i+\rho )\omega ]\) and \(x\in (-{\underline{h}}(t),{\underline{h}}(t))\),
and for \(t\in ((i+\rho )\omega ,(i+1)\omega ]\) and \(x\in (-{\underline{h}}(t),{\underline{h}}(t))\),
in which
Obviously, if \(\Theta (t,x)\le 0\) for \(t\in ((i+\rho )\omega ,(i+1)\omega ]\) and \(x\in (-{\underline{h}}(t),{\underline{h}}(t))\), then (5.6) holds. To check it, we start by the case \(x\in [{\underline{h}}(t)-M,{\underline{h}}(t)]\) and \(t\in ((i+\rho )\omega ,(i+1)\omega ]\). In this case, we can find a large \(L_3\ge L_2\) such that
where \({{\tilde{z}}}=\max _{t\in [0,\omega ]}z^*(t)\). Thus, we have
Notice that \(\Theta (t,-x)=\Theta (t,x)\). Then the above inequality also holds for \(x\in [-{\underline{h}}(t),-{\underline{h}}(t)+M]\) and \(t\in ((i+\rho )\omega ,(i+1)\omega ]\).
As for the case \(x\in [-{\underline{h}}(t)+M,{\underline{h}}(t)-M]\) and \(t\in ((i+\rho )\omega ,(i+1)\omega ]\), (5.7) holds and then
This proves (5.6).
Since \(J(-x)=J(x)\) and \({\underline{u}}(t,-x)={\underline{u}}(t,x)\), by (5.5), there holds that
Finally, we are ready to compare (u, g, h) with \(({\underline{u}},-{\underline{h}},{\underline{h}})\) by a comparison argument. By the assumption that spreading happens for (u, g, h), there is a sufficiently large integer \(m_*>0\) such that
From (5.5), (5.6) and (5.9), an application of the lower solution version of Proposition 3.7 implies that
This leads to that
Letting \(\epsilon \rightarrow 0\), we obtain (5.4). The proof is completed. \(\square \)
Proof of Theorem 1.6
By the above lemmas, there holds that
It remains to show
Notice that \({{\tilde{u}}}(t,x):u(t,-x)\) is the solution of (1.7) by replacing the free boundaries h(t), g(t) as \({{\tilde{h}}}(t):=-g(t),{{\tilde{g}}}(t)=-h(t)\), and the initial datum \(u_0(x)\) as \({{\tilde{u}}}(x):=u_0(-x)\). Thus, we can apply Lemmas 5.1 and 5.2 to conclude that
\(\square \)
Proof of Theorem 1.7
By the proof of Lemmas 5.1 and 5.2, for any given small \(\varrho >0\), we can find positive constants \(m^*,m_*\) and \({{\hat{l}}}\) such that
where
and
where
By shrinking \(\varrho \) to be sufficiently small if necessary, we see that there is a large integer \(M_1\) such that for all \(t\ge M_1\omega \) and \(\varrho \in (0,\epsilon )\),
and
Thus, for \(t\ge M_1\omega \), it holds that
On the other hand, we can see from Theorem 4.9 that the unique \(\omega \)-periodic positive solution \(\Psi ^{k_0}\) of (1.10) satisfies \(\Psi ^{k_0}(t,x)<z^*(t)\) for all \(t\ge 0,x\le 0\) and \(\Psi ^{k_0}(t,x)\rightarrow z^*(t)\) uniformly for \(t\in [0,+\infty )\) as \(x\rightarrow -\infty \). Hence, there exists a large \({{\hat{L}}}>0\) and a large integer \(M_2\ge M_1\) such that
where we have used the fact that \(\varsigma (t)\rightarrow +\infty \) as \(t\rightarrow +\infty \).
Consequently, if
we have
and
which implies that (1.13) holds. \(\square \)
6 Concluding Remarks
In this paper, we pay our attention to the dynamical properties of the nonlocal dispersal logistic model with seasonal succession and double free boundaries. Seasonal cycles periodically change the natural environment, affecting the dispersal behaviour of the population (the species spreads in good season and hibernate in bad season). The free boundaries describe the spreading fronts of the species. Our main conclusions can be summarized as follows.
(A) The long time behaviors imply that if the species spreads successful, then its density will converge to the unique \(\omega \)-periodic positive state \(z^*(t)\); and if the species cannot spread successful, it will survive within a bounded region and dies out in the long run.
(B) The criteria governing spreading and vanishing show that: (i) When the duration of the bad season is too long (namely, \(\rho \) is close to 1), or the season is too bad (for example, bad weather and food shortages contributes to the large death rate \(\delta \)) such that \((1-\rho )a-\rho \delta \le 0\), then the species will die out eventually regardless the initial population size and the initial habitat; (ii) If the bad season is not long, or the food resource a is not small such that \(\rho \delta <(1-\rho )a\le (1-\rho )d+\rho \delta \), then both spreading and vanishing are determined by the size of initial habitat of the species and the moving coefficient \(\mu \) of free boundaries, which is similar to that for the local diffusion model (1.1); (iii) When the good season is very long (i.e., \(\rho \) is close to 0), or the species has enough food such that \((1-\rho )(a-d)-\rho \delta >0\), then the species always spreads and can migrate over the entire space \({\mathbb {R}}\) regardless the initial population size and the initial habitat. This is one of the most distinct differences between the local and nonlocal diffusion model considered in this paper.
(C) When spreading happens, under the “thin tail" condition \(\mathbf{(J1)}\) for J, the asymptotic spreading speed of u is \(c_*(\mu )\in (0,c^*)\), where \(c_*(\mu )\) is defined in Theorem 1.7 and \(c^*\) is the minimal speed of the travel wave to (4.4). It follows from the proof of Theorem 1.5 that \(c_*(\mu )\) is increasing with respect to \(\mu \).
We remark here that due the particularity of system (1.7), one cannot apply the method of Du et al. [8] to prove that there is a unique semi-wave solution to (1.10) if and only if the condition
is satisfied, which also makes it difficult to prove that accelerated propagation occurs when \(\mathbf{(J2)}\) does not hold. We leave these challenging problems for future investigations.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
Theorem 4.5 in [16] only shows that \(\phi (t,\xi )\) is left continuous in \(\xi \). In fact, we can show that \(\phi (t,x-ct)\) is indeed a classical solution of (4.2) by similar argument to [17, Theorem 2.12], and thus \(\phi (t,\xi )\in C({\mathbb {R}}\times {\mathbb {R}})\cap C^{1,1}((i\omega ,(i+\rho )\omega ]\times {\mathbb {R}})\cap C^{1,1}(((i+\rho )\omega ,(i+1)\omega ]\times {\mathbb {R}})\).
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11871475) and the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts040).
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Li, Z., Dai, B. The Dynamics of a Nonlocal Dispersal Logistic Model with Seasonal Succession and Free Boundaries. J Dyn Diff Equat 36, 2193–2238 (2024). https://doi.org/10.1007/s10884-022-10184-9
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DOI: https://doi.org/10.1007/s10884-022-10184-9