- Research
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Optimal diffusion rate of species in flowing habitat
- Benlong Xu^{1}Email authorView ORCID ID profile and
- Nannan Liu^{1}
https://doi.org/10.1186/s13662-017-1326-8
© The Author(s) 2017
- Received: 19 May 2017
- Accepted: 23 August 2017
- Published: 2 September 2017
Abstract
It is widely accepted that diffusive dispersal can permit persistence in an advective environment. This paper studies in some sense the optimal diffusion rate of species in a flowing habitat with hostile downstream boundary conditions. Firstly, we study the dependence of the critical length of the habitat on the dispersal rate d. It is shown that the critical length first decreases and then increases and asymptotically tends to infinity. Then there is a unique optimal diffusion rate \(d_{0}\) for a single species to evolve. Then, by using this observation, we study the competition system of two species which are the same but only with different dispersal rates. We get an open finite interval, which is a neighborhood of \(d_{0}\), such that, if one of the dispersal rates lies within the interval but the other rate falls outside, then competition exclusion occurs. If the two dispersal rates both lie within the interval, the one with an intermediate dispersal rate can always invade the other with its dispersal rate near the ends of the interval.
Keywords
- flowing habitat
- diffusion rate
- globally asymptotically stable
MSC
- 35K57
- 92D25
1 Introduction
Speirs and Gurney [4] obtained a critical length \(L^{*}\) of the habitat and conclude that the species permits persistence if and only if \(q<\sqrt{4dr} \text{ and } L>L^{*}\). So the species can persist if and only if the diffusion rate d is bigger than \(\frac{q^{2}}{4r}\), and the length of the habitat is bigger than the critical length \(L^{*}\).
A natural question arises: is there an optimal diffusion rate d, in the sense that species adopting this diffusion rate will need the smallest critical length \(L^{*}\) to persist?
- (I)no-flux (closed) boundary condition:$$d_{1}u_{x}(L,t)-qu(L,t)=0,\qquad d_{2}v_{x}(L,t)-qv(L,t)=0,\quad t>0, $$
- (II)free flow (open) boundary condition:$$u_{x}(L,t)=0,\qquad v_{x}(L,t)=0,\quad t>0, $$
- (III)hostile (lethal) boundary condition:$$u(L,t)=0,\qquad v(L,t)=0, \quad t>0. $$
It is an interesting question how the diffusion rates \(d_{1}\) and \(d_{2}\) determine the dynamic behavior of system (1.3). For the free flow downstream condition (II), it is proved in [5] that the species with a higher diffusion rate will always prevail, that is, faster diffusion wins. Later on, Lou and Zhou [6] extended this result to a wider class of boundary conditions. Recently, Lam et al. [6] studied the system with no-flux downstream conditions (I), and showed that, when r is a positive constant, the higher diffusion will always prevail, just as in the case of free flow downstream conditions. The reason for these results is that a sufficient intensity of random diffusion can better balance the overmatching of resources or even a net loss of individuals at the boundary.
It is natural to ask about the role of random dispersal in the situation of hostile downstream boundary conditions (III). It turns out that this situation is more complicated. On the one hand, the random dispersal balances the overmatching of resources and loss of individuals caused by drifting, while on the other hand, too strong dispersal also likely drives more individuals to the hostile end. Lou and Lustscher [5] conjectured that an intermediate diffusion rate seems most beneficial.
This paper is motivated by the question mentioned above. The rest of the paper is organized as follows. In Section 2, we state and prove our main results. Firstly, we explore the optimal diffusion rate for a single species to persist. We give a rigorous analytical proof that there exists a unique optimal diffusion rate, in the sense that a species adopting this diffusion rate will need the smallest length of critical habitat for persistence. Then we study the global dynamics of a two-species competition model. By using the optimal results for the single species, we get an optimal interval such that the species with its diffusion rate in this interval always prevails against its competitor with its diffusion rate beyond the interval. In Section 3, some concluding remarks and the discussion are given.
2 Main results
Theorem 2.1
\(L^{*}(d)\) has a unique critical point \(d_{0}\) in \((\frac{q^{2}}{4r},+\infty)\), such that \(L^{*}(d)\) is strictly decreasing in \((\frac {q^{2}}{4r},d_{0})\) and strictly increasing in \((d_{0},+\infty)\). Moreover, \(\lim_{d\rightarrow\frac{q^{2}}{4r}+0}L^{*}(d)=+\infty\), \(\lim_{d\rightarrow+\infty}L^{*}(d)=+\infty\).
Proof
Finally, the assertions \(\lim_{d\rightarrow\frac {q^{2}}{4r}+0}L^{*}(d)=+\infty\) and \(\lim_{d\rightarrow+\infty}L^{*}(d)=+\infty\) can be verified directly by equation (2.3). This completes the proof. □
Remark 2.1
As is well known, a species with diffusion rate \(d>q^{2}/(4r)\) can persist if and only if the habitat length L is bigger than the critical length \(L^{*}(d)\). Theorem 2.1 tells us that, for fixed r and q, there exists an optimal diffusion rate \(d_{0}\) which has a minimal critical length. A species adopting diffusion rate \(d_{0}\) can persist with the smallest length of the habitat.
By Theorem 2.1, \(L^{*}(d)\) has a minimum critical point \(d_{0}\), therefore we discuss the dynamics of (2.8) by two parts: \(d_{0}\in(\frac{q^{2}}{4r},d_{0}) \) and \(d\in(d_{0},+\infty)\).
Theorem 2.2
- (a)
if \(d_{1}\in(\underline{d}, \overline{d})\), the semi-trivial steady state solution \((\tilde{u}, 0)\) exists. Furthermore, if \(d_{2}\) is not in the interval \((\underline{d}, \overline{d})\), \((\tilde{u}, 0)\) is globally asymptotically stable for system (2.8),
- (b)
if \(d_{2}\in(\underline{d}, \overline{d})\), the semi-trivial steady state solution \((0, \tilde{v})\) exists. Furthermore, if \(d_{1}\) is not in the interval \((\underline{d}, \overline{d})\), \((0, \tilde{v})\) is globally asymptotically stable for system (2.8),
- (c)
if \(d_{1}\) and \(d_{2}\) are not in the interval \((\underline{d}, \overline{d})\), then \((0, 0)\) is globally asymptotically stable for system (2.8).
Proof
By Theorem 2.1, \(L^{*}(d)\) is strictly decreasing in \((\frac{q^{2}}{4r},d_{0})\), strictly increasing in \((d_{0}, +\infty)\), and \(\lim_{d\rightarrow \frac{q^{2}}{4r}+0}L^{*}(d)=+\infty\), \(\lim_{d\rightarrow+\infty}L^{*}(d)=+\infty\). Since \(L>L_{0}=L^{*}(d_{0})\), there exist two constants \(\underline{d}\), and d̅ with \(\frac{q^{2}}{4r}<\underline {d}<d_{0}<\overline{d}<+\infty\), such that \(L^{*}(\underline{d})=L^{*}(\overline{d})=L\). It follows that \(L>L^{*}(d)\) if \(d\in(\underline{d}, \overline{d})\), and \(L\leq L^{*}(d)\) if \(\frac{q^{2}}{4r}< d<\underline{d}\), or \(d\geq \overline{d}\).
(a) As is well known, positive solution ũ of steady state equation (2.10) exists if and only if \(d_{1}>\frac{q^{2}}{4r}\) and \(L>L^{*}(d_{1})\), and this condition is actually satisfied if \(d_{1}\in(\underline{d}, \overline{d})\). This establishes the existence of the semi-trivial steady state solution \((\tilde{u}, 0)\).
(b) The argument is exactly the same as that in the proof of part (a).
(c) If \(d_{1}\) and \(d_{2}\) are not in \((\underline{d}, \overline {d})\), then both the semi-trivial steady states \((\tilde{u}, 0)\) and \((\tilde{v}, 0)\) do not exist. Then a similar argument to that in the proof of part (a) implies that \(u(x,t)\rightarrow0\) and \(v(x,t)\rightarrow0\) as \(t\rightarrow\infty \). This completes the proof. □
Theorem 2.2 tells us that a species with its dispersal rate within the interval \((\underline{d}, \overline{d})\) will drive its competitor with its dispersal rate outside this interval to extinction. In fact, this is not surprising since, by the proof of Theorem 2.2, there is only one semi-trival steady state in that situation. The more interesting case is when the dispersal rates of both species lie within the interval \((\underline{d}, \overline{d})\), and then both semi-trivial steady states \((\tilde{u}, 0)\) and \((\tilde {v}, 0)\) exist. To this end, we need some necessary preparations.
Theorem 2.3
- (1)
the single species permits persistence for any non-negative initial value \(u_{0}(x)\not\equiv0\) in (1.1),
- (2)
the steady state equation of (1.1) has a unique positive steady state \(u^{*}\) which is globally asymptotically stable,
- (3)
the dispersal rate d lies within the open interval \((\underline{d}, \overline{d})\),
- (4)
the principal eigenvalue \(\lambda_{1}(d)\) of (2.12) is positive.
We now state our stability result of semi-trivial solutions for \(d_{1}, d_{2}\in(\underline{d}, \overline{d})\) with \(d_{1}< d_{2}\).
Theorem 2.4
Suppose that \(L>L_{0}\equiv\min\{ L^{*}(d): d>\frac{q^{2}}{4r}\}\), \(\underline{d}\) and d̅ are defined as in Theorem 2.2, and \(d_{1}, d_{2}\in(\underline{d}, \overline{d})\) with \(d_{1}< d_{2}\). Then the two semi-trivial steady state solutions \((\tilde{u}, 0)\) and \((0, \tilde{v})\) exist. For any fixed \(d_{2}\in(\underline{d}, \overline{d})\), there exists a constant \(\delta>0\), such that for any \(d_{1}\in(\underline{d}, \underline{d}+\delta)\subset (\underline{d}, d_{2})\), the solutions \((\tilde{u}, 0)\) are linearly unstable.
Similarly, we have the following.
Theorem 2.5
Suppose that \(L>L_{0}\equiv\min\{ L^{*}(d): d>\frac{q^{2}}{4r}\}\), \(\underline{d}\) and d̅ are defined as in Theorem 2.2, and \(d_{1}, d_{2}\in(\underline{d}, \overline{d})\) with \(d_{1}< d_{2}\). Then the two semi-trivial steady state solutions \((\tilde{u}, 0)\) and \((0, \tilde{v})\) exist. For any fixed \(d_{1}\in(\underline{d}, \overline{d})\), there exists a constant \(\delta>0\), such that for any \(d_{2}\in(\overline{d}-\delta, \overline{d})\subset(d_{1}, \overline{d})\), the solutions \((0, \tilde{v})\) are linearly unstable.
Before giving the proofs of Theorem 2.4 and Theorem 2.5, we provide the following lemma.
Lemma 2.6
Suppose that \(L>L_{0}\equiv\min\{L^{*}(d): d>\frac{q^{2}}{4r}\}\), \(\underline{d}\) and d̅ are defined as in Theorem 2.2, and \(d\in(\underline{d}, \overline{d})\). Then the problem (2.1) has a unique positive solution \(u^{*}\). Moreover, \(\int_{0}^{L} {u^{*}}^{2}\rightarrow0\) as \(d\rightarrow\underline {d}+0\) or \(d\rightarrow\overline{d}-0\).
Proof
Proof of Theorem 2.4
The proof of Theorem 2.5 is similar to that of Theorem 2.4 and thus omitted. □
Remark 2.2
Note that Theorem 2.2 deals with the competitive exclusion situation, namely a species with dispersal rate in the interval \((\underline{d}, \overline{d})\) always drives its competitor with dispersal rate outside the interval to extinction. But Theorem 2.4 and Theorem 2.5 just give some competitive invasion results. If the dispersal rates \(d_{1}\) and \(d_{2}\) both lie within the interval \((\underline{d}, \overline{d})\), then any one of the two competitor species can evolve separately (without its competitor). Theorem 2.4 and Theorem 2.5 imply that a species with some intermediate dispersal rate in \((\underline{d}, \overline{d})\) can always invade its competitor with dispersal rate in \((\underline{d}, \overline{d})\) but close to one of the end points, regardless of the non-negative initial values \(u(0, x), v(0,x)\not\equiv0\). In this situation, competitive exclusion as well as coexistence may occur. For example, when \(d_{1}, d_{2}\in(\underline{d}, \overline{d})\) and \(d_{1}\) is close to \(\underline{d}\) and \(d_{2}\) close to d̅, it seems that coexistence may happen, at least in some cases. The exact dynamical behavior in this situation seems subtle, and is likely related to a long time standing conjecture raised by Lou and Lutscher [5], which was also a main motivation for this paper.
3 Concluding remarks
This paper explores the optimal strategy of diffusion of a single-species model and a two-species competition model in a 1-dimensional flowing homogeneous habitat with hostile downstream boundary conditions.
For the single-species model, we show that there exists an optimal diffusion rate \(d_{0}\), in the sense that a species adopting the strategy of diffusion rate \(d_{0}\) can be persistent with the smallest length of habitat.
For the two-species competition mode, given a fixed length of the habitat, there is a living interval \((\underline{d}, \overline{d})\) containing \(d_{0}\) as an interior point, such that a species adopting the diffusion strategy and dispersal rates in this interval will replace its competitor with its dispersal rate outside the interval (Theorem 2.2). If the dispersal rates \(d_{1}\) and \(d_{2}\) both lie within the interval \((\underline{d}, \overline{d})\), then a species with some intermediate dispersal rate can always invade its competitor with its dispersal rate near the end points, regardless of the non-negative initial values \(u(0, x), v(0,x)\not\equiv0\). These are remarkably different phenomena from those with downstream boundary conditions of type (I) and (II); see, for example, [5, 6].
Our work in this paper is mainly motivated by an open problem raised by Lou and Lutscher [5]. Lou and Lutscher [5] conjectured that there may exist an optimal intermediate diffusion rate d̃ so that a species adopting d̃ will replace its competitor with a different diffusion rate. Obviously, if the optimal rate d̃ does exist, then it must be in our living interval \((\underline{d}, \overline{d})\). Some further work is necessary to solve Lou and Lutscher’s original conjecture.
Declarations
Acknowledgements
We thank Prof. Yuan Lou and Dr. Peng Zhou for their helpful discussion and communication. The authors are grateful to the two anonymous reviewers for their excellent suggestions, which have greatly improved the exposition of the paper. This work is supported by the Natural Science Foundation of Shanghai, China (No. 13ZR1430100) and Shanghai Peak Subject Funding.
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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