Dynamical behaviors of a twocompetitive metapopulation system with impulsive control
 Shasha Tian^{1},
 Yepeng Xing^{1}Email authorView ORCID ID profile and
 Tao Ma^{1}
https://doi.org/10.1186/s1366201711844
© The Author(s) 2017
Received: 9 June 2016
Accepted: 31 March 2017
Published: 2 June 2017
Abstract
In this paper, we study the dynamical behaviors of a twocompetitive metapopulation system with impulsive control and focus on the stable coexistence of the superior and inferior species. A Poincaré map is introduced to prove the existence of a periodic solution and its stability. It is also shown that a stably positive periodic solution bifurcates from the semitrivial periodic solution through a transcritical bifurcation.
Keywords
MSC
1 Introduction and preliminaries
Metapopulation is a population in which individuals are spatially distributed in a habitat in two or more subpopulations. Populations of butterflies and coralreef fishes are good examples of metapopulation. Human activities and natural disasters are the main causes of metapopulation as they increase the population that occurs as metapopulatons. Such factors cause the fragmentation of a large habitat into patches. This may be an important reason whereby models of metapopulation dynamics become important methods in the field of conservation biology. Readers can refer to the references [1–15] for details.
It is impossible that only one species exists in a region. Nevertheless, the emergence of other species in the plaque will cause competition, symbiosis, predation and other relations, and these are the potential effects of the two species’ extinction and survival. So the Levins model was developed to the metapopulation model of two interacting species or multi interacting species. That is a study of metacommunity, which is the improvement and perfection of the theory of metapopulation.
By calculation, we know that equilibriums can be grouped into the following three cases:
(1) If \(c > d\), \(a > b\), \( {a^{2}} + bc + ab  ad > 0\), then system (1.3) has one trivial equilibrium \({E_{0}} = ( {0,0} )\), two semitrivial equilibria \({E_{1}} = ( 0,\frac{{c  d}}{{c}} )\) and \({E_{2}} = ( {\frac{{a  b}}{a},0} )\), one positive equilibrium \({E_{*}} = ( {\frac{{a  b}}{a},  {a^{2}} + bc + ab  ad} )\).
(2) If \(c > d\), \(a > b\), \( {a^{2}} + bc + ab  ad < 0\), then \({E_{*}}\) disappears and it leaves only three equilibriums \({E_{0}}\), \({E_{1}}\) and \({E_{2}} \). \({E_{0}}\) is an unstable node, \({E_{1}}\) is a saddle and \({E_{2}}\) is a stable node (see Figure 1(b)).
(3) If \(c > d\), \(a > b\), \( {a^{2}} + bc + ab  ad = 0\), then system (1.3) also has three equilibria, \({E_{0}}\) is an unstable node, \({E_{1}}\) is a saddle and \({E_{2}}\) is a saddlenode (see Figure 1(c)).
Consequently, \(c > d\), \(a > b\), \( {a^{2}} + bc + ab  ad > 0\) are necessary and sufficient conditions for the stable coexistence of a superior and an inferior competitor in a subdivided habitat. However, in the case of (2) and (3), the inferior competitor goes extinct finally. The purpose of this paper is to find control strategy to ensure the stable coexistence of two competitors.
See that the superior species grows logistically and approaches its equilibrial abundance \(\frac{ab}{a}\). Nevertheless, our control strategy will not work if the amount of superior competitor exceeds \(\frac{ab}{a}\). It is reasonable that for protecting the inferior species, we firstly decrease the density of superior species down under its equilibrium \(\frac{ab}{a}\) by catching or poisoning, etc. Also we assume that the threshold value satisfies \(h < \frac{{a  b}}{a}\) for possible stable coexistence of the two species.
The following lemma is used extensively to prove the stability of periodic solutions for impulsive differential equations.
Lemma 1
[26]
Next lemma is employed to prove the existence of bifurcation of a mapping.
Lemma 2
[27]
 (i)
\(F ( {0,\mu} ) = 0\),
 (ii)
\(\frac{{\partial F}}{{\partial x}} ( {0,0} ) = 1\),
 (iii)
\(\frac{{{\partial^{2}}F}}{{\partial x\,\partial\mu}} ( {0,0} ) > 0\),
 (iv)
\(\frac{{{\partial^{2}}F}}{{\partial{x^{2}}}} ( {0,0} ) < 0\).
Then F has two branches of fixed points for μ near zero. The first branch is \({x_{1}} ( \mu ) = 0\) for all μ. The second bifurcation branch \({x_{2}} ( \mu )\) changes its value from negative to positive as μ increases through \(\mu = 0\). The fixed points of the first branch is stable if \(\mu < 0\) and unstable if \(\mu > 0\), while those of the bifurcating branches have the opposite stability.
The rest of this paper is organized as follows. In Section 2, the sufficient conditions for the existence and stability of a semitrivial periodic solution are given. The Poincaré maps are constructed and theoretical results of dynamical behaviors are presented, including the transcritical and flip bifurcations.
2 Dynamical properties
2.1 Poincaré map
Noting that \(\frac{{Q ( {x,y} )}}{{P ( {x,y} )}}\) is continuous on the stripe region \(\{(x,y)\vert (1p)h\leq x \leq h, 0\leq y< \infty\}\), we have the following proposition.
Lemma 3
The Poincaré map \(\hat{P}:R\rightarrow R\), \(\hat{P} ( {{u},q,\tau} )=(1+q)g({u}) + \tau\) is continuous in u. (1.4) has a periodic solution on the stripe region \(\{(x,y)\vert (1p)h\leq x \leq h, 0\leq y< \infty\}\) if and only if P̂ has a fixed point.
Lemma 4
Assume that the Poincaré map P̂ has a fixed point \(y^{*}\). Then the periodic solution is stable \(\hat{P}'(y^{*})<1\) and unstable \(\hat{P}'(y^{*})>1\).
Proof
Similarly, we can prove that the periodic solution is unstable \(\hat {P}'(y^{*})>1\). □
2.2 The case of \(\tau = 0\)
In what follows, the dynamical properties of periodic solutions of system (2.9) are discussed in the case where we assume that q is a control parameter.
Then we discuss the stability of the semitrivial periodic solution in two ways by using Lemma 1 and Lemma 4, respectively.
Case I: \(c > d\) , \(a > b\) , \( {a^{2}} + bc + ab  ad < 0\).
Theorem 1
Proof
Two different ways are introduced to prove the theorem. One is to use Lemma 1 given by Simeonov and Bainov. Another one is to employ the Poincaré map shown in Section 2.1.
It is easy to see that \(\vert {{\mu_{2}}} \vert < 1\) if and only if (2.12) holds. That is the proof of Theorem 1.
Remarks
(1) It follows from the proof that our construction of the Poincaré map is proper. (2) It is clear that a bifurcation may occur at \(q = {q^{*}}\) for \(\vert {{\mu_{2}}} \vert = 1\). As a result, a positive periodic solution may appear when \(q > {q^{*}}\).
In order to discuss the bifurcation of map (2.13), we will use Lemma 2. To apply this lemma, for \(F ( {u,q} )= ( {1 + q} )g ( u )\), we need to compute \(\frac{{\partial F}}{{\partial x}} ( {0,q^{*}} )\), \(\frac {{{\partial^{2}}F}}{{\partial x\,\partial\mu}} ( {0,q^{*}} ) \) and \(\frac{{{\partial^{2}}F}}{{\partial{x^{2}}}} ( {0,q^{*}} )\).
Theorem 2
Suppose that \(h < \frac{{a  b}}{a}\), \(c > d\), \(a > b\) and \( {a^{2}} + bc + ab  ad < 0\). System (2.9) has a stable positive periodic solution if \(q \in ( {{q^{*}},{q^{*}} + \varepsilon} )\) with \(\varepsilon > 0\), where \({q^{*}} =  1 + { ( {1  p} )^{\frac{{c  d}}{{a  b}}}}{ ( {\frac{{a  b  ah}}{{a  b  a ( {1  p} )h}}} )^{\frac{{  {a^{2}} + ab + bc  ad}}{{a ( {a  b} )}}}}\).
Proof
Then all the conditions in Lemma 2 are satisfied and so the proof of Theorem 2 is completed. □
Case II: \(c > d\) , \(a > b\) , \( {a^{2}} + bc + ab  ad =0\) .
 (A):

\(F(\delta,q)>0\), \(F((1+q)\bar{y},q)<(1+q)\bar{y}\), \(\bar{y}>0\), \(0<\delta\ll1\);
 (B):

\(F''(u,q)<0\), \(\forall0< u<+\infty\);
 (C):

\(F'(0,q)>1\).
2.3 The case of \(\tau > 0\)
In this section, we discuss the existence of a positive periodic solution with \(\tau > 0\) by using Poincaré map (2.5). In the case of \(\tau > 0\), it is obvious that system (1.4) has no semitrivial solution. Moreover, we see the fact that \(\tilde{P}(0,q,\tau)=\tau>0\). Next we try to find \(\tilde{y}>0\) such that \(\tilde{P}(\tilde{y},q,\tau)<\tilde{y}\).
 (Ã):

\(\tilde{P}(0,q,\tau)>0\), \(\tilde{P}((1+q)\hat {y},q,\tau)<(1+q)\hat{y}\), \(\hat{y}>0\);
 (B̃):

\(\tilde{P}(u,q,\tau)''<0\), \(\forall 0< u<+\infty\);
 (C̃):

\(\tilde{P}(u,q,\tau)'>1\).
Deducing similarly as in the proof of Theorem 3, we have the following theorem.
Theorem 4
Suppose that \(h < \frac{{a  b}}{a}\), \(c > d\), \(a > b\) and \( {a^{2}} + bc + ab  ad \leq0\). Then the semitrivial solution of system (1.4) has a unique positive periodic solution and this solution is stable.
3 Conclusion
In this paper, we study the dynamical behaviors of a twocompetitive metapopulation system with impulsive control. To show the stability of the semitrivial periodic solution, we use two different methods: one is the theorem developed by Simeonov and Bainov, and the other one is the method of Poincaré map. The first method can be used only when the explicit expression of the solution is given. Undoubtedly, this restriction narrows its application. So when it comes to the stability of a positive periodic solution, we have to use the Poincaré map together with its particular geometric properties with respect to our specific system. Also, in the case of \(h < \frac{{a  b}}{a}\), \(c > d\), \(a > b\) and \( {a^{2}} + bc + ab  ad < 0\), we find the positive periodic solution and obtain its stability through a transcritical bifurcation in which q is taken as a bifurcation parameter.
Declarations
Acknowledgements
This work was supported by NNSF of China No. 11431008 and NNSF of China No.11271261.
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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