A free boundary problem for a ratio-dependent diffusion predator-prey system
- Chenglin Li^{1}Email author
Received: 21 June 2016
Accepted: 20 October 2016
Published: 3 November 2016
Abstract
A ratio-dependent diffusion predator-prey system with free boundary is investigated to understand the impact of free boundary on spreading-vanishing dichotomy and a long time behavior of species. The existence and uniqueness of solutions are verified and the behavior of positive solutions is considered for this system. Moreover, the criteria for spreading-vanishing dichotomy are also derived. The results show that if the length of the initial occupying area is longer than a critical size for the predators or the length of the initial occupying area is shorter than a critical size, but the moving coefficient of free boundary is relatively big, then the spreading of predators always happens under relatively small rate of death for the predator. On the other hand, it is found that if the initial value of free boundary is smaller than a threshold value and the moving coefficient of free boundary is relatively small depending on initial size of predator or the rate of death is relatively big, the predators fail in spreading to new environment.
Keywords
spreading vanishing free boundary1 Introduction
The following phenomena often happen in the real world. In order to control or kill the pest population, one can put some natural enemies (predators) into a certain area (a bounded region) by employing the biological method because this kind of preys (pest population) first gets into a bounded area (initial habitat) at the initial state and develops very quickly. In this initial habitat (a bounded area), there is some kind of pest population (prey) and another kind of population (predator, the new or invasive species) enters this region to predate at some time (initial time).
During the process of the predators being put into a new habitat, the predators have a tendency to move from the boundary to their new habitat, i.e., they will get into a new bounded area along the free boundary (which is an unknown curve) to predate as time increases. It is reasonable to suppose that the predator invades a new habitat at a rate which is proportional to the gradients of the predators there. Such kind of free boundary conditions has been already introduced in [7–11]. For the more ecological backgrounds of free boundary conditions, one can also refer to [12].
- (i)
the predator \(u(x, t)\) spreads successfully to a new environment in the sense that \(h(t)\rightarrow\infty\),
- (ii)
the predator \(u(x,t)\) fails in establishing and vanishes eventually, i.e., \(h_{\infty}<\infty\), \(\|u(x,t)\|_{C[0, h(t)]}\rightarrow0\), and \(v(x,t)\rightarrow1\).
The criteria for spreading and vanishing are obtained as follows. If the length of the initial occupying area is longer than a critical size for the predators or the length of the initial occupying area is shorter than a critical size, but the moving coefficient of free boundary is relatively big, then the spreading of predators always happens under relatively small rate of death for the predator. On the other hand, if \(h_{0}\) is smaller than a threshold value and μ is relatively small depending on the initial size of predator or the rate of death is relatively big, the vanishing of predator happens.
Compared with [15], this work mainly has the following differences: (1) it is proved that if the rate of death for the predator is relatively big, then the vanishing of predator happens (Theorem 3.1); (2) new comparison principle is established and then it is used to investigate the criteria for spreading and vanishing; (3) one initial occupying critical size \(\tilde{h}_{0}\) is found (Theorem 3.3) and this value describes that if \(h_{0}>\tilde{h}_{0}\), spreading always happens regardless of μ and the initial value \((u_{0}, v_{0})\); (4) when \(h_{0}\leq \frac{1}{4}\sqrt{\frac{d_{1}}{ec-a}}\), one critical value \(\mu_{0}=\frac{d_{1}}{8M}\) (\(M=\frac{4}{3}\|u_{0}\|_{\infty}\)) is found and specifically expressed (in [15], the existence of this value is proved, but not expressed specifically), and it shows that if \(\mu<\mu_{0}\), then spreading fails.
This paper is organized into four sections. In the next section, the unique existence of solutions for system (1.3) is established. In Section 3, the spreading-vanishing dichotomy is investigated. In the final section, we make some brief comments and draw conclusions.
2 Existence of solution
Theorem 2.1
Proof
The rest of the proof is similar to that of Theorem 2.1 in [1], which follows from the contraction mapping theorem together with standard \(L^{p}\) theory and Sobolev imbedding, so we omit it here. □
To show that the local solution can being extended to all \(t>0\), the following estimate will be employed.
Lemma 2.1
Proof
Let \((u,v)\) be solution of (1.3), then it follows from the strong maximum principle that \(u(x,t)>0\) in \([0, h(t))\times[0, T_{0}]\) and \(v(x,t)>0\) in \([0, \infty)\times[0, T_{0}]\). In addition, by using the maximum principle, we find that there exist positive constants \(C_{1}\) and \(C_{2}\) such that \(u(x, t)\leq C_{1}\) in \([0, h(t))\times[0, T_{0}]\) and \(v(x,t)\leq C_{2}\) in \([0, \infty)\times[0, T_{0}]\). □
Lemma 2.2
Proof
Employing Lemma 2.1 and Lemma 2.2, the local solution of (1.3) can be extended for all \(t>0\) by the regular argument, that is, one can obtain the following results.
Theorem 2.2
There exists a unique solution of system (1.3) for all \(t>0\).
3 The spreading-vanishing dichotomy
From Lemma 2.2 it follows that \(x=h(t)\) is monotonic increasing. Thus, \(\lim_{t\rightarrow\infty}h(t)=h_{\infty}\in(0, \infty]\). The following result shows that the predator will not spread successfully in the case \(a>ec\).
Theorem 3.1
Suppose that \(a>ec\), then \(\lim_{t\rightarrow\infty}\|u(\cdot, t)\|_{C([0, h(t)])}=0\) and \(h_{\infty}<\infty\). Furthermore, \(\lim_{t\rightarrow\infty}v(r, t)=1\) holds uniformly in any bounded subset of \([0, \infty)\).
Proof
Hence, \(\lim_{t\rightarrow\infty}v(x, t)=1\) holds uniformly in any bounded subset of \([0, \infty)\).
Next, use Lemma 3.2 of [17] to system (3.1), one can obtain \(h_{\infty}<\infty\). This completes the proof. □
The following comparison principle can be used to estimate the solution \((u(x,t), v(x,t))\) and free boundary \(x=h(t)\).
Lemma 3.1
The comparison principle
Proof
First of all, the application of the comparison principle to the equations of v and v̅ yields \(\overline{v}>v\) directly. Since the function \(\frac{uv}{u+v} \) is increasing in u for \(u,v \geq0\), by employing the comparison principle given by [1] for the single equation to u̅ and u, one can get \(\overline{u}>u\) directly. The regular arguments and the detailed proofs are omitted. □
Lemma 3.2
Suppose that \(h_{0}< L^{\ast}\). Then there exists \(\mu_{0}>0\) which depends on \(u_{0}\) such that the predator fails in spreading if \(\mu\leq\mu_{0}\).
Proof
Theorem 3.2
Suppose that \(a< ec\) and \(h_{0}\leq\frac{1}{4}\sqrt{\frac{d_{1}}{ec-a}}\). If \(\mu\leq\frac{d_{1}}{8M}\), then \(h_{\infty}\leq4h_{0}<\infty\), where \(M=\frac{4}{3}\|u_{0}\|_{\infty}\).
Proof
Lemma 3.3
Suppose that \(h_{\infty}<\infty\). Then \(\lim_{t\rightarrow\infty}\|u(\cdot, t)\|_{C([0, h(t)))}=0\) and \(\lim_{t\rightarrow\infty}v(x, t)=1\) hold uniformly in any bounded subset of \([0, \infty)\).
Proof
Remark 3.1
If the predator fails in spreading, then it will be extinct finally.
Theorem 3.3
Suppose that \(a< ec\), then \(h_{\infty}=\infty\) if \(h_{0}>\tilde{h}_{0}\), where \(\lambda_{1}(\tilde{h}_{0})=\frac{1}{d_{1}}(ec-a)>0\).
Proof
Remark 3.2
Since \(\lambda_{1}(\tilde{h}_{0})=\frac{1}{d_{1}}(ec-a)<\frac {ec}{d_{1}}=\lambda_{1}(L^{\ast})\), one can find that \(\tilde{h}_{0}>L^{\ast}\), which is different from the result of [1].
Theorem 3.4
Assume that \(a< ec\) and \(h_{0}\leq \frac{1}{4}\sqrt{d_{1}/(ec-a)}\). Then there exists \(\mu_{1}\) depending on \(u_{0}(x)\) and \(v_{0}(x)\) such that \(h_{\infty}=\infty\) if \(\mu>\mu_{1}\) for system (1.3).
Proof
Using a similar argument to the proof of Theorem 3.2 in [15], one can prove the following theorem, which shows that the predator establishes itself successfully in the new environment in the sense that \(h_{\infty}=\infty\) if the rate of death for the predator is relatively small. Moreover, in this case, both predator and prey can coexist for a long time.
Theorem 3.5
The proof of the theorem is similar to that of Theorem 3.2 in [15], so it is omitted.
Theorem 3.6
Assume that \(ec>a\) and \(h_{0}<\frac{1}{4}\sqrt{d_{1}/(ec-a)}\). There exists \(\mu^{\ast}>\mu_{\ast}>0\) which depends on \(u_{0}(x)\) and \(v_{0}(x)\), such that \(h_{\infty}<4h_{0}\) if \(\mu<\mu_{\ast}\) and \(h_{\infty}=\infty\) if \(\mu>\mu^{\ast}\) for system (1.3).
The proof of Theorem 3.6 is essentially the same as that of Theorem 5.2 in [4] and thus is omitted.
4 Comments and conclusions
In this article, we have investigated a ratio-dependent diffusion predator-prey system with the free boundary \(x=h(t)\), which describes the process of movement for the predator species.
For the successful spreading of predator to a new environment for this model, only one result is derived, that is, the predator \(u(x, t)\) spreads successfully to a new environment in the sense that \(h(t)\rightarrow\infty\) if \(a< ec\) and \(h_{0}>\tilde{h}_{0}\), where \(\lambda_{1}(\tilde{h}_{0}) =\frac{1}{d_{1}}(ec-a)>0\).
Assume one of the following three cases holds: (i) \(a< ec\), \(h_{0}\leq\frac{1}{4}\sqrt{\frac{d_{1}}{ec-a}}\) and \(\mu\leq\frac{d_{1}}{8M}\), then \(h_{\infty}<\infty\), where \(M=\frac{4}{3}\|u_{0}\|_{\infty}\); (ii) \(a>ec\); (iii) \(h_{0}< L^{\ast}\) and \(\mu\leq\mu_{0}\), where \(\mu_{0}>0\) depending on \(u_{0}\). Then the predator \(u(x,t)\) fails in establishing itself and vanishes finally, i.e., \(h_{\infty}<\infty\), \(\|u(x,t)\|_{C[0, h(t)]}\rightarrow0\) and \(v(x,t)\rightarrow1\).
Therefore, the criteria for spreading and vanishing are as follows. If the death rate of predator is relatively small and the length of the initial occupying area is longer than a critical size \(\tilde{h}_{0}\), then the spreading of predator always happens. For vanishing of the predator, there are three criteria: (i) the rate of death is bigger than a critical value ec; (ii) the length of initial occupying area \(h_{0}\) is shorter than a threshold value \(L^{\ast}\) and μ is smaller than the critical value \(\mu_{0}\), depending on \(u_{0}\); (iii) the length of the initial occupying area \(h_{0}\) is shorter than \(\frac{1}{4}\sqrt{\frac{d_{1}}{ec-a}}\) and μ is smaller than \(\frac{d_{1}}{8M}\), depending on \(u_{0}\).
From the above results of the dichotomy, it follows that in order to control the prey population (pest species) one should at least put predator population (natural enemies) into the initial habitat at the initial state in one of four ways: (i) decrease the death rate of predator during the process of putting; (ii) extend the range of predator’s targets; (iii) accelerate putting predators; (iv) choose the natural enemies which have a strong ability for predating.
Declarations
Acknowledgements
The author thanks the anonymous referee very much for the valuable comments and suggestions to improve the contents of this article. This work was supported by the National Natural Science Foundation of China (11461023) and the research funds of PhD for Honghe University (14bs19). The author also thanks Prof. Wang Mingxin very much for providing reference [15] which was important for work in the revision stage.
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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