Open Access

Dynamical behaviors of a two-competitive metapopulation system with impulsive control

Advances in Difference Equations20172017:156

https://doi.org/10.1186/s13662-017-1184-4

Received: 9 June 2016

Accepted: 31 March 2017

Published: 2 June 2017

Abstract

In this paper, we study the dynamical behaviors of a two-competitive 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 semi-trivial periodic solution through a transcritical bifurcation.

Keywords

metapopulation system impulsive state feedback Poincaré map bifurcation

MSC

34D09 34D99

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 coral-reef 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 [115] for details.

The name metapopulation was first used in 1969 [8] by Levins to describe a population dynamics model for insect pests inhabiting crop growing areas; however, the idea has since been most broadly applied to species in fragmented habitats. Let p be the fraction of sites occupied by a species which will be called its abundance. Levins posed a simple, general model for the dynamics of the site occupancy in such a system
$$ \frac{{dp}}{{dt}} = cp ( {1 - p} ) - mp, $$
(1.1)
where c is the colonization rate and m is the morality (local extinction) rate. Propagules dispersed randomly among all sites. The rate of propagules production by the occupied sites cp is multiplied by the proportion of sites that are not occupied \(1-p\) to give the rate of production of newly colonized sites. The morality rate m is multiplied by the proportion of occupied sites p to give the density-independent rate at which occupied sites become vacant. A site becomes vacant when the individual occupying that site dies.

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.

Tilman considered in [15] the co-existence of two competitors. In his paper, the superior competitor (the first species \(p_{1}\)) has the same equation as would a species living by itself and thus is totally unaffected by the inferior competitor (the second species \(p_{2}\)). The inferior competitor can colonize only sites in which both it and the superior species are absent (the term \(1-p_{1}-p_{2}\)). However, superior species can invade into and displace inferior species (the term \(-c_{1}p_{1}p_{2}\)). Thus, this leads to the following system:
$$ \textstyle\begin{cases} \frac{{d{p_{1}}}}{{dt}} = {c_{1}}{p_{1}} ( {1 - {p_{1}}} ) - {e_{1}}{p_{1}},\\ {\frac{{d{p_{2}}}}{{dt}} = {c_{2}}{p_{1}} ( {1 - {p_{1}} - {p_{2}}} ) - {e_{2}}{p_{2}} - {c_{1}}{p_{1}}{p_{2}}}. \end{cases} $$
(1.2)
It is easy to see that the superior species grows logistically and approaches its equilibrial abundance. Once the first species is at or very near to equilibrium, the second species grows logistically to its equilibrium. In this paper, the system is written as
$$ \textstyle\begin{cases} \dot{x} = ax ( {1 - x} ) - bx,\\ \dot{y} = cy ( {1 - x - y} ) - dy - axy, \end{cases} $$
(1.3)
for all \(t > 0\). Obviously, the linear part of system (1.3) at equilibrium is determined by the matrix
$$D ( {x,y} ) = \left ( \begin{matrix} {a - b - 2ax}&0\\ { - cy - ay}&{c - d - ( {a + c} )x - 2cy} \end{matrix} \right ). $$
The dynamics of system (1.3) in the neighborhood of an equilibrium \(( x,y )\) directly depends on the property of eigenvalues of the matrix \(D ( {x,y} )\).

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 semi-trivial equilibria \({E_{1}} = ( 0,\frac{{c - d}}{{c}} )\) and \({E_{2}} = ( {\frac{{a - b}}{a},0} )\), one positive equilibrium \({E_{*}} = ( {\frac{{a - b}}{a}, - {a^{2}} + bc + ab - ad} )\).

\({E_{0}}\) is an unstable node, \({E_{1}}\), \({E_{2}}\) are saddles and \({E_{*}}\) is a stable node (see Figure 1(a)).
Figure 1

The phase portrait of system ( 1.3 ).

(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 saddle-node (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.

In ecological practice, controlling the amount of strong competition is our first consideration. Integrated strategy is a complete control strategy that uses a combination of biological, cultural and chemical tactics that hold the strong competition population on suitable levels with less cost and minimal effect on the environment. It has very important practical significance on how to protect the rare and endangered species (which is always the inferior competitor), maintain biodiversity and maintain ecological balance, and it also has great theoretical significance on the mathematical model of population ecology, genetic drift and biological evolution of the genetics. When the amount of the strong competition population reaches a threshold, integrated strategy may be taken. Consequently, the superior competitor population decreases, while the amount of the inferior competitor increases abruptly. Reader can refer to the references [1625] for details. Now we consider the competition system (1.3) by introducing a state feedback control strategy rather than the usual fixed-time control strategy. This design is often feasible in practice. We can control the population size by catching or poisoning the superior competitor and releasing the inferior competitor when the amount of the strong competition reaches the threshold value. The controlled system is modeled by the following equations:
$$ \textstyle\begin{cases} \left . \textstyle\begin{array}{l} \dot{x} = ax ( {1 - x} ) - bx,\\ \dot{y} = cy ( {1 - x - y} ) - dy - axy \end{array}\displaystyle \right \}x \ne h,\\ \left . \textstyle\begin{array}{l} \Delta x = - px,\\ \Delta y = qy + \tau \end{array}\displaystyle \right \}x = h, \end{cases} $$
(1.4)
where the parameters \(p \in ( {0,1} )\), \(h > 0\), \(q > 0\), \(\tau \ge0\), and \(\Delta x ( t ) = x ( {{t^{+} }} ) - x ( t )\), \(\Delta y ( t ) = y ( {{t^{+} }} ) - y ( t )\). When the amount of the strong competition reaches the threshold value h at the time \({t_{k - 1}}\), the controlling measures are taken and the amount of strong competition and weak competition abruptly turn to \(( {1 - p} )h\) and \(( {1 + q} )y ( {{t_{k - 1}}} ) + \tau\), respectively. In this paper, \(X ( t ) = ( {x ( t ),0} )\) is called a semi-trivial solution for \(y \equiv0\), \(X ( t )\) is called a periodic solution if \(X ( {nT} ) = X ( { ( {n + 1} )T} )\) for all \(n > 0\). According to the actual significance, we can only consider the solutions with nonnegative components, continuously differentiable in \({R^{+} } - \{ {{t_{k - 1}}} \}\).

See that the superior species grows logistically and approaches its equilibrial abundance \(\frac{a-b}{a}\). Nevertheless, our control strategy will not work if the amount of superior competitor exceeds \(\frac{a-b}{a}\). It is reasonable that for protecting the inferior species, we firstly decrease the density of superior species down under its equilibrium \(\frac{a-b}{a}\) 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.

There may be two types of trajectories when the value of x is on the left of h (see Figure 2). With the parameters p, h given, the purpose of this paper is to obtain the control measure of how to release the strong competition to control the weak competition by studying the dynamics of system (1.4). Therefore, it is easy to see that q and τ are control parameters.
Figure 2

The simple phase portrait of system ( 1.3 ).

The following lemma is used extensively to prove the stability of periodic solutions for impulsive differential equations.

Lemma 1

[26]

The T-periodic solution \(( {x,y} ) = ( {\xi ( t ),\eta ( t )} )\) of the system
$$\textstyle\begin{cases} \frac{{dx}}{{dt}} = P ( {x,y} ), \frac{{dy}}{{dt}} = Q ( {x,y} ), &\textit{if }\varphi ( {x,y} ) \ne0,\\ \Delta x = \alpha ( {x,y} ), \Delta y = \beta ( {x,y} ), &\textit{if }\varphi ( {x,y} ) \ne0, \end{cases} $$
is orbitally asymptotically stable if the Floquet multiplier \({\mu_{2}}\) satisfies the condition \(\vert {{\mu_{2}}} \vert < 1\), where
$${\mu_{2}} = \prod _{k = 1}^{q} {{ \Delta_{k}}\exp \biggl( { \int _{0}^{T} { \biggl( {\frac{{\partial P}}{{\partial x}} \bigl( {\xi ( t ),\eta ( t )} \bigr) + \frac{{\partial Q}}{{\partial y}} \bigl( {\xi ( t ),\eta ( t )} \bigr)} \biggr)} \,dt} \biggr)} $$
with
$${\Delta_{k}} = \frac{{{P_{+} } ( {\frac{{\partial\beta}}{{\partial y}}\frac{{\partial\varphi}}{{\partial x}} - \frac{{\partial\beta }}{{\partial x}}\frac{{\partial\varphi}}{{\partial y}} + \frac {{\partial\varphi}}{{\partial x}}} ) + {Q_{+} } ( {\frac {{\partial\alpha}}{{\partial x}}\frac{{\partial\varphi}}{{\partial y}} - \frac{{\partial\alpha}}{{\partial y}}\frac{{\partial\varphi }}{{\partial x}} + \frac{{\partial\varphi}}{{\partial y}}} )}}{{P\frac{{\partial\varphi}}{{\partial x}} + Q\frac{{\partial \varphi}}{{\partial y}}}} $$
and P, Q, \(\frac{{\partial\alpha}}{{\partial x}}\), \(\frac{{\partial \alpha}}{{\partial y}}\), \(\frac{{\partial\beta}}{{\partial x}}\), \(\frac {{\partial\beta}}{{\partial y}}\), \(\frac{{\partial\varphi }}{{\partial x}}\) and \(\frac{{\partial\varphi}}{{\partial y}}\) are calculated at the point \(( {\xi ( {{t_{k}}} ),\eta ( {{t_{k}}} )} )\), \({P_{+} } = P ( {\xi ( {t_{k}^{+} } ),\eta ( {t_{k}^{+} } )} )\) and \({Q_{+} } = Q ( {\xi ( {t_{k}^{+} } ),\eta ( {t_{k}^{+} } )} )\). Here \(\varphi ( {x,y} )\) is a sufficiently smooth function such that grad \(\varphi ( {x,y} ) = 0\), and \({t_{k}}\) (\({k \in N} \)) is the time of the kth jump.

Next lemma is employed to prove the existence of bifurcation of a mapping.

Lemma 2

[27]

Let \(F:R \times R \to R\) be a one-parameter family of \({C^{2}}\) maps satisfying
  1. (i)

    \(F ( {0,\mu} ) = 0\),

     
  2. (ii)

    \(\frac{{\partial F}}{{\partial x}} ( {0,0} ) = 1\),

     
  3. (iii)

    \(\frac{{{\partial^{2}}F}}{{\partial x\,\partial\mu}} ( {0,0} ) > 0\),

     
  4. (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.

To understand this lemma, according to the conditions, we may assume that the mapping \(F(x,\mu)\) has the normal form
$$F(x,\mu)=x+\alpha\mu x-\beta x^{2}, \quad \mbox{where } \alpha>0, \beta>0, $$
and
$$F'_{x}(x,\mu)=1+\alpha\mu-2\beta x. $$
If we let \(F(x,\mu)=x\), then we get \(x_{1}(\mu)=0\) and \(x_{2}(\mu)=\frac {\alpha}{\beta}\). Obviously, \(F'_{x}(0,\mu)=1+\alpha\mu\) and \(F'_{x}(\frac{\alpha}{\beta},\mu)=1-\alpha \mu\) and so that the two fixed points have opposite stability.

The rest of this paper is organized as follows. In Section 2, the sufficient conditions for the existence and stability of a semi-trivial 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

In order to discuss the dynamical behavior of system (1.4), two types of Poincaré maps are constructed. Firstly, we set Poincaré section \({S_{0}} = \{ { ( {x,y} )\vert x = ( {1 - p} )h,y \ge0} \}\) and Poincaré section \({S_{1}} = \{ { ( {x,y} )\vert x = h,y \ge0} \}\). Suppose that system (1.4) has a periodic solution \(( {\varphi ( t ),\phi ( t )} )\) (see Figure 3 and Figure 4).
Figure 3

The phase portrait of system ( 1.4 ).

Figure 4

The simple phase portrait of system ( 1.4 ) with \(\pmb{\tau > 0}\) .

Now we establish a type of Poincaré mapping. Suppose that the periodic trajectory with the initial point \({E^{+} } = ( { ( {1 - p} )h,{y_{0}}} )\) intersects the Poincaré section \({S_{1}}\) at the point \(E = ( {h,{y_{1}}} )\), then jumps to the point \({E^{+} }\) on line \({S_{0}}\) due to the impulsive effects \(\Delta x = - px\) and \(\Delta y = qy + \tau\). Therefore
$$ \varphi ( 0 ) = ( {1 - p} )h,\qquad \phi ( 0 ) = {y_{0}},\qquad \varphi ( T ) = h,\qquad \phi ( T ) = {y_{1}} = \frac{{{y_{0}} - \tau}}{{1 + q}}. $$
(2.1)
Consider another solution \(( {\overline{\varphi}( t ),\overline{\phi}( t )} )\) with initial point \({A_{0}} = ( { ( {1 - p} )h,{y_{0}} + \delta{y_{0}}} )\), where \(\delta{y_{0}}\) is small enough. This disturbed trajectory starting from the point \({A_{0}}\) first intersects the Poincaré section \({S_{1}}\) at the point \({B_{0}} = ( {h,\overline{{y_{1}}} } )\) at moment \(t = T + \delta t\) and then jumps to the point \({A_{1}} = ( { ( {1 - p} )h,\overline{y} } )\) on the line \({S_{0}}\). Hence,
$$ \overline{\varphi}( 0 ) = ( {1 - p} )h,\qquad \overline{\phi}( 0 ) = {y_{0}} + \delta{y_{0}},\qquad \overline{\varphi}( {T + \delta t} ) = h,\qquad \overline{\phi}( {T + \delta t} ) = \overline{{y_{1}}}. $$
(2.2)
Let \(\delta x = \overline{\varphi}( t ) - \varphi ( t )\) and \(\delta y = \overline{\phi}( t ) - \phi ( t )\), then \(\delta{x_{0}} = \overline{\varphi}( 0 ) - \varphi ( 0 ) = ( {1 - p} )h - ( {1 - p} )h = 0\) and \(\delta{y_{0}} = \overline{\phi}( 0 ) - \phi ( 0 ) = \vert {\overline{{A_{0}}{E^{+} }} } \vert \). Set \(\delta{y_{1}} = \vert {\overline{{A_{1}}{E^{+} }} } \vert \) and \(\delta y_{0}^{*} = \vert {\overline{{B_{0}}E} } \vert \), and then the relation between \(\delta{y_{0}}\) and \(\delta {y_{1}}\) determines one type of Poincaré map P:
$$ \delta{y_{1}} = f ( {q,\tau, \delta{y_{0}}} ). $$
(2.3)
We next establish another type of Poincaré map. According to the above discussions, any trajectory through the initial point \(( { ( {1 - p} )h,u} )\) will always intersect with the line \({S_{1}}\) at the point \(( {h,g ( u )} )\). Thus, for any point \((x,y)\) which satisfies \(\dot{x} > 0\), system (1.3) can be transformed as follows:
$$ \frac{{dy}}{{dx}} = \frac{{Q ( {x,y} )}}{{P ( {x,y} )}}, $$
(2.4)
where \(Q ( {x,y} )=cy ( 1 - x - y ) - dy - axy\) and \(P ( {x,y} )=ax ( {1 - x} ) - bx\).
Let \(( {x,y ( {x;{x_{0}},{y_{0}}} )} )\) be an orbit of system (2.5), and set \({x_{0}} = ( {1 - p} )h\), \({y_{0}} = u\), then we have
$$y \bigl( {x; ( {1 - p} )h,u} \bigr) \equiv y ( {x,u} ),\qquad ( {1 - p} )h \le x \le h $$
and
$$ \frac{{\partial y ( {x,u} )}}{{\partial u}} = \exp \biggl\{ { \int_{ ( {1 - p} )h}^{x} {\frac{\partial}{{\partial y}} \biggl( { \frac{{Q ( {s,y ( {s,u} )} )}}{{P ( {s,y ( {s,u} )} )}}} \biggr)} \,ds} \biggr\} >0. $$
(2.5)
Then
$$ \frac{{{\partial^{2}}y ( {x,u} )}}{{\partial{u^{2}}}} = \frac {{\partial y ( {x,u} )}}{{\partial u}} \int_{ ( {1 - p} )h}^{x} {\frac{{{\partial^{2}}}}{{\partial{y^{2}}}} \biggl( { \frac {{Q ( {s,y ( {s,u} )} )}}{{P ( {s,y ( {s,u} )} )}}} \biggr)} \frac{{\partial y ( {s,u} )}}{{\partial u}}\,ds. $$
(2.6)
Taking into account that
$${\frac{{{\partial^{2}}}}{{\partial{y^{2}}}} \biggl( {\frac{{Q ( {s,y ( {s,u} )} )}}{{P ( {s,y ( {s,u} )} )}}} \biggr)}=\frac{-2c}{(a-b)x-ax^{2}}, $$
and \((1-p)h\leq x \leq h<(a-b)/a\), we have that
$$ \frac{{{\partial^{2}}y ( {x,u} )}}{{\partial{u^{2}}}} < 0. $$
(2.7)
Suppose that the point \({A_{k}} ( {(1-h)h,{y_{k}}} )\) is on the Poincaré section \({S_{0}}\), \(B_{k}(h, g(y_{k}))\in S_{1}\). Then \(B_{k}^{+} ( { ( {1 - p} )h, ( {1 + q} )g({y_{k}}) + \tau} ):=A_{k+1}\) is on the line \({S_{0}}\) again due to the impulsive effects. Hence we get the following Poincaré map : \(R \rightarrow R\):
$$ \hat{P} ( {u,q,\tau} )=(1+q)g({u}) + \tau. $$
(2.8)

Noting that \(\frac{{Q ( {x,y} )}}{{P ( {x,y} )}}\) is continuous on the stripe region \(\{(x,y)\vert (1-p)h\leq x \leq h, 0\leq y< \infty\}\), 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 \(\{(x,y)\vert (1-p)h\leq x \leq h, 0\leq y< \infty\}\) if and only if has a fixed point.

From the definition of Poincaré map and (2.5), we have that
$$P'(u)=(1+q)g'(u)=(1+q)\exp \biggl\{ { \int_{ ( {1 - p} )h}^{h} {\frac{\partial}{{\partial y}} \biggl( { \frac{{Q ( {s,y ( {s,u} )} )}}{{P ( {s,y ( {s,u} )} )}}} \biggr)} \,ds} \biggr\} >0. $$

Lemma 4

Assume that the Poincaré map has a fixed point \(y^{*}\). Then the periodic solution is stable \(\hat{P}'(y^{*})<1\) and unstable \(\hat{P}'(y^{*})>1\).

Proof

Suppose that \(\hat{P}'(y^{*})<1\). Then there exist positive constants ρ and δ such that for \(\vert u-y^{*}\vert<\delta\)
$$\begin{aligned} \bigl\vert \hat{P}(u)-y^{*} \bigr\vert &=\bigl\vert \hat{P}\bigl(y^{*}\bigr)+\hat {P}'\bigl(y^{*}\bigr) \bigl(u-y^{*}\bigr)+o \bigl( \bigl\vert u-y^{*} \bigr\vert ^{2}\bigr)-y^{*}\bigr\vert \\ &=\bigl\vert \hat{P}\bigl(y^{*}\bigr) \bigl(u-y^{*}\bigr) +o\bigl(\bigl\vert u-y^{*} \bigr\vert ^{2}\bigr)\bigr\vert < \rho\bigl\vert u-y^{*}\bigr\vert . \end{aligned} $$
Let \(\hat{P}^{2}(u)=\hat{P}(\hat{P}(u))\), \(\hat{P}^{3}(u)=\hat{P}(\hat {P}^{2}(u))\), … , \(\hat{P}^{n}(u)=\hat{P}(\hat{P}^{n-1}(u))\), … . Then we have for \(\vert u-y^{*}\vert<\delta\) that
$$\bigl\vert \hat{P}^{2}(u)-y^{*} \bigr\vert < \rho \bigl\vert \hat{P}(u)-y^{*} \bigr\vert < \rho^{2} \bigl\vert u-y^{*} \bigr\vert , \ldots, \bigl\vert \hat{P}^{n}(u)-y^{*} \bigr\vert < \rho\bigl\vert \hat {P}^{n-1}(u)-y^{*} \bigr\vert < \rho^{n} \bigl\vert u-y^{*} \bigr\vert . $$
Thus,
$$\hat{P}^{n}(u)\rightarrow y^{*},\quad n\rightarrow\infty. $$
By the continuity of solution with respect to the initial values of equation (2.4), we have that \(y(x,h,\hat{P}^{n}(u))\rightrightarrows y(x,h,y^{*}) \) uniformly for \(x\in [(1-p)h, h]\), and therefore the periodic solution \(y(x,h,y^{*})\) is stable.

Similarly, we can prove that the periodic solution is unstable \(\hat {P}'(y^{*})>1\). □

2.2 The case of \(\tau = 0\)

It is easy to see that the semi-trivial periodic solution with \(y = 0\) of system (1.4) exists if and only if \(\tau = 0\). If \(\tau = 0\), system (1.4) has the following form:
$$ \textstyle\begin{cases} \left . \textstyle\begin{array}{l} \dot{x} = ax ( {1 - x} ) - bx, \\ \dot{y} = cy ( {1 - x - y} ) - dy - axy \end{array}\displaystyle \right \}x \ne h,\\ \left . \textstyle\begin{array}{l} \Delta x = - px,\\ \Delta y = qy \end{array}\displaystyle \right \}x = h. \end{cases} $$
(2.9)

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.

Let \(y ( t ) \equiv0\) for \(t \in ( {0,\infty} )\), then we have
$$ \textstyle\begin{cases} \dot{x} = ax ( {1 - x} ) - bx, &x \ne h,\\ \Delta x = - px, &x = h. \end{cases} $$
(2.10)
The solution of equation \(\dot{x} = ax ( {1 - x} ) - bx\) can be calculated as
$$x ( t ) = \frac{{ ( {a - b} ) ( {1 - p} )h}}{{a ( {1 - p} )h + [ { ( {a - b} ) - a ( {1 - p} )h} ]\exp ( { - ( {a - b} ) ( {t - {t_{k}}} )} )}}. $$
Set \(T =\frac{1}{{a - b}}\ln\frac{{a - b - a ( {1 - p} )h}}{{ ( {1 - p} ) ( {a - b - ah} )}}\), \(x ( T ) = h\), \(x ( {{T^{+} }} ) = ( {1 - p} )h\). Then we have the semi-trivial periodic solution of system (2.9) for \(( {k - 1} )T \le t < kT\), \(k = 1,2,3,\ldots\) :
$$ \textstyle\begin{cases} \xi ( t ) = \frac{{ ( {a - b} ) ( {1 - p} )h}}{{a ( {1 - p} )h + [ { ( {a - b} ) - a ( {1 - p} )h} ]\exp ( { - ( {a - b} ) ( {t - ( {k - 1} )T} )} )}},\\ \eta ( t ) = 0 . \end{cases} $$
(2.11)

Then we discuss the stability of the semi-trivial 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

Suppose that \(h < \frac{{a - b}}{a}\), \(c > d\), \(a > b\) and \(- {a^{2}} + bc + ab - ad < 0\). Then the semi-trivial periodic solution \(( {\xi ( t ),0} )\) of system (2.9) is stable if the following condition holds:
$$ 0 < q < {q^{*}}, $$
(2.12)
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

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.

(Method I) See that
$$\begin{aligned}& P ( {x,y} ) = ax ( {1 - x} ) - bx,\qquad Q ( {x,y} ) = cy ( {1 - x - y} ) - dy - axy, \\& \alpha ( {x,y} ) = - px,\qquad \beta ( {x,y} ) = qy,\qquad \varphi ( {x,y} ) = x - h, \\& \bigl( {\xi ( T ),\eta ( T )} \bigr) = ( {h,0} ),\qquad \bigl( {\xi \bigl( {{T^{+} }} \bigr),\eta \bigl( {{T^{+} }} \bigr)} \bigr) = \bigl( { ( {1 - p} )h,0} \bigr). \end{aligned}$$
Then, by virtue of Lemma 1, we can get
$$\begin{aligned}& \frac{{\partial P}}{{\partial x}} = a - b - 2ax,\qquad \frac{{\partial Q}}{{\partial y}} = c - d - ( {a + c} )x - 2cy, \\& \frac{{\partial\alpha}}{{\partial x}} = - P,\qquad \frac{{\partial\alpha }}{{\partial y}} = 0,\qquad \frac{{\partial\beta}}{{\partial x}} = 0, \\& \frac{{\partial\beta}}{{\partial y}} = q,\qquad \frac{{\partial\varphi }}{{\partial x}} = 1,\qquad \frac{{\partial\varphi}}{{\partial y}} = 0, \\& \begin{aligned} {\Delta_{1}} &= \frac{{{P_{+} } ( {\frac{{\partial\beta}}{{\partial y}}\frac{{\partial\varphi}}{{\partial x}} - \frac{{\partial\beta }}{{\partial x}}\frac{{\partial\varphi}}{{\partial y}} + \frac {{\partial\varphi}}{{\partial x}}} ) + {Q_{+} } ( {\frac {{\partial\alpha}}{{\partial x}}\frac{{\partial\varphi}}{{\partial y}} - \frac{{\partial\alpha}}{{\partial y}}\frac{{\partial\varphi }}{{\partial x}} + \frac{{\partial\varphi}}{{\partial y}}} )}}{{P\frac{{\partial\varphi}}{{\partial x}} + Q\frac{{\partial \varphi}}{{\partial y}}}} \\ & = \frac{{{P_{+} } ( {\xi ( {{T^{+} }} ),\eta ( {{T^{+} }} )} ) ( {1 + q} )}}{{P ( {\xi ( T ),\eta ( T )} )}}= \biggl( {\frac {{a - b - ah + aph}}{{a - b - ah}}} \biggr) ( {1 - p} ) ( {1 + q} ). \end{aligned} \end{aligned}$$
Note that \(h < \frac{{a - b}}{a}\) implies \(a-b-ax\) stays positive on the interval \([(1-p)h,h]\). Then we compute directly to get
$$\begin{aligned} \theta &= \exp \biggl( { \int _{0}^{T} { \biggl( {\frac{{\partial P}}{{\partial x}} \bigl( {\xi ( t ),\eta ( t )} \bigr) + \frac{{\partial Q}}{{\partial y}} \bigl( {\xi ( t ),\eta ( t )} \bigr)} \biggr)\,dt} } \biggr) \\ & = \exp \biggl( { \int _{0}^{T} { ( {a - b - 2ax + c - d - ax - cx - 2cy} )\,dt} } \biggr) \\ &= \exp \biggl( { \int _{0}^{T} { \bigl( { ( {a - b + c - d} ) - ( {3a + c} )x} \bigr)\,dt} } \biggr) \\ & = \exp \bigl( { ( {a - b + c - d} )T} \bigr)\exp \biggl( { - ( {3a + c} ) \int_{ ( {1 - p} )h}^{h} {\frac{{x\,dx}}{{ ( {a - b} )x - a{x^{2}}}}} } \biggr) \\ &= {e^{ ( {a - b + c - d} )T}}\exp \biggl( {\frac{{3a + c}}{a} \int_{ ( {1 - p} )h}^{h} {\frac{{ - a\,dx}}{{ - ax + a - b}}} } \biggr) \\ & = { \biggl( {\frac{1}{{1 - p}}} \biggr)^{1 + \frac{{c - d}}{{a - b}}}} { \biggl( { \frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac{{ - 2{a^{2}} + 2ab - ad + bc}}{{a ( {a - b} )}}}}. \end{aligned}$$
Thus the Floquet multiplier \({\mu_{2}}\) can be calculated directly as follows:
$${\mu_{2}} = {\Delta_{1}}\theta = { \biggl( { \frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} { \biggl( {\frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac{{ - {a^{2}} + ab - ad + bc}}{{a ( {a - b} )}}}} ( {1 + q} ). $$

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.

(Method II) See that \(\hat{P}'(0)=(1+q)g'(0)\) and
$$\begin{aligned} g' ( 0 ) &= \exp \biggl\{ { \int_{ ( {1 - p} )h}^{h} {\frac{\partial}{{\partial y}} \biggl( { \frac{{Q ( {s,y ( {s,u} )} )}}{{P ( {s,y ( {s,u} )} )}}} \biggr)} \,ds} \biggr\} = \exp \biggl\{ { \int_{ ( {1 - p} )h}^{h} {\frac{{c - d - ( {a + c} )s}}{{ ( {a - b} )s - a{s^{2}}}}} \,ds} \biggr\} \\ & = \exp \biggl\{ { \int_{ ( {1 - p} )h}^{h} {\frac {{c - d}}{{ ( {a - b} )s - a{s^{2}}}} - \frac{{a + c}}{{a - b - as}}} \,ds} \biggr\} \\ & = \exp \biggl\{ \frac{{{a^{2}} - ab - bc + ad}}{{a ( {a - b} )}}\ln \biggl( {\frac{{a - b - ah}}{{a - b - a ( {1 - p} )h}}} \biggr) + \frac{{c - d}}{{a - b}}\ln \biggl( {\frac{1}{{1 - p}}} \biggr) \biggr\} \\ &= { \biggl( {\frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} { \biggl( { \frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac {{ - {a^{2}} + ab - ad + bc}}{{a ( {a - b} )}}}}. \end{aligned}$$
Thus, \(\hat{P}'(0)<1\) if and only if
$${ \biggl( {\frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} { \biggl( { \frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac {{ - {a^{2}} + ab - ad + bc}}{{a ( {a - b} )}}}}(1+q)< 1. $$
Then the conclusion follows by Lemma 4. The proof is therefore completed. □

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 \(q > {q^{*}}\).

We study the problem of the bifurcation of a nontrivial periodic solution of system (2.9) near the semi-trivial one \(( {\xi ( t ),0} )\). Consider the Poincaré map in (2.5) with \(\tau = 0\). We have
$$ \hat{P} ( y,q,0)\equiv ( {1 + q} )g(y) :=F(y,q). $$
(2.13)

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 \(\frac{{{\partial^{2}}F}}{{\partial{x^{2}}}} ( {0,q^{*}} )\).

Clearly, we can compute \(g' ( 0 )\) and \(g'' ( 0 )\) as follows:
$$g' ( 0 ) = { \biggl( {\frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} { \biggl( {\frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac{{ - {a^{2}} + ab - ad + bc}}{{a ( {a - b} )}}}}. $$
Taking into account the range of p and h, we know \(g' ( 0 ) > 0\). Furthermore, we have
$$\begin{aligned} g'' ( 0 ) &= g' ( 0 ) \int_{ ( {1 - p} )h}^{x} {\frac{\partial}{{\partial y}} \biggl( { \frac{{ - 2cy + c - d - ( {a + c} )s}}{{ ( {a - b} )s - a{s^{2}}}}} \biggr)\frac{{\partial y ( {s,0} )}}{{\partial u}}} \,ds \\ & = g' ( 0 ) \int_{ ( {1 - p} )h}^{x} {\frac{{ - 2c}}{{ ( {a - b} )s - a{s^{2}}}}} \frac{{\partial y ( {s,0} )}}{{\partial u}}\,ds. \end{aligned}$$
It follows by (2.5) that \(g'' ( 0 ) < 0\). By Lemma 2, we know a transcritical bifurcation occurs whenever \(q = {q^{*}}\). Therefore, a stable positive fixed point appears when the parameter q changes through \({q^{*}}\) from left to right.

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

It is easy to verify that
$$\begin{aligned} \begin{aligned} &F ( {0,q} ) = g ( 0 ) = 0,\quad q \in ( {0, + \infty } ), \\ &\frac{{\partial F ( {0,q} )}}{{\partial u}} = ( {1 + q} )g' ( 0 ) = ( {1 + q} ){ \biggl( { \frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} { \biggl( {\frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac{{ - {a^{2}} + ab - ad + bc}}{{a ( {a - b} )}}}}, \end{aligned} \end{aligned}$$
which yields \(\frac{{\partial F ( {0,{q^{*}}} )}}{{\partial u}} = 1\). This means that \(( {0,{q^{*}}} )\) is a fixed point with the eigenvalue 1 of map (2.13).
Then
$$\frac{{{\partial^{2}}F ( {0,{q^{*}}} )}}{{\partial u\,\partial q}} = g' ( 0 ) > 0. $$
Finally,
$$\frac{{{\partial^{2}}F ( {0,{q^{*}}} )}}{{\partial{u^{2}}}} = \bigl( {1 + {q^{*}}} \bigr)g'' ( 0 ) < 0. $$

Then all the conditions in Lemma 2 are satisfied and so the proof of Theorem 2 is completed. □

Case II: \(c > d\) , \(a > b\) , \(- {a^{2}} + bc + ab - ad =0\) .

In this case, for any \(h>0\), \(0< p<1\), we find that
$${ \biggl( {\frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} { \biggl( { \frac{{a - b - a ( {1 - p} )h}}{{a - b - ah}}} \biggr)^{\frac {{ - {a^{2}} + ab - ad + bc}}{{a ( {a - b} )}}}} ( {1 + q} )= { \biggl( { \frac{1}{{1 - p}}} \biggr)^{\frac{{c - d}}{{a - b}}}} ( {1 + q} )>1 $$
is always true. It follows from Lemma 4 that the semi-trivial solution is unstable. This means there exists ȳ, \(0<\bar{y}\leq\delta\ll1\), such that the Poincaré map
$$ F(\bar{y},q)\equiv(1 + q)g(\bar{y})>\bar{y}. $$
(2.14)
On the other hand, we notice that \(Q(x,y)=-cy^{2}+[c-d-(a+c)x]y\rightarrow-\infty\) as \(y\rightarrow +\infty\) and \(P(x,y)= ax(1-x)\) is bounded by \(\lambda< P(x,y)<\frac{(a-b)^{2}}{4a}\), where \(\lambda=\min \{(a-b)(1-p)h-a(1-p)^{2}h^{2}, (a-b)h-ah^{2}\}\). It follows that there must be ȳ such that
$$\frac{Q(x,y)}{P(x,y)}+\frac{q\bar{y}}{ph}< 0,\quad \forall y\geq\bar {y}, (1-p)h \leq x \leq h. $$
We show that the y-coordinate of the solution starting from point \(((1-p)h, (1+q)\bar{y})\) cannot stay above ŷ for \(x\in [(1-p)h,h]\). Otherwise
$$\begin{aligned} y(h)-y\bigl((1-p)h\bigr) =&y(h)-(1+q)\bar{y}= \int_{(1-p)h}^{h}\frac {Q(x,y)}{P(x,y)}\,dx \\ < &- \int_{(1-p)h}^{h}\frac{q\bar{y}}{ph}\,dx< -q\bar{y}. \end{aligned}$$
This implies that
$$y(h)< \bar{y}, $$
which is a contradiction. Therefore there exists such that \(y(\bar{x})=\bar{y}\) and \(y(x)<\bar{y}\), \(\bar{x}< x\leq h\). Thus, \(g((1+q)\bar{y})<\bar{y}\), and so \(((1+q)g((1+q)\bar {y}))<(1+q)\bar{y}\). This is equivalent to saying that the Poincaré map satisfies
$$F\bigl((1+q)\bar{y},q\bigr)\equiv(1 + q)g\bigl((1+q)\bar{y}\bigr)< (1+q)\bar{y}. $$
To sum up, we have for \(F(u,q)\) that
(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): 

\(F'(0,q)>1\).

If follows from (A) that \(F(u,q)\) must have a fixed point. Denote \(y^{*}=\inf\{y\vert y>\delta, F(y,q)=y\}\). We firstly show that \(F'(y^{*},q)<1\). Otherwise, by (B) and (C) we have that
$$F'(u,q)\geq1,\quad 0\leq u \leq y^{*}. $$
This implies by the Lagrange mean value theorem that
$$F\bigl(y^{*},q\bigr)=F(0,q)+F'(\theta) \bigl(y^{*}-0\bigr)>y^{*}, $$
where \(\theta\in(0, y^{*})\). This is a contradiction and so \(F'(y^{*})<1\). We next prove that \(y^{*}\) is the unique fixed point. Otherwise, we have \(y^{**}>y^{*}\), \(F(y^{**},q)=y^{**}\), and also by (B) there holds
$$ F'(u,q)< 1, y^{*}\leq u\leq y^{**}. $$
(2.15)
Then, by using the Lagrange mean value theorem again, we have that there exists ϑ, \(y^{*}<\vartheta<y^{**}\) such that
$$F'(\vartheta,q)=\frac{F(y^{**},q)-F(y^{*},q)}{y^{**}-y^{*}}=1. $$
This contradicts (2.15). To conclude, F has a unique positive fixed point \(y^{*}\) and \(F'(y^{*})<1\). Thus, we obtain the following theorem.

Theorem 3

Suppose that \(h < \frac{{a - b}}{a}\), \(c > d\), \(a > b\) and \(- {a^{2}} + bc + ab - ad =0\). Then the semi-trivial solution of system (2.9) is unstable. Moreover, system (2.9) has a unique positive periodic solution and it is stable.

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 semi-trivial solution. Moreover, we see the fact that \(\tilde{P}(0,q,\tau)=\tau>0\). Next we try to find \(\tilde{y}>0\) such that \(\tilde{P}(\tilde{y},q,\tau)<\tilde{y}\).

We notice that \(Q(x,y)=-cy^{2}+[c-d-(a+c)x]y\rightarrow-\infty\) as \(y\rightarrow+\infty\) and \(P(x,y)= ax(1-x)\) is bounded by \(\lambda< P(x,y)<\frac{(a-b)^{2}}{4a}\), where \(\lambda=\min \{(a-b)(1-p)h-a(1-p)^{2}h^{2}, (a-b)h-ah^{2}\}\). It follows that there must be ŷ such that
$$\frac{Q(x,y)}{P(x,y)}+\frac{q\hat{y}+\tau}{ph}< 0,\quad \forall y\geq \hat{y}, (1-p)h \leq x \leq h. $$
We next show that the y-coordinate of the solution starting from point \(((1-p)h, (1+q)\hat{y}+\tau)\) cannot stay above ŷ for \(x\in [(1-p)h,h]\). Otherwise
$$\begin{aligned} y(h)-y\bigl((1-p)h\bigr)&=y(h)-(1+q)\hat{y}-\tau= \int_{(1-p)h}^{h}\frac {Q(x,y)}{P(x,y)}\,dx-\tau\\ &< - \int_{(1-p)h}^{h}\frac{q\hat{y}+\tau }{ph}\,dx-\tau< q\hat{y}. \end{aligned} $$
This implies that
$$y(h)< \hat{y}, $$
which is a contradiction. Consequently,
$$\tilde{P}\bigl((1+q)\hat{y},q,\tau\bigr)\equiv(1 + q)g\bigl((1+q)\hat{y} \bigr)< (1+q)\hat{y}. $$
To sum up, we have for \(\tilde{P}(u,q,\tau)\) that
(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 semi-trivial 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 two-competitive metapopulation system with impulsive control. To show the stability of the semi-trivial 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

(1)
Department of Mathematics, Shanghai Normal University

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