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PureMathematicsnØêÆ,2021,11(7),1379-1388
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.117155
‘k{zMonod-Haldane.R-M
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yyyššš___§§§ææ朜œ
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Email:2677737928@qq.com
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Á‡
©ïÄ‘k{zMonod-Haldane.Rosenzweig-MacArthurÓ ö- ."Äk
|^Routh-Hurwitzâ?Ø~‡©•§.¤k²ï:•35ÚÛÜ-½5",©Û3 ˜
½^‡e§²ï:E
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'…c
Rosenzweig-MacArthur.§{zMonod-Haldane.õU‡A§Hopf©|
LocalStabilityAnalysiswithSimplified
Monod-HaldaneR-MPredator-Prey
Model
ChenxiaSong,YafeiYang
SchoolofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Email:2677737928@qq.com
Received:Jun.11
th
,2021;accepted:Jul.13
th
,2021;published:Jul.21
st
,2021
©ÙÚ^:yš_,æœ.‘k{zMonod-Haldane.R-MÓ ö- .ÛÜ-½5©Û[J].nØêÆ,
2021,11(7):1379-1388.DOI:10.12677/pm.2021.117155
yš_§æœ
Abstract
Inthispaper,Rosenzweig-MacArthurpredator-preymodelwithsimplifiedMonod-
Haldanetypeisstudied.Firstly,theexistenceandlocalstabilityofallequilibrium
pointsofordinarydifferentialmodelarediscussed.Thenanalyzethatundercertain
conditions,HopfbranchesaregeneratedatthepositiveequilibriumE
∗
.
Keywords
Rosenzweig-MacArthurModel, SimplifiedMonod-HaldaneFunctionalResponse, Hopf
Branch
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
glLotkaÚVolterra óŠ±5,Ó ö†ÔƒpŠ^˜†´êÆÚ)Æ¥9€{
K,¿…„Ы)XÚ´L)Ôõ5.cïÄÓ ö- ƒpŠ^êÆ.kéõ,
Ù¥ƒ˜´Rosenzweig-MacArthur.[1][2][3].1963c,Gause[4]JÑXe.:







dB
dt
= rB(1−B)−
αθ
θ+B
BR,
dR
dt
= −δR+
ζαθ
θ+B
BR,
Ù¥BÚR©OL« ÚÓ ö3t ž•«+—Ý.rL« SO•Ç. φ(B)=
αθB
θ+B
´Holling¼ê[5],L«Ó öÓ Ç.δ•Ó ö«+g,kÇ.ζ•=zÇ.5
Rosenzweig-MacArthur.¤ïÄ9Œ.'uRosenzweig-MacArthurÄ•¡,©z[6] [7]
ºL8 ØÅ›.©z[8]•ÄäkAllee ARosenzweig-MacArthur .,,ïá
²ï:•3^‡9ÛÜ-½5,¿y²3²ï:±Œ•34•‚.‘,©z[9][10]©Û
‘k o«Rosenzweig-MacArthur.-½5ÚHopf ©|•35.…, Javidi Ú
Nyamoradi[11]ïÄ‘k¼‘©êRosenzweig-MacArthur .ÛÜ-½5.
DOI:10.12677/pm.2021.1171551380nØêÆ
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õU‡A3ÄåÆ[12]ïÄ¥åXš~-‡Š^,Ù¥˜«´Holling-II.õU‡A[13]
§´•~^õU‡A¼ê.3©z[14]¥,Beay <é‘kc#(Holling-II.Rosenzweig-
MacArthur .?1ïÄ,©ÛXÚ¤k ²ï:-½5ÚHopf©|•35.•C, Kruff,
LaxÚLiebscher<3©z[15]¥ïá˜‡{ü‘aq[15]~‡©•§|§Ù/ª•















dB
dt
= rB(1−B)−αBH,
dS
dt
= −ηS+γBH,
dH
dt
= βS−δH+ηS−γBH,
(1.1)
Ù¥SÚH©OL«>EÚöÓ ö«+—Ý, rL« SO•Ç, α•Ó öÓ „
Ç,δ•Ó ö«+g,kÇ.ζ•=zÇ,β•·EÓ öÑ)Ç,η•·EÓ öˆ£ö
„Ç,γ•Ó öörÝ,¤këêþ•.‘(½¤kŒU‘ÛÉÄ,uyHopf
©|u)3‘XÚSܲï:?.
w,,3X Ú(1.1) ¥Ó öéÙ õU‡A´‚5.,,3)ÔÆ¿Âe,‚5õ U
‡AéÓ þvk•›,Ïd·‚òXÚ(1.1)¥‚5õU‡AO†•{zMonod-Haldane .,
l/¤±e.















dB
dt
= rB(1−
B
K
)−
αBH
a+B
2
,
dS
dt
= −ηS+
eαBH
a+B
2
,
dH
dt
= βS−δH+ηS−
eαBH
a+B
2
,
(1.2)
Ù¥K•‚¸NBþ, e•=zÇ,
αB
a+B
2
•{zMonod-Haldane.õU‡A.
ÏLÃþjC†
x=
B
K
,y= (β+η)S,z=
αH
r
,t=
τ
r
,
¿EPτ•t,KXÚ(1.2)Œz•















dx
dt
= x(1−x)−
xz
a+x
2
,
dy
dt
= −α
1
y+
α
2
xz
a+x
2
,
dz
dt
= β
1
y−β
2
z−
β
3
xz
a+x
2
.
(1.3)
Ù¥
α
1
=
η
r
,α
2
= eK(β+η),β
1
=
α
r
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3
= e,
XÚ(1.3)¤këêþ•~ê,K•‚¸«1Uå.
©(|„Xe:312!¥,·‚‰ÑXÚ(1.3) ¤k²ï:,¿‰ÑÛÜ-½5
DOI:10.12677/pm.2021.1171551381nØêÆ
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^‡.313!¥,y²3²ï:E
∗
?)Hopf©|.314!¥Ñ(Ø.
2.²ï:•35Ú-½5
2.1.²ï:•35
éu.(1.3),
(i)o•3²…²ï:E
0
= (0,0,0)ÚŒ²…²ï:E
1
= (1,0,0);
(ii)
(H
1
)
α
2
α
1
>
β
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2
(1+a)
β
1
¤á, …4= (α
1
β
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1
α
2
)
2
−4α
2
1
β
2
2
a= 0ž, XÚ(1.3) •3•˜~ê²ï:E
∗
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∗
,y
∗
,z
∗
),
Ù¥
x
∗
=
α
2
β
1
−α
1
β
3
2α
1
β
2
,y
∗
=
α
2
α
1
x
∗
(1−x
∗
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∗
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∗
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∗2
).
•{zE
∗
?JacobiÝ9ƒA A•§L«†©Û,Pm=α
2
β
1
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1
β
3
>0 (d(H
1
)).
u´,x
∗
=
m
2α
1
β
2
.
2.2.²ï:ÛÜ-½5
XÚ(1.3)3(x,y.z)?JacobiÝXe
J=







1−2x+
(x
2
−a)(1−x)
a+x
2
0−
x
a+x
2
−α
2
(x
2
−a)(1−x)
a+x
2
−α
1
α
2
x
a+x
2
β
3
(x
2
−a)(1−x)
a+x
2
β
1
−β
2
−
β
3
x
a+x
2







,(1)
e¡ÏLOŽXÚ(1.3)3z‡²ï:?JacobiÝAŠ,5(½ù²ï:-½5.
½n1(i)²…²ï:E
0
= (0,0,0)´Ã^‡Ø-½.
(ii)e^‡(H
1
)¤á,KŒ²…²ï:E
1
= (1,0,0)Ø-½;ÄKE
1
´ÛÜìC-½.
y²(i)XÚ(1.3)3²ï:E
0
?JacobiÝ•
J
E
0
=






100
0−α
1
0
0β
1
−β
2






.(2)
DOI:10.12677/pm.2021.1171551382nØêÆ
yš_§æœ
Ý(2)AŠ•λ
1
= 1,λ
2
= −α
1
Úλ
3
= −β
2
.Ïd,²ï:E
0
´Ø-½.
(ii)XÚ(1.3)3²ï:E
1
?JacobiÝ•
J
E
1
=








−10−
1
1+a
0−α
1
α
2
1+a
0β
1
−β
2
−
β
3
1+a








.(3)
Ý(3)A•§•
(λ+1)

λ
2
+

α
1
+β
2
+
β
3
1+a

λ+α
1
β
2
+
α
1
β
3
−α
2
β
1
1+a

= 0.
¤±,Ý(3)AŠ•λ
1
= −1,λ
2
Úλ
3
,¿…
λ
2
+λ
3
= −

α
1
+β
2
+
β
3
1+a

<0,
λ
2
λ
3
= α
1
β
2
+
α
1
β
3
−α
2
β
1
1+a
.
(H
1
)¤áž,k
α
1
β
2
(1+a)+α
1
β
3
−α
2
β
1
<0,
λ
2
λ
3
<0,l²ï:E
1
´Ø-½;‡ƒ,E
1
´ÛÜìC-½.
½n2b2x−
d+2β
2
c
2β
2
c
>0,…
α
1
α
2
>
β
1
β
3
¤áž.e÷v
(H
2
)(s+t)
h
st+α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c
+
mβ
3
d
c
2
i
>
h
tα
1
β
2
+
m(α
1
β
3
−α
2
β
1
)(2tcβ
2
+d)
c
2
i
,
Ù¥
s= α
1
+β
2
+
2mβ
2
β
3
c
>0,t= 2x−
d+2β
2
c
2β
2
c
>0.
KXÚ(1.3)~ê²ï:E
∗
´ÛÜìC-½.‡L5,e(H
2
)ؤá,KE
∗
´Ø-½.
y²XÚ(1.3)3~ê²ï:E
∗
?JacobiÝ•
J
E
∗
=








−2x
∗
+
d+2β
2
c
2β
2
c
0−
2mβ
2
c
−α
2
d
2β
2
c
−α
1
2mα
2
β
2
c
β
3
d
2β
2
c
β
1
−β
2
−
2mβ
2
β
3
c








.(4)
DOI:10.12677/pm.2021.1171551383nØêÆ
yš_§æœ
Ù¥c=
m
2
α
1
+4aα
1
β
2
2
>0,d=
2m
2
β
2
α
1
−8aα
1
β
3
2
−
m
3
α
2
1
+4amβ
2
2
Ý(4)A•§•
λ
3
+N
1
λ
2
+N
2
λ+N
3
= 0.(5)
Ù¥
N
1
= s+t,
N
2
= st+α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c
+
mdβ
3
c
2
,
N
3
= t

α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c

+
md(α
1
β
3
−α
2
β
1
)
c
2
,
w,,N
1
>0.d^‡(H
1
)Υ,N
3
>0.dž,
N
1
N
2
−N
3
=(s+t)

st+α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c
+
mβ
3
d
c
2

−t

α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c

−
md(α
1
β
3
−α
2
β
1
)
c
2
=(s+t)

st+α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c
+
mβ
3
d
c
2

−

tα
1
β
2
+
m(α
1
β
3
−α
2
β
1
)(2tcβ
2
+d)
c
2

.
(6)
Ïd,^‡(H
2
)¤á ž, N
1
>0,N
3
>0,N
1
N
2
−N
3
>0,dRouth-Hurwitz⌕,A
•§(5)ŠþäkK¢Ü,?XÚ(1.3)~ê²ï:E
∗
´ÛÜìC-½.‡ƒ,A
•§(5)ŠØkK¢Ü…¢ÜØ•",Ïd~ê²ï:E
∗
´Ø-½.
3.Hopf©|•35
!Àëêβ
1
5ïÄXÚ(1.3)3~ê²ï:E
∗
?Hopf©|•35.
Ún3b^‡(H
1
)¤á,K•3β
∗
1
>0,¦β
1
= β
∗
1
ž,N
1
(β
∗
1
)N
2
(β
∗
1
)−N
3
(β
∗
1
) = 0.
y²-<(s) = (N
1
N
2
−N
3
)(s) = 0.
<(s) = b
1
(β
1
)s
2
+b
2
(β
1
)s+b
3
(β
1
) = 0,(7)
DOI:10.12677/pm.2021.1171551384nØêÆ
yš_§æœ
Ù¥
b
1
(β
1
) = t>0,
b
2
(β
1
) = t
2
+α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c
+
mβ
3
d
c
2
,
b
3
(β
1
) =
md[β
3
(t−α
1
)+β
1
α
2
]
c
2
.

β
1
β
3
<
α
1
−t
α
2
ž,kb
3
(β
1
)<0.Ïd,d‰ˆ½nÚsLˆªŒ•,•3β
∗
1
>0,¦(7)¤á,
…÷v
s=
−b
2
(β
∗
1
)+
p
b
2
(β
∗
1
)
2
−4b
1
(β
∗
1
)b
3
(β
∗
1
)
2b
1
(β
∗
1
)
.
K,db
1
,b
3
ÉÒŒ•,β
∗
1
´•˜.
½n4b^‡(H
1
) ¤á,kβ
∗
1
>0, β
1
= β
∗
1
ž,XÚ(1.3) 3•²ï:E
∗
?)Hopf
©|.
y²dÚn3•,•3•˜β
∗
1
>0,¦N
1
(β
∗
1
)N
2
(β
∗
1
)−N
3
(β
∗
1
) = 0.Ïd,β
1
= β
∗
1
ž,
A•§(7)Œ-•
(λ+N
1
)(λ
2
+N
2
) = 0.(8)
d½n2ÚÚn3Œ•,N
1
>0,N
2
(β
∗
1
)>0.l•§(8)AŠ©O•λ
1
=−N
1
,λ
2
=
±i
p
N
2
(β
∗
1
).éÙ§β
1
∈
˚
U(β
∗
1
),N
1
(β
∗
1
)N
2
(β
∗
1
)−N
3
(β
∗
1
) 6=0,Ïd,bλ= κ(β
1
)+iθ(β
1
)
•A•§(7)˜‡EŠ,“\•§(7),
(κ+iθ)
3
+N
1
(κ+iθ)
2
+N
2
(κ+iθ)+N
3
= 0,
ü>Óž'uβ
1
¦,k
D
1
(β
1
)κ
0
(β
1
)−D
2
(β
1
)θ
0
(β
1
)+D
3
(β
1
) = 0,(9)
D
2
(β
1
)κ
0
(β
1
)+D
1
(β
1
)θ
0
(β
1
)+D
4
(β
1
) = 0.(10)
Ù¥
D
1
(β
1
)=3(κ
2
−θ
2
)+2N
1
κ+N
2
,
D
2
(β
1
)=6κθ+2N
1
θ,
D
3
(β
1
)=N
0
1
(κ
2
−θ
2
)+N
0
2
κ+N
0
3
,
D
4
(β
1
)=2N
0
1
κθ+N
0
2
θ,
N
0
1
(β
1
)=
2β
2
β
3
(m
0
c−mc
0
)
c
2
+2x
0
−
d
0
c−dc
0
2β
2
c
2
,
DOI:10.12677/pm.2021.1171551385nØêÆ
yš_§æœ
N
0
2
(β
1
)=
2β
2
(m
0
c−mc
0
)
h
α
1
β
3
−α
2
β
1
+β
3
(2x−
d
2β
2
c
+1)
i
c
2
+

β
2
+
2mβ
2
β
3
c
+α
1

·

2x
0
−
d
0
c−dc
0
2β
2
c
2

+
(m
0
d+md
0
)β
3
c−2mdβ
3
c
0
−2mc
2
α
2
β
2
c
3
,
N
0
3
(β
1
)=

2x
0
−
d
0
c−dc
0
2β
2
c
2

α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c

+(2x−
d
2β
2
c
−1)·

2β
2
(m
0
c−mc
0
)(α
1
β
3
−α
2
β
1
)−2mα
2
β
2
c
c
2

+
(α
1
β
3
−β
1
α
2
)c[m
0
d+md
0
−2c
0
md]−mdβ
1
α
2
c
c
4
,
u
∗
1
0
(β
1
)=α
2
>0,
m
0
(β
1
)=α
2
,
c
0
(β
1
)=
2mm
0
α
1
,
d
0
(β
1
)=
4mm
0
β
2
α
1
−
3mm
0
α
2
1
+4aβ
2
2
m
0
,
(H3)m
0
c−mc
0
>0,2x
0
−
d
0
c−dc
0
2β
2
c
2
>0¤áž, kN
0
1
(β
1
) >0.Ïd,D
1
(β
1
)D
3
(β
1
)+D
2
(β
1
)D
4
(β
1
) 6=
0.d(9),(10)Œ
d(Reλ)
dβ
1




β
∗
1
= κ
0
(β
∗
1
) = −
D
1
(β
∗
1
)D
3
(β
∗
1
)+D
2
(β
∗
1
)D
4
(β
∗
1
)
N
2
1
(β
∗
1
)+N
2
2
(β
∗
1
)
=

(N
3
−N
1
N
2
)
0
2(N
2
1
+N
2
)

β
1
=β
∗
1
.
?˜Ú,d½n2Ú(H3),´
s
0
(β
1
) =
2β
2
β
3
(m
0
c−mc
0
)
c
2
>0,t
0
(β
1
) = 2x
0
−
d
0
c−dc
0
2β
2
c
2
>0.
d(6),
d(N
1
N
2
−N
3
)
dβ
1
=(s
0
+t
0
)

st+α
1
β
2
+
2mβ
2
(α
1
β
3
−α
2
β
1
)
c
+
mβ
3
d
c
2

+(s+t)
·

s
0
t+st
0
+
2β
2
(m
0
c−mc
0
)(α
1
β
3
−α
2
β
1
)−2mα
2
β
2
c
c
2
+
(m
0
d+md
0
)β
3
c−2mdβ
3
c
0
c
3

−t
0
α
1
β
2
−
(α
1
β
3
−α
2
β
1
)[m
0
(2tcβ
2
+d)+m(2t
0
cβ
2
+d
0
+2tc
0
β
2
)]−α
2
m(2tcβ
2
+d)
c
4
.
(ܽn2,Ún3Ús
0
,t
0
Lˆª,Œ
d(N
1
N
2
−N
3
)
dβ
1




β
1
=β
∗
1
>0.κ
0
(β
∗
1
) <0.î5^‡
¤á,XÚ(1.3)3²ï:E
∗
?)Hopf©|.
3)ÔÆ¿Âþ,Hopf ©|»€XÚžm-½5,3žmþ—˜mþ!Ú±Ï
.
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1
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∗
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∗
´ÛÜì?-½…=^‡(H
2
)¤
á.,rβ
∗
1
wŠ©|ëê,Šâ½n4 XÚ(1.3) 3˜½^‡eŒ±)Hopf ©|,ÑyØ
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ë•©z
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