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A diffusive stage-structured model with a free boundary
- Jingfu Zhao^{1}Email authorView ORCID ID profile,
- Changming Song^{1} and
- Hongtao Zhang^{1}
- Received: 10 May 2018
- Accepted: 2 September 2018
- Published: 14 September 2018
Abstract
In this paper we mainly consider a free boundary problem for a single-species 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.
Keywords
- Stage-structured model
- Free boundary
- Spreading–vanishing dichotomy
- Long time behavior
- Criteria for spreading and vanishing
1 Introduction
- (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 \(t\to \infty \).
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.
Stage-structured 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 single-species growth model with stage structure represents a mathematically more careful and biologically meaningful formulation approach.
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 well-known 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
Then, we have the following existence and uniqueness result and a priori estimates for the solution of the problem (4).
Theorem 1.2
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:
- (i)vanishing: \(h_{\infty }\leq R^{*}\) andor$$ \lim_{t\to \infty } \bigl\Vert u(t,\cdot ) \bigr\Vert _{C([0,h(t)])}=\lim_{t\to \infty } \bigl\Vert v(t,\cdot ) \bigr\Vert _{C([0,h(t)])}=0; $$
- (ii)spreading: \(h_{\infty }=\infty \), anduniformly for r in any bounded set of \([0,\infty )\).$$ \lim_{t\to \infty }\bigl(u(t,r),v(t,r)\bigr)=(\tilde{u},\tilde{v}) $$
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
We use a squeezing method as in [28] to prove the following theorem.
Theorem 2.1
Proof
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).
Theorem 2.2
Proof
Proof of Theorem 1.1
3 Existence and uniqueness
Theorem 3.1
Proof
Theorem 3.2
Proof
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)
Proof
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
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
Proof
The proof is identical to that of Proposition 3.1 in [23], so we leave out the details. □
Theorem 4.1
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 \(t\to \infty \).
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 4.3
Proof
5 The criteria governing spreading and vanishing
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.
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.
Proof of Theorem 1.4
6 Discussion
We have examined a free boundary problem of a single-species stage-structured 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 single-species stage-structured 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 two-species competitive system with staged structure? We leave these problems to our future work.
Declarations
Acknowledgements
The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
Availability of data and materials
Not applicable.
Funding
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).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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