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Persistence of delayed cooperative models: Impulsive control method
Applied Mathematics and Computation 342 (2019) 130–146
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Persistence of delayed cooperative models: Impulsive control methodR Xiaodi Li a,b,∗, Xueyan Yang a, Tingwen Huang c a
School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, PR China Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, School of Physics and Electronics, Shandong Normal University, Ji’nan Shandong, China c Texas A&M University at Qatar, Doha 5825, Qatar b
a r t i c l e
i n f o
MSC: 34K45 92B05 Keywords: Persistence Cooperative models Impulsive control Time-varying delays Ultimate boundedness
a b s t r a c t In this paper, the problem of impulsive control for persistence of N-species cooperative models with time-varying delays are studied. A method on impulsive control is introduced to delayed cooperative models and some sufficient conditions for the persistence of the addressed models are derived, which are easy to check in real problems. The results show that proper impulsive control strategy may contribute to the persistence of cooperative populations and maintain the balance of an ecosystem. Conversely, the undesired impulsive control such as impulsive harvesting too frequently or impulsive harvesting too drastically may destroy the persistence of populations and leads to the extinction of some species. In addition, some discussions and comparisons with the recent works in the literature are given. Finally, the proposed method is applied to two numerical examples to show the effectiveness and advantages of our results. © 2018 Elsevier Inc. All rights reserved.
1. Introduction It is well known that one of the important problems in mathematical biology is to find some conditions which ensure all species in a multispecies community can be persistent [1–3]. It is necessary to keep persistence of populations to maintain the balance of an ecosystem in the real world. During the past two decades, many research work has been done for the persistence of various biological models, see [4–8] and the references cited therein. Cooperation is one of the important interactions among species, which is commonly seen in social animals and in human society [9]. The corresponding mathematical modelling which is called cooperative model usually has a common property that each state variable has a nonnegative influence on the other state variables [10,11]. Many interesting results on dynamics, especially on persistence problem, of various cooperative models have been reported in the past years, see [12–22]. In particular, May [14] proposed
R This work was supported by National Natural Science Foundation of China (11301308, 61673247), and the Research Fund for Distinguished Young Scholars and Excellent Young Scholars of Shandong Province (JQ201719, ZR2016JL024). The paper has not been presented at any conference. ∗ Corresponding author at: School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, PR China. E-mail address: [email protected] (X. Li).
https://doi.org/10.1016/j.amc.2018.09.003 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
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a class of cooperative models to describe a pair of mutualist as follows:
⎧ u(t ) ⎪ ˙ u ( t ) = r u ( t ) 1 − − c u ( t ) , ⎪ 1 1 ⎨ a1 + b1 v(t ) ⎪ v(t ) ⎪ ⎩v˙ (t ) = r1 v(t ) 1 − − c2 v(t ) , a2 + b2 u(t )
where ri , ai , bi , ci , i = 1, 2, are some positive constants; u and v denote the densities of two species at time t, respectively. Some results on persistence of the models were initially presented from mathematical point of view when u and v are positive variables. And then, various generalized forms of the above models which reflect more realistic dynamic in nature the are being proposed and investigated, for instance, [15,16,19] for the cases of nonautonomous systems and [22] for the discrete cases. In particular, as we know, time delays occur naturally in just about every interaction of the real world which are natural components of the dynamic processes of biology, ecology, physiology, economics, epidemiology and mechanics. For cooperative models, it is more reasonable and realistic to introduce the time delay into the models, since the population of a species in cooperative community depends on not only itself current and past history, but also the current and past history of another species due to the intraspecific cooperative relationship. There have been many studies in literatures that investigate the population dynamics of cooperative models with time delays [20,21]. Systems with impulsive effects describing many evolution processes are characterized by the fact that they are subjected to instantaneous perturbations and experience abrupt changes at certain moments of time which cannot be considered continuously, and there have been significant studies of such systems, see [23–25,30,32,33] and the references therein. Especially, impulsive control as a discontinuous control method has been applied to many practical problems such as orbital transfer of satellite, drug administration, stabilization and synchronization in chaotic secure communication systems, ecological systems and population models see [26–29]. Its extraordinary superiority lie in that, impulses control may change the dynamics of systems, even reverse the dynamics via the perturbations only in some discrete instants, thus control cost and the amount of transmitted information can be reduced greatly. In many cases, moreover, even only impulsive control can be used for control purpose. For instance, a central bank can not change its interest rate everyday in order to regulate the money supply in a financial market; A deep-space spacecraft can not leave its engine on continuously if it has only limited fuel supply [26]; To make the rocket transfer to a higher energy orbit, increments in velocity are given impulsively when the rocket reaches the position of peri-apse and apo-apse [31]. As we know, in the real world, many species in ecosystem are often disturbed by some internal or external factors such as birthrate and deathrate in itself, natural enemies or human activity, that exhibit the impulsive effects, which may lead to the decrease or extinct of some species, see [34–37]. One of the practical problem in ecology is that whether or not an ecosystem can withstand those disturbances and keep persistence over a long period of time. In this sense, the proper impulsive control strategies may contribute to the persistence of populations and maintain the balance of an ecosystem. Conversely, the improper impulsive effects may lead to ecological unbalance and extinction of some species. In recent years, many interesting results on impulsive control of persistence of various biological models have been reported, see [38–44]. For instance, the authors [38] proposed Holling II functional response predator-prey system by periodic impulsive immigration of natural enemies and derived some conditions for extinction of pest and permanence of the system caused by the impulsive control; In [39], a class of impulsive control strategies were given to ensure the permanence and stability of an Ivlev-type predatorprey system based on Floquet theory and comparison principle; In [41], impulsive control for Lotka-Volterra competitive models with time delays were studied by comparison principle and the Lyapunov method. Although there are so many works on impulsive control of biological models, one may note that there is little work on impulsive control of cooperative models with/witout time delays. In fact, extending those impulsive control methods such as those in [38–44] to cooperative systems is quite hard, the reason being that the existing theories of impulsive differential equations is of little help for cooperative systems since each state variable in such systems has a nonnegative influence on the other state variables. Recently, Stamova [45] considered a class of N-species cooperative models with time delays and impulsive control as follows:
⎧ ⎪ ⎨x˙ i (t ) = xi (t ) ri (t ) − ⎪ ⎩
xi (t − τii (t ))
ai (t ) +
xi (tk ) = xi (tk− ) + Iik (xi (tk− )),
N
j=1, j=i
b j (t )x j (t − τi j (t ))
− ci (t )xi (t ) ,
k ∈ Z+ .
Some results for persistence and uniform asymptotic stability of the models were derived via Lyapunov Razumikhin method. It shows that the proper impulsive effects such as stocking and harvesting can control the cooperative system’s population dynamics. Unfortunately, we have to point out that one of these contributions [45] contains a result which is not correct. In this paper, we shall present some results on impulsive control of cooperative models with time-varying delays. More precisely, a method on impulsive control is introduced in the context of delayed cooperative models. The rest of this paper is organized as follows: In Section 2, we shall introduce some preliminary knowledge and problem formulation. In Section 3, some new sufficient conditions for persistence of the addressed models are presented. It is shown that proper impulsive control strategies may contribute to the persistence of cooperative populations and maintain the balance of an ecosystem. Also, some comparison with recent works are given in this Section. Two examples and their simulations are illustrated to show the effectiveness and advantages of the result in Section 4. Finally, we shall make concluding remarks in Section 5.
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2. Preliminaries Notations. Let R denote the set of real numbers, R+ the set of positive real numbers, Z+ the set of posi tive integers, RN the N-dimensional Euclidean space equipped with the norm x = N for any x ∈ RN and i=1 |xi | T ∈ RN : x ∈ R , i = 1, . . . , N }. [·]∗ denotes the integral function. α ∨β denotes the maximum value of RN = { ( x , x , . . . , x ) + 1 2 N i + α and β . For any interval J ⊆ R, set S ⊆ Rk (1 ≤ k ≤ N ), C (J, S ) = {ϕ : J → S is continuous} and PC (J, S ) = {ϕ : J → S is continuous everywhere except at finite number of points t, at which ϕ (t + ), ϕ (t − ) exist and ϕ (t + ) = ϕ (t )}. In particular, let Cτ be an open set in C ([−τ , 0], RN + ). Consider the following N-species cooperative models:
⎧ ⎪ ⎪ ˙ ⎪ ⎨xi (t ) = ri (t )xi (t ) 1 −
xi (t − τii (t ))
ai (t ) +
N
j=1, j=i b j
(t )x j (t − τi j (t ))
− ci (t )xi (t − τii (t )) ,
t ∈ [tk−1 , tk ), (1)
⎪ xi (tk ) = xi (tk− ) + Iik (xi (tk− )), k ∈ Z+ , ⎪ ⎪ ⎩ xi (t0 + s ) = φi (s ), −τ ≤ s ≤ 0, i ∈ ,
where = {1, 2, . . . , N}, φ = (φ1 , φ2 , . . . , φN )T ∈ Cτ and 0 ≤ τ ij (t) ≤ τ ij , i, j ∈ , τ ij are given constants, τ ࣊max i, j ∈ τ ij ; the impulse times tk , k ∈ Z+ , satisfy 0 ≤ t0 < t1 < . . . < tk → ∞, as k → ∞; the numbers xi (tk− ) and xi (tk ) denote the population densities of ith species before and after impulsive perturbations at the moment tk , respectively; ri , ai , bi and ci : R+ → R+ , i ∈ , are some continuous functions with positive upper and lower bounds; Iik : R+ → R are continuous weight functions which characterize the magnitude of the impulse effects on the ith species at the moments tk and satisfy Iik (s ) + s > 0 for any s ∈ R+ , i ∈ , k ∈ Z+ . Under the above conditions, the existence-uniqueness of solution of model (1) can be derived, see [37] for detailed information. In the following, denote by x(t ) = x(t, t0 , φ ) the solution of model (1) with initial value (t0 , φ ). Given a continuous bounded function f which is defined on J ∈ R, we set
. f I = inf f (s ),
. f S = sup f (s ).
s∈J
s∈J
Definition 1. Model (1) is said to be persistent, if there exist constants M > 0 and m > 0 such that each positive solution x(t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T of model (1) satisfies
0 < m ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ M. t→+∞
t→+∞
Remark 1. In the following, we shall present some results on persistence of model (1). Especially, we shall consider the effects of impulse intervals tk − tk−1 and impulse weights Iik . It will show that, under the designed impulsive control, the persistence of model (1) can be guaranteed and moreover, in this case the upper and lower bound of model (1) can be evaluated which is dependent on impulses. But out of the designed one, the persistence cannot be guaranteed and in such case it is possible that the solution of model (1) tend to zero. 3. Main results First, we introduce two lemmas which will be used in the proofs of the main results. T Lemma 1. The set RN + = { (x1 , x2 , . . . , xN ) : xi ∈ R+ , i ∈ } is the positively invariant set of model (1).
Proof. First, it is obvious that
xi (t ) = xi (t0 ) exp where
t
t0
i (t ) = ri (t ) 1 −
i (s )ds , t ∈ [t0 , t1 ), xi (t − τii (t ))
ai (t ) +
N
j=1, j=i b j
(t )x j (t − τi j (t ))
− ci (t )xi (t − τii (t )) .
Since xi (t0 ) = φi (0 ) ∈ Cτ , it is clear that xi (t) > 0 for t ∈ [t0 , t1 ). If there is no impulsive point, then by the continuity of xi we know that xi (t1 ) = xi (t1− ) > 0. Note that for the case that t1 is an impulsive point, the value of xi (t1 ) will be replaced by Ii1 (xi (t1− )) + xi (t1− ). It then follows from the assumption on Iik that xi (t1 ) = Ii1 (xi (t1− )) + xi (t1− ) > 0. Considering model (1) on interval [t1 , t2 ), we have
xi (t ) = xi (t1 ) exp
t
t1
i (s )ds , t ∈ [t1 , t2 ).
Obviously, xi (t) > 0 for t ∈ [t1 , t2 ) in view of xi (t1 ) > 0. In this way, it can be easily deduced that xi (t) > 0 for t ∈ [t0 , ∞). The proof is completed.
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Lemma 2. Assume that there exist two positive constants ρ > 1 and δ ∈ (0, 1) such that
⎧1 − ρ ⎪ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0;
(2)
ln ρ ⎪ {tk − tk−1 } > . ⎪ ⎩kinf ∈Z + (1 − δ ) min riI i∈
Then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (1), where Mi , mi are any constants satisfying
Mi > mi =
ρ riS riI ciI
exp(riS τii ),
1 [ τμii ]∗ +4 ρ
riI aIi δ
riS (1 + aIi ciS )
exp
riI
−
riS
−
riS (1 + aIi ciS ) aIi
Mi τii ,
where μ = infk∈Z+ {tk − tk−1 }. Proof. Define two auxiliary constants:
Mi =
ρ riS
, I
riI ci
m i =
riI aIi δ
riS (1 + aIi ciS )
.
Choose a constant ε > 0 small enough such that
(Mi + ε ) exp(riS τii ) < Mi .
(3)
In the following, we shall prove that there exists a T1 ≥ t0 such that xi (t) < Mi , t ≥ T1 . First, we claim that there exists a T0 ≥ t0 such that
xi (T0− ) < Mi + ε . Suppose on the contrary that xi (t − ) ≥ Mi + ε for all t ≥ t0 . In this case, no matter whether t is an impulsive point or not, it can be easily deduced from (2) that
xi (t ) ≥
Mi + ε
ρ
,
t ≥ t0 .
It then follows from model (1) and Lemma 1 that
x˙ i (t ) ≤ ri (t )xi (t ) 1 − ci (t )xi (t − τii (t ))
≤ xi (t ) riS − riI ciI xi (t − τii (t ))
≤ xi (t ) riS − ≤−
riI ciI (Mi + ε )
ρ
riI ciI ε (Mi + ε )
ρ2
, t ∈ [tk−1 , tk ) ∩ [t0 + τii , ∞ ), k ∈ Z+ .
Integrating the above inequality from t0 + τii to t, we get
xi (t − ) − xi (t0 + τii ) ≤ − +
riI ciI ε (Mi + ε )
ρ2
t0 +τii ≤tk
≤−
(t − τii − t0 )
[xi (tk ) − xi (tk− )]
riI ciI ε (Mi + ε )
ρ2
(t − τii − t0 ) → −∞, t → +∞,
which is a contradiction and thus there exists a T0 ≥ t0 such that xi (T0− ) < Mi + ε . Now we show that
xi (t − ) < Mi ,
t ≥ T0 .
− If this assertion is false, then there exists some t > T0 such that xi (t ) ≥ Mi . Let
t = inf{t ∈ [T0 , t ) : xi (t − ) ≥ Mi }.
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Clearly,
xi (t − ) = xi (t + ) = Mi
t > T0 , and
xi (t − ) < Mi , t ∈ [T0 , t ) in view of the fact
xi (T0− ) < Mi + ε < Mi . In this case, we further define
t¯ = sup{t ∈ [T0 , t ) : xi (t − ) ≤ Mi + ε}. Since xi (t − ) = Mi , we know that
t¯ < t ,
xi (t¯− ) = Mi + ε
and
Mi + ε < xi (t − ) < Mi , t ∈ (t¯, t ). Moreover, we claim that t¯ can not be an impulsive point, i.e., xi (t¯+ ) = xi (t¯− ) . Or else, by (2), we know that
xi (t¯+ ) < xi (t¯− ) = Mi + ε , which is obviously a contradiction with the definition of t¯. Thus, we obtain that
xi (t¯+ ) = xi (t¯− ) = Mi + ε ,
xi (t + ) = xi (t − ) = Mi
(4)
and
Mi + ε < xi (t − ) < Mi ,
t ∈ (t¯, t ).
(5)
Then we claim that
t > t¯ + τii . Or else, t ≤ t¯ + τii . It follows from model (1) and Lemma 1 that
x˙ i (t ) ≤ ri (t )xi (t ){1 − ci (t )xi (t − τii )} ≤ xi (t )riS , t ∈ [tk−1 , tk ),
k ∈ Z+ .
(6)
If there is no impulsive point on interval [t¯, t ), then by (4), we get
Mi = xi (t − ) ≤ xi (t¯ ) exp(riS τii ) ≤ (Mi + ε ) exp(riS τii ), which is a contradiction with (3). If there exist some impulsive points on interval [t¯, t ) generality that
and assume without loss of
t¯ < ti1 < ti2 < · · · < tin0 < t , where n0 denotes the number of impulsive points at interval [t¯, t ). Then it can be deduced from (6) that
xi (ti1 − ) ≤
xi (t¯ ) exp riS (ti1 − t¯ ) ,
xi (ti2 − ) ≤ xi (ti1 ) exp riS (ti2 − ti1 ) , .. .
xi (t − ) ≤ xi (tin0 ) exp riS (t − tin0 ) . Note that xi (ti j ) ≤ xi (ti j − ), j = 1, . . . , n0 . We arrive at
Mi = xi (t − ) ≤ xi (t¯ ) exp(riS τii ) ≤ (Mi + ε ) exp(riS τii ), which is also a contradiction and so it holds that t > t¯ + τii . Now we consider the interval [t¯ + τii , t ). Integrating the former inequality of (6) at the interval [t¯ + τii , t ) considering (4) and (5), one may derive that
xi (t − ) − xi (t¯ + τii ) ≤
t¯+τii
≤
t
t
t¯+τii
xi (s ) riS − riI ciI xi s − τii (t )
xi (s )ds riS −
riI ciI (Mi + ε )
ds +
ρ
(M + ε )riI ciI ε ≤− i (t − t¯ − τii ) < 0, ρ2
t¯+τii ≤tk
[xi (tk ) − xi (tk− )]
and
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
135
which implies that
xi (t¯ + τii ) > xi (t − ) = Mi . In this case, no matter whether t¯ + τii is an impulsive point or not, it always holds that
xi (t¯ + τii − ) ≥ xi (t¯ + τii ) > Mi , which is a contradiction with (5) and thus we have proven that xi (t − ) < Mi . xi (t) < Mi for all t ≥ T1 = T0 + τ . Next, we shall prove that there exists a T2 ≥ T1 such that mi ≤ xi (t), t ≥ T2 . First, by (2), one may choose a constant ε 0 > 0 small enough such that
ln ρ
inf {tk − tk−1 } >
[1 − δ (1 + ε0 )] min riI
k∈Z +
for all t ≥ T0 . Thus, we finally obtain that
> 0.
(7)
i∈
For above given ε 0 , one define two auxiliary constants:
[ τμii ]∗ +3
1 m i = ρ
B =
m i (1 + ε0 ) exp(−B τii ),
riS (1 + aIi ciS )
Mi − riI > 0.
aIi
We first show that there exists a T3 ≥ T1 such that
xi (T3 ) > m i (1 + ε0 ). If this assertion is false, then xi (t ) ≤ m i (1 + ε0 ) for all t ≥ T1 . It follows from model (1) and Lemma 1 that
x˙ i (t ) ≥ xi (t ) riI −
≥ xi (t )
riI
−
riS (1 + aIi ciS ) aIi riS (1 + aIi ciS ) aIi
xi (t − τii (t ))
m i
( 1 + ε0 )
= xi (t )riI [1 − δ (1 + ε0 )], t ∈ [tk−1 , tk ) ∩ [T1 , ∞ ), k ∈ Z+ .
(8)
Let l = min{k ∈ Z+ : tk ≥ T1 }. Then by (7) and (8), it can be deduced that
xi (tl − ) ≥ xi (T1 ) exp riI [1 − δ (1 + ε0 )](tl − T1 ) ,
xi (tl+1 − ) ≥ xi (tl ) exp riI [1 − δ (1 + ε0 )](tl+1 − tl ) 1
≥
ρ
xi (T1 ) exp riI [1 − δ (1 + ε0 )](tl − T1 )
× exp riI [1 − δ (1 + ε0 )](tl+1 − tl )
exp mini∈ riI [1 − δ (1 + ε0 )]μ
≥
ρ
xi (T1 ),
xi (tl+2 − ) ≥ xi (tl+1 ) exp riI [1 − δ (1 + ε0 )](tl+2 − tl+1 ) ≥
1
ρ
≥
xi (tl+1 − ) exp min riI [1 − δ (1 + ε0 )]μ i∈
exp(mini∈ riI [1 − δ (1 + ε0 )]μ )
xi (tl+k ) ≥ −
exp(mini∈ riI [1 − δ (1 + ε0 )]μ )
ρ
2 xi (T1 ),
ρ
···
k xi (T1 ) → ∞,
k → ∞,
(9)
which implies that
xi (tl+k ) ≥
1
ρ
xi (tl+k − ) → ∞,
k → ∞,
which is a contradiction and thus there exists a T3 ≥ T1 such that xi (T3 ) > m i (1 + ε0 ). Now we show that xi (t) > mi for all t ≥ T3 . If this assertion is false, then there exists a t ≥ T3 such that xi (t ) ≤ mi . Obviously, t > T3 . Let
t ∗ = inf{t ∈ [T3 , t
] : xi (t ) ≤ mi },
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
then it is clear that
t ∗ > T3 , xi (t ∗− ) ≥ mi , xi (t ∗+ ) ≤ mi and
xi (t ) > mi , t ∈ [T3 , t ∗ ). In this case, one further define
t = sup{t ∈ [T3 , t ∗ ] : xi (t ) ≥ m i (1 + ε0 )}, then it holds that
t ≤ t ∗ , xi (t − ) ≥ m i (1 + ε0 ), xi (t + ) ≤ m i (1 + ε0 ) and
xi (t ) ≤ m i (1 + ε0 ), t ∈ [t , t ∗ ]. Moreover, we can show that t < t∗ . In fact, if t = t ∗ , then it follows from (2) that:
m i (1 + ε0 ) ≤ xi (t − ) ≤ ρ xi (t ) = ρ xi (t ∗ ) = ρ xi (t ∗+ ) ≤ ρ mi , which contradicts the definitions of mi and m i . Thus we obtain that
xi (t + ) ≤ m i (1 + ε0 ) ≤ xi (t − ),
xi (t ∗+ ) ≤ mi ≤ xi (t ∗− )
(10)
and
mi ≤ xi (t ) ≤ m i (1 + ε0 ), t ∈ [t , t ∗ ].
(11)
Furthermore, we claim that
xi (t + τii ) ≥ m i .
(12)
In fact, from (8) and the known assertion that
t ≥ t ≥ T3 ≥ T0 + τ ,
xi (t ) < Mi ,
it can be deduced that
x˙ i (t ) ≥ xi (t )
riI
−
≥ xi (t ) riI −
riS (1 + aIi ciS ) aIi riS (1 + aIi ciS ) aIi
xi (t − τii (t ))
Mi
≥ −B xi (t ), t ∈ [tk−1 , tk ) ∩ [t , ∞ ), k ∈ Z+ .
(13)
τ Since tk − tk−1 ≥ μ, k ∈ Z+ , there exist at most [ μii ]∗ + 1 impulsive points at interval [t , t + τii ) and assume without loss
of generality that
t < tk1 < tk2 < · · · < tk
τ [ μii ]∗ +1
Thus by (13), we get
< t + τii .
xi (tk− ) ≥ xi (t + ) exp −B (tk1 − t ) , 1
(
xi tk− 2
) ≥ xi (tk1 ) exp −B (tk2 − tk1 ) 1 ≥ xi (t + ) exp −B (tk2 − t ) , ρ ···
xi (t + τii ) ≥ −
1 [ τμii ]∗ +1 ρ
xi (t + ) exp(−B τii ),
which together with (2) and (10) yields that
xi (t + τii ) ≥
1
ρ
xi (t + τii ) ≥ −
1 [ τμii ]∗ +3
Hence, (12) holds. Now we will consider two cases: Case 1. t + τii < t ∗ , i ∈ .
ρ
m i (1 + ε0 ) exp(−B τii ) = m i .
(14)
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137
In this case, by (11), we know that
xi (t − τii ) ≤ m i (1 + ε0 ),
t ∈ [t + τii , t ∗ ).
It then follows from (8) that
x˙ i (t ) ≥ xi (t )riI [1 − δ (1 + ε0 )], t ∈ [t + τii , t ∗ ). If there is no impulsive point at interval [t + τii , t ∗ ), then from (2), (10) and (12), we have
ρ mi ≥ xi (t ∗− ) ≥ xi (t + τii ) exp riI [1 − δ (1 + ε0 )](t ∗ − t − τii ) > xi (t + τii ) ≥ m , i
which is a contradiction with the definitions of mi and m . If there exist some impulsive points at interval [t + τii , t ∗ ), i then similar to the analysis of (9), we can finally obtain that
xi (t ∗− ) > xi (t + τii ), which is also a contradiction. Hence, Case 1 is impossible. Case 2. t + τii ≥ t ∗ , i ∈ . In this case, it follows from (10) and (14) that:
xi (t ∗ ) = xi (t ∗+ ) ≤ mi < m i ≤ xi (t + τii ). Thus it must be
t + τii > t ∗ . Let
t˜ = inf{t ∈ [t ∗ , t + τii ] : xi (t ) ≥ m i }. Obviously,
t˜ > t ∗ , xi (t˜+ ) = xi (t˜− ) = m i ,
∗ ˜ xi (t ) < m i , t ∈ [t , t ).
Furthermore, let
tˇ = sup{t ∈ [t ∗ , t˜) : xi (t ) ≤ mi }. Then we get
xi (tˇ+ ) = xi (tˇ− ) = mi ,
tˇ < t˜,
xi (t ) > mi , t ∈ (tˇ, t˜).
Hence, we obtain that
xi (t˜) = m i ,
ˇ ˜ xi (tˇ) = mi , mi < xi (t ) < m i , t ∈ (t , t ).
Then no matter whether there are some impulsive points on interval (tˇ, t˜) or not, it can be finally deduced from (6) and the definition of mi that
S ˜ S ∗ ˇ ˜ ˇ mi exp(riS τii ) ≤ m i = xi (t ) ≤ xi (t ) exp ri (t − t ) ≤ mi exp ri (t + τii − t ) ,
which implies that t ≥ t∗ . This is a contradiction and thus Case 2 is also impossible. To this, we have proven that mi < xi (t) < Mi for all t ≥ T3 . The proof of Lemma 2 is completed. Remark 2. From the proof of Lemma 2, one may observe that mi can be actually replaced by the following weaker one:
[ τμii ]∗ +3
mi = ρ1
riI aIi δ
riS (1 + aIi ciS )
exp
riI −
riS (1 + aIi ciS ) aIi
Mi
Based on Remark 2, we can derive the following results: Corollary 1. Assume that there exist two positive constants
ρ > exp(max τii riS ), and δ ∈ (0, 1 ) i∈
such that (2) holds, then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (1), where
Mi =
ρ 2 riS riI ciI
,
1 . τii min exp(−riS τii ), ρ
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
1 [ τμii ]∗ +4
mi =
ρ
riI aIi δ
riS (1 + aIi ciS )
riI −
exp
riS (1 + aIi ciS ) aIi
Mi
τii .
Corollary 2. Assume that there exist two positive constants
ρ < exp(min τii riS ) and δ ∈ (0, 1 ) i∈
such that (2) holds, then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (1), where
Mi = exp(2τii riS )
1 [ τμii ]∗ +3
mi =
ρ
riS riI ciI
,
riI aIi δ
riS (1 + aIi ciS )
exp
riI − riS −
riS (1 + aIi ciS ) aIi
Mi
τii .
In particular, when there is no delay effect, model (1) becomes:
⎧ xi (t ) ⎪ ⎪ ˙ x ( t ) = r ( t ) x ( t ) 1 − − c ( t ) x ( t ) , t ∈ [tk−1 , tk ), ⎪ i i i i ⎨ i ai (t ) + Nj=1, j=i b j (t )x j (t ) ⎪ xi (tk ) = xi (tk− ) + Iik (xi (tk− )), k ∈ Z+ , ⎪ ⎪ ⎩ xi (t0 ) = xi0 , i ∈ ,
For model (15), we have Corollary 3. Assume that there exist two positive constants ρ > 1 and δ ∈ (0, 1) such that
⎧1 − ρ ⎪ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0;
ln ρ ⎪ {tk − tk−1 } > . ⎪ ⎩kinf ∈Z + (1 − δ ) min riI i∈
Then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (15), where
mi =
4
1 ρ
riI aIi δ
riS
(1 + aIi ciS )
and Mi is any given constant satisfying
Mi >
ρ riS riI ciI
.
Now we are in a position to present the main results of this paper. Theorem 1. Model (1) is persistent if there exists a constant ρ > 1 such that
⎧1 − ρ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0; ⎪ ⎩ inf {tk − tk−1 } > k∈Z +
ln ρ . mini∈ riI
Proof. Since inf {tk − tk−1 } > k∈Z +
inf {tk − tk−1 } >
k∈Z +
ln ρ mini∈ riI
, one may choose a constant δ > 0 such that
ln ρ . (1 − δ ) mini∈ riI
By Lemma 2, we can obtain the above theorem easily.
(15)
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
139
Corollary 4. Model (1) is persistent if there exists a constant ρ > 1 such that
⎧ 1−ρ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0; ⎪ ⎩ inf {tk − tk−1 } ≥ k∈Z +
ρ
mini∈ riI
.
Corollary 5. Assume that μ = infk∈Z+ {tk − tk−1 } > 0. Model (1) is persistent if
1
μ mini∈ riI
≤
Iik (s ) + s ≤ 1, s
s > 0.
Remark 3. One may observe from Theorem 1 and Corollary 4 that the designed impulsive control strategy is dependent on the lower bounds of parameters ri , i ∈ drastically. Given model (1) and assume that impulsive interval tk − tk−1 = μ (μ > 0 is called impulsive period), if the ratio between impulsive period μ and impulsive amplitude ρ is no less than a specified 1 value (i.e., μ I ), then the persistence of model (1) can be guaranteed. ρ ≥ mini∈ ri
Remark 4. The persistence of model (1) has been investigated in [45] by using Lyapunov Razumikhin method under the assumption that
−s ≤ Iik (s ) ≤ 0,
s > 0.
Here we point out that such an assumption is not enough for the persistence. In fact, when the impulsive interval is small enough and/or Iik /s near to −1, the persistence of model (1) cannot be guaranteed effectively. The idea behind it is that, for fish farming, if the harvesting operation periods become shorter or harvesting quantities become larger such that the fish don’t have enough time to propagate and grow, then the number of fish populations will reduce by degrees and may lead to the extinction of some species, which will be illustrated in Section 4. Remark 5. In [20,21,41,45], the authors studied the stability problems of periodic solution for model (1) with or without impulsive effects via Lyapunov function method. It is worth pointing out that the assumption that
[xi (s − τii ) − yi (s − τii )]sgn(xi (s ) − yi (s )) > 0 is uniformly used in the main proof of those results [20,21] and
ln M/m ≤ V (tk ) is used in [41,45]. Obviously, those assertions are false. But how to overcome those difficulties and find desirable conditions guaranteeing the stability of periodic solution for model (1) with impulsive effects still is an open problem. Remark 6. It has been shown that the symbiosis between populations can be expressed by a rational term [46–48]. In the literature, more classically, it has been done through the use of the more simple quadratic terms. For example, various ecosystems containing always a symbiotic pair have been extensively studied in [46]. Mathematical models and dynamics of diseases spreading in symbiotic communities have been studied in [47,48]. Ref. [48] extended the analysis of predatorprey or competing models subject to a disease spreading among one of the species to populations mutually benefiting from interactions. Inspired by this work, an interesting future topic is to extend the idea in this paper to mathematical models of diseases spreading in symbiotic communities. 4. Applications In this section, we employ two numerical examples to demonstrate the effectiveness and advantages of the obtained results. Example 1. Consider a simple 2-species cooperative model:
⎧ ⎪ ⎪ x˙ 1 (t ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x˙ (t ) = 2
x1 (t − 1 ) 1 3 x1 (t ) 1 − − x1 (t − 1 ) , t ∈ [tk−1 , tk ), 5 1 + 2x2 (t − 1 ) 2 x2 (t − 1 ) 9 3 x2 (t ) 1 − − x2 (t − 1 ) , t ∈ [tk−1 , tk ), 4 2 + x1 (t − 1 ) 10
⎪ 1 ⎪ ⎪ ⎪ x ( t ) = − 1 x1 (tk− ), ⎪ 1 k ⎪ ρ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x2 (tk ) = 1 − 1 x2 (t − ), k ρ2
(16)
k ∈ Z+ , k ∈ Z+ ,
where tk = μk, k ∈ Z+ , ρ 1 > 1, ρ 2 > 1 and μ > 0 are some given constants. By Theorem 1, we can obtain the following result:
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Property 1. Model (16) is persistent if
μ
ln(ρ1 ∨ ρ2 )
> 0.75.
Remark 7. In particular, when there is no impulsive effects, i.e., ρ1 = ρ2 = 1, μ∀, the persistence of model (16) is obvious, which is shown in Fig. 1(a). If there exist some impulsive effects, for instance, ρ1 = 2, ρ2 = 1.5, then by Property 1 model (16) is still persistent if μ > 0.5199. The corresponding numerical simulations are shown in Fig. 1(b)–(d) with μ = 18, 3 and 0.52, respectively. From those numerical simulations, one may observe that the solutions of model (16) can keep persistence if the ratio between impulsive interval μ and impulsive weight ρ 1 ∨ρ 2 is larger than a specified value. Moreover, the persistence becomes weaker when the impulsive interval become smaller under the same impulsive weight. In particular, when the impulsive interval μ = 0.4 or 0.3 < 0.5199, that is, our result cannot guarantee the persistence, Fig. 1(e, f) tell us that the variables x1 or/and x2 will tend to zero, which implies that model (16) becomes no-persistent and some of species will extinct. The numerical results greatly reflect the advantages of our development control strategies. It also implies that the result for persistence in [45] is not correct. The idea behind it is that, such as in a two-species fish cooperative system, when one/two species of fish are caught or hunted in appropriate quantities and appropriate time periods, the fish populations can keep persistence. Under the same harvest quantities, if the operation periods become shorter such that the fish do not have enough time to propagate and grow, then the number of fish populations will reduce by degrees. When the operation periods are less than some threshold, some species of fish will nearly extinct, which is a very real problem. Example 2. Consider a 3-species cooperative model:
⎧ 47 ⎪ ⎪ x˙ 1 (t ) = + 0.1 sin 0.4t x1 (t ) 1 − (0.8 + 0.2 sin t )x1 (t − τ11 (t )) ⎪ ⎪ 40 ⎪ ⎪
⎪ ⎪ x1 (t − τ11 (t )) ⎪ ⎪ − , ⎪ ⎪ 3 + (2.5 + 0.2 cos t )x2 (t − τ12 (t )) + (1.5 + 0.3 cos 1.5t )x3 (t − τ13 (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 17 ⎪ ⎪ ˙ x ( t ) = + 0 . 2 cos 1 . 2 t x ( t ) 1 − (1.1 + 0.5 cos t )x2 (t − τ22 (t )) ⎪ 2 2 ⎪ 38 ⎪ ⎪
⎪ ⎨ x2 (t − τ22 (t )) − , 2 + (1 + 0.33 cos 0.7t )x1 (t − τ21 (t )) + (1.5 + 0.3 cos 1.5t )x3 (t − τ23 (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 11 ⎪ x˙ 3 (t ) = + 0.01 cos t x3 (t ) 1 − (0.7 + 0.2 cos 0.4t )x3 (t − τ33 (t )) ⎪ ⎪ 47 ⎪ ⎪
⎪ ⎪ ⎪ x3 (t − τ33 (t )) ⎪ ⎪− , ⎪ ⎪ 3 + (1 + 0.33 cos 0.7t )x1 (t − τ31 (t )) + (2.5 + 0.2 cos t )x2 (t − τ32 (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ − 1 xi (tk− ), i = 1, 2, 3, k ∈ Z+ , ⎩xi (tk ) = ρi
(17)
where τi j (t ) = 1.2 + [sin(i + j )t]∗ , i, j = 1, 2, 3; tk = μk, k ∈ Z+ , ρ j > 1 and μ > 0 are some given constants. By Theorem 1, we can obtain the following result: Property 2. Model (17) is persistent if
μ
ln(ρ1 ∨ρ2 ∨ρ3 )
> 4.4634.
Remark 8. In the simulations, the persistence of model (17) is obvious when there is no impulsive effects, which is shown in Fig. 2(a). If there exist some impulsive effects such as μ = 8, by Property 2 we know that model (17) is persistent if ρ 1 ∨ρ 2 ∨ρ 3 < 6.5035. The corresponding numerical simulation is given in Fig. 2(b) for the case that ρ1 = 4, ρ2 = 3 and ρ3 = 2. In such case, if we replace ρ3 = 2 by ρ3 = 7 such that ρ1 ∨ ρ2 ∨ ρ3 = 7 > 6.5035, where our result cannot guarantee the persistence, it is interesting to see from Fig. 2(c) that model (17) becomes no-persistent since the variable x3 tends to zero. However, if we enlarge the impulsive interval via Property 2, i.e, let μ > 8.3143 for the case that ρ1 ∨ ρ2 ∨ ρ3 = 7, then the persistence of model (17) can be guaranteed effectively, which is shown in Fig. 2(d,e). This matches our results greatly. Moreover, one may observe that the persistence becomes better and better along with the increasing of the impulsive interval. Conversely, under the same impulsive constants, when the impulsive interval is less than a special threshold, it can be seen from Fig. 2(f) that all of the variables x1 , x2 and x3 will tend to zero and model (17) is no-persistent. The idea behind it is that, in a three-species fish cooperative system, when some species of fish are caught or hunted at a certain time periods, some species of fish will nearly extinct if the harvest quantities beyond the allowed range at each operation time. Under the same harvest quantities, however, if the operation periods can be properly enlarged, then the persistence of fish populations can also be guaranteed effectively. Hence, it is of great benefit for us to design the proper control strategies such that the fish populations can keep persistence.
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
a
4 3.5
x
= =1,
1
2
1
x2
3 solution x
141
2.5 2 1.5 1 0.5 0
b
0
20
40
time t
60
80
4 =2,
3.5
1
=1.5,
2
100
x
1
=18
x
2
solution x
3 2.5 2 1.5 1 0.5 0
c
0
20
40
time t
60
80
4 3.5
=2,
1
=1.5,
2
100
x
1
=3
x
2
solution x
3 2.5 2 1.5 1 0.5 0
0
20
40
time t
60
80
100
Fig. 1. (a) State trajectory of model (16) without impulsive effects. (b) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 18. (c) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 3. (d) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 0.52. (e) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 0.4. (f) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 0.3.
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
d
4 =2,
3.5
1
=1.5,
2
x1
=0.52
x2
solution x
3 2.5 2 1.5 1 0.5 0
e
0
20
40
time t
60
80
4 =2,
3.5
1
=1.5,
2
100
x
1
=0.4
x
2
solution x
3 2.5 2 1.5 1 0.5 0
f
0
20
40
time t
60
80
4 =2,
3.5
1
=1.5,
2
100
x
1
=0.3
x
2
solution x
3 2.5 2 1.5 1 0.5 0
0
20
40
time t
Fig. 1. Continued
60
80
100
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
a
8
x1
= = =1,
7
1
2
3
x2 x3
6 solution x
143
5 4 3 2 1 0
0
50
100 time t
150
b8 7
=4, 1
=3, 2
=2, 3
x1
=8
x2 x3
6 solution x
200
5 4 3 2 1 0
c
0
50
100 time t
150
8 7
=4, 1
=3, 2
=7, 3
x1
=8
x2 x
6 solution x
200
3
5 4 3 2 1 0
0
50
100 time t
150
200
Fig. 2. (a) State trajectory of model (16) without impulsive effects. (b) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 2, μ = 8. (c) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 8. (d) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 9. (e) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 19. (f) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 1.2.
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
d
8 =4,
7
1
=3,
2
=7,
3
x
=9
x x
solution x
6
1 2 3
5 4 3 2 1 0
e
0
50
100 time t
150
8 =4,
7
1
=3,
2
=7,
3
x1
=19
x x
6 solution x
200
2 3
5 4 3 2 1 0
f
0
50
100 time t
150
8 =4,
7
1
=3,
2
=7,
3
x
=1.2
x x
6 solution x
200
1 2 3
5 4 3 2 1 0
0
50
100 time t Fig. 2. Continued
150
200
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
145
5. Conclusion In this paper, we have investigated the impulsive control for persistence of N-species cooperative models with timevarying delays. A method on impulsive control was introduced to delayed cooperative models. Some sufficient conditions for the persistence of such models were presented, which improves and updates some recent works. The results show that impulsive control may not only contribute to the persistence of cooperative populations but also destroy the persistence and leads to the extinction of some species. To show it, two numerical examples and their simulations were given. Unfortunately, so far these techniques cannot be extended to provide conditions to ensure the stability of periodic solution for cooperative models, in view of the theoretical difficulties in such kind of models, due to the fact that each state variable has a nonnegative influence on the other state variables. Our results for cooperative models are nevertheless significant and important to design the impulsive control strategy that keeps the ecosystem populations persistent. References [1] H. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. [2] J. Wu, X. Zhao, Permanence and convergence in multi-species competition systems with delay, Proceedings of the American Mathematical Society 126 (1998) 1709–1714. [3] J. Hale, P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (2012) 388–395. [4] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, 2003. Princeton and Oxford [5] H. Freedman, S. Ruan, Uniform persistence in functional differential equations, J. Differ. Equ. 115 (1995) 173–192. [6] Y. Takeuchi, Global Dynamical Properties of Loteka-Volterra Systems, World Scientific, Singapore, 1996. [7] H. Freedman, J. 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Liu, L. Chen, Extinction and permanence of a two-prey one-predator system with impulsive effect, IMA J. Math. Appl. Med. Biol. 20 (2003) 309–325. [35] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, New York, 1993. [36] X. Liu, K. Rohlf, Impulsive control of a Lotka-Volterra system, IMA J. Math. Control Inf. 15 (1998) 269–284. [37] I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter, Berlin, 2009. [38] X. Liu, L. Chen, Complex dynamics of holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos Solitons Fractals 16 (2003) 311–320. [39] H. Baek, S. Kim, P. Kim, Permanence and stability of an IVLEV-type predator-prey system with impulsive control strategies, Math. Comput. Model. 50 (2009) 1385–1393. [40] K. Negi, S. Gakkhar, Dynamics in a Beddington-Deangelis prey-predator system with impulsive harvesting, Ecol. 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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Persistence of delayed cooperative models: Impulsive control methodR Xiaodi Li a,b,∗, Xueyan Yang a, Tingwen Huang c a
School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, PR China Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, School of Physics and Electronics, Shandong Normal University, Ji’nan Shandong, China c Texas A&M University at Qatar, Doha 5825, Qatar b
a r t i c l e
i n f o
MSC: 34K45 92B05 Keywords: Persistence Cooperative models Impulsive control Time-varying delays Ultimate boundedness
a b s t r a c t In this paper, the problem of impulsive control for persistence of N-species cooperative models with time-varying delays are studied. A method on impulsive control is introduced to delayed cooperative models and some sufficient conditions for the persistence of the addressed models are derived, which are easy to check in real problems. The results show that proper impulsive control strategy may contribute to the persistence of cooperative populations and maintain the balance of an ecosystem. Conversely, the undesired impulsive control such as impulsive harvesting too frequently or impulsive harvesting too drastically may destroy the persistence of populations and leads to the extinction of some species. In addition, some discussions and comparisons with the recent works in the literature are given. Finally, the proposed method is applied to two numerical examples to show the effectiveness and advantages of our results. © 2018 Elsevier Inc. All rights reserved.
1. Introduction It is well known that one of the important problems in mathematical biology is to find some conditions which ensure all species in a multispecies community can be persistent [1–3]. It is necessary to keep persistence of populations to maintain the balance of an ecosystem in the real world. During the past two decades, many research work has been done for the persistence of various biological models, see [4–8] and the references cited therein. Cooperation is one of the important interactions among species, which is commonly seen in social animals and in human society [9]. The corresponding mathematical modelling which is called cooperative model usually has a common property that each state variable has a nonnegative influence on the other state variables [10,11]. Many interesting results on dynamics, especially on persistence problem, of various cooperative models have been reported in the past years, see [12–22]. In particular, May [14] proposed
R This work was supported by National Natural Science Foundation of China (11301308, 61673247), and the Research Fund for Distinguished Young Scholars and Excellent Young Scholars of Shandong Province (JQ201719, ZR2016JL024). The paper has not been presented at any conference. ∗ Corresponding author at: School of Mathematics and Statistics, Shandong Normal University, Ji’nan 250014, PR China. E-mail address: [email protected] (X. Li).
https://doi.org/10.1016/j.amc.2018.09.003 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
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a class of cooperative models to describe a pair of mutualist as follows:
⎧ u(t ) ⎪ ˙ u ( t ) = r u ( t ) 1 − − c u ( t ) , ⎪ 1 1 ⎨ a1 + b1 v(t ) ⎪ v(t ) ⎪ ⎩v˙ (t ) = r1 v(t ) 1 − − c2 v(t ) , a2 + b2 u(t )
where ri , ai , bi , ci , i = 1, 2, are some positive constants; u and v denote the densities of two species at time t, respectively. Some results on persistence of the models were initially presented from mathematical point of view when u and v are positive variables. And then, various generalized forms of the above models which reflect more realistic dynamic in nature the are being proposed and investigated, for instance, [15,16,19] for the cases of nonautonomous systems and [22] for the discrete cases. In particular, as we know, time delays occur naturally in just about every interaction of the real world which are natural components of the dynamic processes of biology, ecology, physiology, economics, epidemiology and mechanics. For cooperative models, it is more reasonable and realistic to introduce the time delay into the models, since the population of a species in cooperative community depends on not only itself current and past history, but also the current and past history of another species due to the intraspecific cooperative relationship. There have been many studies in literatures that investigate the population dynamics of cooperative models with time delays [20,21]. Systems with impulsive effects describing many evolution processes are characterized by the fact that they are subjected to instantaneous perturbations and experience abrupt changes at certain moments of time which cannot be considered continuously, and there have been significant studies of such systems, see [23–25,30,32,33] and the references therein. Especially, impulsive control as a discontinuous control method has been applied to many practical problems such as orbital transfer of satellite, drug administration, stabilization and synchronization in chaotic secure communication systems, ecological systems and population models see [26–29]. Its extraordinary superiority lie in that, impulses control may change the dynamics of systems, even reverse the dynamics via the perturbations only in some discrete instants, thus control cost and the amount of transmitted information can be reduced greatly. In many cases, moreover, even only impulsive control can be used for control purpose. For instance, a central bank can not change its interest rate everyday in order to regulate the money supply in a financial market; A deep-space spacecraft can not leave its engine on continuously if it has only limited fuel supply [26]; To make the rocket transfer to a higher energy orbit, increments in velocity are given impulsively when the rocket reaches the position of peri-apse and apo-apse [31]. As we know, in the real world, many species in ecosystem are often disturbed by some internal or external factors such as birthrate and deathrate in itself, natural enemies or human activity, that exhibit the impulsive effects, which may lead to the decrease or extinct of some species, see [34–37]. One of the practical problem in ecology is that whether or not an ecosystem can withstand those disturbances and keep persistence over a long period of time. In this sense, the proper impulsive control strategies may contribute to the persistence of populations and maintain the balance of an ecosystem. Conversely, the improper impulsive effects may lead to ecological unbalance and extinction of some species. In recent years, many interesting results on impulsive control of persistence of various biological models have been reported, see [38–44]. For instance, the authors [38] proposed Holling II functional response predator-prey system by periodic impulsive immigration of natural enemies and derived some conditions for extinction of pest and permanence of the system caused by the impulsive control; In [39], a class of impulsive control strategies were given to ensure the permanence and stability of an Ivlev-type predatorprey system based on Floquet theory and comparison principle; In [41], impulsive control for Lotka-Volterra competitive models with time delays were studied by comparison principle and the Lyapunov method. Although there are so many works on impulsive control of biological models, one may note that there is little work on impulsive control of cooperative models with/witout time delays. In fact, extending those impulsive control methods such as those in [38–44] to cooperative systems is quite hard, the reason being that the existing theories of impulsive differential equations is of little help for cooperative systems since each state variable in such systems has a nonnegative influence on the other state variables. Recently, Stamova [45] considered a class of N-species cooperative models with time delays and impulsive control as follows:
⎧ ⎪ ⎨x˙ i (t ) = xi (t ) ri (t ) − ⎪ ⎩
xi (t − τii (t ))
ai (t ) +
xi (tk ) = xi (tk− ) + Iik (xi (tk− )),
N
j=1, j=i
b j (t )x j (t − τi j (t ))
− ci (t )xi (t ) ,
k ∈ Z+ .
Some results for persistence and uniform asymptotic stability of the models were derived via Lyapunov Razumikhin method. It shows that the proper impulsive effects such as stocking and harvesting can control the cooperative system’s population dynamics. Unfortunately, we have to point out that one of these contributions [45] contains a result which is not correct. In this paper, we shall present some results on impulsive control of cooperative models with time-varying delays. More precisely, a method on impulsive control is introduced in the context of delayed cooperative models. The rest of this paper is organized as follows: In Section 2, we shall introduce some preliminary knowledge and problem formulation. In Section 3, some new sufficient conditions for persistence of the addressed models are presented. It is shown that proper impulsive control strategies may contribute to the persistence of cooperative populations and maintain the balance of an ecosystem. Also, some comparison with recent works are given in this Section. Two examples and their simulations are illustrated to show the effectiveness and advantages of the result in Section 4. Finally, we shall make concluding remarks in Section 5.
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2. Preliminaries Notations. Let R denote the set of real numbers, R+ the set of positive real numbers, Z+ the set of posi tive integers, RN the N-dimensional Euclidean space equipped with the norm x = N for any x ∈ RN and i=1 |xi | T ∈ RN : x ∈ R , i = 1, . . . , N }. [·]∗ denotes the integral function. α ∨β denotes the maximum value of RN = { ( x , x , . . . , x ) + 1 2 N i + α and β . For any interval J ⊆ R, set S ⊆ Rk (1 ≤ k ≤ N ), C (J, S ) = {ϕ : J → S is continuous} and PC (J, S ) = {ϕ : J → S is continuous everywhere except at finite number of points t, at which ϕ (t + ), ϕ (t − ) exist and ϕ (t + ) = ϕ (t )}. In particular, let Cτ be an open set in C ([−τ , 0], RN + ). Consider the following N-species cooperative models:
⎧ ⎪ ⎪ ˙ ⎪ ⎨xi (t ) = ri (t )xi (t ) 1 −
xi (t − τii (t ))
ai (t ) +
N
j=1, j=i b j
(t )x j (t − τi j (t ))
− ci (t )xi (t − τii (t )) ,
t ∈ [tk−1 , tk ), (1)
⎪ xi (tk ) = xi (tk− ) + Iik (xi (tk− )), k ∈ Z+ , ⎪ ⎪ ⎩ xi (t0 + s ) = φi (s ), −τ ≤ s ≤ 0, i ∈ ,
where = {1, 2, . . . , N}, φ = (φ1 , φ2 , . . . , φN )T ∈ Cτ and 0 ≤ τ ij (t) ≤ τ ij , i, j ∈ , τ ij are given constants, τ ࣊max i, j ∈ τ ij ; the impulse times tk , k ∈ Z+ , satisfy 0 ≤ t0 < t1 < . . . < tk → ∞, as k → ∞; the numbers xi (tk− ) and xi (tk ) denote the population densities of ith species before and after impulsive perturbations at the moment tk , respectively; ri , ai , bi and ci : R+ → R+ , i ∈ , are some continuous functions with positive upper and lower bounds; Iik : R+ → R are continuous weight functions which characterize the magnitude of the impulse effects on the ith species at the moments tk and satisfy Iik (s ) + s > 0 for any s ∈ R+ , i ∈ , k ∈ Z+ . Under the above conditions, the existence-uniqueness of solution of model (1) can be derived, see [37] for detailed information. In the following, denote by x(t ) = x(t, t0 , φ ) the solution of model (1) with initial value (t0 , φ ). Given a continuous bounded function f which is defined on J ∈ R, we set
. f I = inf f (s ),
. f S = sup f (s ).
s∈J
s∈J
Definition 1. Model (1) is said to be persistent, if there exist constants M > 0 and m > 0 such that each positive solution x(t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T of model (1) satisfies
0 < m ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ M. t→+∞
t→+∞
Remark 1. In the following, we shall present some results on persistence of model (1). Especially, we shall consider the effects of impulse intervals tk − tk−1 and impulse weights Iik . It will show that, under the designed impulsive control, the persistence of model (1) can be guaranteed and moreover, in this case the upper and lower bound of model (1) can be evaluated which is dependent on impulses. But out of the designed one, the persistence cannot be guaranteed and in such case it is possible that the solution of model (1) tend to zero. 3. Main results First, we introduce two lemmas which will be used in the proofs of the main results. T Lemma 1. The set RN + = { (x1 , x2 , . . . , xN ) : xi ∈ R+ , i ∈ } is the positively invariant set of model (1).
Proof. First, it is obvious that
xi (t ) = xi (t0 ) exp where
t
t0
i (t ) = ri (t ) 1 −
i (s )ds , t ∈ [t0 , t1 ), xi (t − τii (t ))
ai (t ) +
N
j=1, j=i b j
(t )x j (t − τi j (t ))
− ci (t )xi (t − τii (t )) .
Since xi (t0 ) = φi (0 ) ∈ Cτ , it is clear that xi (t) > 0 for t ∈ [t0 , t1 ). If there is no impulsive point, then by the continuity of xi we know that xi (t1 ) = xi (t1− ) > 0. Note that for the case that t1 is an impulsive point, the value of xi (t1 ) will be replaced by Ii1 (xi (t1− )) + xi (t1− ). It then follows from the assumption on Iik that xi (t1 ) = Ii1 (xi (t1− )) + xi (t1− ) > 0. Considering model (1) on interval [t1 , t2 ), we have
xi (t ) = xi (t1 ) exp
t
t1
i (s )ds , t ∈ [t1 , t2 ).
Obviously, xi (t) > 0 for t ∈ [t1 , t2 ) in view of xi (t1 ) > 0. In this way, it can be easily deduced that xi (t) > 0 for t ∈ [t0 , ∞). The proof is completed.
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Lemma 2. Assume that there exist two positive constants ρ > 1 and δ ∈ (0, 1) such that
⎧1 − ρ ⎪ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0;
(2)
ln ρ ⎪ {tk − tk−1 } > . ⎪ ⎩kinf ∈Z + (1 − δ ) min riI i∈
Then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (1), where Mi , mi are any constants satisfying
Mi > mi =
ρ riS riI ciI
exp(riS τii ),
1 [ τμii ]∗ +4 ρ
riI aIi δ
riS (1 + aIi ciS )
exp
riI
−
riS
−
riS (1 + aIi ciS ) aIi
Mi τii ,
where μ = infk∈Z+ {tk − tk−1 }. Proof. Define two auxiliary constants:
Mi =
ρ riS
, I
riI ci
m i =
riI aIi δ
riS (1 + aIi ciS )
.
Choose a constant ε > 0 small enough such that
(Mi + ε ) exp(riS τii ) < Mi .
(3)
In the following, we shall prove that there exists a T1 ≥ t0 such that xi (t) < Mi , t ≥ T1 . First, we claim that there exists a T0 ≥ t0 such that
xi (T0− ) < Mi + ε . Suppose on the contrary that xi (t − ) ≥ Mi + ε for all t ≥ t0 . In this case, no matter whether t is an impulsive point or not, it can be easily deduced from (2) that
xi (t ) ≥
Mi + ε
ρ
,
t ≥ t0 .
It then follows from model (1) and Lemma 1 that
x˙ i (t ) ≤ ri (t )xi (t ) 1 − ci (t )xi (t − τii (t ))
≤ xi (t ) riS − riI ciI xi (t − τii (t ))
≤ xi (t ) riS − ≤−
riI ciI (Mi + ε )
ρ
riI ciI ε (Mi + ε )
ρ2
, t ∈ [tk−1 , tk ) ∩ [t0 + τii , ∞ ), k ∈ Z+ .
Integrating the above inequality from t0 + τii to t, we get
xi (t − ) − xi (t0 + τii ) ≤ − +
riI ciI ε (Mi + ε )
ρ2
t0 +τii ≤tk
≤−
(t − τii − t0 )
[xi (tk ) − xi (tk− )]
riI ciI ε (Mi + ε )
ρ2
(t − τii − t0 ) → −∞, t → +∞,
which is a contradiction and thus there exists a T0 ≥ t0 such that xi (T0− ) < Mi + ε . Now we show that
xi (t − ) < Mi ,
t ≥ T0 .
− If this assertion is false, then there exists some t > T0 such that xi (t ) ≥ Mi . Let
t = inf{t ∈ [T0 , t ) : xi (t − ) ≥ Mi }.
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Clearly,
xi (t − ) = xi (t + ) = Mi
t > T0 , and
xi (t − ) < Mi , t ∈ [T0 , t ) in view of the fact
xi (T0− ) < Mi + ε < Mi . In this case, we further define
t¯ = sup{t ∈ [T0 , t ) : xi (t − ) ≤ Mi + ε}. Since xi (t − ) = Mi , we know that
t¯ < t ,
xi (t¯− ) = Mi + ε
and
Mi + ε < xi (t − ) < Mi , t ∈ (t¯, t ). Moreover, we claim that t¯ can not be an impulsive point, i.e., xi (t¯+ ) = xi (t¯− ) . Or else, by (2), we know that
xi (t¯+ ) < xi (t¯− ) = Mi + ε , which is obviously a contradiction with the definition of t¯. Thus, we obtain that
xi (t¯+ ) = xi (t¯− ) = Mi + ε ,
xi (t + ) = xi (t − ) = Mi
(4)
and
Mi + ε < xi (t − ) < Mi ,
t ∈ (t¯, t ).
(5)
Then we claim that
t > t¯ + τii . Or else, t ≤ t¯ + τii . It follows from model (1) and Lemma 1 that
x˙ i (t ) ≤ ri (t )xi (t ){1 − ci (t )xi (t − τii )} ≤ xi (t )riS , t ∈ [tk−1 , tk ),
k ∈ Z+ .
(6)
If there is no impulsive point on interval [t¯, t ), then by (4), we get
Mi = xi (t − ) ≤ xi (t¯ ) exp(riS τii ) ≤ (Mi + ε ) exp(riS τii ), which is a contradiction with (3). If there exist some impulsive points on interval [t¯, t ) generality that
and assume without loss of
t¯ < ti1 < ti2 < · · · < tin0 < t , where n0 denotes the number of impulsive points at interval [t¯, t ). Then it can be deduced from (6) that
xi (ti1 − ) ≤
xi (t¯ ) exp riS (ti1 − t¯ ) ,
xi (ti2 − ) ≤ xi (ti1 ) exp riS (ti2 − ti1 ) , .. .
xi (t − ) ≤ xi (tin0 ) exp riS (t − tin0 ) . Note that xi (ti j ) ≤ xi (ti j − ), j = 1, . . . , n0 . We arrive at
Mi = xi (t − ) ≤ xi (t¯ ) exp(riS τii ) ≤ (Mi + ε ) exp(riS τii ), which is also a contradiction and so it holds that t > t¯ + τii . Now we consider the interval [t¯ + τii , t ). Integrating the former inequality of (6) at the interval [t¯ + τii , t ) considering (4) and (5), one may derive that
xi (t − ) − xi (t¯ + τii ) ≤
t¯+τii
≤
t
t
t¯+τii
xi (s ) riS − riI ciI xi s − τii (t )
xi (s )ds riS −
riI ciI (Mi + ε )
ds +
ρ
(M + ε )riI ciI ε ≤− i (t − t¯ − τii ) < 0, ρ2
t¯+τii ≤tk
[xi (tk ) − xi (tk− )]
and
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
135
which implies that
xi (t¯ + τii ) > xi (t − ) = Mi . In this case, no matter whether t¯ + τii is an impulsive point or not, it always holds that
xi (t¯ + τii − ) ≥ xi (t¯ + τii ) > Mi , which is a contradiction with (5) and thus we have proven that xi (t − ) < Mi . xi (t) < Mi for all t ≥ T1 = T0 + τ . Next, we shall prove that there exists a T2 ≥ T1 such that mi ≤ xi (t), t ≥ T2 . First, by (2), one may choose a constant ε 0 > 0 small enough such that
ln ρ
inf {tk − tk−1 } >
[1 − δ (1 + ε0 )] min riI
k∈Z +
for all t ≥ T0 . Thus, we finally obtain that
> 0.
(7)
i∈
For above given ε 0 , one define two auxiliary constants:
[ τμii ]∗ +3
1 m i = ρ
B =
m i (1 + ε0 ) exp(−B τii ),
riS (1 + aIi ciS )
Mi − riI > 0.
aIi
We first show that there exists a T3 ≥ T1 such that
xi (T3 ) > m i (1 + ε0 ). If this assertion is false, then xi (t ) ≤ m i (1 + ε0 ) for all t ≥ T1 . It follows from model (1) and Lemma 1 that
x˙ i (t ) ≥ xi (t ) riI −
≥ xi (t )
riI
−
riS (1 + aIi ciS ) aIi riS (1 + aIi ciS ) aIi
xi (t − τii (t ))
m i
( 1 + ε0 )
= xi (t )riI [1 − δ (1 + ε0 )], t ∈ [tk−1 , tk ) ∩ [T1 , ∞ ), k ∈ Z+ .
(8)
Let l = min{k ∈ Z+ : tk ≥ T1 }. Then by (7) and (8), it can be deduced that
xi (tl − ) ≥ xi (T1 ) exp riI [1 − δ (1 + ε0 )](tl − T1 ) ,
xi (tl+1 − ) ≥ xi (tl ) exp riI [1 − δ (1 + ε0 )](tl+1 − tl ) 1
≥
ρ
xi (T1 ) exp riI [1 − δ (1 + ε0 )](tl − T1 )
× exp riI [1 − δ (1 + ε0 )](tl+1 − tl )
exp mini∈ riI [1 − δ (1 + ε0 )]μ
≥
ρ
xi (T1 ),
xi (tl+2 − ) ≥ xi (tl+1 ) exp riI [1 − δ (1 + ε0 )](tl+2 − tl+1 ) ≥
1
ρ
≥
xi (tl+1 − ) exp min riI [1 − δ (1 + ε0 )]μ i∈
exp(mini∈ riI [1 − δ (1 + ε0 )]μ )
xi (tl+k ) ≥ −
exp(mini∈ riI [1 − δ (1 + ε0 )]μ )
ρ
2 xi (T1 ),
ρ
···
k xi (T1 ) → ∞,
k → ∞,
(9)
which implies that
xi (tl+k ) ≥
1
ρ
xi (tl+k − ) → ∞,
k → ∞,
which is a contradiction and thus there exists a T3 ≥ T1 such that xi (T3 ) > m i (1 + ε0 ). Now we show that xi (t) > mi for all t ≥ T3 . If this assertion is false, then there exists a t ≥ T3 such that xi (t ) ≤ mi . Obviously, t > T3 . Let
t ∗ = inf{t ∈ [T3 , t
] : xi (t ) ≤ mi },
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
then it is clear that
t ∗ > T3 , xi (t ∗− ) ≥ mi , xi (t ∗+ ) ≤ mi and
xi (t ) > mi , t ∈ [T3 , t ∗ ). In this case, one further define
t = sup{t ∈ [T3 , t ∗ ] : xi (t ) ≥ m i (1 + ε0 )}, then it holds that
t ≤ t ∗ , xi (t − ) ≥ m i (1 + ε0 ), xi (t + ) ≤ m i (1 + ε0 ) and
xi (t ) ≤ m i (1 + ε0 ), t ∈ [t , t ∗ ]. Moreover, we can show that t < t∗ . In fact, if t = t ∗ , then it follows from (2) that:
m i (1 + ε0 ) ≤ xi (t − ) ≤ ρ xi (t ) = ρ xi (t ∗ ) = ρ xi (t ∗+ ) ≤ ρ mi , which contradicts the definitions of mi and m i . Thus we obtain that
xi (t + ) ≤ m i (1 + ε0 ) ≤ xi (t − ),
xi (t ∗+ ) ≤ mi ≤ xi (t ∗− )
(10)
and
mi ≤ xi (t ) ≤ m i (1 + ε0 ), t ∈ [t , t ∗ ].
(11)
Furthermore, we claim that
xi (t + τii ) ≥ m i .
(12)
In fact, from (8) and the known assertion that
t ≥ t ≥ T3 ≥ T0 + τ ,
xi (t ) < Mi ,
it can be deduced that
x˙ i (t ) ≥ xi (t )
riI
−
≥ xi (t ) riI −
riS (1 + aIi ciS ) aIi riS (1 + aIi ciS ) aIi
xi (t − τii (t ))
Mi
≥ −B xi (t ), t ∈ [tk−1 , tk ) ∩ [t , ∞ ), k ∈ Z+ .
(13)
τ Since tk − tk−1 ≥ μ, k ∈ Z+ , there exist at most [ μii ]∗ + 1 impulsive points at interval [t , t + τii ) and assume without loss
of generality that
t < tk1 < tk2 < · · · < tk
τ [ μii ]∗ +1
Thus by (13), we get
< t + τii .
xi (tk− ) ≥ xi (t + ) exp −B (tk1 − t ) , 1
(
xi tk− 2
) ≥ xi (tk1 ) exp −B (tk2 − tk1 ) 1 ≥ xi (t + ) exp −B (tk2 − t ) , ρ ···
xi (t + τii ) ≥ −
1 [ τμii ]∗ +1 ρ
xi (t + ) exp(−B τii ),
which together with (2) and (10) yields that
xi (t + τii ) ≥
1
ρ
xi (t + τii ) ≥ −
1 [ τμii ]∗ +3
Hence, (12) holds. Now we will consider two cases: Case 1. t + τii < t ∗ , i ∈ .
ρ
m i (1 + ε0 ) exp(−B τii ) = m i .
(14)
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137
In this case, by (11), we know that
xi (t − τii ) ≤ m i (1 + ε0 ),
t ∈ [t + τii , t ∗ ).
It then follows from (8) that
x˙ i (t ) ≥ xi (t )riI [1 − δ (1 + ε0 )], t ∈ [t + τii , t ∗ ). If there is no impulsive point at interval [t + τii , t ∗ ), then from (2), (10) and (12), we have
ρ mi ≥ xi (t ∗− ) ≥ xi (t + τii ) exp riI [1 − δ (1 + ε0 )](t ∗ − t − τii ) > xi (t + τii ) ≥ m , i
which is a contradiction with the definitions of mi and m . If there exist some impulsive points at interval [t + τii , t ∗ ), i then similar to the analysis of (9), we can finally obtain that
xi (t ∗− ) > xi (t + τii ), which is also a contradiction. Hence, Case 1 is impossible. Case 2. t + τii ≥ t ∗ , i ∈ . In this case, it follows from (10) and (14) that:
xi (t ∗ ) = xi (t ∗+ ) ≤ mi < m i ≤ xi (t + τii ). Thus it must be
t + τii > t ∗ . Let
t˜ = inf{t ∈ [t ∗ , t + τii ] : xi (t ) ≥ m i }. Obviously,
t˜ > t ∗ , xi (t˜+ ) = xi (t˜− ) = m i ,
∗ ˜ xi (t ) < m i , t ∈ [t , t ).
Furthermore, let
tˇ = sup{t ∈ [t ∗ , t˜) : xi (t ) ≤ mi }. Then we get
xi (tˇ+ ) = xi (tˇ− ) = mi ,
tˇ < t˜,
xi (t ) > mi , t ∈ (tˇ, t˜).
Hence, we obtain that
xi (t˜) = m i ,
ˇ ˜ xi (tˇ) = mi , mi < xi (t ) < m i , t ∈ (t , t ).
Then no matter whether there are some impulsive points on interval (tˇ, t˜) or not, it can be finally deduced from (6) and the definition of mi that
S ˜ S ∗ ˇ ˜ ˇ mi exp(riS τii ) ≤ m i = xi (t ) ≤ xi (t ) exp ri (t − t ) ≤ mi exp ri (t + τii − t ) ,
which implies that t ≥ t∗ . This is a contradiction and thus Case 2 is also impossible. To this, we have proven that mi < xi (t) < Mi for all t ≥ T3 . The proof of Lemma 2 is completed. Remark 2. From the proof of Lemma 2, one may observe that mi can be actually replaced by the following weaker one:
[ τμii ]∗ +3
mi = ρ1
riI aIi δ
riS (1 + aIi ciS )
exp
riI −
riS (1 + aIi ciS ) aIi
Mi
Based on Remark 2, we can derive the following results: Corollary 1. Assume that there exist two positive constants
ρ > exp(max τii riS ), and δ ∈ (0, 1 ) i∈
such that (2) holds, then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (1), where
Mi =
ρ 2 riS riI ciI
,
1 . τii min exp(−riS τii ), ρ
138
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
1 [ τμii ]∗ +4
mi =
ρ
riI aIi δ
riS (1 + aIi ciS )
riI −
exp
riS (1 + aIi ciS ) aIi
Mi
τii .
Corollary 2. Assume that there exist two positive constants
ρ < exp(min τii riS ) and δ ∈ (0, 1 ) i∈
such that (2) holds, then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (1), where
Mi = exp(2τii riS )
1 [ τμii ]∗ +3
mi =
ρ
riS riI ciI
,
riI aIi δ
riS (1 + aIi ciS )
exp
riI − riS −
riS (1 + aIi ciS ) aIi
Mi
τii .
In particular, when there is no delay effect, model (1) becomes:
⎧ xi (t ) ⎪ ⎪ ˙ x ( t ) = r ( t ) x ( t ) 1 − − c ( t ) x ( t ) , t ∈ [tk−1 , tk ), ⎪ i i i i ⎨ i ai (t ) + Nj=1, j=i b j (t )x j (t ) ⎪ xi (tk ) = xi (tk− ) + Iik (xi (tk− )), k ∈ Z+ , ⎪ ⎪ ⎩ xi (t0 ) = xi0 , i ∈ ,
For model (15), we have Corollary 3. Assume that there exist two positive constants ρ > 1 and δ ∈ (0, 1) such that
⎧1 − ρ ⎪ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0;
ln ρ ⎪ {tk − tk−1 } > . ⎪ ⎩kinf ∈Z + (1 − δ ) min riI i∈
Then set
= {(x1 , x2 , . . . , xN )T ∈ RN+ : 0 < mi < xi < Mi , i ∈ } is the ultimately bounded set of model (15), where
mi =
4
1 ρ
riI aIi δ
riS
(1 + aIi ciS )
and Mi is any given constant satisfying
Mi >
ρ riS riI ciI
.
Now we are in a position to present the main results of this paper. Theorem 1. Model (1) is persistent if there exists a constant ρ > 1 such that
⎧1 − ρ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0; ⎪ ⎩ inf {tk − tk−1 } > k∈Z +
ln ρ . mini∈ riI
Proof. Since inf {tk − tk−1 } > k∈Z +
inf {tk − tk−1 } >
k∈Z +
ln ρ mini∈ riI
, one may choose a constant δ > 0 such that
ln ρ . (1 − δ ) mini∈ riI
By Lemma 2, we can obtain the above theorem easily.
(15)
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
139
Corollary 4. Model (1) is persistent if there exists a constant ρ > 1 such that
⎧ 1−ρ ⎪ ⎨ ρ s ≤ Iik (s ) ≤ 0, s > 0; ⎪ ⎩ inf {tk − tk−1 } ≥ k∈Z +
ρ
mini∈ riI
.
Corollary 5. Assume that μ = infk∈Z+ {tk − tk−1 } > 0. Model (1) is persistent if
1
μ mini∈ riI
≤
Iik (s ) + s ≤ 1, s
s > 0.
Remark 3. One may observe from Theorem 1 and Corollary 4 that the designed impulsive control strategy is dependent on the lower bounds of parameters ri , i ∈ drastically. Given model (1) and assume that impulsive interval tk − tk−1 = μ (μ > 0 is called impulsive period), if the ratio between impulsive period μ and impulsive amplitude ρ is no less than a specified 1 value (i.e., μ I ), then the persistence of model (1) can be guaranteed. ρ ≥ mini∈ ri
Remark 4. The persistence of model (1) has been investigated in [45] by using Lyapunov Razumikhin method under the assumption that
−s ≤ Iik (s ) ≤ 0,
s > 0.
Here we point out that such an assumption is not enough for the persistence. In fact, when the impulsive interval is small enough and/or Iik /s near to −1, the persistence of model (1) cannot be guaranteed effectively. The idea behind it is that, for fish farming, if the harvesting operation periods become shorter or harvesting quantities become larger such that the fish don’t have enough time to propagate and grow, then the number of fish populations will reduce by degrees and may lead to the extinction of some species, which will be illustrated in Section 4. Remark 5. In [20,21,41,45], the authors studied the stability problems of periodic solution for model (1) with or without impulsive effects via Lyapunov function method. It is worth pointing out that the assumption that
[xi (s − τii ) − yi (s − τii )]sgn(xi (s ) − yi (s )) > 0 is uniformly used in the main proof of those results [20,21] and
ln M/m ≤ V (tk ) is used in [41,45]. Obviously, those assertions are false. But how to overcome those difficulties and find desirable conditions guaranteeing the stability of periodic solution for model (1) with impulsive effects still is an open problem. Remark 6. It has been shown that the symbiosis between populations can be expressed by a rational term [46–48]. In the literature, more classically, it has been done through the use of the more simple quadratic terms. For example, various ecosystems containing always a symbiotic pair have been extensively studied in [46]. Mathematical models and dynamics of diseases spreading in symbiotic communities have been studied in [47,48]. Ref. [48] extended the analysis of predatorprey or competing models subject to a disease spreading among one of the species to populations mutually benefiting from interactions. Inspired by this work, an interesting future topic is to extend the idea in this paper to mathematical models of diseases spreading in symbiotic communities. 4. Applications In this section, we employ two numerical examples to demonstrate the effectiveness and advantages of the obtained results. Example 1. Consider a simple 2-species cooperative model:
⎧ ⎪ ⎪ x˙ 1 (t ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x˙ (t ) = 2
x1 (t − 1 ) 1 3 x1 (t ) 1 − − x1 (t − 1 ) , t ∈ [tk−1 , tk ), 5 1 + 2x2 (t − 1 ) 2 x2 (t − 1 ) 9 3 x2 (t ) 1 − − x2 (t − 1 ) , t ∈ [tk−1 , tk ), 4 2 + x1 (t − 1 ) 10
⎪ 1 ⎪ ⎪ ⎪ x ( t ) = − 1 x1 (tk− ), ⎪ 1 k ⎪ ρ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x2 (tk ) = 1 − 1 x2 (t − ), k ρ2
(16)
k ∈ Z+ , k ∈ Z+ ,
where tk = μk, k ∈ Z+ , ρ 1 > 1, ρ 2 > 1 and μ > 0 are some given constants. By Theorem 1, we can obtain the following result:
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
Property 1. Model (16) is persistent if
μ
ln(ρ1 ∨ ρ2 )
> 0.75.
Remark 7. In particular, when there is no impulsive effects, i.e., ρ1 = ρ2 = 1, μ∀, the persistence of model (16) is obvious, which is shown in Fig. 1(a). If there exist some impulsive effects, for instance, ρ1 = 2, ρ2 = 1.5, then by Property 1 model (16) is still persistent if μ > 0.5199. The corresponding numerical simulations are shown in Fig. 1(b)–(d) with μ = 18, 3 and 0.52, respectively. From those numerical simulations, one may observe that the solutions of model (16) can keep persistence if the ratio between impulsive interval μ and impulsive weight ρ 1 ∨ρ 2 is larger than a specified value. Moreover, the persistence becomes weaker when the impulsive interval become smaller under the same impulsive weight. In particular, when the impulsive interval μ = 0.4 or 0.3 < 0.5199, that is, our result cannot guarantee the persistence, Fig. 1(e, f) tell us that the variables x1 or/and x2 will tend to zero, which implies that model (16) becomes no-persistent and some of species will extinct. The numerical results greatly reflect the advantages of our development control strategies. It also implies that the result for persistence in [45] is not correct. The idea behind it is that, such as in a two-species fish cooperative system, when one/two species of fish are caught or hunted in appropriate quantities and appropriate time periods, the fish populations can keep persistence. Under the same harvest quantities, if the operation periods become shorter such that the fish do not have enough time to propagate and grow, then the number of fish populations will reduce by degrees. When the operation periods are less than some threshold, some species of fish will nearly extinct, which is a very real problem. Example 2. Consider a 3-species cooperative model:
⎧ 47 ⎪ ⎪ x˙ 1 (t ) = + 0.1 sin 0.4t x1 (t ) 1 − (0.8 + 0.2 sin t )x1 (t − τ11 (t )) ⎪ ⎪ 40 ⎪ ⎪
⎪ ⎪ x1 (t − τ11 (t )) ⎪ ⎪ − , ⎪ ⎪ 3 + (2.5 + 0.2 cos t )x2 (t − τ12 (t )) + (1.5 + 0.3 cos 1.5t )x3 (t − τ13 (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 17 ⎪ ⎪ ˙ x ( t ) = + 0 . 2 cos 1 . 2 t x ( t ) 1 − (1.1 + 0.5 cos t )x2 (t − τ22 (t )) ⎪ 2 2 ⎪ 38 ⎪ ⎪
⎪ ⎨ x2 (t − τ22 (t )) − , 2 + (1 + 0.33 cos 0.7t )x1 (t − τ21 (t )) + (1.5 + 0.3 cos 1.5t )x3 (t − τ23 (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 11 ⎪ x˙ 3 (t ) = + 0.01 cos t x3 (t ) 1 − (0.7 + 0.2 cos 0.4t )x3 (t − τ33 (t )) ⎪ ⎪ 47 ⎪ ⎪
⎪ ⎪ ⎪ x3 (t − τ33 (t )) ⎪ ⎪− , ⎪ ⎪ 3 + (1 + 0.33 cos 0.7t )x1 (t − τ31 (t )) + (2.5 + 0.2 cos t )x2 (t − τ32 (t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ − 1 xi (tk− ), i = 1, 2, 3, k ∈ Z+ , ⎩xi (tk ) = ρi
(17)
where τi j (t ) = 1.2 + [sin(i + j )t]∗ , i, j = 1, 2, 3; tk = μk, k ∈ Z+ , ρ j > 1 and μ > 0 are some given constants. By Theorem 1, we can obtain the following result: Property 2. Model (17) is persistent if
μ
ln(ρ1 ∨ρ2 ∨ρ3 )
> 4.4634.
Remark 8. In the simulations, the persistence of model (17) is obvious when there is no impulsive effects, which is shown in Fig. 2(a). If there exist some impulsive effects such as μ = 8, by Property 2 we know that model (17) is persistent if ρ 1 ∨ρ 2 ∨ρ 3 < 6.5035. The corresponding numerical simulation is given in Fig. 2(b) for the case that ρ1 = 4, ρ2 = 3 and ρ3 = 2. In such case, if we replace ρ3 = 2 by ρ3 = 7 such that ρ1 ∨ ρ2 ∨ ρ3 = 7 > 6.5035, where our result cannot guarantee the persistence, it is interesting to see from Fig. 2(c) that model (17) becomes no-persistent since the variable x3 tends to zero. However, if we enlarge the impulsive interval via Property 2, i.e, let μ > 8.3143 for the case that ρ1 ∨ ρ2 ∨ ρ3 = 7, then the persistence of model (17) can be guaranteed effectively, which is shown in Fig. 2(d,e). This matches our results greatly. Moreover, one may observe that the persistence becomes better and better along with the increasing of the impulsive interval. Conversely, under the same impulsive constants, when the impulsive interval is less than a special threshold, it can be seen from Fig. 2(f) that all of the variables x1 , x2 and x3 will tend to zero and model (17) is no-persistent. The idea behind it is that, in a three-species fish cooperative system, when some species of fish are caught or hunted at a certain time periods, some species of fish will nearly extinct if the harvest quantities beyond the allowed range at each operation time. Under the same harvest quantities, however, if the operation periods can be properly enlarged, then the persistence of fish populations can also be guaranteed effectively. Hence, it is of great benefit for us to design the proper control strategies such that the fish populations can keep persistence.
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
a
4 3.5
x
= =1,
1
2
1
x2
3 solution x
141
2.5 2 1.5 1 0.5 0
b
0
20
40
time t
60
80
4 =2,
3.5
1
=1.5,
2
100
x
1
=18
x
2
solution x
3 2.5 2 1.5 1 0.5 0
c
0
20
40
time t
60
80
4 3.5
=2,
1
=1.5,
2
100
x
1
=3
x
2
solution x
3 2.5 2 1.5 1 0.5 0
0
20
40
time t
60
80
100
Fig. 1. (a) State trajectory of model (16) without impulsive effects. (b) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 18. (c) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 3. (d) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 0.52. (e) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 0.4. (f) State trajectory of model (16) with ρ1 = 2, ρ2 = 1.5, μ = 0.3.
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
d
4 =2,
3.5
1
=1.5,
2
x1
=0.52
x2
solution x
3 2.5 2 1.5 1 0.5 0
e
0
20
40
time t
60
80
4 =2,
3.5
1
=1.5,
2
100
x
1
=0.4
x
2
solution x
3 2.5 2 1.5 1 0.5 0
f
0
20
40
time t
60
80
4 =2,
3.5
1
=1.5,
2
100
x
1
=0.3
x
2
solution x
3 2.5 2 1.5 1 0.5 0
0
20
40
time t
Fig. 1. Continued
60
80
100
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
a
8
x1
= = =1,
7
1
2
3
x2 x3
6 solution x
143
5 4 3 2 1 0
0
50
100 time t
150
b8 7
=4, 1
=3, 2
=2, 3
x1
=8
x2 x3
6 solution x
200
5 4 3 2 1 0
c
0
50
100 time t
150
8 7
=4, 1
=3, 2
=7, 3
x1
=8
x2 x
6 solution x
200
3
5 4 3 2 1 0
0
50
100 time t
150
200
Fig. 2. (a) State trajectory of model (16) without impulsive effects. (b) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 2, μ = 8. (c) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 8. (d) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 9. (e) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 19. (f) State trajectory of model (16) with ρ1 = 4, ρ2 = 3, ρ3 = 7, μ = 1.2.
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X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
d
8 =4,
7
1
=3,
2
=7,
3
x
=9
x x
solution x
6
1 2 3
5 4 3 2 1 0
e
0
50
100 time t
150
8 =4,
7
1
=3,
2
=7,
3
x1
=19
x x
6 solution x
200
2 3
5 4 3 2 1 0
f
0
50
100 time t
150
8 =4,
7
1
=3,
2
=7,
3
x
=1.2
x x
6 solution x
200
1 2 3
5 4 3 2 1 0
0
50
100 time t Fig. 2. Continued
150
200
X. Li et al. / Applied Mathematics and Computation 342 (2019) 130–146
145
5. Conclusion In this paper, we have investigated the impulsive control for persistence of N-species cooperative models with timevarying delays. A method on impulsive control was introduced to delayed cooperative models. Some sufficient conditions for the persistence of such models were presented, which improves and updates some recent works. The results show that impulsive control may not only contribute to the persistence of cooperative populations but also destroy the persistence and leads to the extinction of some species. To show it, two numerical examples and their simulations were given. Unfortunately, so far these techniques cannot be extended to provide conditions to ensure the stability of periodic solution for cooperative models, in view of the theoretical difficulties in such kind of models, due to the fact that each state variable has a nonnegative influence on the other state variables. 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