On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response
UDC 517.9 In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one $w$-peri...
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In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one $w$-periodic solution. This result is mainly based on the continuation theorem in the coincidence degree theory and to get the globally attractive $w$-periodic solution of the given system, an inequality is given as the necessary and sufficient condition by using the analytic structure of the system.
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DOI: 10.37863/umzh.v73i4.619
UDC 517.9
N. N. Pelen (Ondokuz Mayıs Univ., Math. Dep., Kurupelit, Samsun, Turkey)
ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS
WITH BEDDINGTON – DEANGELIS TYPE FUNCTIONAL RESPONSE
ПРО ДИНАМIКУ IМПУЛЬСНИХ СИСТЕМ ТИПУ ХИЖАК–ЗДОБИЧ
IЗ ФУНКЦIОНАЛЬНОЮ ВIДПОВIДДЮ ТИПУ БЕДДIНГТОНА – ДЕАНГЕЛIСА
In this study, the two-dimensional predator-prey system with Beddington – DeAngelis type functional response with impulses
is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the
considered system when it has at least one w-periodic solution. This result is mainly based on the continuation theorem in
the coincidence degree theory and to get the globally attractive w-periodic solution of the given system, an inequality is
given as the necessary and sufficient condition by using the analytic structure of the system.
Вивчається двовимiрна система типу хижак–здобич iз функцiональною вiдповiддю типу Беддiнгтона – ДеАнгелiса
та iмпульсами у перiодичному середовищi. Для цього спецiального випадку знайдено необхiднi та достатнi умови
для того, щоб система мала принаймнi один w-перiодичний розв’язок. Цей результат базується головним чином
на теоремi продовження з теорiї степенiв збiгу, а для того, щоб знайти глобально притягуючий w-перiодичний
розв’язок розглядуваної системи, за допомогою аналiтичної структури системи отримано нерiвнiсть, яка вiдiграє
роль необхiдної та достатньої умови.
1. Introduction. Population dynamics is an important branch of the mathematical ecology and
biomathematics. Predator-prey systems is one of the research field of this subject and many studies
have been done on the these type of dynamical systems. Studying on these systems is important
because it helps us to understand the future of the considered species.
In this paper, we have investigate the impulsive predator-prey dynamic systems, since giving
impulse to a system has many important examples in the real life. For instance, if you have used a
pesticide against to a insect species, then there is an immediate decrease in the population or if there
is an immigration from one territory to another for the same species, then there is also an immediate
increase in the population. All of these can be expressed mathematically by using impulses and
these type of equations are said to be impulsive differential equations. There are many studies on
this type of differential and difference equations and especially, its theory has been investigated in
[1, 17 – 19, 21, 25].
The other significant notion that is important for this study, is being in a periodic environment
or not, since many things in real life has a periodic structure. Therefore, to consider the dynamical
systems in a periodic environment becomes important. Global existence and the existence of the
positive periodic solutions are significant aspects of the periodic predator-prey systems and the studies
[7 – 11, 14, 15, 20, 24] investigated these problems on the nonautonomous predator-prey systems by
using coincidence degree theory and the continuation theorem.
Another important notion for this work is functional responses. In this study, we have used
the Beddington – DeAngelis type functional response because of some advantages of this one to the
other functional responses like Holling type, ratio dependent, semi ratio dependent, mono type and
etc. Beddington and DeAngelis uses Beddington – DeAngelis type functional response, according to
c\bigcirc N. N. PELEN, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4 523
524 N. N. PELEN
their observation on the populations on the fishes in Adriatic. Also the advantages of this type of
functional response can be found in their work [2, 6].
Especially, the singularity problem in Holling type and similar functional responses when predator
or prey goes to extinction is solved by using Beddington – DeAngelis functional response. Because
of that singularity problem, in [22], they can not use results of coincidence degree theory in their
system directly. They have divided their system, applied continuation theorem to some part of it
and obtained their goal with Brouwer’s fixed point theorem. In addition to these, they need to use
constant impulses in the variable that symbolize predator to obtain the globally stable w-periodic
solution. In our system, with the advantage of Beddington – DeAngelis type functional response, we
have used different impulses for both prey and predator which is more meaningful in the real life.
We apply continuation theorem directly to our system to get the w-periodic solution and to show the
global stability of that solution, we find a connection between the extiction of prey and predator and
its consequences. Nevertheless, there exist some difficulties in the application of the continuation
theorem to the whole system and these are solved by some analytic technics.
2. Preliminaries. The following informations are obtained from [3]. Let L : \mathrm{D}\mathrm{o}\mathrm{m}L \subset X \rightarrow Y
be a linear mapping, C : X \rightarrow Y be a continuous mapping where X, Y be normed vector spaces.
If \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{K}\mathrm{e}\mathrm{r}L = \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m} \mathrm{I}\mathrm{m}L < +\infty and \mathrm{I}\mathrm{m}L is closed in Y, then the mapping L will be called a
Fredholm mapping of index zero. There exist continuous projections U : X \rightarrow X and V : Y \rightarrow Y
when L is a Fredholm mapping of index zero such that \mathrm{I}\mathrm{m}U = \mathrm{K}\mathrm{e}\mathrm{r}L, \mathrm{I}\mathrm{m}L = \mathrm{K}\mathrm{e}\mathrm{r}V = \mathrm{I}\mathrm{m}(I - V ),
then it follows that L| DomL\cap KerU : (I - U)X \rightarrow \mathrm{I}\mathrm{m}L is invertible. The inverse of that map
is denoted as KU . The mapping C will be called L-compact on \Omega if V C(\Omega ) is bounded and
KU (I - V )C : \Omega \rightarrow X is compact, where \Omega is an open bounded subset of X. Since \mathrm{I}\mathrm{m}V is
isomorphic to \mathrm{K}\mathrm{e}\mathrm{r}L, the isomorphism J : \mathrm{I}\mathrm{m}V \rightarrow \mathrm{K}\mathrm{e}\mathrm{r}L is exist and the above informations are
important for the continuation theorem that we give below.
Definition 1 [5]. The codimension (or quotient or factor dimension) of a subspace L of a vector
space V is the dimension of the quotient space V/L; it is denoted by \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}V L or simply by
\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}L and is equal to the dimension of the orthogonal complement of L in V, and one has
\mathrm{d}\mathrm{i}\mathrm{m}L+ \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}L = \mathrm{d}\mathrm{i}\mathrm{m}V.
Theorem 1 [12] (continuation theorem). Suppose that L is a Fredholm mapping of index zero
and C is L-compact on \Omega . Assume that:
(a) For any y that satisfies Ly = \lambda Cy is not on \partial \Omega , for each \lambda \in (0, 1);
(b) V Cy \not = 0 and the Brouwer degree \mathrm{d}\mathrm{e}\mathrm{g}\{ JV C, \partial \Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L, 0\} \not = 0 for each y \in \partial \Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L.
Then Ly = Cy has at least one solution lying in \mathrm{D}\mathrm{o}\mathrm{m}L \cap \partial \Omega .
Definition 2 [26]. A w-periodic semiflow F (t) : X \rightarrow X (X is the initial value space) in the
sense that F (t)x is continuous in (t, x) \in [0,+\infty )\times X, F (0) = I and F (t+ w) = F (t)F (w) for
all t > 0 is generated by the solutions of a w-periodic system.
Definition 3 [26]. If there exists \eta > 0 such that, for any x \in X0,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
t\rightarrow \infty
d(F (t)x, \partial X0) \geq \eta ,
then the periodic semiflow F (t) is said to be uniformly persistent with respect to (X0, \partial X0).
Definition 4 [13]. Suppose that F : \BbbR n \rightarrow \BbbR n. If there exists a bounded set B such that, for
each x \in \BbbR n, there is an integer n0 = n0(x,B) such that Fn(x) \in B for each n \geq n0, then the
map F is called point dissipative.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 525
Lemma 1 [26]. Assume that S : X \rightarrow X is continuous such that S(X0) \subset X0. Suppose that
S is uniformly persistent with respect to (X0, \partial X0), compact and point dissipative. Then, for S in
X0 relative to strongly bounded sets in X0, there exists a global attractor A0 and S has coexistence
state x0 \in A0.
Definition 5 [11]. The system (3) is called permanent if there exist positive constants r1, r2,
R1, and R2 such that solution
\bigl(
\~x(t), \~y(t)
\bigr)
of system (3) satisfies
r1 \leq \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f} \~x(t) \leq \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p} \~x(t) \leq R1,
r2 \leq \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f} \~y(t) \leq limt\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p} \~y(t) \leq R2.
Lemma 2 [22]. Consider the system
\~x\prime (t) = a(t)\~x(t) - b(t)\~x2(t), t \not = tk,
\~x(t+k ) = (1 + gk)\~x(tk).
(1)
Then system (1) admits a unique, positive, w-periodic solution if and only if
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi) > 0, (2)
which, moreover, is globally asymptotically stable.
3. Main result. The equation that we investigate is
\~x\prime (t) = a(t)\~x(t) - b(t)\~x2(t) - c(t)\~x(t)E(t, \~y(t), \~x(t), \~y(t)), t \not = tk,
\~y\prime (t) = - d(t)\~y(t) + f(t)\~y(t)E(t, \~x(t), \~x(t), \~y(t)), t \not = tk,
\~x(t+k ) = (1 + gk)\~x(tk),
\~y(t+k ) = (rk)\~y(tk),
(3)
where E
\bigl(
t, u(t), x(t), y(t)
\bigr)
=
u(t)
\alpha (t) + \beta (t)x(t) +m(t)y(t)
. In (3), if we take \~x(t) = \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) and
\~y(t) = \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), then we obtain the following system:
x\prime (t) = a(t) - b(t) \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) - c(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
, t \not = tk,
y\prime (t) = - d(t) + f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
, t \not = tk,
\Delta x(tk) = \mathrm{l}\mathrm{n}(1 + gk),
\Delta y(tk) = \mathrm{l}\mathrm{n}(rk),
(4)
where tk+q = tk + w, a(t + w) = a(t), b(t + w) = b(t), c(t + w) = c(t), d(t + w) = d(t),
f(t + w) = f(t), \alpha (t + w) = \alpha (t), \beta (t + w) = \beta (t), m(t + w) = m(t), k, gk and rk are the
constants with 1 > gk > - 1 and rk > 0. Here, x(tk+q) = x(tk) + w, \~x(tk+q) = \~x(tk) + w,
y(tk+q) = y(tk)+w, \~y(tk+q) = \~y(tk)+w. In equations (3) and (4) each coefficient function is from
continuous functions class and all the coefficient fuctions are positive.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
526 N. N. PELEN
Definition 6. In system (4), we say that x(t) (y(t)) (prey (predator)) goes to extinction if and
only if \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) (\mathrm{e}\mathrm{x}\mathrm{p}(y(t))) tends to 0 as t tends to infinity, for all solutions of x(t) (y(t)).
Equivalently, we also say that prey (predator) goes to extinction if and only if \~x(t) (\~y(t)) tends to
zero as t tends to infinity, for all solutions of system (3).
Lemma 3. Assume that
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri > 0 (5)
is satisfied. If y(t) does not go to extinction, then neither x(t) does.
Proof. The statement of the above lemma is the same with the statement: assume that (5) is
satisfied. Then if x(t) goes to extinction, then y(t) also goes to extinction. By using the second
equation in system (4) and taking the integral of that equation from 0 to t, we obtain
\mathrm{e}\mathrm{x}\mathrm{p}(y(t)) = \mathrm{e}\mathrm{x}\mathrm{p}(y(0))
\prod
ti<t
(ri) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
- d(s) + f(s)E(t, \mathrm{e}\mathrm{x}\mathrm{p}(x(s)), \mathrm{e}\mathrm{x}\mathrm{p}(x(s)), \mathrm{e}\mathrm{x}\mathrm{p}(y(s)))ds
\right) .
(6)
If x(t) goes to extinction, then \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) tends to 0 as t tends to infinity. Since all the coefficient
functions are positive, f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
also tends to 0 as t tends to infinity.
For sufficiently large t the integral
t\int
0
- d(s) + \mathrm{l}\mathrm{n}
q\prod
i=1
ri + f(s)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(s)), \mathrm{e}\mathrm{x}\mathrm{p}(x(s)), \mathrm{e}\mathrm{x}\mathrm{p}(y(s))
\bigr)
ds
becomes negative and the right-hand side of the equation (6) tends to 0 as t tends to infinity which
means \mathrm{e}\mathrm{x}\mathrm{p}(y(t)) tends to 0 as t tends to infinity. Thus, y(t) goes to extinction. Hence, we are done.
3.1. Permanence and extinction of the solutions.
Lemma 4. If inequalities (2) and (5) are satisfied, then for the given system (3)
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}t\rightarrow \infty \~x(t) \geq r1 for some \~r1 > 0.
Proof. Assume that prey goes to extiction, then by Lemma 3 predator also goes to extinction.
Then for sufficiently large T > 0 there exists \epsilon 0 > 0 such that, for each t > T,
\~y(t) < \epsilon 0.
If
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
k=1
(1 + gk) > 0
for sufficiently small \epsilon 0, we have
w\int
0
a(t) - \epsilon 0c(t)
\alpha (t) + \epsilon 0m(t)
- b(t)\~x(t)dt+ \mathrm{l}\mathrm{n}
q\prod
k=1
(1 + gk) > 0. (7)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 527
Additionally, for sufficiently small \epsilon 0, the following inequality becomes also true:
\~x\prime (t) > \~x(t)
\biggl(
a(t) - \epsilon 0c(t)
\alpha (t) + \epsilon 0m(t)
- b(t)\~x(t)
\biggr)
.
Then consider the system
\=x\prime (t) = \=x(t)
\biggl(
a(t) - \epsilon 0c(t)
\alpha (t) + \epsilon 0m(t)
- b(t)\=x(t)
\biggr)
,
\=x(t+k ) = (1 + gk)\=x(tk).
(8)
For system (8), since inequality (7) is true, we can apply Lemma 2. Then system (8) has glo-
bally attractive, w-periodic solution \v x\ast (t). By the comparison theorem for impulsive differential
equations \~x(t) > \v x\ast (t). Therefore prey does not go to extiction which is a contradiction. Hence,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}t\rightarrow \infty \~x(t) \geq \~r1, for some \~r1 > 0 is true.
Lemma 5. Asume that inequality (2) is satisfied. Predator in system (3) goes to extinction if
and only if
w\int
0
- d(t) +
f(t)x\ast (t)
\alpha (t) + \beta (t)x\ast (t)
dt+ \mathrm{l}\mathrm{n}
\Biggl(
q\prod
k=1
(rk)
\Biggr)
\leq 0 (9)
holds, where x\ast (t) is the unique, positive, globally attractive, w-periodic solution of the system (1).
Proof. By taking the contrapositive of the necessary part of the lemma, we have if (9) does not
holds then predator does not go to extiction. From now on, it is a proof by contradiction. Therefore,
assume that (9) does not holds and predator goes to extiction. If we can find a contradiction, then
we are able to get the desired result for the first side of the lemma. Here suppose that system (3)
satisfies the equation
w\int
0
- d(t) +
f(t)x\ast (t)
\alpha (t) + \beta (t)x\ast (t)
dt+ \mathrm{l}\mathrm{n}
\Biggl(
q\prod
k=1
(rk)
\Biggr)
> 0. (10)
Then there exists \~\epsilon > 0 such that
w\int
0
- d(t) +
f(t)(x\ast (t) - \~\epsilon )
\alpha (t) + \beta (t)(x\ast (t) - \~\epsilon ) +m(t)\~\epsilon
dt+ \mathrm{l}\mathrm{n}
\Biggl(
q\prod
k=1
(rk)
\Biggr)
> 0. (11)
Consider the system
\~x\prime (t) = \~x(t)(a(t) - 2\gamma c(t)
\alpha (t) + 2\gamma m(t)
- b(t)\~x(t)),
\~x(t+k ) = (1 + gk)
\bigl(
\~x(tk)
\bigr)
.
(12)
where \gamma is a positive constant. It is obvious that for sufficiently small \gamma , a(t) - 2\gamma c(t)
\alpha (t) + 2\gamma m(t)
> 0.
Thus, system (12) has a globally attractive, unique, w-periodic solution from Lemma 2 for sufficiently
small \gamma . Assume that x\gamma be the globally attractive solution of the system (12). Thus, x\gamma (t) \rightarrow x\ast (t)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
528 N. N. PELEN
as \gamma \rightarrow 0. Then there exists \^\gamma such that x\^\gamma (t) \geq x\ast (t) - \~\epsilon /2 and 2\^\gamma < \~\epsilon . Since predator goes to
extinction, then
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
t\rightarrow \infty
\~y(t) < \^\gamma .
So, there exists T such that, for any t > T,
\~y(t) < 2\^\gamma < \~\epsilon .
Since \~y(t) < 2\^\gamma , then \~x\prime (t) > \~x(t)(a(t) - 2\gamma c(t)
\alpha (t) + 2\gamma m(t)
- b(t)\~x(t)). By the comparison theorem
for impulsive differential equations \~x(t) > x\ast (t) - \~\epsilon . Therefore, we get the system
\~y\prime (t) \geq \~y(t)
\biggl(
- d(t) +
f(t)(x\ast (t) - \~\epsilon )
\alpha (t) + \beta (t)(x\ast (t) - \~\epsilon ) +m(t)\~\epsilon
\biggr)
,
\~y(t+k ) = (rk)(\~y(tk)).
Here,
\~y(t) \geq \~y(0) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
- d(s) +
f(s)(x\ast (s) - \~\epsilon )
\alpha (s) + \beta (s)(x\ast (s) - \~\epsilon ) +m(s)\~\epsilon
ds+ \mathrm{l}\mathrm{n}
\left( \prod
0<tk<t
(rk)
\right) \right) . (13)
Since inequality (11) is satisfied, the right-hand side of inequality (13) is always positive for suffi-
ciently large t and does not go to zero as t tends to infinity. Therefore \~y(t) becomes always positive
and does not go to zero. In other words, predator does not go to extinction. Hence, we have proved
that if predator goes to extinction then inequality (9) holds.
For converse, to prove the result, we use contradiction again. Assume that inequality (9) holds
and predator does not go to extinction. Then \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}t\rightarrow \infty \~y(t) \geq \~r2. Since (2) is true, then by
Lemma 6, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}t\rightarrow \infty \~x(t) \leq R1. Thus, we have
\~x\prime (t) \leq \~x(t)
\biggl(
a(t) - c(t)\~r2
\alpha (t) + \beta (t)R1 +m(t)\~r2
- b(t)\~x(t)
\biggr)
,
\~x(t+k ) = (1 + gk)
\bigl(
\~x(tk)
\bigr)
.
By the comparison theorem for impulsive differential equations \~x(t) \leq x\ast (t), therefore, the following
inequality is true:
\~y\prime (t) \leq \~y(t)
\biggl(
- d(t) +
f(t)x\ast (t)
\alpha (t) + \beta (t)x\ast (t) +m(t)\~r2
\biggr)
,
\~y(t+k ) = (rk)(\~y(tk)).
(14)
Since in this system each coefficient functions are positive, then
f(t)x\ast (t)
\alpha (t) + \beta x\ast (t) +m(t)\~r2
\leq f(t)x\ast (t)
\alpha (t) + \beta (t)x\ast (t)
- \mu
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 529
for some \mu > 0. Then
\~y(t) \leq y(0) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
- d(t) +
f(t)x\ast (t)
\alpha (t) + \beta (t)x\ast (t)
- \mu dt+ \mathrm{l}\mathrm{n}
\left( \prod
0\leq tk\leq t
(rk)
\right) \right) . (15)
Since inequality (9) holds, for sufficiently large t, the inside of the exponential function in inequa-
lity (15) is negative. Thus, as t tends to infinity, \~y(t) tends to zero, which means predator goes to
extinction which is a contradiction.
Lemma 5 is proved.
Lemma 6. If inequalities (2) and (5) are satisfied, then there exist positive constants R1 and
R2 such that
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}x(t) \leq R1, \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p} y(t) \leq R2. (16)
Proof. First consider the system (3), then the following inequality is true:
\~x\prime (t) \leq a(t)\~x(t) - b(t)\~x2(t), t \not = tk,
\~x(t+k ) = (1 + gk)\~x(tk).
(17)
Suppose that (2) holds and consider the equations
\~u\prime (t) = a(t)\~u(t) - b(t)\~u2(t), t \not = tk,
\~u(t+k ) = (1 + gk)\~u(tk).
(18)
By Lemma 2, system (18) has unique, positive, globally attractive (or globally asypmtotically stable),
w-periodic solution \=u(t). By using comparison theorem for impulsive differential equations from [1],
we obtain that
\~x(t) \leq u(t).
The attractivity of \=u(t) implies that there exists T > 0 such that
u(t) \leq \=u(t) + 1 for t > T.
Therefore, it is clear that \~x(t) is bounded above with a positive constant R1. Secondly, consider the
system (3). The coefficient functions in system (3) is bounded, positive and w-periodic, then the
following inequality is true:
\~y\prime (t) \leq - d(t)\~y(t) +
f(t)\~x(t)
m(t)
\leq fMR1
mL
- d(t)\~y(t), t \not = tk,
\~y(t+k ) = (rk)\~y(tk).
(19)
Then, we get
\~y(t) \leq \~y(0)
\prod
0<tk<t
rk \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
- d(s)ds
\right) +
t\int
0
\prod
s<tk<t
rk \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
s
- d(\sigma )d\sigma
\right) fMR1
mL
ds.
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530 N. N. PELEN
We can rewrite the last inequality as
\~y(t) \leq \~y(0) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
- d(s)ds+ \mathrm{l}\mathrm{n}
\left( \prod
0<tk<t
rk
\right) \right) +
+
fMR1
mL
t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
s
- d(\sigma )d\sigma + \mathrm{l}\mathrm{n}
\Biggl( \prod
s<tk<t
rk
\Biggr) \right) ds. (20)
For sufficently large t, inside of the exponential function for the first term and the second term
of inequality (20) is negative. So if we take
fMR1
mL
= M, then
\~y(t) \leq \~y(0)e1+Dw - ct +Me1+Dw
t\int
0
ec(s - t)ds \leq
\leq \~y(0)
\biggl(
e1+Dw - ct +
Me1+Dw
c
(1 - e - ct)
\biggr)
\leq
\leq \~y(0)
\biggl(
e1+Dw +
Me1+Dw
c
\biggr)
.
Here, D = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
| d(t)| : t \in [0, w]
\bigr\}
and
c = \mathrm{m}\mathrm{i}\mathrm{n}
\left\{
\left( w\int
0
d(s)ds - \mathrm{l}\mathrm{n}
\Bigl( \prod q
k=1
rk
\Bigr) \right) \Bigg/ w, 1/w
\right\} .
Thus, we have a positive constant R2 such that \~y(t) is bounded above with a positive con-
stant R2.
Lemma 7. If (2) and (10) for x\ast (t) is the unique, positive, globally attractive, w-periodic
solution of the system (1) is satisfied, then \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow \infty \~y(t) > \~r2 for some positive \~r2.
Proof. This result is the immediate consequence of Lemma 5.
Lemma 8. Assume that inequalities (2) and (5) are satisfied. Then system (3) is permanent
if and only if inequality (10) is satisfied. Therefore, from Theorem 2, this system has at least one
w-periodic solution.
Proof. This is the immediate consequence of the Lemmas 3, 6, and 5.
Lemma 9. In system (3), assume that (2) is satisfied. If at least one solution of \~y(t), does not
tend to 0 as t tends to infinity, then for all solutions of \~y(t), does not tend to 0 as t tends to infinity.
Proof. This is a proof by contradiction. Let us assume that there exist two solutions for
system (3),
\bigl(
\~x(t), \~y(t)
\bigr)
and (\^x(t), \^y(t)) such that \~y(t), does not tend to 0 as t tends to infinity and
\^y(t) tends to 0 as t tends to infinity. Since \^y(t) tends to 0 as t tends to infinity, then \^x(t) tends
to x\ast as t tends to infinity. According to Definition 6 predator does not go to extinction, and as a
consequence of Lemma 5,
w\int
0
- d(t) +
f(t)x\ast (t)
\alpha (t) + \beta (t)x\ast (t)
dt+ \mathrm{l}\mathrm{n}
\Biggl(
q\prod
k=1
(rk)
\Biggr)
> 0
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ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 531
is satisfied. Then, by using Lemma 7, we have \^y(t) > \^r, for some positive \^r which is a contradiction.
Hence, proof is completed.
3.2. \bfitw -Periodicity of the solutions.
Theorem 2. Assume that all the coefficient functions in system (4) are bounded, positive, w-
periodic, from C(\BbbR ,\BbbR 2) and inequalities (2) and (5) are satisfied. Then there exists at least one
w-periodic solution if and only if y(t) does not go to extinction.
Proof. X :=
\bigl\{
(p, z)\intercal \in PC(\BbbR ,\BbbR 2) : p(t+ w) = p(t), z(t+ w) = z(t)
\bigr\}
with the norm
\| (p, z)\intercal \| = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,w]
\bigl(
| p(t)| , | z(t)|
\bigr)
and
Y :=
\Bigl\{ \bigl[
(p, z)\intercal , (d1, f1)
\intercal , . . . , (dq, fq)
\intercal \bigr] \in PC(\BbbR ,\BbbR 2)\times (\BbbR 2)q, p(t+ w) = p(t), z(t+ w) = z(t)
\Bigr\}
with the norm\bigm\| \bigm\| \bigm\| \bigl[ (p, z)\intercal , (d1, f1)\intercal , . . . , (dq, fq)\intercal \bigr] \bigm\| \bigm\| \bigm\| = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,w]
\Bigl(
\| (p, z)\intercal \| ,
\bigm\| \bigm\| (d1, f1)\intercal \bigm\| \bigm\| , . . . ,\bigm\| \bigm\| (dq, fq)\intercal \bigm\| \bigm\| \Bigr) .
Let us define the mappings L and C by L : \mathrm{D}\mathrm{o}\mathrm{m}L \subset X \rightarrow Y such that
L
\bigl(
(p, z)\intercal
\bigr)
=
\bigl(
(p\prime , z\prime )\intercal , (\Delta p(t1),\Delta z(t1))
\intercal , . . . , (\Delta p(tq),\Delta z(tq))
\intercal \bigr)
and C : X \rightarrow Y such that
C
\bigl(
(p, z)\intercal
\bigr)
=
=
\Biggl( \Biggl[
a(t) - b(t) \mathrm{e}\mathrm{x}\mathrm{p}(p(t)) - c(t)E
\bigl(
t, z(t), p(t), z(t)
\bigr)
- d(t) + f(t)E
\bigl(
t, p(t), p(t), z(t)
\bigr) \Biggr]
,
\Biggl[
\mathrm{l}\mathrm{n}(1 + g1)
\mathrm{l}\mathrm{n}(p1)
\Biggr]
, . . . ,
\Biggl[
\mathrm{l}\mathrm{n}(1 + gq)
\mathrm{l}\mathrm{n}(pq)
\Biggr] \Biggr)
.
Then \mathrm{K}\mathrm{e}\mathrm{r}L =
\bigl\{
(p, z)\intercal such that (p, z)\intercal = (c1, c2)
\intercal
\bigr\}
, c1 and c2 are constants,
\mathrm{I}\mathrm{m}L =
\left\{
\bigl[
(p, z)\intercal , (d1, f1)
\intercal , . . . , (dq, fq)
\intercal \bigr] :
\left[
w\int
0
p(s)ds+
q\sum
i=1
di
w\int
0
z(s)ds+
q\sum
i=1
fi
\right] = (0, 0)\intercal
\right\}
.
\mathrm{I}\mathrm{m}L is closed in Y and \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{K}\mathrm{e}\mathrm{r}L = \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m} \mathrm{I}\mathrm{m}L = 2. We can show this as follows. It is
obvious that summation of any element from \mathrm{I}\mathrm{m}L and \mathrm{K}\mathrm{e}\mathrm{r}L is in Y. Without loss of generality,
take p \in Y and
\int w+\kappa
\kappa
p(t)dt+
\sum q
i=1
di = I \not = 0. Let us define a new function
g = p - I
w
.
Then
I
w
is constant because, for all \kappa ,
\int w+\kappa
\kappa
p(t)dt is always same by the definition of periodic
time scales and the impulses are constant and there are same number of impulses in the interval
[\kappa ,w + \kappa ] for all \kappa . If we take the integral of g from \kappa to w + \kappa , we get
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532 N. N. PELEN
w+\kappa \int
\kappa
g(t)dt+
q\sum
i=1
di =
w+\kappa \int
\kappa
p(t)dt+
q\sum
i=1
di - I = 0.
Then p \in Y can be written as the summation of g \in \mathrm{I}\mathrm{m}L and
I
w
\in \mathrm{K}\mathrm{e}\mathrm{r}L, since
I
w
is constant.
Similar steps are used for z. (p, z)\intercal \in Y can be written as the summation of an element from \mathrm{I}\mathrm{m}L
and an element from \mathrm{K}\mathrm{e}\mathrm{r}L. Also, it is easy to show that any element in Y is uniquely expressed as
the summation of an element \mathrm{K}\mathrm{e}\mathrm{r}L and an element from \mathrm{I}\mathrm{m}L. So \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m} \mathrm{I}\mathrm{m}L is also 2, we get the
desired result. Therefore, L is a Fredholm mapping of index zero.
There exist continuous projectors U : X \rightarrow X and V : Y \rightarrow Y such that
U
\bigl(
(p, z)\intercal
\bigr)
=
1
w
\left[
w\int
0
p(s)ds
w\int
0
z(s)ds
\right]
and
V
\bigl(
(p, z)\intercal , (d1, f1)
\intercal , . . . , (dq, fq)
\intercal \bigr) = 1
w
\left(
\left[
w\int
0
p(s)ds+
q\sum
i=1
di
w\int
0
z(s)ds+
q\sum
i=1
fi
\right] ,
\biggl[
0
0
\biggr]
, . . . ,
\biggl[
0
0
\biggr]
\right) .
The generalized inverse KU = \mathrm{I}\mathrm{m}L \rightarrow \mathrm{D}\mathrm{o}\mathrm{m}L \cap \mathrm{K}\mathrm{e}\mathrm{r}U is given,
KU
\bigl(
(p, z)\intercal , (d1, f1)
\intercal , . . . , (dq, fq)
\intercal \bigr) =
=
\left[
t\int
0
p(s)ds+
\sum
t>ti
di -
1
w
w\int
0
t\int
0
p(s)dsdt -
q\sum
i=1
di +
1
w
q\sum
i=1
diti
t\int
0
z(s)ds+
\sum
t>ti
fi -
1
w
w\int
0
t\int
0
z(s)dsdt -
q\sum
i=1
fi +
1
w
q\sum
i=1
fiti
\right]
,
V C
\bigl(
(p, z)\intercal
\bigr)
=
=
1
w
\left(
\left[
w\int
0
a(s) - b(s) \mathrm{e}\mathrm{x}\mathrm{p}(p(s)) - c(s)E(s, z(s), p(s), z(s))ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
w\int
0
- d(s) + f(s)E(s, p(s), p(s), z(s))ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(ri)
\right] , . . . ,
\Biggl[
0
0
\Biggr]
\right) .
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ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 533
Let
a(t) - b(t) \mathrm{e}\mathrm{x}\mathrm{p}(p(t)) - c(t)E
\bigl(
t, z(t), p(t), z(t)
\bigr)
= C1(t),
- d(t) + f(t)E
\bigl(
t, p(t), p(t), z(t)
\bigr)
= C2(t),
1
w
w\int
0
a(s) - b(s) \mathrm{e}\mathrm{x}\mathrm{p}(p(s)) - c(s)E(s, z(s), p(s), z(s))ds = \=C1,
and
1
w
w\int
0
- d(s) + f(s)E(s, p(s), p(s), z(s))ds = \=C2,
KU (I - V )C
\bigl(
(p, z)\intercal
\bigr)
= KU
\Biggl( \Biggl[
C1(t) - \=C1
C2(t) - \=C2
\Biggr]
,
\Biggl[
\mathrm{l}\mathrm{n}(1 + g1)
\mathrm{l}\mathrm{n}(p1)
\Biggr]
, . . . ,
\Biggl[
\mathrm{l}\mathrm{n}(1 + gq)
\mathrm{l}\mathrm{n}(pq)
\Biggr] \Biggr)
=
=
\left[
t\int
0
C1(s) - \=C1ds+ \mathrm{l}\mathrm{n}
\prod
t>ti
(1 + gi) -
1
w
w\int
0
t\int
0
C1(s) -
- \=C1dsdt - \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi) +
1
w
q\sum
i=1
\mathrm{l}\mathrm{n}(1 + gi)ti
t\int
0
C2(s) - \=C2ds+ \mathrm{l}\mathrm{n}
\prod
t>ti
ri -
1
w
w\int
0
t\int
0
C2(s) -
- \=C2dsdt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri +
1
w
q\sum
i=1
\mathrm{l}\mathrm{n}(ri)ti
\right]
.
Clearly, V C and KU (I - V )C are continuous. Since X and Y are Banach spaces, then by using
Arzela – Ascoli theorem we can find \=KU (I - V )C(\=\Omega ) is compact for any open bounded set \Omega \subset X.
Additionally, V C(\=\Omega ) is bounded. Thus, C is L-compact on \=\Omega with any open bounded set \Omega \subset X.
Now, continuation theorem will be used. To be able to use this theorem, we should investigate
the following system:
x\prime (t) = \lambda
\bigl[
a(t) - b(t) \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) - c(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr) \bigr]
, t \not = tk,
y\prime (t) = \lambda
\bigl[
- d(t) + f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr) \bigr]
, t \not = tk,
\Delta x(tk) = \lambda \mathrm{l}\mathrm{n}(1 + gk),
\Delta y(tk) = \lambda \mathrm{l}\mathrm{n}(rk),
(21)
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi) =
w\int
0
b(t) \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) + c(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
dt,
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
(ri) =
w\int
0
f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
dt.
(22)
By using (21) and (22), we obtain
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534 N. N. PELEN
w\int
0
| x\prime (t)| dt \leq
\leq \lambda
\left[ w\int
0
| a(t)| dt+
w\int
0
b(t) \mathrm{e}\mathrm{x}\mathrm{p}(x(t)) + c(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
dt
\right] \leq
\leq \lambda
\left[ w\int
0
| a(t)| dt+
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
\right] \leq M1, (23)
where M1 := 2
\int w
0
a(t)dt+ \mathrm{l}\mathrm{n}
\prod q
i=1
(1 + gi), and
w\int
0
| y\prime (t)| dt \leq \lambda
\left[ w\int
0
| d(t)| dt+
w\int
0
f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
dt
\right] \leq
\leq \lambda
\left[ w\int
0
| d(t)| dt+
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri
\right] \leq M2, (24)
where M2 := 2
\int w
0
d(t)dt - \mathrm{l}\mathrm{n}
\prod q
i=1
ri.
Since (x, y)\intercal \in X and there are q impulses which are constant, then we can say that there exist
\eta i, \xi i, i = 1, 2, such that
x(\xi 1) = \mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\mathrm{i}\mathrm{n}\mathrm{f}
t\in [0,t1]
x(t), \mathrm{i}\mathrm{n}\mathrm{f}
t\in (t1,t2]
x(t), . . . , \mathrm{i}\mathrm{n}\mathrm{f}
t\in (tq ,w]
x(t)
\Bigr\}
,
x(\eta 1) = \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,t1]
x(t), \mathrm{s}\mathrm{u}\mathrm{p}
t\in (t1,t2]
x(t), . . . , \mathrm{s}\mathrm{u}\mathrm{p}
t\in (tq ,w]
x(t)
\Bigr\}
,
(25)
y(\xi 2) = \mathrm{m}\mathrm{i}\mathrm{n}
\Bigl\{
\mathrm{i}\mathrm{n}\mathrm{f}
t\in [0,t1]
y(t), \mathrm{i}\mathrm{n}\mathrm{f}
t\in (t1,t2]
y(t), . . . , \mathrm{i}\mathrm{n}\mathrm{f}
t\in (tq ,w]
y(t)
\Bigr\}
,
y(\eta 2) = \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,t1]
y(t), \mathrm{s}\mathrm{u}\mathrm{p}
t\in (t1,t2]
y(t), . . . , \mathrm{s}\mathrm{u}\mathrm{p}
t\in (tq ,w]
y(t)
\Bigr\}
.
(26)
By the first equation of (22) and (23), we get x(\xi 1) < l1, where
l1 : = \mathrm{l}\mathrm{n}
\left(
\int w
0
a(t)dt+ \mathrm{l}\mathrm{n}
\prod q
i=1
(1 + gi)\int w
0
b(t)dt
\right) .
Since x(\xi 1) is the infimum of x(t) for t \in [0, w], then there exists t1 \in [0, w] such that
x(\xi 1) \leq x(t1) < l1. By using the first inequality in Lemma 1, we have
x(t) \leq x(t1) +
w\int
0
| x\prime (t)| dt \leq x(t1) +
\left( 2
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
\right) < H1 := l1 +M1. (27)
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ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 535
From the second equation of (22), we get x(\eta 1) \geq l2, where
l2 : = \mathrm{l}\mathrm{n}
\left(
\int w
0
d(t)dt - \mathrm{l}\mathrm{n}
\prod q
i=1
ri\int w
0
\bigl(
f(t)/\alpha (t)
\bigr)
dt
\right) .
Since x(\eta 1) is the supremum of x(t) for t \in [0, w], then there exists t2 \in [0, w] such that x(\eta 1) \geq
\geq x(t2) > l2. By using second inequality in Lemma 1, we obtain
x(t) \geq x(t2) -
w\int
0
| x\prime (t)| dt \geq x(t2) -
\left( 2
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
\right) >
> H2 := l2 - M1. (28)
By (27) and (28) \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,w] | x(t)| \leq B1 := \mathrm{m}\mathrm{a}\mathrm{x}\{ | H1| , | H2| \} . By using
f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
=
= f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
\mathrm{e}\mathrm{x}\mathrm{p}(x(t) - y(t)),
we get
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri <
w\int
0
(f(t)/m(t))[\mathrm{e}\mathrm{x}\mathrm{p}(x(t) - y(t))]dt \leq
\leq
\bigl[
\mathrm{e}\mathrm{x}\mathrm{p}(x(\eta 1) - y(\xi 2))
\bigr] w\int
0
(f(t)/m(t))dt.
We have the following inequality, since (27) is true, for each t \in [0, w]:
y(\xi 2) < H1 - \mathrm{l}\mathrm{n}
\left(
\int w
0
d(t)dt - \mathrm{l}\mathrm{n}
\prod q
i=1
ri\int w
0
(f(t)/m(t))dt
\right) := l3.
Since y(\xi 2) is the infimum of y(t) for t \in [0, w], then there exists t3 \in [0, w] such that y(\xi 2) \leq
\leq y(t3) < l3. By using first equation of Lemma 1, we obtain
y(t) \leq y(t3) +
w\int
0
| y\prime (t)| dt \leq y(t3) +
\left( 2
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri
\right) <
< H3 := l3 +M2. (29)
Here, all the coefficient functions in f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
are positive and y(t)
does not go to extinction and by Lemma 9, since systems (3) and (4) are equivalent, for all solutions
of y(t) as t tends to infinity \mathrm{e}\mathrm{x}\mathrm{p}(y(t)) does not tend to 0, then we obtain
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536 N. N. PELEN
f(t)
m(t)
> f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
> 0.
Then there exists k \in \BbbN such that
f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
>
1
k
f(t)
m(t)
> 0,
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri =
w\int
0
f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr) \bigl[
\mathrm{e}\mathrm{x}\mathrm{p}(x(t) - y(t))
\bigr]
dt \geq
\geq
\bigl[
\mathrm{e}\mathrm{x}\mathrm{p}(x(\xi 1) - y(\eta 2))
\bigr] w\int
0
f(t)E
\bigl(
t, \mathrm{e}\mathrm{x}\mathrm{p}(y(t)), \mathrm{e}\mathrm{x}\mathrm{p}(x(t)), \mathrm{e}\mathrm{x}\mathrm{p}(y(t))
\bigr)
dt >
>
\bigl[
\mathrm{e}\mathrm{x}\mathrm{p}(x(\xi 1) - y(\eta 2))
\bigr] 1
k
w\int
0
f(t)
m(t)
dt.
Then, we get
y(\eta 2)) > x(\xi 1) - \mathrm{l}\mathrm{n}
\left(
\int w
0
d(t)dt - \mathrm{l}\mathrm{n}
\prod q
i=1
ri
1/k
f(t)
m(t)
\right) .
By (28), we have
y(\eta 2)) > H2 - \mathrm{l}\mathrm{n}
\left(
\int w
0
d(t)dt - \mathrm{l}\mathrm{n}
\prod q
i=1
ri
1/k
f(t)
m(t)
\right) := l4.
Since y(\eta 2) is the supremum of y(t) for t \in [0, w], then there exists t4 \in [0, w] such that y(\eta 2) \geq
\geq y(t4) > l4. If we use second inequality of Lemma 1, we get
y(t) \geq y(t4) -
w\int
0
| x\prime (t)| dt \geq
\geq y(t4) -
\left( 2
w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
ri
\right) > H4 := l4 - M2. (30)
By (29) and (30) we obtain \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,w] | y(t)| \leq B2 : = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
| H3| , | H4|
\bigr\}
. Obviously, B1 and
B2 are both independent of \lambda . Let M = B1 + B2 + 1. Then \mathrm{m}\mathrm{a}\mathrm{x}t\in [0,w]
\bigm\| \bigm\| (x, y)\intercal \bigm\| \bigm\| < M. Let
\Omega =
\bigl\{ \bigm\| \bigm\| (x, y)\intercal \bigm\| \bigm\| \in X :
\bigm\| \bigm\| (x, y)\intercal \bigm\| \bigm\| < M
\bigr\}
and \Omega verifies the requirement (a) in Theorem 1. If\bigm\| \bigm\| (x, y)\intercal \bigm\| \bigm\| \in \mathrm{K}\mathrm{e}\mathrm{r}L \cap \partial \Omega ,
\bigm\| \bigm\| (x, y)\intercal \bigm\| \bigm\| is a constant with
\bigm\| \bigm\| (x, y)\intercal \bigm\| \bigm\| = M, then
V C
\bigl(
(x, y)\intercal
\bigr)
=
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ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 537
=
\left(
\left[
w\int
0
a(s) - b(s) \mathrm{e}\mathrm{x}\mathrm{p}(x) - c(s)E(s, y(s), x(s), y(s))ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
w\int
0
- d(s) + f(s)E(s, x(s), x(s), y(s))ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(ri)
\right] , . . . ,
\biggl[
0
0
\biggr]
\right) \not =
\not = ((0, 0)\intercal , . . . , (0, 0)\intercal ) ,
JV C
\bigl(
(x, y)\intercal
\bigr)
=
\left[
w\int
0
a(s) - b(s) \mathrm{e}\mathrm{x}\mathrm{p}(x(s)) - c(s)E(s, y(s), x(s), y(s))ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
w\int
0
- d(s) + f(s)E(s, x(s), x(s), y(s))ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(ri)
\right] ,
where J : \mathrm{I}\mathrm{m}V \rightarrow \mathrm{K}\mathrm{e}\mathrm{r}L such that J
\bigl(
(x, y)\intercal , (0, 0)\intercal , . . . , (0, 0)\intercal
\bigr)
= (x, y)\intercal .
Define the homotopy H\nu = \nu (JV C) + (1 - \nu )G, where
G
\bigl(
(x, y)\intercal
\bigr)
=
\left[
w\int
0
a(s) - b(s) \mathrm{e}\mathrm{x}\mathrm{p}(x)ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
w\int
0
d(s) - f(s)E(s, x, x, y)ds+ \mathrm{l}\mathrm{n}
q\prod
i=1
(ri)
\right] .
Since H\nu is a homotopy, then for each \nu \in [0, 1] the Brouwer degree of \mathrm{d}\mathrm{e}\mathrm{g}(JV C,\Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L, 0),
\mathrm{d}\mathrm{e}\mathrm{g}(G,\Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L, 0) and \mathrm{d}\mathrm{e}\mathrm{g}(\nu (JV C) + (1 - \nu )G,\Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L, 0) are equal. Then, it is enough to
find the Brouwer degree of one of them.
Take DJG as the determinant of the Jacobian of G. Since (x, y)\intercal \in \mathrm{K}\mathrm{e}\mathrm{r}L, then Jacobian of G
is \left[
- ex
w\int
0
b(s)ds 0
w\int
0
- f(s)E(s, ex, ex, ey)ds+
+
w\int
0
(ex)2f(s)\beta (s)
(\alpha (s) + \beta (s)ex +m(s)ey)2
ds
-
w\int
0
exeyf(s)m(s)
(\alpha (s) + \beta (s)ex +m(s)ey)2
ds
\right]
.
All the functions in Jacobian of G is positive, then \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}DJG is always positive. Hence
\mathrm{d}\mathrm{e}\mathrm{g}(JV C,\Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L, 0) = \mathrm{d}\mathrm{e}\mathrm{g}(G,\Omega \cap \mathrm{K}\mathrm{e}\mathrm{r}L, 0) =
=
\sum
(x,y)\intercal \in G - 1((0,0)\intercal )
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}DJG
\bigl(
(x, y)\intercal
\bigr)
\not = 0.
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538 N. N. PELEN
Thus, all the conditions of Theorem 1 are satisfied. Therefore, system (4) has at least one positive
w-periodic solution.
If the given system (4) has at least one periodic solution, then for at least one solution of y(t),
\mathrm{e}\mathrm{x}\mathrm{p}(y(t)) does not go to zero as t goes to infinity which means y(t) does not go to extinction.
Then by using Lemma 9, since systems (3) and (4) are equivalent, we get for all solutions of y(t),
\mathrm{e}\mathrm{x}\mathrm{p}(y(t)) does not go to zero as t tends to infinity. Hence we are done.
Since systems (4) and (3) are equivalent, to if one of them has at least one w-periodic solution,
then the other one also has.
3.3. Some simple results obtained from Subsections 3.1 and 3.2.
Remark 1. Assume that (5) and
w\int
0
a(t) - c(t)
m(t)
dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi) > 0 (31)
are satisfied. Consider the system
\~v\prime (t) = (a(t) - c(t)
m(t)
)\~v(t) - b(t)\~v2(t), t \not = tk,
\~v(t+k ) = (1 + gk)\~v(tk).
(32)
By using system (32) and Lemma 2, we get
w\int
0
a(t) - c(t)
m(t)
dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi) =
w\int
0
b(t) \mathrm{e}\mathrm{x}\mathrm{p}(v(t))dt.
Here, \~v(t) = \mathrm{e}\mathrm{x}\mathrm{p}(v(t)). Therefore,
l1 :=
\int w
0
a(t)dt+ \mathrm{l}\mathrm{n}
\prod q
i=1
(1 + gi)\int w
0
b(t)dt
\leq \~v(\xi 1),
where \~v(\xi 1) is the supremum of \~v.
If we use system (32) and in this system take \~v(t) = \mathrm{e}\mathrm{x}\mathrm{p}(v(t)), then
w\int
0
| v\prime (t)| dt \leq
\left[ w\int
0
a(t)dt+
w\int
0
b(t) \mathrm{e}\mathrm{x}\mathrm{p}(u(t))
\right] \leq
\leq
\left[ w\int
0
a(t)dt+
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
\right] .
By Lemma 2, supremum of \~v(t), therefore supremum of v(t) exists. Since \~v(\xi 1) is the supremum
of \~v, by the definition of v(t), v(\xi 1) is the supremum of v(t) for t \in [0, w], then there exists
t1 \in [0, w] such that v(\xi 1) \geq v(t1) > l1. By using Lemma 2.4 in [4], we have
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ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 539
x(t) \geq v(t) \geq v(t1) -
w\int
0
| v\prime (t)| dt \geq
\geq \mathrm{l}\mathrm{n}
\left(
\int w
0
a(t)dt+ \mathrm{l}\mathrm{n}
\prod q
i=1
(1 + gi)\int w
0
b(t)dt
\right) -
\left( 2
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
\right) .
The following corollary is obtained from Lemma 5.
Corollary 1. In addition to (31) and (5), if the following inequality is also satisfied:\left(
\int w
0
a(t) - c(t)
m(t)
dt+ \mathrm{l}\mathrm{n}
\prod q
i=1
(1 + gi)\int w
0
b(t)dt
\right) \times
\times \mathrm{e}\mathrm{x}\mathrm{p}
\left[ -
\left( 2
w\int
0
a(t)dt+ \mathrm{l}\mathrm{n}
q\prod
i=1
(1 + gi)
\right) \right] \times
\times
\left( w\int
0
f(t)dt - \beta u
\left( w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
(ri)
\right) \right) -
- \alpha u
\left( w\int
0
d(t)dt - \mathrm{l}\mathrm{n}
q\prod
i=1
(ri)
\right) > 0,
then system (3) has at least one w-periodic solution.
This result is same with Theorem 2 in [16] for continuous case.
3.4. Some examples.
Example 1:
x\prime (t) = (2 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 3) - (0.2 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 0.4) \mathrm{e}\mathrm{x}\mathrm{p}(x) -
- (5 + 2 \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t)) \mathrm{e}\mathrm{x}\mathrm{p}(y)
(\mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 1.2) + (1 + 0.5 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t)) \mathrm{e}\mathrm{x}\mathrm{p}(x) + \mathrm{e}\mathrm{x}\mathrm{p}(y)
,
y\prime (t) = - (0.5 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 1.5) +
(0.8 \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t) + 4.45) \mathrm{e}\mathrm{x}\mathrm{p}(x)
(\mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 1.2) + (1 + 0.5 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t)) \mathrm{e}\mathrm{x}\mathrm{p}(x) + \mathrm{e}\mathrm{x}\mathrm{p}(y)
,
\Delta x(t+k ) = \mathrm{l}\mathrm{n}(1 + gk),
\Delta y(t+k ) = \mathrm{l}\mathrm{n}(rk).
(33)
Impulse points: t1 = 2k + 1/4, t2 = 2k + 3/4 for k = 1, 2, 3, . . . and q = 2. g1 = e1 - 1,
g2 = e1 - 1, r1 = e0.4, r2 = e0.4.
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540 N. N. PELEN
Fig. 1
Fig. 2
First consider the system
x\prime (t) = (2 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 3) - (0.2 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 0.4) \mathrm{e}\mathrm{x}\mathrm{p}(x),
\Delta x(t+k ) = \mathrm{l}\mathrm{n}(1 + gk),
g1 = e1 - 1, g2 = e1 - 1.
Then, by using the program Mathlab, x\ast > 6.5 can be found. Then, by doing some simple calcula-
tions, it is easy to find that system (33) satisfies inequality (10) and by Lemma 8 system (33) has at
least one 1-periodic solution and Fig. 1 (x(0) = 0.1, y(0) = 0.5) also supports this result.
In Example 1, system (33) if we take g1 = e0.3 - 1, g2 = e0.3 - 1, r1 = e - 1.7, r2 = e - 1.7, then
the inequality (9) is satisfied and by Lemma 5 we obtain the Fig. 2 (x(0) = 0.3, y(0) = 0.7).
This result shows us the importance of the impulses. When we take impulses as g1 = e0.3 - 1,
g2 = e0.3 - 1, r1 = e - 1.7, r2 = e - 1.7; although the system without impulses is same, since
system (33) does not satisfies the inequality (10), predator goes to extinction.
The following example is for Corollary 1.
Example 2:
x\prime = (0.2 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 0.3) - (0.2 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 0.2) \mathrm{e}\mathrm{x}\mathrm{p}(x) -
- (0.1 + 0.1 \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t)) \mathrm{e}\mathrm{x}\mathrm{p}(y)
(0.5 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 0.7) + (1 + 0.5 \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t)) \mathrm{e}\mathrm{x}\mathrm{p}(x) + \mathrm{e}\mathrm{x}\mathrm{p}(y)
, t \not = tk,
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ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 541
Fig. 3
y\prime (t) = - (0.3 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 1)+
+
(4 \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t) + 6.5) \mathrm{e}\mathrm{x}\mathrm{p}(x)
(0.5 \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) + 0.7) + (1 + 0.5 \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t)) \mathrm{e}\mathrm{x}\mathrm{p}(x) + \mathrm{e}\mathrm{x}\mathrm{p}(y)
, t \not = tk,
\Delta x(tk) = \mathrm{l}\mathrm{n}(1 + gk),
\Delta y(tk) = \mathrm{l}\mathrm{n}(rk).
Impulse points: t1 = 2k + 1/4, t2 = 2k + 3/4, and q = 2, g1 = e - 0.01 - 1, g2 = e - 0.01 - 1,
p1 = e0.1, p2 = e0.1.
Example 2 satisfies the condition of Corollary 1, therefore it has at least one w-periodic solution
and Fig. 3 (x(0) = 0.1, y(0) = 0.3) supports this result.
3.5. Global attractivity of the solutions.
Theorem 3. If inequalities (2), (5) and (10) are satisfied, then the w-periodic solution of the
system (3) is globally attractive (globally asymptotically stable).
Proof. Proof is very similar to the proof of Theorem 4.4 in [22]. To get the result, we apply
Lemma 1. Let us consider the following ordinary differential equation:
z\prime (t) = F (t, z(t)),
z(t+k ) - z(tk) = Ik(z(tk)),
z(0) = \phi .
(34)
Here, F \in C([0,\infty ) \times \BbbR 2,\BbbR 2), \phi \in \BbbR 2, F (t + w, u) = F (t, u), Ik \in C(\BbbR 2,\BbbR 2) and there exists
an integer q such that Ik+q = Ik, tk+q = tk + w. Then, the operator that solves system (34) can be
written as
\^T (t)z = ze - \lambda t +
t\int
0
e - \lambda (t - s)
\bigl[
F (s, \^T (s)z) + \lambda \^T (s)z
\bigr]
ds+
\sum
0<tk<t
e - \lambda (t - tk)Ik( \^T (tk)z),
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542 N. N. PELEN
where \lambda is a positive constant. It is obvious that T (0) = I. Also, we can verify that
u(s) =
\left\{ T (s)z, 0 \leq s \leq w,
T (s - w)T (w)z, w \leq s \leq t+ w,
is the solution of system (34) with the initial value u(0) = z, where s \in [0, t + w]. By uniqueness
theorem, system (34) has a unique solution, therefore T (t+ w)z = u(t+ w) = T (t)T (w)z. This is
true when t \not = tk. For t = tk,
T (t+k + w)z = T (tk + w)z + Ik(T (tk + w)z) =
= T (tk)T (w)z + Ik(T (tk)T (w)z) = T (t+k )T (w)z.
To apply Lemma 1, let S = T (w), S2 = SoS = T (w)oT (w) = T (2w). Here, the considered
system (34) is a periodic system, therefore we can apply Arzela – Ascoli theorem for impulsive
differential equations and the result from [1]. Hence, we obtain that T (t) is a compact operator.
If we take X+
i = \{ zi : zi \in \BbbR , zi \geq 0\} for i = 1, 2 and X+
i0
= \{ zi : zi \in \BbbR , zi > 0\} for
i = 1, 2, then X = X+
1 \times X+
2 , X = X+
10
\times X+
20
and \delta X0 = X/X0. When system (3) satisfies
inequality (2), (5), (10), the system becomes permanent. Therefore, S satisfies the conditions of
Lemma 1. Hence, S admits a global attractor which means the system has globally asymptotically
stable or globally attractive w-periodic solution.
Corollary 2. Assume that all the coefficient functions in system (3) are bounded, positive, w-
periodic, from PC(\BbbR ,\BbbR 2). Then there exists globally attractive w-periodic solution for system (3) if
and only if inequalities (2), (5), and (10) are satisfied.
Proof is immediate from Theorems 2 and 3.
Example 1 satisfies all the inequalities (2), (5) and (10), therefore it has a w-periodic, globally
asymptotically stable solution.
References
1. D. Bainov, P. Simeonov, Impulsive differential equations: periodic solutions and applications, Pitman Monogr. and
Surveys Pure and Appl. Math., vol. 66, Longman Sci. and Techn., Harlow, UK (1993).
2. J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficency, J. Animal
Ecology, 44, 331 – 340 (1975).
3. M. Bohner, A. Peterson, Dynamic equations on times scales: an introduction with applications, Birkhäuser, Basel
etc. (2001).
4. M. Bohner, Meng Fan, Jimin Zhang, Existence of periodic solutions in predator-prey and competition dynamic
systems, Nonlinear Anal.: Real World Appl., 7, 1193 – 1204 (2006).
5. N. Bourbaki, Elements of mathematics. Algebra: algebraic structures. Linear algebra, 1, Addison-Wesley (1974).
6. D. L. DeAngelis, R. A. Goldstein, R. V. O’Neill, A model for trophic interaction, Ecology, 56, 881 – 892 (1975).
7. M. Fan, S. Agarwal, Periodic solutions for a class of discrete time competition systems, Nonlinear Stud., 9, № 3,
249 – 261 (2002).
8. M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin – Ayala competition model, Comput.
Math. Appl., 40, № 10-11, 1141 – 1151 (2000).
9. M. Fan, K. Wang, Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. and Appl., 262,
№ 1, 179 – 190 (2001).
10. M. Fan, Q. Wang, Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey
systems, Discrete Contin. Dyn. Syst. Ser. B, 4, № 3, 563 – 574 (2004).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
ON THE DYNAMICS OF THE IMPULSIVE PREDATOR-PREY SYSTEMS . . . 543
11. Q. Fang, X. Li, M. Cao, Dynamics of a discrete predator-prey system with Beddington – DeAngelis function response,
Appl. Math., 3, 389 – 394 (2012).
12. R. E. Gaines, J. L. Mawhin, Coincidence degree and non-linear differential equations, Springer, Berlin (1977).
13. J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys and Monogr., vol. 25, Amer. Math. Soc.,
Providence, R.I. (1988).
14. H. F. Huo, Periodic solutions for a semi-ratio-dependent predator-prey system with functional responses, Appl. Math.
Lett., 18, 313 – 320 (2005).
15. Y. K. Li, Periodic solutions of a periodic delay predator-prey system, Proc. Amer. Math. Soc., 127, № 5, 1331 – 1335
(1999).
16. A. F. Guvenilir, B. Kaymakcalan, N. N. Pelen, Impulsive predator-prey dynamic systems with Beddington –
DeAngelis type functional response on the unification of discrete and continuous systems, Appl. Math., 8 (2015),
10.4236/am.2015.69147.
17. A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Sci. Ser. Nonlinear Sci. Ser. A:
Monographs and Treatises, vol. 14, World Sci., River Edge, NJ, USA (1995).
18. V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Vol. 6, Ser. Modern
Appl. Math., World Sci., Teaneck, NJ, USA (1989).
19. S. Tang, Y. Xiao, L. Chen, R. A. Cheke, Integrated pest management models and their dynamical behaviour, Bull.
Math. Biol., 67, № 1, 115 – 135 (2005).
20. Q. Wang, M. Fan, K. Wang, Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with
functional responses, J. Math. Anal. and Appl., 278, № 2, 443 – 471 (2003).
21. P. Wang, Boundary value problems for first order impulsive difference equations, Int. J. Difference Equat., 1, № 2,
249 – 259 (2006).
22. W. Wang, J. Shen, J. Nieto, Permanence and periodic solution of predator-prey system with holling type functional
response and impulses, Discrete Dyn. Nat. and Soc., 2007, Article ID 81756 (2007), 15 p.
23. C. Wei, L. Chen, Periodic solution of prey-predator model with Beddington – DeAngelis functional response and
impulsive state feedback control, J. Appl. Math., 2012 (2012), 17 p.
24. R. Xu, M. A. J. Chaplain, F. A. Davidson, Periodic solutions for a predator-prey model with Holling-type functional
response and time delays, Appl. Math. and Comput., 161, № 2, 637 – 654 (2005).
25. Z. Xiang, Y. Li, X. Song, Dynamic analysis of a pest management SEI model with saturation incidence concerning
impulsive control strategy, Nonlinear Anal., 10, № 4, 2335 – 2345 (2009).
26. X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with
applications, Can. Appl. Math. Quart., 3, № 4, 473 – 495 (1995).
Received 10.04.18,
after revision — 03.02.21
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
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| id | umjimathkievua-article-619 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:16Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/67/a7445f02ccddba65dfd793e7b1b0ff67.pdf |
| spelling | umjimathkievua-article-6192025-03-31T08:48:15Z On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response Pelen , N. N. Pelen , N. N. Predator-prey systems impulsive differential equations stability periodicity Predator-prey systems impulsive differential equations stability periodicity UDC 517.9 In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one $w$-periodic solution. This result is mainly based on the continuation theorem in the coincidence degree theory and to get the globally attractive $w$-periodic solution of the given system, an inequality is given as the necessary and sufficient condition by using the analytic structure of the system. &nbsp; УДК 517.9 Про динамiку iмпульсних систем типу хижак–здобич iз функцiональною вiдповiддю типу Беддiнгтона–де Ангелiса Вивчається двовимірна система типу хижак--здобич із функціональною відповіддю типу Беддінгтона–ДеАнгеліса та імпульсами у періодичному середовищі. Для цього спеціального випадку знайдено необхідні та достатні умови для того, щоб система мала принаймні один $w$-періодичний розв'язок. Цей результат базується головним чином на теоремі продовження з теорії степенів збігу, а для того, щоб знайти глобально притягуючий $w$-періодичний розв'язок розглядуваної системи, за допомогою аналітичної структури системи отримано нерівність, яка відіграє роль необхідної та достатньої умови. Institute of Mathematics, NAS of Ukraine 2021-04-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/619 10.37863/umzh.v73i4.619 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 4 (2021); 523 - 543 Український математичний журнал; Том 73 № 4 (2021); 523 - 543 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/619/9006 |
| spellingShingle | Pelen , N. N. Pelen , N. N. On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title | On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title_alt | On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title_full | On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title_fullStr | On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title_full_unstemmed | On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title_short | On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response |
| title_sort | on the dynamics of the impulsive predator-prey systems with beddington – deangelis type functional response |
| topic_facet | Predator-prey systems impulsive differential equations stability periodicity Predator-prey systems impulsive differential equations stability periodicity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/619 |
| work_keys_str_mv | AT pelennn onthedynamicsoftheimpulsivepredatorpreysystemswithbeddingtondeangelistypefunctionalresponse AT pelennn onthedynamicsoftheimpulsivepredatorpreysystemswithbeddingtondeangelistypefunctionalresponse |