A reaction-diffusion-advection logistic model with a free boundary in heterogeneous environment
- Jianxiu Liang^{1},
- Lili Liu^{2} and
- Zhen Jin^{2}Email author
Received: 8 April 2016
Accepted: 6 July 2016
Published: 13 July 2016
Abstract
The aim of this paper is to investigate the dynamics of the solution for a class of reaction-diffusion-advection logistic model with a free boundary in heterogeneous environment. The species undergoes diffusion and advection in a one dimensional heterogeneous environment, and it invades the environment with a spreading front evolving as the free boundary. To understand the effects of the advection rate α and the expansion capacity μ on the dynamics of this model, we derive a spreading-vanishing dichotomy and obtain the sharp criterion for the spreading and vanishing by choosing α and μ as variable parameters. That is, the invasion species can unconditionally survive for a slow advection rate, while, for a fast advection rate, whether it can survive or not depends on the expansion capacity and initial values of the invasion species.
Keywords
reaction-diffusion-advection free boundary condition heterogeneous environment spreading vanishing1 Introduction
Due to the serious threat of invasive species to bio-diversity conservation and the global economy, mathematical modeling has become an important tool in analyzing the prediction and prevention of biological invasions [1, 2]. Recently, a great deal of attention has been paid to developing more realistic mathematical models for the invasion dynamics. There have been a number of works on modeling the invasion of a species described by a reaction-diffusion system; see [3–7] and the references cited therein for further details.
The rest of our paper is arranged as follows. In the next section, we present some preliminaries including the global existence and uniqueness of the solution of model (1.3) and the comparison principle in the moving domain. In order to show the results of the spreading-vanishing dichotomy, the eigenvalue problems associated with model (1.3) are given in Section 3, and a sharp criterion for spreading and vanishing is established in Section 4. In Section 5, the asymptotic spreading speed of the free boundary is estimated if spreading of the invasion species occurs. In Section 6, we give a brief discussion.
2 Preliminaries
In this section, we first present the local existence and uniqueness of the solution of model (1.3) and then use suitable estimates to show that the solution is global.
First, we show that the local existence and uniqueness of the solution of model (1.3). Similar to those presented in [20] and [32], the proof can be shown via minor modifications. Thus, we omit it here.
Theorem 2.1
Next, we show the global existence of the solution of model (1.3) gained in Theorem 2.1. The following lemma is needed.
Lemma 2.2
Proof
Following the standard proof in [36], we obtain the following theorem for the global existence of the solution of model (1.3).
Theorem 2.3
The solution of model (1.3) exists and is unique for \(t\in (0,\infty)\).
Finally, we introduce the comparison principle for model (1.3).
Lemma 2.4
(The comparison principle [37])
Here, we refer to the pair \((\overline{u},\overline{h})\) as the upper solution of model (1.3). Similarly, the lower solution can be defined by reversing all the inequalities.
To investigate the dependence of the solution of model (1.3) on the expanding capability μ, we rewrite the solution \((u,h)\) of model (1.3) as \((u_{\mu},h_{\mu})\). As a direct consequence of Lemma 2.4, we have the following corollary.
Corollary 2.5
Let \((u_{\mu},h_{\mu})\) be the solution of model (1.3). For fixed \(u_{0}\), α, \(h_{0}\), \(r(x)\), if \(\mu_{1}\leq\mu_{2}\), then \(u_{\mu _{1}}(x,t)\leq u_{\mu_{2} }(x,t)\) for \((x,t)\in[0,h_{\mu_{1}}(t)]\times (0,\infty)\), and \(h_{\mu_{1}}(t)\leq h_{\mu_{2}}(t)\) for \(t\in(0,\infty)\).
3 An eigenvalue problem
In this section, we mainly study the principal eigenvalue problem and the properties of its principal eigenvalue. These results are significant for later sections.
First, we fix \(r(x)\) and α. Note that α is bounded. Then we have the following theorem.
Theorem 3.1
- (i)
\(\tau_{1}(h_{0})\) is strictly increasing function of \(h_{0}\);
- (ii)
\(\lim_{h_{0}\to0}\tau_{1}(h_{0})=-\infty\);
- (iii)
\(\lim_{h_{0}\to\infty}\tau_{1}(h_{0})=\max_{x\in [0,h_{0}]}{r(x)}-\frac{\alpha^{2}}{4}>0\).
Proof
For the proof of part (i), it is similar to [27], Theorem 3.2, with some minor modifications. Here, we omit it.
As a consequence of the above theorem, we have the following corollary.
Corollary 3.2
Next, we fix \(r(x)\) and \(h_{0}\). To consider the effects of advection on dynamics of model (1.3), we introduce the following theorem, which is the counterpart of Theorem 3.1, and we refer to [16], Lemmas 4.5, 4.6 and Theorem 4.1, for a detailed proof.
Theorem 3.3
- (i)
\(\tau_{1}(\alpha)\) is a strictly decreasing function of α;
- (ii)
\(\lim_{\alpha\to\infty}\tau_{1}(\alpha)=-\infty\);
- (iii)
\(\lim_{\alpha\to0}\tau_{1}(\alpha)>0\).
The above theorem implies the following corollary.
Corollary 3.4
Thus, Lemma 2.2 implies that the following theorem holds true.
Theorem 3.5
\(\tau_{1}(h(t))\) is a strictly monotone increasing function of t, it is equivalent to \(\tau_{1}(h(t))\) being a strictly monotone increasing function of \(h(t)\).
4 Spreading and vanishing of an invasive species
This section is devoted to the proofs of the spreading-vanishing dichotomy, and the sharp criteria for spreading and vanishing.
4.1 Spreading-vanishing dichotomy
It follows from Lemma 2.2 that \(x=h(t)\) is a monotone increasing function of t. There exists \(h_{\infty}\in(h_{0},\infty]\) such that \(\lim_{t\rightarrow\infty}h(t)=h_{\infty}\). The spreading-vanishing dichotomy is a consequence of the following three theorems.
First, we show that the species will die out if the species cannot spread to the whole domain. Thus, the following theorem holds true.
Theorem 4.1
Assume that \(\alpha< c_{0}\). If \(h_{\infty}<\infty\), then \(h_{\infty}\leq h^{\ast}\) and \(\lim_{t\to\infty}\|u(x,t)\|_{C([0,h(t)])}=0\).
Proof
Theorem 4.1 can be proved by the following two steps.
Next, we show the case that the species can spread to the whole domain. Thus, we have the following theorem.
Theorem 4.2
If \(0<\alpha\leq\alpha^{\ast}\), then \(\lim_{t\to\infty}\|u(x,t)\| _{C([0,h(t)])}>0\) and \(h_{\infty}=\infty\), i.e., spreading occurs.
Proof
Here, we still use \(\tau_{1}\) and \(\phi_{1}\) to denote the principal eigenvalue and the corresponding eigenfunction of problem (3.1), respectively.
Consider the case \(\alpha=\alpha^{\ast}\). By Corollary 3.4, one has \(\tau_{1}=0\). The monotonicity of \(h(t)\) shows that \(h(t_{0})>h_{0}\) for small \(t_{0}>0\). It follows from Theorem 3.1 that we have \(\tau _{1}(\alpha^{\ast},r,h(t_{0}))>\tau_{1}(\alpha^{\ast},r,h_{0})\). Hence, replacing \(h_{0}\) by \(h(t_{0})\), we can repeat the same process as above to construct a lower solution over \([0,h(t_{0})]\times[t_{0},\infty)\). And so, the desired result follows. The proof is finished. □
Finally, we study the long time behavior of the spreading species. The proof is similar with those in [38], Theorem 2.3, and [39], Propositions 3, 4. Here, we omit the proof for brevity.
Theorem 4.3
Combining Theorems 4.1, 4.2 and 4.3, we obtain the following spreading-vanishing dichotomy theorem.
Theorem 4.4
- (i)
Spreading: \(h_{\infty}=\infty\) and \(\lim_{t\to\infty }u(x,t)=u^{\ast}(x)\) uniformly in any bounded subset of \((0,\infty)\);
- (ii)
Vanishing: \(h_{\infty}<\infty\) and \(\lim_{t\to\infty}\| u(x,t)\|_{C([0,h(t)])}=0\).
4.2 Sharp criteria for spreading and vanishing
From Theorem 4.2, we see that vanishing is possible only when \(\alpha>\alpha^{\ast}\). The following theorem reveals the sharp criteria for spreading and vanishing, which provides some sufficient conditions for vanishing.
Theorem 4.5
Proof
As a direct consequence of Theorem 4.5, we are able to show the following theorem.
Theorem 4.6
The next result implies that spreading occurs for large expansion capacity and the proof will be leaved out, since it is similar to that in [32], Lemma 3.7, or [33], Lemma 2.8.
Theorem 4.7
For \(\alpha>\alpha^{\ast}\) and any given \(u_{0}\) satisfying model (1.3), if μ is sufficiently large then \(h_{\infty}=\infty\).
Combining Theorems 4.5-4.7, we can derive the sharp criteria for spreading-vanishing for the invasion species.
Theorem 4.8
(Sharp criteria)
For given \(h_{0}\), α, and \(u_{0}\) satisfying model (1.3), there exists \(\mu^{\ast}\in[0,\infty)\) (depending on \(u_{0}\) and α) such that spreading occurs if \(\mu>\mu^{\ast}\); and vanishing occurs if \(0<\mu\leq\mu^{\ast}\). Furthermore, \(\mu^{\ast}=0\) if \(0<\alpha\leq\alpha^{\ast}\), while \(\mu^{\ast}>0\) if \(\alpha>\alpha^{\ast}\).
Proof
First, for the case \(0<\alpha\leq\alpha^{\star}\), it follows from Theorem 4.2 that spreading always occurs if \(0<\alpha\leq\alpha^{\ast}\). Therefore, we can choose \(\mu^{\ast}=0\).
Finally, we show that vanishing happens if \(\mu=\mu^{\ast}\). Otherwise, \(h_{\infty}=\infty\) for \(\mu=\mu^{\ast}\). It follows from Corollary 3.2 that there exists \(T_{0}>0\) such that \(h(T_{0})>h^{\ast}\). Now, the symbol \((u_{\mu},h_{\mu})\) is used to emphasize the dependence of solution \((u,h)\) of model (1.3) on μ. Hence, \(h_{\mu }(T_{0})>h^{\ast}\) follows. Since \((u_{\mu},h_{\mu})\) has a continuous dependence on μ, we see that there exists small \(\varepsilon>0\) such that \(\mu=\mu^{\ast}-\varepsilon\), which implies that the species spreads, contradicting the definition of \(\mu^{\ast}\). Hence, vanishing occurs if \(\mu=\mu^{\ast}\). The proof is completed. □
5 Asymptotic spreading speed
This section is devoted to rough estimates of the asymptotic spreading speed. Following [40], Proposition 2.1, we have the following proposition.
Proposition 5.1
([40])
The following proposition is useful for the proof of the main result.
Proposition 5.2
([41])
- (i)
Problem (5.1) has exactly one solution \((k^{\ast},q^{\ast})\) such that \(\mu(q^{\ast})'(0)=k^{\ast}\). Moreover, \(k^{\ast}=k^{\ast}(\alpha ,r)\in(0,2\sqrt{\overline{r}} +\alpha)\);
- (ii)
\(0<\overline{k}^{\ast}<k^{\ast}\), where \(\overline{k}^{\ast}\) is the speed of problem (5.1) with \(\alpha=0\);
- (iii)\(k^{\ast}\) is strictly increasing of parameter r, i.e., for any \(r_{1}>r_{2}>0\), we have$$k^{\ast}(\alpha, r_{1})>k^{\ast}(\alpha,r_{2}), \qquad \lim_{\varepsilon \rightarrow0} k^{\ast}(\alpha,r+ \varepsilon)=k^{\ast}(\alpha,r). $$
Now, we have the following estimates of the asymptotic spreading speed for model (1.3) when spreading happens.
Theorem 5.3
Proof
6 Discussion
In this paper, we incorporated the free boundary and heterogenous environment into the reaction-diffusion-advection logistic model, which is more realistic for describing the invasion dynamics of a new species, investigated the influence of the advection term and spatial heterogeneous environment features on the dynamics of the invading species, and gave a rough estimates of the asymptotic spreading speed.
According to the analysis, we determined the spreading-vanishing dichotomy and the sharp criteria for the spreading and vanishing by choosing the advection rate α and the expansion capacity μ as variable parameters. Therefore, whether the invasion species will spread or vanish depends on the advection rate α, the expansion capacity μ, and the initial function \(u_{0}(x)\). More specifically, we found a positive threshold \(\alpha^{\ast}\) such that the invasive species always spreads if \(0<\alpha\leq\alpha^{\ast}\), which is consistent with the result of [16]. However, for \(\alpha>\alpha^{\ast}\), there exists a critical criterion, \(\mu^{\ast}>0\), such that species spreads if \(\mu>\mu^{\ast}\) and species vanishes if \(0<\mu\leq\mu^{\ast}\), which is distinct from that in [16]. Biologically, these results mean that the invasive species spreads either under a small advection rate, or under a big advection rate when it is favored by its expansion capacity and initial values. Furthermore, species with a large expansion capacity will benefit to survive.
Now, we make some comparisons with the previous work. Compared with the work of [32, 33], we considered the influence of the advection term on the dynamics of the invading species. Compared with the work of [37], we considered the influence of spatial heterogeneous environment features on the dynamics of the invasive population. Compared with the work of [9, 16], our work implied that spreading or vanishing of invasive population not only depends on advection term and no-flux boundary condition but it also relates to free boundary condition. It should be noticed that we found two thresholds, \(\alpha^{\ast}\) and \(\mu^{\ast}\), which are different from the previous work. Therefore, our work and the corresponding conclusions are more general. It should be noticed that, due to the assumption of the heterogeneous environment, we only obtain the weaker result on the asymptotic behavior of the invasive species.
Declarations
Acknowledgements
This work is Supported by National Natural Science Foundation of China (No. 11331009) and Shanxi Scholarship (No. 901013-1/7). We are grateful to the editors and the anonymous referees for their careful reading, valuable comments, and helpful suggestions, which have helped us to improve the presentation of this work significantly.
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.
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