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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: