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A free boundary problem for a ratiodependent diffusion predatorprey system
Boundary Value Problems volumeÂ 2016, ArticleÂ number:Â 192 (2016)
Abstract
A ratiodependent diffusion predatorprey system with free boundary is investigated to understand the impact of free boundary on spreadingvanishing 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 spreadingvanishing 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.
1 Introduction
In mathematical ecology, the invasion of immigration for the new species is one of the most important topics. From a viewpoint of mathematical ecology, the various invasion models have been recently put forward and investigated by many ecological mathematicians. For instance, [1â€“6] proposed reactiondiffusion population models with free boundary to understand the process of the new or invasive population. In [1], the free boundary model is proposed with a logistic diffusion equation:
where \(x=h(t)\) is the free boundary which will be determined, a, b, d, Î¼ and \(h_{0}\) are positive constants, and \(u_{0}\) is a nonnegative initial function. The authors in [1] have considered the uniqueness and existence of global solutions, and derived some interesting results for the dynamics of solution. The vanishingspreading dichotomy of the population is one of the most remarkable and important results, that is, the solution \((u, h)\) fulfills: \(h(t)\rightarrow\infty\), \(u(x, t)\rightarrow a/b\) as \(t\rightarrow\infty\) or \(h(t)\rightarrow h_{\infty}\leq \frac{\pi}{2}\sqrt{d/a} \), \(u(x, t)\rightarrow0\) as \(t\rightarrow \infty\).
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 [7â€“11]. For the more ecological backgrounds of free boundary conditions, one can also refer to [12].
Recently, in [3], the authors have considered the following double free boundary predatorprey problem \(\mathbb{R}^{1}\):
where \(\mathbb{R}=(\infty, \infty)\), \(x=h(t) \) and \(x=g(t)\) stand for the right and left moving boundaries, respectively, a, b, c, D, \(h_{0}\) and Î¼ are positive constants. The existence and uniqueness of global solutions of (1.2) and a spreadingvanishing dichotomy have been established. Moreover, the criteria for spreading and vanishing have been obtained in this paper, that is, a spreading critical size \(h_{0}=\frac{\pi}{2}\sqrt{1/(1+ab)}\) has been derived. Wang in [4] has examined three preypredator models with free boundary in \(\mathbb{R}^{1}\): DFB, NFB, and TFB. The spreadingvanishing dichotomy, criteria governing spreadingvanishing, and the long time behavior of solution have been provided. For more detailed results, one can refer to [3, 4]. In higher dimension space, Zhao and Wang [13] have considered a LotkaVolterra competition system incorporating two free boundary with signchanging coefficients, derived some sufficient conditions for species spreading success and spreading failure, and derived the long time behavior of solutions.
In the process of spreading for the predators, some of them die of starvation, cold and illness. We want to understand how the rate of death impacts on spreading. The behavior of predating always changes by the change of the size of preys and many ecologists observe that the ratiodependent functional response is more reasonable to describe the process of predating for some predators. Based on these facts, we consider the following ratiodependent reactiondiffusion predatorprey system with free boundary including a death term:
where \(x=h(t)\) represents the moving boundary to be determined; u expresses the population density of the predator species while v stands for the population density of the prey species. \(h_{0}\), a, b, c, \(d_{i}\) (\(i=1,2\)) and e are positive constants. For what these coefficients stand for the detailed meaning, one can refer to [14]. The initial functions \(u_{0}(x)\) and \(v_{0}(x)\) correspondingly satisfy
In this article, we shall show that system (1.3) admits a unique solution and a spreadingvanishing dichotomy holds for this system, namely, as \(t\rightarrow\infty\), either

(i)
the predator \(u(x, t)\) spreads successfully to a new environment in the sense that \(h(t)\rightarrow\infty\),
or

(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 Î¼ 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)Â it is proved that if the rate of death for the predator is relatively big, then the vanishing of predator happens (TheoremÂ 3.1); (2)Â new comparison principle is established and then it is used to investigate the criteria for spreading and vanishing; (3)Â one initial occupying critical size \(\tilde{h}_{0}\) is found (TheoremÂ 3.3) and this value describes that if \(h_{0}>\tilde{h}_{0}\), spreading always happens regardless of Î¼ and the initial value \((u_{0}, v_{0})\); (4)Â when \(h_{0}\leq \frac{1}{4}\sqrt{\frac{d_{1}}{eca}}\), 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 SectionÂ 3, the spreadingvanishing dichotomy is investigated. In the final section, we make some brief comments and draw conclusions.
2 Existence of solution
Theorem 2.1
Assume that \(u_{0}\) and \(v_{0}\) satisfies (1.4) for some \(h_{0} > 0\). Then, for \(0\leq t< T\) and any \(\alpha\in(0, 1)\), there exists a unique solution,
for system (1.3), furthermore,
where \(D_{T}=\{(x, t)\in \mathbb{R}^{2}: x\in[0, h(t)], t\in[0, T) \}\), \(D_{T}^{\infty}=\{(x, t)\in \mathbb{R}^{2}: x\in[0, \infty), t\in[0, T) \}\), c and T only depend on \(h_{0}\), Î±, \(\u_{0}\_{C^{2}[0, h_{0}]}\) and \(\v_{0}\_{C^{2}([0, \infty))}\).
Proof
As in [16], it needs to straighten the free boundary. Assume that \(\zeta(s)\) is a function in \(C^{3}[0, \infty)\) fulfilling
Let us introduce a transformation
which yields the transformation
For \(t\geq0\), if
the above transformation \(x\rightarrow y\) is a diffeomorphism from \(\mathbb{R}^{1}\) onto \(\mathbb{R}^{1}\) and the induced transformation \(s\rightarrow x\) is also a diffeomorphism from \([0, \infty)\) onto \([0, \infty)\). Furthermore, it transforms the free boundary \(x=h(t)\) into the line \(s=h_{0}\). Straightforward computations show that
Setting
then the free boundary system (1.3) can be rewritten as follows:
where \(A=A(h(t), s)\), \(B=B(h(t), s)\), \(C=C(h(t), s)\).
The rest of the proof is similar to that of TheoremÂ 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.â€ƒâ–¡
To show that the local solution can being extended to all \(t>0\), the following estimate will be employed.
Lemma 2.1
Assume that \((u,v)\) is a bounded solution of (1.3) for \(t\in(0, T_{0})\) and \(T_{0}\in(0, +\infty]\). Then there exist positive constants \(C_{1}\) and \(C_{2}\) which are independent of \(T_{0}\) such that
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 \([0, \infty)\times[0, T_{0}]\).â€ƒâ–¡
Lemma 2.2
Assume that \((u, v)\) is a defined solution of (1.3) for \(t\in(0, T_{0})\) and \(T_{0}\in(0, +\infty]\). Then there exists a positive constant \(C_{3}\) which is independent of \(T_{0}\) such that
Proof
By using Hopf Lemma to the equation of u, we immediately obtain
for \(0< t< T_{0}\), and \(0\leq x< h(t)\). It follows from the Stefan condition that \(h'(t)>0\) for \(t\in(0, T_{0})\).
We define
as in [2] and establish an auxiliary function
We shall show that A can be chosen so that \(w(x,t)\geq u(x,t)\) holds over Î©. It is easy to check that, for \((x,t)\in\Omega\),
It follows that
if \(A^{2}\geq\frac{ec}{2 d_{1}}\). Next, we have
Thus, for \(0< t< T_{0}\) and \(x\in[h_{0}A^{1}, h_{0}]\), if we can take A such that \(u_{0}(x)\leq w(x, 0)\), then, by applying the maximum principle to \(wu\) over Î©, we have \(u(x, t)\leq w(x,t)\) for \((x, t)\in\Omega\), which yields
Therefore, it is necessary to find certain A independent of \(T_{0}\) such that \(u_{0}(x)\leq w(x, 0)\) for \(x\in[h_{0}A^{1}, h_{0}]\).
By direct calculation, we get
for \(x\in[h_{0}(2A)^{1}, h_{0}]\). Then, by choosing
for \(x\in[h_{0}(2A)^{1}, h_{0}]\), we have
Since \(w(h_{0}, 0)=u_{0}(h_{0})=0\), the above inequality yields \(w(x, 0)\geq u_{0}(x)\). Furthermore, for \(x\in[h_{0}A^{1}, h_{0}(2A)^{1}]\), one can easily find that
Therefore, \(u_{0}(x)\leq w(x, 0)\) for \(x\in[h_{0}A^{1}, h_{0}]\). This completes the proof.â€ƒâ–¡
Employing LemmaÂ 2.1 and LemmaÂ 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 spreadingvanishing dichotomy
From LemmaÂ 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
By the comparison principle, one can get \(v(x, t)\leq\overline{v}(t)\) for \(x\geq0\) and \(t\in(0, \infty)\), where
is the solution of the ODE problem
Since \(\lim_{t\rightarrow\infty}\overline{v}(t)=1\), it is deduced that \(\lim\sup_{t\rightarrow\infty} v(t)\leq1\) holds uniformly in any bounded subset of \([0, \infty)\). From the condition \(a>ec\) it follows that there exists a small Îµ such that \(a>\frac{ec(1+\varepsilon)}{u+b(K+\varepsilon)}\). On the other hand, for this Îµ, there exists \(T_{0}\) such that \(v(x,t)\leq 1+\varepsilon\) in \([0, +\infty)\times[T_{0}, \infty)\). Then \(u(x,t)\) satisfies
Using comparison again, we find that \(\lim_{t\rightarrow \infty}\u(\cdot, t)\_{C([0, h(t)])}=0\). This, together with the condition \(u(x, t)=0\) for \(t>0\), \(x\geq h(t)\), shows that there exists \(T_{\varepsilon}>0\) such that \(u(x, t)<\varepsilon\) for any given \(0<\varepsilon\ll1\), \(t>T_{\varepsilon}\) and \(x>0\). Then the function \(v(x, t)\) satisfies
By using the comparison principle again, we obtain
holds uniformly in any bounded subset of \([0, +\infty)\). Since \(\varepsilon>0\) is arbitrary, it follows that \(\lim\inf_{t\rightarrow\infty} v(x, t)\geq1\) uniformly in any bounded subset of \([0, \infty)\).
Hence, \(\lim_{t\rightarrow\infty}v(x, t)=1\) holds uniformly in any bounded subset of \([0, \infty)\).
Next, use LemmaÂ 3.2 of [17] to system (3.1), one can obtain \(h_{\infty}<\infty\). This completes the proof.â€ƒâ–¡
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
Suppose that \(T\in(0, \infty)\), \(\overline{h}\in C^{1}([0, T])\), \(\overline{v}\in C((0, \infty)\times[0,T])\cap C^{2,1}((0, \infty)\times(0, T])\), \(\overline{u}\in C(\overline{D}^{\ast}_{T})\cap C^{2,1}(D^{\ast}_{T})\) with \(D^{\ast}_{T}=\{(x,t)\in \mathbb{R}^{2}:0< x<\overline{h}(t), 0<t\leq T\}\), and
Then the solution \((u(r,t), v(r,t), h(t))\) of the free boundary problem (1.3) fulfills
Proof
First of all, the application of the comparison principle to the equations of v and vÌ… 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 uÌ… and u, one can get \(\overline{u}>u\) directly. The regular arguments and the detailed proofs are omitted.â€ƒâ–¡
It is well known that the principal eigenvalue \(\lambda_{1}(L)\) of the problem
is a strictly decreasing continuous function and
Hence, there exists a unique \(L^{\ast}\) (\(L^{\ast}>0\)) such that
\(\lambda_{1}(L)<\frac{ec}{d_{1}}\) for \(L>L^{\ast}\) and \(\lambda_{1}(L)>\frac{ec}{d_{1}}\) for \(L< L^{\ast}\).
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
For \(t>0\) and \(x\in(0, \sigma(t))\), define
where M, Î´, Î³ are positive constants to be taken later and \(W(x)\) is the first eigenfunction of the problem
with \(W\geq0\) and \(\W\_{\infty}=1\). From \(h_{0}< L^{\ast}\), it follows that
By (3.5), it is obvious that \(W'(0)=0\). Therefore, it is deduced that \(W'(x)<0\) for \(0< x\leq h_{0}\).
Set \(\tau(t)=1+\delta\frac{\delta}{2}e^{\gamma t}\) so that \(\sigma(t)=h_{0}\tau(t)\). By direct calculations, it is derived that
Hence, by \(\lambda_{1}(h_{0})>ec\), one can take \(\gamma< a\) such that, for \(t>0\), \(x\in[0, \sigma(t)]\),
which indicates
for \(t>0\), \(x\in[0, \sigma(t)]\).
Now, take \(M>0\) large enough such that
Note that
Hence, by taking
we find that \(\sigma'(t)\geq\mu\omega_{x}(t, \sigma(t))\) for any \(0<\mu\leq\mu_{0}\).
Let vÌ… be a unique positive solution of
and thus \((\omega, \overline{v}, \sigma)\) satisfies
Hence, by the comparison principle (LemmaÂ 3.1), one can conclude that
for \(0\leq x\leq h(t)\) and \(t>0\). It follows that
This completes the proof.â€ƒâ–¡
Theorem 3.2
Suppose that \(a< ec\) and \(h_{0}\leq\frac{1}{4}\sqrt{\frac{d_{1}}{eca}}\). If \(\mu\leq\frac{d_{1}}{8M}\), then \(h_{\infty}\leq4h_{0}<\infty\), where \(M=\frac{4}{3}\u_{0}\_{\infty}\).
Proof
First, we construct a suitable upper solution to system (1.3) by defining
where \(M_{1}=\max\{\v_{0}\_{\infty}, 1 \}\), and
where \(M_{1}\), Î³ and M are positive constants to be taken later.
Straightforward computation yields
for all \(0< x<\overline{h}(t)\) and \(t>0\). In addition, one can easily check that \(\overline{h}'(t)=2h_{0}\gamma e^{\gamma t}\) and \(\mu\overline{u}_{x}(h(t), t)=2M\mu\overline{h}^{1}e^{\gamma t}\). Furthermore, we note that \(\overline{u}(x, 0)=M(1x^{2}/4h_{0}^{2})\geq\frac{3}{4} M\), \(\overline{v}(x,0)\geq v_{0}(x)\) for \(x\in(0, h_{0}]\). Since \(\overline{h}(t)\leq4h_{0}\), we choose \(M=\frac{4}{3}\u_{0}\_{\infty}\) and \(\gamma=\frac{d_{1}}{16 h_{0}^{2}}\), \(\mu\leq\frac{d_{1}}{8M}\), where \(h_{0}\leq\frac{1}{4}\sqrt{\frac{d_{1}}{eca}}\). Then we find that (3.3) holds. Hence, by LemmaÂ 3.1, we find that \(h(t)\leq\overline{h}(t)\) for \(t>0\) and \(h_{\infty}\leq \lim_{t\rightarrow\infty}\overline{h}(t)=4h_{0}<\infty\). This completes the proof.â€ƒâ–¡
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
Suppose for contradiction that
Then, for all \(m\in\mathbb{N}\), there exists a sequence \((x_{m}, t_{m})\) in \([0, h(t))\times(0, \infty)\) such that \(u(x_{m}, t_{m})\geq\sigma/2\) and \(t_{m}\rightarrow\infty\) as \(m\rightarrow\infty\). From \(0\leq x_{m}< h(t)< h_{\infty}<\infty\) it follows that a subsequence of \(\{x_{n}\}\) converges to \(x_{0}\in[0, h_{\infty})\). Without loss of generality, denote this subsequence still by \(\{x_{n}\}\).
Define \(u_{m}(x,t)=u(x, t_{m}+t)\) and \(v_{m}(x,t)=v(x, t_{m}+t)\) for \((x,t)\in(0, h(t_{m}+t))\times(t_{m}, \infty)\). By the parabolic regularity, we find that there exists a subsequence \(\{(u_{m_{i}}, v_{m_{i}})\}\) of \(\{(u_{m}, v_{m})\}\) such that \((u_{m_{i}}, v_{m_{i}})\rightarrow(\tilde{u}, \tilde{v})\) as \(i\rightarrow\infty\) and \((\tilde{u}, \tilde{v})\) satisfies
From \(\tilde{u}(x_{0}, 0)\geq\sigma/2\), it follows that \(\tilde{u}>0\) in \((0, h_{\infty})\times(\infty, +\infty)\). Since \(a +\frac{ec \tilde{v}}{\tilde{u}+\tilde{v}}\) is bounded by \(R:=a+ec\), by using Hopf lemma to the equation \(\tilde{u}_{t}d_{1}\tilde{u}_{xx}\geqR\tilde{u}\) at the point \((h_{\infty}, 0)\), one can find that \(\tilde{u}_{x}(h_{\infty}, 0)\leq\delta_{0}\) for some \(\delta_{0}>0\).
As in [18], we define
then straightforward calculations yield
Therefore, \(\vartheta(s,t)\) fulfils
The free boundary \(x=h(t)\) is straightened as the fixed line \(s=h_{0}\). By Proposition A in [19], there exists a positive constant \(K_{0}\) such that
which shows that there exists a constant K such that
By \(h'(t)=\mu u_{x}(h(t), t)\), \(0< h'(t)< M_{3}\) and \(\\vartheta_{s}(s, t)\_{C^{\frac{\alpha}{2}}([1, \infty))}< K_{0}\), one can obtain
where \(M_{4}\) depends on \(K_{0}\) and \(M_{3}\). Then, by using PropositionÂ 3.2 of [3], one can directly obtain \(\lim_{t\rightarrow\infty}\u\_{C([0, h(t)])}=0\). This completes the proof.â€ƒâ–¡
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 \(\lambda_{1}(\tilde{h}_{0})=\frac{1}{d_{1}}(eca)>0\).
Proof
Assume, for contradiction, that \(h_{\infty}<\infty\). Then it follows from LemmaÂ 3.3 that
uniformly in any bounded subset \([0,h_{0}]\). Hence, for any \(\varepsilon>0\), there exists \(\widetilde{T}>0\) such that \(u(x, t)\leq\varepsilon\) and \(v(x,t)\geq1\varepsilon\) for \(t\geq\widetilde{T}\), \(x\in[0, h(t)]\).
Note that \(u(x,t)\) satisfies
Let \(\underline{u}(x,t)\) satisfy the following problem:
Then \(\underline{u}(x,t)\) is a lower solution of \(u(x,t)\). Since \(h_{0}>\tilde{h}_{0}\), one can choose Îµ small enough such that
By the condition \(a< ec\), it follows from the wellknown result that \(\underline{u} \) is unbounded in \((0, h_{0})\times[T^{\ast}, \infty)\), which contradicts that \(\lim_{t\rightarrow\infty}\u(\cdot, t)\_{C([0, h(t)])}=0\). This completes the proof.â€ƒâ–¡
Remark 3.2
Since \(\lambda_{1}(\tilde{h}_{0})=\frac{1}{d_{1}}(eca)<\frac {ec}{d_{1}}=\lambda_{1}(L^{\ast})\), 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 \(h_{0}\leq \frac{1}{4}\sqrt{d_{1}/(eca)}\). 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
The proof is similar to that of LemmaÂ 3.2 in [20]. For convenience of the reader, it is included here. It follows from (1.3) that there exists a constant \(\sigma_{\ast}>0\) such that
Construct and consider the following auxiliary free boundary system:
By using the comparison principle, one can find that \(z(x, t)\leq u(x, t)\) and \(r(t)\leq h(t)\) for all \(t\geq0\) and \(0\leq x \leq r(t)\). By using similar argument to the proof of LemmaÂ 3.2 in [20], one can find that there exists a constant \(\mu_{1}>0\) such that
for all \(\mu\geq\mu_{1}\). Therefore, it is derived that
This, together with TheoremÂ 3.2, yields the desired result.â€ƒâ–¡
Using a similar argument to the proof of TheoremÂ 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
Assume that \(a<(1c)ec\) and \(h_{\infty}=\infty\). Then the solution \((u, v, h)\) of system (1.3) satisfies
uniformly in any compact subset of \([0, \infty)\).
The proof of the theorem is similar to that of TheoremÂ 3.2 in [15], so it is omitted.
Theorem 3.6
Assume that \(ec>a\) and \(h_{0}<\frac{1}{4}\sqrt{d_{1}/(eca)}\). 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 systemÂ (1.3).
The proof of TheoremÂ 3.6 is essentially the same as that of TheoremÂ 5.2 in [4] and thus is omitted.
4 Comments and conclusions
In this article, we have investigated a ratiodependent diffusion predatorprey 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 \(\lambda_{1}(\tilde{h}_{0}) =\frac{1}{d_{1}}(eca)>0\).
Assume one of the following three cases holds: (i)Â \(a< ec\), \(h_{0}\leq\frac{1}{4}\sqrt{\frac{d_{1}}{eca}}\) and \(\mu\leq\frac{d_{1}}{8M}\), then \(h_{\infty}<\infty\), where \(M=\frac{4}{3}\u_{0}\_{\infty}\); (ii)Â \(a>ec\); (iii)Â \(h_{0}< L^{\ast}\) 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)Â the rate of death is bigger than a critical value ec; (ii)Â the length of initial occupying area \(h_{0}\) is shorter than a threshold value \(L^{\ast}\) and Î¼ is smaller than the critical value \(\mu_{0}\), depending on \(u_{0}\); (iii)Â the length of the initial occupying area \(h_{0}\) is shorter than \(\frac{1}{4}\sqrt{\frac{d_{1}}{eca}}\) and Î¼ 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)Â decrease the death rate of predator during the process of putting; (ii)Â extend the range of predatorâ€™s targets; (iii)Â accelerate putting predators; (iv)Â choose the natural enemies which have a strong ability for predating.
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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.
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Li, C. A free boundary problem for a ratiodependent diffusion predatorprey system. Bound Value Probl 2016, 192 (2016). https://doi.org/10.1186/s1366101607019
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DOI: https://doi.org/10.1186/s1366101607019
Keywords
 spreading
 vanishing
 free boundary