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A diffusive stagestructured model with a free boundary
Boundary Value Problems volume 2018, Article number: 138 (2018)
Abstract
In this paper we mainly consider a free boundary problem for a singlespecies model with stage structure in a radially symmetric setting. In our model, the individuals of a new or invasive species are classified as belonging either to the immature or to the mature cases. We firstly study the asymptotic behavior of the solution to the corresponding initial problem, then obtain a spreading–vanishing dichotomy and give sharp criteria governing spreading and vanishing for the free boundary problem.
1 Introduction
Recently, Du and Lin [1] have proposed the new mathematical model
to describe the expanding of a new or invasive species u, where \(x=h(t)\) is the moving boundary to be determined by Stefanlike condition \(h'(t)=\mu u_{x}(t,h(t))\), a, b, d, μ and \(h_{0}\) are given positive constants, and \(u_{0}\) is a given nonnegative initial function.
Du and Lin [1] established various interesting results about the solution \((u,h)\) of (1). One of very remarkable results is a spreading–vanishing dichotomy of the species, i.e., the solution \((u,h)\) of (1) satisfies one of the following properties:

(i)
Spreading of the species: \(h(t)\to \infty \), \(u(t,x)\to a/b\) as \(t\to \infty \);

(ii)
Vanishing of the species: \(h(t)\to h_{\infty }\leq (\pi /2) \sqrt{d/a}\), and \(u(t,x)\to 0\) as \(t\to \infty \).
Another is that the spreading speed approaches to a positive constant if the spreading occurs. On the other hand, they derived the criteria for spreading and vanishing. Later on, Du and Guo [2, 3], studied a free boundary problem similar to (1) in higher dimension space, Kaneko and Yamada [4] discussed (1) and the case of bistable nonlinearity with \(u_{x}(t,0)=0\) replaced by \(u(t,0)=0\).
In problem (1), it is assumed that during the whole life histories the individual’s characteristics are broadly similar to each other. In the real world, almost all animals have stage structure of the immature and mature cases. For many animals whose babies are raised by their parents or are dependent on the nutrition from the eggs they stay in, the babies are much weaker than the mature. It is important and practical to introduce the stage structure into the model.
Stagestructured models have received much attention in recent years (see for example [5–11] and the references therein). The pioneering work of Aiello and Freedman [5] (1990) on a singlespecies growth model with stage structure represents a mathematically more careful and biologically meaningful formulation approach.
The ODE version on a singlespecies growth model with stage structure takes the form
where u and v are the population densities of immature and mature, respectively, a, α, β, b are given positive constants, and c is a nonnegative constant. In the immature stage, both the birth rate and the death rate obey the Malthus rule, while in the mature stage, the birth rate obeys the Malthus rule and the death rate logistic type.
In the present paper, we firstly consider diffusion in the above ODE model and discuss the long time behavior of the solution to PDE model
with \(\theta >\beta \). This problem admits a positive steady state if and only if
in which case the positive constant steady state is uniquely given by \((\tilde{u},\tilde{v})\), where
Then, motivated by the work of [1–3] and [5–7, 9, 10], we investigate the diffusive stagestructured model (2) with a free boundary, which reads as follows:
where \(\Delta u=u_{rr}+\frac{n1}{r}u_{r}\), \(\Delta v=v_{rr}+ \frac{n1}{r}v_{r}\) (\(r= \vert x \vert \), \(x\in \mathbb{R}^{n}\)), \(d_{1}\), \(d_{2}\), μ, ρ and \(h_{0}\) are given positive constants, and the initial functions \(u_{0}(r)\), \(v_{0}(r)\) satisfy
Ecologically, this problem (4) describes the spreading of a new or invasive species with immature population density \(u(t, \vert x \vert )\) and mature population density \(v(t, \vert x \vert )\) over a radially symmetric setting, which exists initially in the ball \(r< h_{0}\), disperses through random diffusion over an expanding ball \(r< h(t)\), whose boundary \(r=h(t)\) is the invading front, and evolves according to the free boundary condition \(h'(t)=\mu [u_{r}(t,h(t))+\rho v_{r}(t,h(t))]\).
The wellknown Stefan free boundary condition aries in many other applications. For instance, it was used to describe the melting of ice in contact with water [12], the modeling of oxygen in the muscle [13], the wound healing [14], the tumor growth [15], and so on. As far as population models are concerned, Wang and Zhao [16] used such a condition for a predator–prey system with double free boundaries in one dimension, in which the prey lives in the whole space but the predator lives in a bounded area at the initial state; in [17, 18], a Stefan condition was used for a competition system and a predator–prey system in radially symmetric setting, respectively, in which one species lives in the whole space but the other lives in a bounded area at the initial state. They established the spreading–vanishing dichotomy, long time behavior of the solution and sharp criteria for spreading and vanishing. For more biological discussion, we refer to [19–27] and the references therein.
We now describe the main results of this paper as follows. Hereafter, (3) is always assumed. First, it is proved that the positive constant steady state \((\tilde{u},\tilde{v})\) of problem (2) is globally asymptotically stable.
Theorem 1.1
Suppose that \((u_{0},v_{0})\in [C_{b}(\mathbb{R}^{n})]^{2}\) and \((u(t,x),v(t,x))\) is the solution of the problem (2) with
then
uniformly in any bounded subset of \(\mathbb{R}^{n}\).
Then, we have the following existence and uniqueness result and a priori estimates for the solution of the problem (4).
Theorem 1.2
For any given \((u_{0},v_{0})\) satisfying (5) and any \(\nu \in (0,1)\), the problem (4) admits a unique global solution \((u,v,h)\in C^{(1+\nu )/2,1+\nu }(\Omega )\times C^{(1+ \nu )/2,1+\nu }(\Omega )\times C^{1+\nu /2}([0,\infty ))\) where
such that the following inequalities hold:
for \(t>0\), \(0< r< h(t)\) with \(M_{1},M_{2}>0\) depending on \(d_{1}\), \(d_{2}\), a, θ, β, b, c and \(\Vert u_{0} \Vert _{\infty }\), \(\Vert v_{0} \Vert _{\infty }\). Moreover,
where the constant \(C>0\) depends on ν, the parameters in (4) and \(\Vert u_{0} \Vert _{C^{2}([0,h_{0}])}\), \(\Vert v_{0} \Vert _{C^{2}([0,h _{0}])}\).
Next, a spreading–vanishing dichotomy is given.
Theorem 1.3
Assume that (3) holds and \((u,v,h)\) is the solution of (4), then there exists \(R^{*}>0\) such that the following alternative holds:
Either

(i)
vanishing: \(h_{\infty }\leq R^{*}\) and
$$ \lim_{t\to \infty } \bigl\Vert u(t,\cdot ) \bigr\Vert _{C([0,h(t)])}=\lim_{t\to \infty } \bigl\Vert v(t,\cdot ) \bigr\Vert _{C([0,h(t)])}=0; $$or

(ii)
spreading: \(h_{\infty }=\infty \), and
$$ \lim_{t\to \infty }\bigl(u(t,r),v(t,r)\bigr)=(\tilde{u},\tilde{v}) $$uniformly for r in any bounded set of \([0,\infty )\).
From \(h'(t)>0\) for \(t>0\) and Theorem 1.3, we easily see that \(h_{0}\geq R^{*}\) implies \(h_{\infty }=\infty \). Hence, we last only need to discuss the case \(h_{0}< R^{*}\). Whether spreading or vanishing occurs is dependent on \((u_{0},v_{0})\) and coefficient μ with the other parameters fixed.
Theorem 1.4
Suppose that \(h_{0}< R^{*}\), then there exists \(\mu^{*}>0\) depending on \((u_{0},v_{0})\) and \(h_{0}\), such that \(h_{\infty }\leq R^{*}\) if \(\mu \leq \mu^{*}\), and \(h_{\infty }=\infty \) if \(\mu >\mu^{*}\).
The rest of this paper is organized in the following way. In Sect. 2, we firstly discuss the problem (2). We study a problem corresponding to (2) with fixed boundary and then prove Theorem 1.1. Sections 3, 4 and 5 are devoted to investigating the free boundary problem (4). In Sect. 3, we show Theorem 1.2 and give a comparison principle. Section 4 is applied to the long time behavior of solution \((u,v)\) to the problem (4). From those results we can also get the spreading–vanishing dichotomy (Theorem 1.3). In Sect. 5, the sharp criteria for spreading and vanishing (Theorem 1.4) will be given. The last section is a brief discussion.
2 Global stability
In this section, Theorem 1.1 will be proved. We firstly study the following initialboundary value problem:
where \(B_{R}\) is a ball with center 0 and radius R, \(u_{0}\), \(v_{0}\) are positive functions satisfying
In the sequel, we will use some characteristics of the principal eigenvalue \(\lambda_{1}(d,R)\) of the problem
where \(d>0\) is a constant. It is well known that \(\lambda_{1}(d,R)\) is a strictly decreasing continuous function in R and satisfies
Therefore, for fixed \(d>0\) and any given \(L\in (0,\infty )\), there is a unique \(R_{L}(d)\) such that
and
We use a squeezing method as in [28] to prove the following theorem.
Theorem 2.1
If (3) holds, then there exists \(R^{*}>0\) such that the problem
has a unique positive solution \((u_{\lambda },v_{\lambda })\) for every \(R>R^{*}\) and \(\lambda \geq 1\); moreover,
uniformly on any compact subset of \(B_{R}\) as \(\lambda \longrightarrow \infty \).
Proof
Step 1 Existence The existence follows from a upper and lower solutions argument. Clearly, \((\tilde{u},\tilde{v})\) is an upper solution. Let \((\underline{u},\underline{v})=(\delta_{1} \phi ,\delta_{2}\phi )\), where ϕ satisfying \(\Vert \phi \Vert _{\infty }=1\) is a positive eigenfunction corresponding to \(\lambda_{1}(d_{1},R)\), and \(\lambda_{1}(d_{1},R)\), \(\delta_{1}\), \(\delta _{2}\) are positive constants to be determined later. By direct calculations, we have
Take \(R^{*}\) such that
For every \(R>R^{*}\), the following inequality holds:
Hence
Take
where \(L>2\) is any constant. Then we can obtain \(\delta_{1}<\tilde{u}\), \(\delta_{2}<\tilde{v}\) and
Therefore, \((\underline{u},\underline{v})\) satisfies \(\underline{u}< \tilde{u}\), \(\underline{v}<\tilde{v}\) and
It follows from \(\lambda_{1}(d_{1},R)\underline{u}\leq a\underline{v} \theta \underline{u}\) that \(a\underline{v}\theta \underline{u} \leq \lambda (a\underline{v}\theta \underline{u})\) for \(\lambda \geq 1\). Similarly, \(\beta \underline{u}b\underline{v}^{2}c \underline{v}\leq \lambda (\beta \underline{u}b\underline{v}^{2}c \underline{v})\) for \(\lambda \geq 1\). Thus
So, \((\underline{u},\underline{v})\) is a lower solution. Thus, the problem (10) has at least one positive solution.
Step 2 Uniqueness Now we verify the uniqueness of positive solution to (10). Fix \(R>R^{*}\) and suppose that (10) has two positive solutions \((u_{1},v_{1})\) and \((u_{2},v_{2})\). Then \((u_{i},v_{i})\) satisfies
With the help of the maximum principle and the Hopf boundary lemma for elliptic equations, we can see that \(u_{i}>0\), \(v_{i}>0\) in \(B_{R}\) and \(\partial_{\nu }u_{i}<0\), \(\partial_{\nu }v_{i}<0\), \(i=1,2\) on \(\partial B_{R}\). Hence there exists \(M>1\) such that
It is easily seen that \((M u_{1},Mv_{1})\) is a upper solution of (10) and \((M^{1}u_{1},M^{1}v_{1})\) is a lower solution. As a result, there exist a minimal and a maximal solution to (10) in the order interval \([M^{1}u_{1},Mu_{1}]\times [M^{1}v_{1},Mv_{1}]\) which is denoted by \((u_{*},v_{*})\) and \((u^{*},v^{*})\), respectively. Thus \((u_{*},v_{*})\leq (u_{i},v_{i})\leq (u^{*},v^{*})\), for \(i = 1, 2\). Hence it suffices to show that \((u_{*},v_{*})=(u^{*},v ^{*})\). To achieve this goal, let us define
Clearly \(\sigma_{*}\geq 1\) and \(u^{*}\leq \sigma_{*}u_{*}\), \(v^{*} \leq \sigma_{*} v_{*}\). We next prove \(\sigma_{*}=1\), which will therefore yield \((u_{*},v_{*})=(u^{*},v^{*})\). Suppose for contradiction that \(\sigma_{*}>1\). Then, for
it is easy to check that \(w,z\geq 0, \not \equiv 0\), and \((w,z)\) satisfies
and \(w=z=0\) on \(\partial B_{R}\). Hence, we can use the strong maximum principle and the Hopf boundary lemma for elliptic equations to deduce that \(w,z>0\) in \(B_{R}\), and \(\partial_{\nu }w, \partial_{\nu }z < 0\) on \(\partial B_{R}\). It follows that \(w\geq \varepsilon u^{*}\) and \(z\geq \varepsilon v^{*}\) for some \(\varepsilon >0\) small, and hence \(u^{*}\leq (1+\varepsilon )^{1}\sigma_{*} u_{*}\), \(v^{*}\leq (1+ \varepsilon )^{1}\sigma_{*} v_{*}\), which contradict the definition of \(\sigma_{*}\). Consequently, we must have \(\sigma_{*}=1\), and the uniqueness conclusion is proved.
Step 3 Asymptotic behavior In what follows, let us denote by \((u_{\lambda },v_{\lambda })\) the unique positive solution of (10) for \(\lambda \geq 1\). We then want to show (11).
Given any compact subset K of \(B_{R}\) and any small \(\varepsilon >0\) such that \(\varepsilon <\min \{ \frac{\theta }{a}\tilde{u}, \tilde{v} \} \). Observe that \((\bar{u}_{\varepsilon },\bar{v}_{ \varepsilon })=(\tilde{u}+\frac{a}{\theta }\varepsilon ,\tilde{v}+ \varepsilon )\) is a upper solution of (10) for every \(\lambda >0\). On the other hand, we choose a small neighborhood U of \(\partial B_{R}\) in \(B_{R}\) such that \(\overline{U}\cap K=\emptyset \) and \((\delta_{1}\phi (x),\delta_{2}\phi (x))<(\tilde{u}\frac{a}{ \theta }\varepsilon ,\tilde{v}\varepsilon )\) for \(x\in U\). Define
where \(\delta_{1}\), \(\delta_{2}\) is defined as in (13), ϕ is a positive eigenfunction corresponding to \(\lambda_{1}(d _{1},R)\) with \(\Vert \phi \Vert _{\infty }=1\), \(w_{\varepsilon }(x)\), \(z_{\varepsilon }(x)\) are smooth positive functions with \((w_{\varepsilon }(x),z_{ \varepsilon }(x))\leq (\tilde{u}\frac{a}{\theta }\varepsilon , \tilde{v}\varepsilon )\) in \(B_{R}\setminus (U\cup K)\) such that \(\underline{u}_{\varepsilon }\), \(\underline{v}_{\varepsilon }\) are smooth functions in \(B_{R}\) and satisfy
We can obtain
for all large λ. Since \((\underline{u}_{\varepsilon }, \underline{v}_{\varepsilon })\leq (\bar{u}_{\varepsilon },\bar{v}_{ \varepsilon })\), we deduce that \((\underline{u}_{\varepsilon }, \underline{v}_{\varepsilon })\leq (u_{\lambda },v_{\lambda })\leq ( \bar{u}_{\varepsilon },\bar{v}_{\varepsilon })\) in \(B_{R}\) for all large λ. In particular,
on K for all large λ. This shows that \((u_{\lambda },v_{ \lambda })\longrightarrow (\tilde{u},\tilde{v})\) as \(\lambda \longrightarrow \infty \) uniformly on K, as required. The proof is complete. □
Theorem 2.2
Assume that \((u_{0},v_{0})\) satisfies (9) and \(R>R^{*}\). If \((u,v)\) is the solution of (8), then
where \((u^{*}(x),v^{*}(x))\) is the unique positive solution of (10) with \(\lambda =1\).
Proof
Since \(R>R^{*}\), (10) has a unique positive solution \((u^{*}(x),v^{*}(x))\). With the help of the maximum principle and the Hopf boundary lemma for elliptic equations, we can find \({M>1}\) such that
It is easy to verify that \((Mu^{*},Mv^{*})\) is an upper solution of (10) when \(\lambda =1\). One can still use the same analysis as in proof of Theorem 8 to deduce that \((\delta_{1}\phi ,\delta _{2}\phi )\) is a lower solution of the problem (10), where ϕ is the positive eigenfunction corresponding to \(\lambda_{1}(d _{1},R)\) with \(\Vert \phi \Vert _{\infty }=1\) and \(\delta_{1}\), \(\delta_{2}\) is defined as in (13), choosing \(\delta_{1}\), \(\delta_{2}\) sufficiently small (i.e. L sufficiently large) such that \((\delta_{1}\phi (x), \delta_{2}\phi (x))\leq (u_{0}(x),v_{0}(x))\) for \(x\in \overline{B _{R}}\) if necessary. We consider the following two auxiliary problems:
and
We easily see that
due to the comparison principle. From the theory of monotone dynamical systems (see [29, Corollary 3.6]), we see that \((\underline{u}(t,x), \underline{v}(t,x))\) is increasing while \((\bar{u}(t,x),\bar{v}(t,x))\) is decreasing in t, and
uniformly in \(\overline{B_{R}}\), where \((\underline{u}^{*}, \underline{v}^{*})\) and \((\bar{u}^{*},\bar{v}^{*})\) satisfy (10) with \(\lambda =1\) and
So \((\underline{u}^{*},\underline{v}^{*})=(\bar{u}^{*},\bar{v}^{*})=(u ^{*},v^{*})\), since the positive solution of (10) is unique. Due to (14) and (15), we have \(\lim_{t\to \infty }(u(t,x),v(t,x))=(u ^{*}(x),v^{*}(x))\) uniformly in \(\overline{B}_{R}\). □
Proof of Theorem 1.1
First we recall that the comparison principle gives \((u(t,x),v(t,x))\leq (\bar{u}(t),\bar{v}(t))\) for \(t>0\), \(x\in \mathbb{R}^{n}\), where \((\bar{u}(t),\bar{v}(t))\) is the solution of the problem
Similarly to the proof of Theorem 2.1 in [30], we see that the positive constant steady state \((\tilde{u},\tilde{v})\) of (16) is globally asymptotically stable. So \(\lim_{t\to \infty }(\bar{u}(t), \bar{v}(t)) =(\tilde{u},\tilde{v})\), moreover,
Next, we show
locally uniformly for \(x\in \mathbb{R}^{n}\).
Let \((u_{R}(t,x), v_{R}(t,x))\) be the unique solution of (8) with any \(R>R^{*}\). It follows from comparison principle that \((u_{R}(t,x), v_{R}(t,x))\leq (u(t,x),v(t,x))\) for any R. By using of Theorem 2.2, we easily see that \((u_{R}(t,x), v_{R}(t,x)) \to (u_{R}^{*}(x), v_{R}^{*}(x))\) as \(t\to \infty \) uniformly in \(B_{R}\), where \((u_{R}^{*}(r),v_{R}^{*}(r))\) is the unique positive solution of (8) with \(\lambda =1\). It follows that
uniformly in \(B_{R}\). Let
then \((w_{R}(y),z_{R}(y))\) satisfies
Using Theorem 2.1, we easily see that \((w_{R}(y),z_{R}(y)) \to (\tilde{u},\tilde{v})\) as \(R\to \infty \) uniformly in any compact subset of \(B_{2R^{*}}\). Therefore \((u_{R}^{*}(x),v_{R}^{*}(x))\to ( \tilde{u},\tilde{v})\) as \(R\to \infty \) uniformly in any compact subset of \(\mathbb{R}^{n}\) and then
locally uniformly for \(x\in \mathbb{R}^{n}\), due to (17). This completes the proof of the desired result. □
3 Existence and uniqueness
For the existence and uniqueness of local solution, in previous work, for example in the references [17, 18, 24], the embedding theorem
was used, where \(Q_{T}=\{(t,x) 0\leq t\leq T, x\in Q\}\), Q is a bounded domain of \(\mathbb{R}^{n}\). This is not appropriate because the embedding constant \(C=C(T^{1})\) depends on \(T^{1}\) and \(C(T^{1}) \to \infty \) as \(T\to 0\). For example, for the function \(u\equiv 1\), \(\Vert u \Vert _{C^{(1+\nu )/2,1+\nu }(Q_{T})}=1\) and \(\Vert u \Vert _{W^{1,2}(Q_{T})}= \vert Q \vert ^{1/p}T ^{1/p}\to 0\) as \(T\to 0\). Very recently, Wang has overcome this loophole and given a strict proof in [31]. So the existence and uniqueness of local solution can be obtained from the proof of Theorem 1.1 in [31], and the local solution can be extended to Ω by Theorem 2.4 in [17], we omit the details. Therefore, we will only show the inequalities (6) and (7). Then some comparison results for (4) are given.
Theorem 3.1
Let \((u,v,h)\) be a solution to the problem (4) defined for \(t\in (0,\infty )\). Then there exist constants \(M_{1}\) and \(M_{2}\) such that
for all \(t>0\) and \(0\leq r\leq h(t)\).
Proof
Using the strong maximum principle, we are easy to see that \(u>0\) in \((0,\infty )\times [0,h(t))\) and \(v>0\) in \((0,\infty ) \times [0,h(t))\) as long as the solution exists. It follows from the comparison principle that \((u(t,r),v(t,r))\leq (\bar{u}(t),\bar{v}(t))\) for \(t>0\) and \(r\in [0,h(t)]\), where \((\bar{u}(t),\bar{v}(t))\) solves the initial value problem (16). Let \(U(t)=\bar{u}(t)+\bar{v}(t)\), then U satisfies \(U(0)= \Vert u_{0} \Vert _{\infty }+ \Vert v_{0} \Vert _{\infty }\) and
By using of the comparison principle again, we have
Thus
By straightening the free boundaries with \(s=\frac{h_{0}r}{h(t)}\), we can obtain a fixed boundary problem. Then the strong maximum principle yields the inequalities \(u_{r}(t,h(t))<0\) and \(v_{r}(t,h(t))<0\), and therefore \(h'(t)>0\) in \((0,\infty )\). It remains to show that \(h'(t)\leq M_{2}\) for some positive constant \(M_{2}\). To this end, let M be a positive constant, define
and construct an auxiliary function
We will choose M so that \((w(t,r),w(t,r))\geq (u(t,r),v(t,r))\) holds over \(\Omega_{M}\).
Direct calculations show that, for \((t,r)\in \Omega_{M}\),
It follows that
if \(M^{2}\geq \max \{\frac{a}{2d_{1}},\frac{\beta }{2d_{2}} \}\). On the other hand,
Note that
we know that if \(MM_{1}\geq \Vert u_{0}' \Vert _{{C([0,h_{0}])}}\), then
Similarly, if \(MM_{1}\geq \Vert v_{0}' \Vert _{{C([0,h_{0}])}}\) then
Let
Applying the maximum principle to \(wu\) and \(wv\) over \(\Omega_{M}\) we can deduce that \(u(t,r),v(t,r)\leq w(t,r)\) for \((t,r)\in \Omega_{M}\). Thanks to \(w(t,h(t))=0=u(t,h(t))=v(t,h(t))\), it would then follow that
The proof is complete. □
Theorem 3.2
If \((u,v,h)\) is a solution to problem (4) defined for \(t\in (0,\infty )\), then, for any \(\nu \in (0,1)\), the estimate (7) holds, i.e.
where
the constant \(C>0\) depends on ν, the parameters in (4) and \(\Vert u_{0} \Vert _{C^{2}([0,h_{0}])}\), \(\Vert v_{0} \Vert _{C^{2}([0,h_{0}])}\).
Proof
We first consider the case \(h_{\infty }<\infty \). Define
Hence \(w(t,s)\) satisfies
This is an initialboundary value problem over a fixed ball \(\{s< h_{0}\}\). Since \(h_{0}\leq h(t)< h_{\infty }<\infty \), the differential operator is uniformly parabolic. It follows from Theorem 3.1 that
Therefore, we can apply the standard \(L^{p}\) theory and the Sobolev embedding theorem ([32, 33]) to obtain, for any \(\nu \in (0,1)\),
Furthermore, by virtue of a similar analysis to Proposition A.1 of [16] we can conclude that
where \(C_{1}\) and \(C_{2}\) are constants depending on ν, \(h_{0}\), \(h_{\infty }\), \(M_{1}\), \(M_{2}\) and \(\Vert u_{0} \Vert _{C^{1+\nu }([0,h _{0}])}\). Thus
Similarly, we may obtain
where \(C_{3}\) is a positive constant depending on ν, \(h_{0}\), \(h_{\infty }\), \(M_{1}\), \(M_{2}\) and \(\Vert v_{0} \Vert _{C^{1+\nu }([0,h_{0}])}\).
It follows that there exists a constant C depending on ν, \(h_{0}\), \(h_{\infty }\), \(M_{1}\), \(M_{2}\), \(\Vert u_{0} \Vert _{C^{1+\nu }([0,h _{0}])}\) and \(\Vert v_{0} \Vert _{C^{1+\nu }([0,h_{0}])}\) such that
When \(h_{\infty }=\infty \), similarly to the arguments in Theorem 2.2 of [34] we can obtain (7). So we omit the details.
The proof is complete. □
We now present some comparison principles which will be used in the following sections to estimate the solution \((u(t,r),v(t,r))\) and the free boundary \(r=h(t)\) of (4).
Theorem 3.3
(The comparison principle)
Assume that \(T\in (0,\infty )\), \(\bar{u},\bar{v}\in C(\overline{D}_{T})\cap C^{1,2}(D_{T})\) with \(D_{T}= \{(t,r)\in \mathbb{R}^{2}:t\in (0,T],r\in [0,\bar{h}(t)] \}\), and \((\bar{u},\bar{v},\bar{h})\) satisfies
Let \((u,v,h)\) be the unique positive solution of (4). Then \(h(t)\leq \bar{h}(t)\) in \((0,T]\) and
Proof
First assume that \(h_{0}<\bar{h}(0)\). We claim that \(h(t)<\bar{h}(t)\) for all \(t\in (0,T]\). Clearly, this is true for small \(t>0\). If our claim does not hold, then we can find a first \(\tau \leq T\) such that \(h(t)<\bar{h}(t)\) for all \(t\in (0,\tau )\) and \(h(\tau )=\bar{h}(\tau )\). It follows that
We now show that
Let \(U=\bar{u}u\), \(V=\bar{v}v\), we obtain
The strong maximum principle yields \(U(t,r)>0\) and \(V(t,r)>0\) for \((t,r)\in (0,\tau ]\times [0,h(t))\). It follows from the Hopf boundary lemma that \(U_{r}(\tau ,h(\tau ))<0\) and \(V_{r}(\tau ,h(\tau ))<0\), i.e.
We then deduce that \(h'(\tau )<\bar{h}'(\tau )\). But this contradicts (19). This proves our claim that \(h(t)<\bar{h}(t)\) for all \(t\in (0,T]\). We may now apply the above procedure over \([0,T]\times [0,h(t))\) to conclude (18). Moreover, \((u,v)<(\bar{u},\bar{v})\) for \(t\in (0,T]\) and \(r\in [0,h(t))\).
If \(\bar{h}(0)=h_{0}\), we use approximation. For small \(\varepsilon >0\), let \((u^{\varepsilon },v^{\varepsilon },h^{\varepsilon })\) denote the unique solution of (4) with \(h_{0}\) replaced by \(h_{0}(1 \varepsilon )\). Since the unique solution of (4) depends continuously on the parameters in (4), as \(\varepsilon \to 0\), \((u^{\varepsilon },v^{\varepsilon },h^{\varepsilon })\) converges to \((u,v,h)\), the unique solution of (4). The desired result then follows by letting \(\varepsilon \to 0\) in the inequalities \((u^{ \varepsilon },v^{\varepsilon })<(\bar{u},\bar{v})\) and \(h^{\varepsilon }<\bar{h}\). □
Remark 3.1
The pair \((\bar{u},\bar{v},\bar{h})\) in Theorem 3.3 is called an upper solution of the problem (4). We can define a lower solution by reversing all the inequalities in the above places. Moreover, one can easily prove an analog of Theorem 3.3 for lower solutions.
We next fix \(u_{0}\), \(v_{0}\), \(d_{1}\), \(d_{2}\), a, θ, β, b, c and \(h_{0}\) to examine the dependence of the solution on μ. The solution is denoted as \((u^{\mu },v^{\mu },h^{\mu })\) to emphasize this dependence. As a consequence of Theorem 3.3, we have the following result.
Corollary 3.1
For fixed \(u_{0}\), \(v_{0}\), \(d_{1}\), \(d_{2}\), a, θ, β, b, c and \(h_{0}\). If \(\mu_{1}\leq \mu_{2}\), then \(h^{\mu_{1}}(t)\leq h^{\mu _{2}}(t)\) in \((0,\infty )\) and
for \(t\in (0,\infty )\), \(r\in [0,h^{\mu_{1}}(t)]\).
4 Spreading and vanishing
In this section, we will prove Theorem 1.3. Precisely, we can deduce Theorem 1.3 directly from the following three theorems. To discuss the asymptotic behavior of u and v for the vanishing case (\(s_{\infty }<\infty \)), we first give the following proposition.
Proposition 4.1
Let D, η, σ and \(s_{0}\) be positive constants and C be any real number. Suppose
Assume that \(s(t)\in C^{1+\frac{\sigma }{2}}([0,\infty ))\), \(s(t)>0\) for \(0\leq t<\infty \), \(\lim_{t\to \infty } s(t)=s_{\infty }<\infty \), \(\lim_{t\to \infty } s'(t)=0\); and that \(w\in C^{ \frac{1+\sigma }{2},1+\sigma }([0,\infty )\times [0,s(t)])\), \(w(t,r)>0\) for \(0\leq t<\infty \) and \(0< r< s(t)\), \(\Vert w(t,\cdot ) \Vert _{C^{1}[0, r(t)]}\leq M\) for any \(t\geq 1\) and some \(M>0\). If \((w,r)\) satisfies
then \(\lim_{t\to \infty } \max_{0\leq x\leq r(t)}w(t,x)=0\).
Proof
The proof is identical to that of Proposition 3.1 in [23], so we leave out the details. □
Theorem 4.1
Assume that \((u,v,s)\) is the solution of problem (2). If \(h_{\infty }<\infty \), then
Proof
By the estimate of (7) we know that \(\Vert h' \Vert _{C ^{\frac{\nu }{2}}([1,\infty ))}\leq C\). Combining this with \(h'(t)>0\) and \(h_{\infty }<\infty \) implies \(h'(t)\to 0\) as \(t\to \infty \).
From the proof of Theorem 3.1 we can find that \(u,v>0\) for \(t>0\), \(0< r< h(t)\) and \(u_{r}(t,h(t))<0\), \(v_{r}(t,h(t))<0\). Thus it follows that
By virtue of (5), (7) and Proposition 4.1 it is derived that
In the same way we immediately get \(\lim_{t\to \infty } \max_{0\leq x\leq s(t)}v(t,x)=0\).
This proof is completed. □
The result of Theorem 4.1 shows that if the new or invasive species cannot spread into the whole space, then it will die out eventually. In the following theorem, we will give a necessary condition for vanishing.
Theorem 4.2
Let \((u,v,h)\) be any solution of (4). If \(h_{\infty }< \infty \), then \(h_{\infty }\leq R^{*}\), where \(R^{*}\) is defined as in (12).
Proof
Theorem 3.1 implies that, if \(h_{\infty }<\infty \), then (20) holds. We assume \(h_{\infty }>R^{*}\) to get a contradiction. It is easy to see that there exists \(\tau \gg 1\) such that \(R^{*}< h(\tau ):=R\). Let \((w,z)\) be the positive solution of the following initialboundary value problem with fixed boundary:
By the comparison principle,
Since \(R>R^{*}\) and Theorem 2.2, one can easily see that \((w(t,r),z(t,r))\to (u_{R}^{*}(r), v_{R}^{*}(r))\) as \(t\to \infty \) uniformly in \([0,R]\), where \((u_{R}^{*}(r),v_{R}^{*}(r))\) is the unique positive solution of (10) with \(\lambda =1\). Therefore,
This is a contradiction to Eq. (20). Therefore, \(h_{\infty } \leq R^{*}\). □
Theorem 4.3
If \(h_{\infty }=\infty \), then
uniformly in any bounded subset of \([0,\infty )\).
Proof
First, we consider the ODE problem (16). By the comparison principle, we can easily see that \((u(t,r),v(t,r))\leq ( \bar{u}(t),\bar{v}(t))\) for \(t>0\) and \(r\in [0,h(t)]\). Similarly to the proof of Theorem 2.1 in [30], we see that the positive constant steady state \((\tilde{u},\tilde{v})\) of (16) is globally asymptotically stable. So \(\lim_{t\to \infty }(\bar{u}(t),\bar{v}(t))=( \tilde{u},\tilde{v})\), moreover,
Next, we show
locally uniformly for \(r\in [0,\infty )\). For any \(R>\max \{h_{0},R ^{*}\}\), there exists \(t_{R}>0\) such that \(h(t_{R})=R\). By use of the comparison principle we have \((u(t,r),v(t,r))\geq (\underline{u}_{R}(t,r), \underline{v}_{R}(t,r))\) in \((t_{R},\infty )\times (0,R)\), where \((\underline{u}_{R}(t,r),\underline{v}_{R}(t,r))\) is the solution of the following problem with fixed boundary:
By virtue of \(R>R^{*}\) and Theorem 2.2, we easily see that \((\underline{u}_{R}(t,r),\underline{v}_{R}(t,r))\to (u_{R}^{*}(r), v _{R}^{*}(r))\) as \(t\to \infty \) uniformly in \([0,R]\), where \((u_{R}^{*}(r),v_{R}^{*}(r))\) is the unique positive solution of (10) with \(\lambda =1\). It follows that
uniformly in compact subsets of \([0,R)\). Similarly to the process of the last half proof of Theorem 1.1, we see that \((u_{R}^{*}(r),v _{R}^{*}(r))\to (\tilde{u},\tilde{v})\) as \(R\to \infty \) uniformly in any compact subset of \([0,\infty )\) and then
locally uniformly for \(r\in [0,\infty )\), due to (21). This completes the proof of the desired result. □
5 The criteria governing spreading and vanishing
Now we decide exactly when each of the two alternative occurs. The discussion will be divided into two cases:
For the case (a), due to \(h'(t)>0\) for \(t>0\), we must have \(h_{\infty }>R^{*}\). Hence Theorem 4.2 implies that if \(h_{0}\geq R^{*}\), then \(h_{\infty }=\infty \).
Next we discuss the case (b).
Theorem 5.1
If \(h_{0}< R^{*}\), then there exists \(\mu^{0}>0\) depending on \((u_{0},v_{0})\) such that \(h_{\infty }=\infty \) if \(\mu \geq \mu^{0}\).
Proof
This proof is similar to [24, Lemma 3.6]. We give the details below for completeness.
We see from (6) that there exists a constant \(\delta^{*}>0\) such that the solution \((u,v,h)\) of problem (4) satisfies
for all \((t,r)\in \Omega \). We next consider the auxiliary free boundary problem
Arguing as in proving the existence and uniqueness of the solution to (4), one will easily see that (22) also admits a unique solution \((w,z,g)\) which is well defined for all \(t>0\). Moreover, due to the Hopf boundary lemma, \(g'(t)>0\) for \(t>0\). To stress the dependence of the solutions on the parameter μ, in the sequel, we always write \((u^{\mu },v^{\mu },h^{\mu })\) and \((w^{\mu },z^{\mu },g ^{\mu })\) instead of \((u,v,h)\) and \((w,z,g)\). By Lemma 11, we have
In what follows, we are going to prove that, for all large μ,
To the end, we first choose a smooth function \(\underline{g}(t)\) with \(\underline{g}(0)=h_{0}/2\), \(\underline{g}'(t)>0\) and \(\underline{g}(2)=2R ^{*}\). We then consider the following initialboundary value problem:
Here, for the smooth initial value \((\underline{w}_{0},\underline{z} _{0})\), we require
The standard theory for parabolic equations ensures that (25) has a unique positive solution \((\underline{w},\underline{z})\), and \(\underline{w}_{r}(t,\underline{g}(t))<0\), \(\underline{z}_{r}(t, \underline{g}(t))<0\) for all \(t\in [0,2]\) due to the Hopf boundary lemma. According to our choice of \(\underline{g}(t)\) and \(( \underline{w}_{0},\underline{z}_{0})\), there is a constant \(\mu^{0}>0\) such that, for all \(\mu \geq \mu^{0}\),
On the other hand, for system (25), we can establish the comparison principle analogous with lower solution to Theorem 3.3 by the same argument. Thus, note that \(\underline{g}(0)=h _{0}/2< h_{0}\), it follows from (25), (26) and (27) that
This particularly implies \(g^{\mu }(2)\geq \underline{g}(2)=2R^{*}\), and so (24) holds. Hence, in view of (23) and (24), we find
This, together with Theorem 4.2, yields the desired result. □
Theorem 5.2
If \(h_{0}< R^{*}\), then there exists \(\mu_{0}>0\) depending on \((u_{0},v_{0})\) such that \(h_{\infty }<\infty \) if \(\mu \leq \mu_{0}\).
Proof
We are going to construct a suitable supper solution to (4) and then apply Theorem 3.3.
Inspired by [2], for \(t>0\) and \(r\in [0,\eta (t)]\), we define
where δ, γ, \(K_{1}\) and \(K_{2}\) are positive constants to be chosen later and \(\phi ( \vert x \vert )\) is the first eigenfunction of the problem
with \(\phi >0\) in \(B_{h_{0}}\) and \(\Vert \phi \Vert _{\infty }=1\). Since \(h_{0}< R^{*}\), we have
We also observe that \(\phi '(0)=0\) and
It follows that
Set \(\sigma (t)=\eta (t)/h_{0}\), then \(\eta (t)=h_{0}\sigma (t)\). Direct computations yield
We easily see that
Hence, due to (28), we can choose \(\gamma ,\delta >0\) sufficiently small such that
Then
Let \(\frac{K_{1}}{K_{2}}\) satisfy
It can be derived that
We now choose \(K_{1},K_{2}>0\) satisfying (29) and sufficiently large such that
The direct calculation yields
Therefore, if we take
then, for any \(0<\mu \leq \mu_{0}\), we have \(\eta '(t)\geq \mu [ \bar{u}_{r}(t,\eta (t))+\rho \bar{v}_{r}(t,\eta (t))]\). Thus, \((\bar{u},\bar{v}, \eta )\) satisfies
We can apply Theorem 3.3 to conclude that \(h(t)\leq \eta (t)\), \(u(t,r)\leq \bar{u}(t,r)\) and \(v(t,r)\leq \bar{v}(t,r)\) for \((t,r)\in \Omega \). Therefore, \(h_{\infty }\leq \lim_{t\to \infty }\eta (t)\leq h_{0}(1+\delta )<\infty \). □
Proof of Theorem 1.4
The proof is similar to that of Theorem 5.2 of [16] and Theorem 4.11 of [4]. For convenience of the reader we shall give the details. Denote \((u_{\mu }, v_{\mu }, h_{\mu })\) in place of \((u, v, h)\) to clarify the dependence of the solution of (4) on μ. Set
By Theorem 5.2 and Theorem 3.3, \((0,\mu_{0}]\subset \Sigma \). In view of Theorem 5.1, \(\Sigma \cap [\mu^{0}, \infty )=\emptyset \). Therefore, \(\mu^{*}:=\sup \Sigma \in [\mu_{0}, \mu^{0}]\). By this definition and Corollary 3.1 we find that \(h_{\mu ,\infty }\leq R^{*}\) when \(\mu <\mu^{*}\) and \(h_{\mu ,\infty }=\infty \) when \(\mu >\mu^{*}\).
We will show that \(\mu^{*}\in \Sigma \). Otherwise, \(h_{\mu^{*},\infty }=\infty \). There exists \(T>0\) such that \(h_{\mu^{*}}(T)>R^{*}\). By the continuous dependence of \((u_{\mu }, v_{\mu }, h_{\mu })\) on μ, there is \(\varepsilon >0\) such that \(h_{\mu }(T)>R^{*}\) for \(\mu \in (\mu^{*}\varepsilon ,\mu^{*}+\varepsilon )\). It follows that, for all such μ,
This implies that \([\mu^{*}\epsilon ,\mu^{*}+\epsilon ]\cap \Sigma = \emptyset \), and \(\sup \Sigma \leq \mu^{*}\epsilon \). This contradicts the definition of \(\mu^{*}\). The proof is completed. □
6 Discussion
We have examined a free boundary problem of a singlespecies stagestructured model with higher space dimensions and heterogeneous environment for the special case that the environment and solution are radially symmetric. If the environment or solution is not radially symmetric, then the boundary of the spreading domain would not still be a sphere and the Stefan condition \(h'(t)=\mu [u_{r}(t,h(t))+ \rho v_{r}(t,h(t)) ]\) would become very complicated. Similar to the classical Stefan problem, smooth solutions to these free boundary problems need not exist even if the initial data are smooth. It is necessary to make use of other methods to discuss these problems.
In this paper, we firstly discuss the model on \(\mathbb{R}^{n}\), prove that the positive constant steady state is globally asymptotically stable (Theorem 1.1). Then we investigate a free boundary problem of the singlespecies stagestructured model. Our results about vanishing and spreading of the model generalize and unify the previous Theorems 1.2–1.4, which are the existence and uniqueness of solution, the spreading–vanishing dichotomy, the long time behavior of the solution and sharp criteria for spreading and vanishing.
Biologically, the model with stage structure is more realistic than the model without stage structure. From our results, one can control vanishing and spreading of the species more flexibly by the introduction of stage structure. Note that with (12) and Theorem 1.3, if \((u_{0},v_{0})\) and \(h_{0}\) are fixed, then \(R^{*}\) is a direct factor determining the vanishing and spreading. To help the species spreading to infinity, it can be realized by enlarging the birth rate a of the immature or the birth rate β of the mature.
There are some problems left unsolved in our work. When spreading happens, can we find the spreading speed? Can we extend system (4) into the twospecies competitive system with staged structure? We leave these problems to our future work.
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Acknowledgements
The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
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This work was supported by the National Natural Science Foundation of China (No. 11601542, 11671367) and Key Research Projects of Henan Higher Education Institutions (No. 18A110036, 18A110038).
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Zhao, J., Song, C. & Zhang, H. A diffusive stagestructured model with a free boundary. Bound Value Probl 2018, 138 (2018). https://doi.org/10.1186/s1366101810605
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DOI: https://doi.org/10.1186/s1366101810605