Stationary solutions and spatial-temporal dynamics of a shadow system of LV competition models
- Yuanyuan Zhang^{1} and
- Li Xia^{2, 3}Email author
https://doi.org/10.1186/s13662-017-1081-x
© The Author(s) 2017
Received: 5 October 2016
Accepted: 6 January 2017
Published: 25 January 2017
Abstract
The concern of the paper is nonconstant positive solutions of a class of Lotka-Volterra competition systems over 1D domains. We prove the existence of a positive monotonous solution to the shadow system for each small diffusion rate \(\epsilon >0\). Our theoretical results provide a foundation for further theoretical analysis on the shadow system and give insights on how diffusion and advection rates affect the pattern formation in the advective Lotka-Volterra competition systems. The second part of this paper includes numerical simulations of the nontrivial patterns to the shadow system and its original model. It is demonstrated that nontrivial patterns can develop from small perturbations of the homogeneous solution. Our numerics suggest that this system admits very interesting and complicated spatial-temporal dynamics even over 1D domains.
Keywords
1 Introduction
In this paper, we investigate the existence of a nonconstant positive solution to system (1.1). To this end, we analyze the global bifurcation properties of system (1.1) and show that the global continuum of the first branch must be noncompact in certain Banach spaces. In particular, we prove that, for each small \(\epsilon>0\), (1.1) always admits a nonconstant positive solution \(v_{\epsilon}(x)\), which is monotone in \((0,L)\); see Theorem 1.1. The global bifurcation is important especially when nonconstant positive solutions are concerned, and our results provide a foundation for further analysis on the shadow system (1.1), compared to the local branches, which have been investigated in detail in [1].
From the view point of mathematical modeling, a fundamental problem in mathematical ecology is to study the spatial-temporal evolutions of mutually interacting species. In particular, one of the most interesting phenomena is the well-observed segregation of species through interspecific competition. It is well known that the classical diffusive Lotka-Volterra competition system (i.e., (1.2) with \(\chi=0\)) only admits constant stable steady states except when Ω is nonconvex and the diffusion rates \(d_{i}\) are large [2–7]. We refer to [1] for further discussions. In order to take into account population pressures due to the presence of interacting species, various cross-diffusion models have been proposed and extensively studied over the past few decades. One celebrated system was proposed by Shigesada, Kawasaki, and Teramoto [8] in 1979, now often referred to as the SKT model, and it can be used to describe the aforementioned segregation phenomena. We refer to [9–11] for works on the SKT competition model.
It is necessary to point out that the process of cross-diffusion is very similar to that of chemotaxis, in which cellular organisms direct their dispersals toward or against the concentration gradient of stimulating chemicals in the environment. The mathematical modeling of chemotaxis dates back to the pioneering works of Keller and Segel [12]. Very recently, in [1] and [13], there was independently proposed and studied the Keller-Segel type chemotaxis model (1.1), for which it is assumed that species u takes a combination of random and directed dispersal strategy, whereas species v only disperses randomly. In particular, both papers investigated the existence of nontrivial solutions to (1.2) through its shadow system, and transition layer and boundary spike solutions to (1.1) have been established. These nontrivial positive solutions to (1.1) describe the coexistence and segregation of the competing species in the limit of large diffusion rate \(d_{1}\) and advection rate χ.
We would like to mention that the stability or instability of these bifurcating solutions (1.6) are also investigated in [1], Theorem 4.4. They showed that, among all the local bifurcating solutions, only those on the first branch \(\Gamma_{1}(s)\), \(s\in (-\delta,\delta)\), can be stable, provided that \(\mathcal {K}>0\), whereas all the remaining local branches are unstable. This corresponds to [15], where it is stated that the nonconstant stable solutions to a classical of shadow systems must be spatially monotone. These results make the current study of the first bifurcation branch more realistic for further applications.
One of the motivations of this paper is to study the topological behaviors of the local branches \(\Gamma_{n}(s)\) to (1.1). To this end, we perform the global bifurcation analysis for (1.1); moreover, we shall show that this shadow system always admits nonconstant positive solutions for each small \(\epsilon>0\). To be precise, the theoretical result of the paper states as follows.
Theorem 1.1
Assume that (1.4) is satisfied and \(\epsilon_{n}\) is given by (1.5). Then, for each \(\epsilon\in(0,\epsilon_{1})\), there exists a positive solution \((v_{\epsilon}(x),\lambda_{\epsilon})\) of (1.1) satisfying \(v'_{\epsilon}<0\) on \((0,L)\).
Remark 1
Theorem 1.1 indicates that (1.1) admits decreasing solutions for every small ϵ and the proof can be carried over to show that this system also admits increasing solutions. Indeed, let \(v_{\epsilon}(x)\) be a decreasing solution of (1.1). Then, thanks to the Neumann boundary condition, \(v_{\epsilon}(L-x)\) is also a solution, and it is increasing in x. Therefore, we can construct nonmonotone solutions by reflecting and periodically extending the monotone ones at the end points.
According to Theorem 1.1, the shadow system (1.1) admits nonconstant monotone solutions for any small \(\epsilon>0\). Therefore, from the view point of singular perturbations, we may expect that (1.2) also admits nontrivial solutions when χ and \(d_{1}\) are sufficiently large, and although rigorous mathematical analysis is quite technical and is out of our scope, we perform extensive numerical simulations to verify this observation in Section 4.
2 Nonconstant positive solutions to the shadow system
We first collect the following facts in [1] on the operator \(\mathcal{F}\) before using the bifurcation theory.
Lemma 2.1
[1] The operator \(\mathcal{F}(v,\lambda,\epsilon)\) defined in (2.1) satisfies the following properties:
(1) \(\mathcal{F}(\bar{v},\bar{\lambda},\epsilon)=0\) for any \(\epsilon\in\mathbb{R}^{+}\);
(2) \(\mathcal{F}: \mathcal{X} \times\mathbb{R}^{+} \times\mathbb {R}^{+} \rightarrow\mathcal{Y} \times\mathcal{Y}\) is analytic, where \(\mathcal{Y}=L^{2}(0,L)\);
(4) \(D_{(v,\lambda)}\mathcal{F}(v_{0},\lambda_{0},\epsilon): \mathcal {X} \times\mathbb{R}^{+} \rightarrow\mathcal{Y} \times\mathbb{R}\) is a Fredholm operator with zero index.
Remark 2
3 Global bifurcation analysis
We now proceed to extend the local bifurcation curves obtained in Theorem 1.1 by the global bifurcation theory of Rabinowitz [16] and its developed version in [17]. In particular, we will only study the first bifurcation branch \(\Gamma_{1}\) since all the remaining (local) branches are unstable.
Lemma 3.1
Assume that (1.4) holds. Then, for each \((v,\lambda,\epsilon) \in\mathcal{C}\), \(v(x)>0\) on \([0,L]\), \(\lambda>0\), and \((v,\lambda )\) is a solution of (1.1).
Proof
To prove that \(\mathcal{P}_{0}\) is open in \(\mathcal{C}\), we pick any \((v,\lambda,\epsilon) \in\mathcal{P}_{0}\) and assume that there exists a sequence \((v_{k},\lambda_{k},\epsilon_{k})\) that converges to \((v,\lambda,\epsilon)\) in \(\mathcal{X}\times\mathbb{R} \times \mathbb{R}\). Therefore \(v_{k}\) converges to v in \(C^{2}([0,L])\), and hence \(v_{k}(x)>0\) on \([0,L]\) for all large k since \(v(x)>0\) on \([0,L]\). Moreover, \(\lambda_{k},\epsilon_{k}>0\) since \(\lambda,\epsilon>0\).
We now prove that \(v(x)>0\) on \([0,L]\). If not, then suppose that \(v(x_{0})=0\) for some \(x_{0}\in[0,L]\). Then we can apply the strong maximum principle and Hopf’s lemma to (1.1) to show that \(v\equiv 0\) for all \(x\in[0,L]\) and therefore \(\lambda=\frac{a_{1}}{b_{1}}\). However, this is impossible since it is easy to check that bifurcation does not occur at \((0,\frac{a_{1}}{b_{1}})\). This is a contradiction, and we must have that \(v(x)>0\) on \([0,L]\). □
Remark 3
According to Lemma 3.1 (and the forthcoming discussion), we know that the global continuum \(\mathcal {C}\) cannot intersect with the ϵ-axis. However, we are not able to rule out the possibility that it intersects with the V-axis, that is, \(\mathcal {X}\times \mathbb {R}^{+}\times\{0\}\). Details on the limiting structures are needed for this purpose. Our main results in Theorem 1.1 establish the existence of nonconstant solutions to (1.1) for any small ϵ, and it is out of the scope of this paper to analyze their limiting profiles.
We proceed to show that \(\mathcal {C}\) consists of two disjoint components and each component contains solution v that is spatially monotone. Let \(\mathcal{C}_{u}\) to be the component of \(\mathcal{C} \backslash\{(v_{1}(s,x),\lambda_{1}(s),\epsilon_{1}(s)) \vert s \in (-\delta,0) \}\) containing \(\{(v_{1}(s,x),\lambda_{1}(s),\epsilon_{1}(s)) \vert s\in[0 ,\delta) \}\) and, correspondingly, \(\mathcal{C}_{l}=\mathcal {C} \backslash\{ (v_{1}(s,x),\lambda_{1}(s),\epsilon_{1}(s)) \vert s \in(0,\delta)\} \) containing \(\{(v_{1}(s,x),\lambda_{1}(s),\epsilon_{1}(s)) \vert s\in(-\delta,0]\}\). Then we can readily see that \(\mathcal{C}=\mathcal{C}_{u} \cup\mathcal{C}_{l}\) and \(\mathcal{C}_{u} \cap\mathcal{C}_{l}=\{(\bar{v},\bar{\lambda},\epsilon_{1})\}\). We have the following results.
Lemma 3.2
\(\mathcal{C}_{u}\backslash(\bar{v},\bar{\lambda},\epsilon_{1})\) consists of \((v,\lambda,\epsilon)\) with \(v'(x)>0\) on \((0,L)\), and \(\mathcal{C}_{l} \backslash(\bar{v},\bar{\lambda},\epsilon_{1})\) consists of \((v,\lambda,\epsilon)\) with \(v'(x)<0\) on \((0,L)\) and \(\lambda, \epsilon>0\).
Proof
We only prove the first part, and the second one can be treated in the same way. We note that \(\mathcal{C}_{u}^{0}\) is nonempty since any solution \((v,\lambda,\epsilon)\) of (1.1) near \((\bar{v},\bar {\lambda},\epsilon_{1})\) is in the set \(\mathcal{C}_{u}^{0} \) thanks to (1.4). Since \(\mathcal{C}_{u}^{0}\) is a connected subset of \(\mathcal{X} \times\mathbb{R}^{+} \times\mathbb{R}^{+}\), we only need to show that \(\mathcal{C}_{u}^{0} \cap\mathcal{P}_{1}^{+}\) is both open and closed with respect to the topology of \(\mathcal{C}_{u}^{0} \), and we divide our proof into two parts.
Case 2. \(x_{\infty}\in(0,L)\). This is also impossible by the continuity of \(\tilde{v}'(x)\). Therefore, we have that \(\tilde{v}'(x)>0\) in \((0,l)\) for k large, and hence the openness is proved.
3.1 Global extension of the first bifurcation branch
Finally, we study the extension of the local bifurcation branch \(\Gamma _{1}(s)\), and we present a proof of Theorem 1.1.
Proof of Theorem 1.1
According to Theorem 4.4 in [17], \(\mathcal{C}_{u}\) satisfies one of the following three alternatives: (A1) it is not compact in \(\mathcal{X} \times\mathbb{R}^{+} \times\mathbb{R}^{+}\); (A2) it contains a point \((\bar{v},\bar{\lambda},\epsilon_{*})\) with \(\epsilon_{*} \neq\epsilon_{1}\); (A3) it contains a point \((\bar {v}+\hat{v},\bar{\lambda}+\hat{\lambda},\epsilon)\), where \(0\neq (\hat{v},\hat{\lambda}) \in\mathcal{Z}\), and \(\mathcal{Z}\) is a closed complement of \(\mathcal{N} (D_{(v,\lambda)}\mathcal {F}(\bar{v},\bar{\lambda},\epsilon_{1}) )=\text{span}\{ (\cos \frac{\pi x}{L},0)\}\). We first claim that only alternative (A1) can occur.
If (A2) occurs, then \(\epsilon_{*}\) must be one of the bifurcation values \(\epsilon_{k}\), \(k\geq2\); therefore, \(v(x)=\bar{v}+s \cos\frac {k\pi x}{L}+o(s)\), \(s\in(-\delta,\delta)\), according to (1.4). However, this contradicts to the monotonicity of \(v(x)\), and therefore (A2) is impossible.
Now we proceed to show that the projection \(\mathcal{C}_{u}\) onto the ϵ-axis is of the form \((0,\bar{\epsilon}]\) for some \(\bar {\epsilon} \geq\epsilon_{1}\). We argue by contradiction and assume that there exists \(\underline{\epsilon}>0\) such that \((\underline{\epsilon},\bar{\epsilon})\) is contained in this projection, but this projection does not contain any \(\epsilon<\underline{\epsilon}\). Then we have from the uniform boundedness of \(\Vert v_{\epsilon}(x)\Vert _{\infty}\) and Sobolev embedding that, for each \(\epsilon>0\), \(\Vert v_{\epsilon}\Vert_{C^{3}([0,L])} \leq C\) for all \((v_{\epsilon},\lambda _{\epsilon},\epsilon) \in\mathcal{C}_{u}\). However, this implies that \(\mathcal{C}_{u}\) is compact in \(\mathcal{X} \times\mathbb{R}^{+} \times\mathbb{R}^{+}\), which is a contradiction to alternative (A1). Therefore, \(\mathcal{C}_{u}\) extends to infinity vertically in \(\mathcal {X} \times\mathbb{R}^{+} \times\mathbb{R}^{+}\). This finishes the proof of Theorem 1.1. □
We have from Theorem 1.1 that there exist positive and monotone solutions \(v_{\epsilon}(\lambda_{\epsilon},x)\) to (1.1) for all \(\epsilon\in(0,\epsilon_{1})\). If \(v(x)\) is an increasing solution to (1.1), then \(v(L-x)\) is a decreasing solution. Then we can construct infinitely many nonmonotone-solutions of (1.1) by reflecting and periodically extending \(v(x)\) and \(v(L-x)\) at \(x=0,\pm L, \pm2L,\dots\).
4 Numerical simulations
We proceed to investigate (1.1) and (1.2) by numerical studies. Our simulation illustrates and supports our theoretical finding in the previous sections, that is, (1.2) admits nontrivial positive constant steady states when χ and \(d_{1}\) are large and \(d_{2}\) is small. Moreover, system (1.2) is able to model the well-observed phenomenon of segregation through competition.
4.1 Simulations over one-dimensional domains
4.2 Simulations over two-dimensional domains
Declarations
Acknowledgements
YZ supported by Department of Education, Sichuan (15ZB0473). LX supported by Guangdong Natural Science Foundation (2015A030313623) and Guangdong Training Program for Young College Teachers (YQ2015077).
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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