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European Journal of Applied Sciences – Vol. 11, No. 6
Publication Date: December 25, 2023
DOI:10.14738/aivp.116.15793
Feng, C. (2023). Existence of Positive Periodic Oscillation for a Competitive-Cooperative Model with Feedback Controls. European
Journal of Applied Sciences, Vol - 11(6). 40-53.
Services for Science and Education – United Kingdom
Existence of Positive Periodic Oscillation for a Competitive- Cooperative Model with Feedback Controls
Chunhua Feng
Department of Mathematics and Computer Science,
Alabama State University,Montgomery, USA
ABSTRACT
In this paper, a class of nonlinear competitive-cooperative modes with feedback
controls is considered. By employing the mathematical analysis method, two
sufficient conditions for the existence of positive periodic solutions are obtained.
Computer simulation is provided to verify the criteria. From the simulation, we see
that time delays affect the oscillatory frequency.
Keywords: competitive-cooperative model, instability, positive periodic solution
AMS Mathematical Subject Classification: 34K13
INTRODUCTION
Recently, many researchers considered various competitive or competitive-cooperative
models. For example, Li et al. considered the following model [1]:
!
�!
"
(�) = �!(�)[�! − �!�!(�) − �!#�#(�) − �!�!(�)�#(�)],
�#
" = �#(�)[�# − �#�!(�) − �#!�!(�) − �#�!(� − �)�#(�)]
(1)
The authors investigated the bifurcation and the global stability of the positive equilibrium
point. Mo and Lo dealt with the following competitive system with general a toxic production
system [2]:
0
�!
"
(�) = �!(�)[�! − �!�!(�) − �!�#(�) − �!�(� − �!)],
�#
"
(�) = �#(�)[�# − �#�#(�) − �#�!(�) − �#�(� − �#)],
�"
(�) = �8�!(�), �#(�)9 − �$�(�)
(2)
where �(�1(�), �2(�)) represents the production rate of toxicant from the interaction of
two species. The stability bifurcating periodic solution and the direction of Hopf bifurcation of
system (2) were determined. Li et al. investigated a three-species food chain model [3]:
0
�!
" (�) = �!(�)[�! − �!!�!(�) − �!#�#(�)],
�#
" (�) = �#(�)[−�# + �#!�#(�) − �#$�$(� − �!)]
�$
" (�) = �$(�)[−�$ + �#(� − �#)]
(3)
The authors used the recently proposed frequency-sweeping approach to study the stability
problem of the system (3). Gui and Yan [4] discussed the stability and the Hopf bifurcation of
thesystem (3). For the following competitor-mutualist system:
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41
Feng, C. (2023). Existence of Positive Periodic Oscillation for a Competitive-Cooperative Model with Feedback Controls. European Journal of Applied
Sciences, Vol - 11(6). 40-53.
URL: http://dx.doi.org/10.14738/aivp.116.15793
0
�!
" (�) = �!(�)[�!(�) − �!!
! �!(� − �) − �!!
# �!(� − 2�) − �!#�#(� − 2�) + �!$�$(� − �)],
�#
" (�) = �#(�)[�#(�) − �#!�!(� − 2�) − �##
! �#(� − �) − �##
# �#(� − 2�) + �#$�$(� − �)],
�$
" (�) = �$(�)[�$(�) + �$!�!(
− �) + �$#�#(� − �) − �$$
! �$(�) + �$$
# �$(� − �)].
(4)
The boundedness, permanence, and global attraction of the solutions for system (4) were
obtained[5]. Xu and Chen considered the following time delay system with feedback control [6]:
⎩
⎨
⎧�!
" (�) = �!(�)[�!(�) − �!(�) − �!!(�)�!(� − �) + �!#(�)�#(� − �) − �!(�)�!(� − �!)],
�#
" (�) = �#(�)[�#(�) − �#(�) + �#!(�)�#(� − �) − �##(�)�#(� − �) − �#(�)�#(� − �#)],
�!
" (�) = �!(�) �!(�) + �!(�)�!(� − �!),
�#
" (�) = �#(�) �#(�) + �#(�)�#(� − �#).
(5)
Some sufficient conditions ensured the solutions of the system (5) to be permanent were
provided. Wang et al. extended system (5) to the following:
⎩
⎪⎪
⎨
⎪⎪
⎧ �!
" (�) = �!(�)[�!(�) − �!!
! �!(� − �) − �!!
# �!(� − 2�) − �!#�#(� − 2�) + �!$�$(� − �) − �!�!(�)],
�#
" (�) = �#(�)[�#(�) − −�#!�!(� − 2�) − �##
! �#(� − �) − �##
# �#(� − 2�) + �#$�$(� − �) + �#�#(�)]
�$
" (�) = �$(�)[�$(�) + �$!�!(� − �) + �$#�#(� − �) − �$$
! �$(�) − �$$
# �$(� − �) + �$�$(�)]
,
�!
" (�) = �!(�) − �!(�)�!(�) + �!(�)�!(�),
�#
" (�) = �#(�) − �#(�)�#(�) − �#(�)�#(�),
�$
" (�) = �$(�) − �$(�)�$(�) − �$(�)�$(�).
(6)
By introducing a Lyapunov function, the authors were able to show that the unique coexisting
point of the system (6) is globally stable. For various competitive-cooperative systems, one can
see [8-17]. Motivated by the above models, in this paper, we investigate the following three
species of competitive-cooperative model with feedback control, general toxic production, and
delayed toxic effects as the follows
⎩
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎧�!
" (�) = �!(�) J
�! − �!!�!(�) − �!#�#(�) − �!$�$(�) − �!!�!(� − �!) − �!#�#(� − �#)
+ �!$�$(� − �$) − �!%�(� − �) − �!&�!(� − �!) M,
�#
" (�) = �#(�) J
�# − �#!�!(�) − �##�#(�) − �#$�$(�) − �#!�!(� − �!) − �##�#(� − �#)
+ �#$�$(� − �$) − �#%�(� − �) + �#&�#(� − �#) M
�$
" (�) = �$(�) J
�$ − �$!�!(�) − �$#�#(�) − �$$�$(�) + �$!�!(� − �!) + �$#�#(� − �#)
− �$$�$(� − �$) − �$%�(� − �) + �$&�$(� −
$) M,
,
�"
(�) = �%!�!(�)�#(�)�$(�) − �%#�(�),
�!
" (�) = �! − �&!�!(�) − �&!�!(� − �!),
�#
" (�) = �# − �'#�#(�) − �'#�#(� − �#),
�$
" (�) = �$ − �($�$(�) − �($�$(� − �$),
(7)
where the initial conditions ��(�) = ��(�) ≥ 0, � ∈ [− max �), 0], ��(0) > 0, �(�) = �(�) ≥ 0, � ∈
[−�, 0], � (0) > 0, ��(�) = ��(�) ≥ 0, � ∈ [− max ��, 0], ��(0) > 0, � = 1, 2, 3. The parameters ��, ��,
���, ��� ��� all are positive constants. System (7) includes seven delays. If the seven delays are
different each other, the bifurcating method is hard to study the oscillatory behavior of the
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Services for Science and Education – United Kingdom 42
European Journal of Applied Sciences (EJAS) Vol. 11, Issue 6, December-2023
solutions due to the complexity of the bifurcation equation. In this paper, by meansof the
extended Chafee's criterion, the existence of periodic oscillatory solutions of the system (7) is
investigated.
PRELIMINARIES
Since ��, ��(i=1, 2, 3) are positive real numbers, so system (7) has a positive equilibrium point.
Assume that (�!
∗, �#
∗, �$
∗, �∗, �!
∗, �#
∗ , �$
∗ )+ is a positive equilibrium point of system (7), then
make the change of variables as �1(�) → �1(�) −�!
∗, �2(�) → �2(�) −�#
∗, �3(�) → �3(�) −�$
∗, �(�) →
�(�) − �∗, �1(�) → �1(�) − �!
∗, �2(�) → �2(�) − �#
∗ , �3(�) → �3(�) − �$
∗ . System (7) changes to the
following model:
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎧
�!
" (�) = �!!�!(�) + �!#�#(�) + �!$�$(�) + �!!�!(� − �!) + �!#�#(� − �#) + �!$�$(� − �$),
+�!!�(� − �) + �!!�!(� − �!) − �!!�!
#(�) − �!#�!(�)�#(�) − �!$�!(�)�$(�)
−�!!�!(�)�! (� − �!) − �!#�!(�)�# (� − �#) − �!$�!(�)�$ (� − �$)
−�!%�!(�) � (� − �) − �!&�!(�) �! (� − �!)
�#
" (�) = �#!�!(�) + �##�#(�) + �#$�$(�) + �#!�!(� − �!) + �##�#(� − �#) + �#$�$(� − �$)
+�##�(� − �) + �##�#(� − �#) − �#!�!(�)�#(�) − �##�#
#(�) − �#$�#(�)�$(�)
−�#!�#(�)�! (� − �!) − �##�#(�)�# (� − �#) − �#$�#(�)�$ (� − �$)
−�#%�#(�) � (� − �) − �#&�#(�) �# (� − �#),
�$
" (�) = �$!�!(�) + �$#�#(�) + �$$�$(�) + �$!�!(� − �!) + �$#�#(� − �#) + �$$�$(� − �$)
+�$$�(� − �) + �$$�$(� − �$) − �$!�!(�)�$(�) − �$#�#(�)�$(�) − �$$�$
#(�)
−�$!�$(�)�! (� − �!) − �$#�$(�)�# (� − �#) − �$$�$(�)�$ (� − �$)
−�$%�$(�) � (� − �) − �$&�$(�) �$ (� − �$),
�"
(�) = �!�!(�) + �#�#(�) + �$�$(�) − �%!�$
∗�!(�)�#(�) − �%!�#
∗�!(�)�$(�)
−�%!�!
∗�#(�)�$(�) − �%!�!(�)�#(�)�$(�) − �%#�(�),
�!
" (�) = − �&!�!(�) + �&!�!(� − �!),
�#
" (�) = − �'#�#(�) − �'#�#(� − �#),
�$
" (�) = − �($�$(�) − �($�$(� − �$),
(8)
where �11 = �1 + 2p11�!
∗, + p12�#
∗, + �13�$
∗, + �11�!
∗,+ �12�#
∗ − �13�$
∗ +�14�* + �15�!
∗ , �12 = �12 �!
∗, �13
= �13 �!
∗, �11 = �11�!
∗, �12 = �12�!
∗, �13 = −�13�!
∗, �11 = �14 �!
∗, �11 = �15 �!
∗, �21 = �21�#
∗ , �22 = �2 + �21
�!
∗, + 2�22�#
∗ + �23�$
∗ + �21�!
∗, + �22�#
∗ − �23�$
∗ +�24 �* +�25�#
∗ , �23 = �23�#
∗ , �21 = �21�#
∗ , �22 = �22�#
∗
, �23 = −�23�#
∗ , �22 = �24�#
∗ , �22 = �25 �#
∗, �31 = �31�$
∗ , �32 = �32�$
∗ , �33 = �3 + �31 �!
∗, + �32�#
∗ + 2�33�$
∗
− �31�!
∗, − �32�#
∗ + �33�$
∗ +�34 �* + �35�$
∗ , �31 = �31�$
∗ , �32 = −�32�$
∗ , �33 = �33�$
∗ , �33 = �34�$
∗ , �33 =
�35�$
∗ , �1 = �41�#
∗ �$
∗ , �2 = �41�!
∗, �$
∗ , �3 = �41�!
∗, �#
∗ . The linearized system of (8) is the follows: