设为首页 加入收藏 期刊导航 网站地图
  • 首页
  • 期刊
    • 数学与物理
    • 地球与环境
    • 信息通讯
    • 经济与管理
    • 生命科学
    • 工程技术
    • 医药卫生
    • 人文社科
    • 化学与材料
  • 会议
  • 合作
  • 新闻
  • 我们
  • 招聘
  • 千人智库
  • 我要投稿
  • 办刊

期刊菜单

  • ●领域
  • ●编委
  • ●投稿须知
  • ●最新文章
  • ●检索
  • ●投稿

文章导航

  • ●Abstract
  • ●Full-Text PDF
  • ●Full-Text HTML
  • ●Full-Text ePUB
  • ●Linked References
  • ●How to Cite this Article
PureMathematicsnØêÆ,2023,13(4),902-916
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.134096
äk'~Ô;J¤HollingI.
Ó - .ÛÄåÆ
ÛÛÛûûû
•ânóŒÆêƆÚOÆ§H•â
ÂvFϵ2023c318F¶¹^Fϵ2023c419F¶uÙFϵ2023c426F
Á‡
3©¥§ÏLÚ\KŠüѧïĘaäk'~Ô;J¤HollingI.Ó - .¿
éÙ?1Û©Û§±(½.ÛÄåÆ"y|^FilippovnاLyapunov¼ê{Ú‚
úª•{§3ü‡fXÚÛÄåÆÄ:þ§éKŠüÑeÓ - .§·‚ïÄ
ÙwÄåÆÚÛÄåÆ"•ÏLêŠ[énØ(J?1y"
'…c
š1wXÚ§Ó - .§ØëY§²ï:§-½5
GlobalDynamicsofHollingTypeI
Predator-PreyModelwithEqual
ProportionofPreyRefuge
DanLuo
SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha
Hunan
Received:Mar.18
th
,2023;accepted:Apr.19
th
,2023;published:Apr.26
th
,2023
©ÙÚ^:Ûû.äk'~Ô;J¤HollingI.Ó - .ÛÄåÆ[J].nØêÆ,2023,13(4):
902-916.DOI:10.12677/pm.2023.134096
Ûû
Abstract
Inthispaper,TheobjectiveofthispaperistoinvestigateaGlobaldynamicsof
HollingtypeI predator-preymodelwithequalproportion ofpreyrefugeby introduc-
ingthresholdstrategy.Herewe provide aglobalqualitative analysistodeterminethe
global dynamics of the model.Makinguse of Filippov theory, Lyapunov functions and
Greenformula,onthebasisofglobaldynamicsoftwosubsystems,forthepredator-
preymodelunderthresholdstrategy,weexaminetheslidingmodedynamicsandthe
global dynamics.Finally,the theoreticalresultsare verified bynumericalsimulation.
Keywords
Non-SmoothSystem,Predator-PreyModel,Discontinuous,Equilibrium,Stability
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
)Ó öÚÓ öƒmƒp'Xég,¥•)ÄÔ+nÚ)Ôõ5o–'-‡.
ïÄÓ ö- .̇8´£ãÚ©ÛÓ öÚÔƒmƒpŠ^,l‡Ný¢g
,y–.3y¢)¹¥,Ô2o¤éÓ öÚÔ«+•k-‡K•.Šâ•#ïÄ,Ô
¦^;J¤Œ±ÏLO\‚¸E,5!Jp•Ô—ÝÚò•Ó ö-ÔƒpŠ^[1–3]5J
ø;J,Ô•Œ±ÏL¯„#a½~Թı9Û-[4–6]•ªïá˜m½žm;
J¤±ˆø;Ó ö8,±d ;•Ô«ý.Ïd,•ÄÔ;J¤´š~k^.±e 
.´Ó ö- .^5ïÄð½'~½ð½êþÔ¦^2o¤K•[1–5,7,8].
31926c,Maynard[9]uL˜'uÔ-Ó ö«+nØ(J.Øȃ,pd3[10]¥
ÏL¢y¦‚ýÿ.k˜‡¢Ó ö-ÔXÚ´dúÁÁ(Ó ö)Ú1[œ(
Ô)|¤,¦‚ÏL*úÁÁØU±$— Ý1[œ• Ñ±e(Ø:1[œ3$—Ý
ž,§‚3®.Ü/¤˜‡ œÈÔ,Ïd§‚Ã{ÑE3YÎþúÁÁ>.Ïd,
Ô•3˜‡.—Ý,3ù‡—Ýe,Ô¢Sþò3˜‡;J¤±“ŽÓ ,˜Ô—Ý
‡L.Š,ÔÒ¬-#Ñy3®L¡,,Œ±Ó öÓ .
DOI:10.12677/pm.2023.134096903nØêÆ
Ûû
©1Ü©ò©z[10]¥ëYÓ ö- .?1í2,í2©¡1wÓ ö-
.,¿Ú\'ÇÔ;J¤,±ïÄ'ÇÔ;J¤éÓ ö- .K•±9©
ÛT.ü‡fXÚÄåÆ9wÄåÆ,1nÜ©´ÛÄåÆ©Û.3•˜!,é¤
(Ø?1o(.†ëY‡©ÄåXÚƒ'T.äkwÄåÆ!–²ï:!B 4•
‚!w4•‚ÕA5Ÿ,ÙïÄ•äk]Ô5.
2..09ý•£
3Gauseetal.[10]ÚMaynardSmith[9]Ó - .¥Ú\•¹;J¤Ô«+ê
þR
x
,·‚?UõU‡A¼ê(Holling1959),¦ÙC¤:



dx
dt
= rx

1−
x
k

−c(x−R
x
)y
dy
dt
= ec(x−R
x
)y−dy
(2.1)
Ù¥,x= x(t),y= y(t)©OL«ÔÚÓ ö3tž•êþ,rL«Ô«+SO•Ç,cL
«Ó ö«+Ó¼Ç,eL«Ó ö«+=†Ç,dL«Ó ö«+k Ç,kL«Ô«+
•Œ‚¸Nþ.
Ù¥R
x
êþŒ±lü‡Ý•Ä[4,8]:
(i)R
x
= mx,ÛõÔêþ†Ô—ݤ';
(ii)R
x
= mx
c
,ÛõÔêþ•~ê,x
c
•Ô.—Ý.
y3•Ä1˜«Ý,q Ï•Ó ö† ƒmƒpŠ^† êþ•k—ƒ'X,
êþ'žÿ, ¬ø;å5,ùÒ¦Ó öÓ JÝ\Œ.·‚Œ±±eù
‡äk1˜aõU‡A¼êØëY -Ó .:



dx
dt
= rx

1−
x
k

−c(1−m)xy
dy
dt
= ec(1−m)xy−dy
(2.2)
Ù¥¤këêþ´ê.
éXÚ(2.2)?1ÃþjzÚžmºÝCz,-ex= kx,ey= ky,
e
t=
1
r
t.



dex
d
e
t
=ex(1−ex)−ec(1−m)exey
dey
d
e
t
= eec(1−m)exey−
e
dey
-
ck
r
= ec,
d
r
=
e
d,=



dx
dt
= x(1−x)−c(1−εm)xy
dy
dt
= ec(1−εm)xy−dy
(2.3)
Ù¥ε•››¼ê,σ>0•››KŠ,L«Ôu››KŠσž,Ô¬Ûõå5,ÄKØÛ
DOI:10.12677/pm.2023.134096904nØêÆ
Ûû
õ.
ε=



1,x<σ,
0,x>σ,
PR
2
+
= {(x,y) ∈R
2
: x>0,y>0},ò(x,y) ∈
R
2
+
©•±en‡Ü©:
Σ =

(x,y)


(x,y) ∈
R
2
+
,x= σ}
G
1
=

(x,y)


(x,y) ∈
R
2
+
,x<σ}
G
2
=

(x,y)


(x,y) ∈
R
2
+
,x>σ}
P
F
1
(x,y) = (x(1−x)−c(1−m)xy,ec(1−m)xy−dy)
F
2
(x,y) = (x(1−x)−cxy,ecxy−dy)
u´,XÚ(2.3)Œ±¤e¡FilippovXÚ:
(
dx
dt
,
dy
dt
) =



F
1
(x,y),(x,y) ∈G
1
F
2
(x,y),(x,y) ∈G
2
(2.4)
du•þ|ØëY5,©ò?Ø(2.4)Filippov¿Âe)[11],Ù½ÂXe:
½Â2.1.e•þ¼ê(x(t),y(t))3(0,T)?¿k.4f«m[t
1
,t
2
]þýéëY(0<T≤+∞),÷
vx(0) = x
0
Úy(0) = y
0
,…•3Œÿ¼êγ= γ(t):[0,T) →[0,1]¦éA¤kt∈[0,T)k



dx
dt
= x(1−x)−c(1−γm)xy
dy
dt
= ec(1−γm)xy−dy
(2.5)
@o¡•þ¼ê(x(t),y(t))´XÚ(2.3)LЩŠ(x
0
,y
0
)).
••Ä.(2.5))ÔÆ¿Â,I‡y.)5Úk.5,5†k.5y²d e
¡ü‡·K‰Ñ:
·K2.2.-(x(t),y(t))•XÚ(2.5)þ÷vЩ^‡x(0)=x
0
>0Úy(0)=y
0
>0),½Â«m
•[0,T),Ù¥T∈(0,+∞],Ké¤kt∈[0,T),kx(t) >0Úy(t) >0.
y².b•3t
1
,¦x(t
1
) <0,K•30 <t
∗
<t
1
,¦x(t
∗
) = 0,…t∈(0,t
∗
)ž,kx(t) >0¤
á,dXÚ(2.5)1˜‡f•§Œ.
dx
dt
= x(1−x)−c(1−γm)xy
≥x[−x−c(1−γm)y]
Kt∈(0,t
∗
)k
x(t) ≥x
0
exp(
Z
t
0
[−x−c(1−γm)y]dt
DOI:10.12677/pm.2023.134096905nØêÆ
Ûû
AO/,t= t
∗
žk
x(t
∗
) ≥x
0
exp(
Z
t
∗
0
[−x−c(1−γm)y]dt>0
†x(t
∗
) = 0gñ,Ø•3t
1
,¦x(t
1
) <0,ÓnŒ,é¤kt∈[0,T),ky(t) >0.
·K2.3.XÚ(2.5)lR
2
+
Ñu)´k..
y².PW= x+
1
e
y,÷XÚ(2.5)éW¦,k
dW
dt
=
dx
dt
+
1
e
dy
dt
= x(1−x)−
d
e
y
≤x(1+
d
2
−x)−
d
2
W
≤

1+
d
2

2
4
−
d
2
W
-ρ=
(
1+
d
2
)
2
4
,φ=
d
2
,Kk
W(t) <W(t
0
)exp(−φt)+
ρ
φ
(1−exp(−φt))
≤max

W(t
0
),
ρ
φ

Ú
lim
t→∞
supW(t) ≤
ρ
φ
Ïd,XÚ(2.5)lR
2
+
Ñu)•ª¬33«•Ω =
n
(x,y) ∈R
2



x+
1
e
y≤
ρ
φ
+θ
o
¥,y..
3«•G
1
þ.•



dx
dt
= x(1−x)−c(1−m)xy
dy
dt
= ec(1−m)xy−dy
(2.6)
±en‡²ï:
E
+
0
= (0,0),E
+
1
= (1,0),E
+
2
=

x
+
2
,y
+
2

=

d
ec(1−m)
,
ec(1−m)−d
ec
2
(1−m)
2

3«•G
2
þ.•



x(1−x)−cxy
ecxy−dy
(2.7)
kn‡²ï:,©O•
E
−
0
= (0,0),E
−
1
= (1,0),E
−
2
=

x
−
2
,y
−
2

=

d
ec
,
ec−d
ec
2

éuXÚ (2.5),²ï:E
+
0
E
−
0
•¢(J)²ï:;é>.²ï:E
+
1
E
−
1
,e1<σKE
+
1
E
−
1
•
¢(J)²ï:,e1>σKE
+
1
E
−
1
•J(¢)²ï:;é/•5²ï:E
+
2
E
−
2
,ex
+
2
(x
−
2
)<σKE
+
2
DOI:10.12677/pm.2023.134096906nØêÆ
Ûû
E
−
2
•¢(J)²ï:,ex
+
2
(x
−
2
) >σKE
+
2
E
−
2
•J(¢)²ï:.
½n2.4.3fXÚ(2.6)¥²ï:E
+
0
= (0,0),²ï:E
+
0
´Q:.
y².3fXÚ(2.6)3²ï:E
+
0
= (0,0)?JacobiÝ•
J
G
1

E
+
0

=





10
0−d





Ù¥,Šâp−qO{,q= −d<0,²ï:E
+
0
´Q:.
½n2.5.eec(1 −m)<d,3G
1
S,>.²ï:E
+
1
´ÛìC-½,XJec(1−m)>d,
3G
1
S,/•5²ï:E
+
2
´ÛìC-½.
y².eec(1−m) <d,•ÄLyapunov¼ê
V
1
(x,y) = e(x−1−lnx)+y
du
dV
1
(x,y)
dt
= e

1−
1
x

dx
dt
+
dy
dt
= −e(x−1)
2
+(ec−d)y
≤0
ŠâLaSalleØCn,Œ>.²ï:E
+
1
´ÛìC-½.
eec(1−m) >d,•ÄLyapunov¼ê
V
2
(x,y) = e[(x−x
+
2
)−x
+
2
ln
x
x
+
2
]+(y−y
+
2
)−y
+
2
ln
y
y
+
2
du
dV
2
(x,y)
dt
= e(
x−x
+
2
x
)
dx
dt
+(
y−y
+
2
y
)
dy
dt
= −e

x−x
+
2

≤0
ŠâLaSalleØCn,Œ/•5²ï:E
+
2
´ÛìC-½.
aq/,fXÚ(2.7)ÛÄåÆŒ±ÏLe¡·K¼.
½n2.6.3fXÚ(2.7)¥²ï:E
−
0
= (0,0),²ï:E
−
0
´Q:.
½n2.7.eec<d,3G
2
S,>.²ï:E
−
1
´ÛìC-½,XJec>d,3G
2
S,/•5²ï
:E
−
2
´ÛìC-½.
y².eec<d,•ÄLyapunov¼ê
V
1
(x,y) = e(x−1−lnx)+y
du
dV
1
(x,y)
dt
= −e(x−1)
2
+(ec−d)y≤0
DOI:10.12677/pm.2023.134096907nØêÆ
Ûû
ŠâLaSalleØCn,Œ>.²ï:E
−
1
´ÛìC-½.
eec>d,•ÄLyapunov¼ê
V
2
(x,y) = e[(x−x
−
2
)−x
−
2
ln
x
x
−
2
]+(y−y
−
2
)−y
−
2
ln
y
y
−
2
du
dV
2
(x,y)
dt
= −e

x−x
−
2

≤0
ŠâLaSalleØCn,Œ/•5²ï:E
−
2
´ÛìC-½.
4·‚|^[11]¥˜Vg5©ÛFilippovXÚ(2.5)wÄåÆ1•,•)w•Ú–²
ï:•35.
b∇H••G
1
·L
F
i
H=h∇H,F
i
iL«•þF
i
3H••ê,h,i•IOSÈ,L
m
F
i
H=

∇

L
m−1
F
i
H

,F
i

L«mLieê,Ù¥m≥2.ÏL{üOŽ,·‚k
L
F
1
H= h∇H,F
1
i= x(1−x)−c(1−m)xy
L
F
2
H= h∇H,F
2
i= x(1−x−cy).
dL
F
1
H>0…L
F
2
H<0Υ,
1−σ
c
<y<
1−σ
c(1−m)
Py
1
=
1−σ
c
,y
2
=
1−σ
c(1−m)
.••Bå„,-T
1
= (σ,y
1
),T
2
= (σ,y
2
),KT
1
ÚT
2
Ñ´ƒ:.Ïdƒ†
‚Hþw••
Σ
S
=

(x,y) ∈R
2
+
|y
1
<y<y
2

B«••
Σ
C
1
=

(x,y) ∈R
2
+
|0 <y<y
1

Ú
Σ
C
2
=

(x,y) ∈R
2
+
|y
2
<y

¦^XeFilippovà•{[11],
dZ
dt
= F
S
(Z) = (1−λ)F
G
1
(Z)+λF
G
2
(Z),
XÚ(2.5)wÄåÆ•§Œ£ã•
dZ
dt
= F
S
(Z) =
0
ex(1−x)−dy
!
Ù¥x= σ.K•§••3•˜Šy
p
=
eσ(1−σ)
d
.
ÏdéuXÚ(2.5)ŒU•3•˜–²ï:•E
p
=(σ,y
p
),Šâ[11],E
p
•3…=y
1
<
y
p
<y
2
,qϕ
∂G
∂y


y
p
= −d<0,ÏdeE
P
•3,7½´-½.
DOI:10.12677/pm.2023.134096908nØêÆ
Ûû
3.̇(J9Ùy²
3ù˜!,·‚̇?ØXÚ(2.5)¢²ï:Ú–²ï:Û-½5.¯¢þ,•y²²
ï:Û-½5,·‚I‡üØ4;•3.
½n3.1.3XÚ(2.5)¥,Ø•3 uGi(i= 1,2)«•S4Ü;.
y².|^Bendixon-DulacOK,B(x,y) =
1
xy
,
∂B(x,y)f
11
∂x
+
∂B(x,y)f
12
∂y
= −
1
y
≤0
∂B(x,y)f
21
∂x
+
∂B(x,y)f
22
∂y
= −
1
y
≤0
ddŒ•Ø•3 uGi(i= 1,2)«•S4;‚.
½n3.2.3XÚ(2.5)¥,Ø•3•¹Ü©wÄãAB4•‚.
y².æ^‡y{5y².Ø”˜„5,Ø”b•E
+
2
¢,E
−
2
•Jž,XÚ(2.5)•3•¹Σ
S
4
;Γ.KΓ˜½lƒ:T
1
Ñu¿…ˆΣ
S
,dž4;¡);‚ØU?\4;SÜ(Xã1¤«),
ù†E
+
2
3«•G
1
ÛìC-½5´gñ.ÏdlT
1
Ñu;‚جˆΣ
S
.
Figure1.Thereisnoslidingringaroundthe
slidingsegmentAB
ã1.wÄãAB±Œvkw‚
½n3.3.3XÚ(2.5)¥,Ø•3Œ7Σ
S
4Ü;,ùpΣ
S
´Σ
S
4•.
y².bΣ
S
±Œ•34;‚L=L
1
+L
2
,Ù¥L
1
= L
T
G
1
,L
2
= L
T
G
2
.^KL«dLŒ¤
k.«•,…K
1
∆
= K∩G
1
,K
2
∆
= K∩G
2
^
f
K
i
(i= 1,2)L«L
i
ÚP
i
¤Œ¤k.«•(Xã2),÷
v
f
K
i
→K
i
,(ε→0),Ù¥P
1
ÚP
2
©OL«†‚I= σ−εÚI= σ+ε(∀ε)-f
+
= (f
11
,f
12
)Úf
−
=
(f
21
,f
22
)
ZZ
K

∂(Bf
+
)
∂x
+
∂(Bf
−
)
∂y

dxdy=
2
X
i=1
ZZ
K
i

∂(Bf
i1
)
∂x
+
∂(Bf
i2
)
∂y

dxdy
=
2
X
i=1
ZZ
K
i

−
1
y

dxdy= −2 <0
DOI:10.12677/pm.2023.134096909nØêÆ
Ûû
Figure2.Thereisnolimitcyclearoundthesliding
segmentAB
ã2.wÄãAB±Œvk4•‚
∀(x,y) ∈R
2
+
,ε→0,
f
K
i
→K
i
,K
ZZ
K
i

∂(Bf
i1
)
∂x
+
∂Bf
i2
∂y

dxdy=lim
ε→0
ZZ
f
K
i

∂(Bf
i1
)
∂x
+
∂Bf
i2
∂y

dxdy
÷XL
1
ž,dx= f
11
dt,dy= f
12
dt.3«•
f
K
i
A^GreenúªŒ
ZZ
f
K
1

∂(Bf
11
)
∂x
+
∂Bf
12
∂y

dxdy=
I
∂
f
k
1
B(f
11
dy−f
12
dx)
=
Z
L
1
(Bf
11
)dy−(Bf
12
)dx+
Z
p
1
(Bf
11
)dy−(Bf
12
)dx
=
Z
p
1
(Bf
11
)dy−(Bf
12
)dx
ÓnŒ
ZZ
f
K
2

∂(Bf
21
)
∂x
+
∂Bf
22
∂y

dxdy=
I
∂
f
k
2
B(f
21
dy−f
22
dx)
=
Z
p
2
(Bf
21
)dy−(Bf
22
)dx
?˜Úk
0 >
2
X
i=1
ZZ
K
i

∂(Bf
i1
)
∂x
+
∂(Bf
i2
)
∂y

dxdy
=lim
ε→0
2
X
i=1
ZZ
f
K
i

∂(Bf
i1
)
∂x
+
∂(Bf
i2
)
∂y

dxdy
=lim
ε→0
[
Z
p
1
(Bf
11
)dy−(Bf
12
)dx+
Z
p
2
(Bf
21
)dy−(Bf
22
)dx]
NÚQ•4;L††‚I=σþeü‡:‹I.N
1
ÚQ
1
•4;L
1
††‚I=σ−
DOI:10.12677/pm.2023.134096910nØêÆ
Ûû
ε(∀ε)þeü‡:‹I,N
2
ÚQ
2
•4;L
2
††‚I=σ+ ε(∀ε)þeü‡:‹I,Ù
¥N
2
= N+ε.
0 >lim
ε→0
(
Z
N
1
Q
1
(Bf
11
)dy−(Bf
12
)dx+
Z
Q
2
N
2
(Bf
21
)dy−(Bf
22
)dx)
=lim
ε→0
(
Z
N
1
Q
1
[
1−x
y
−c(1−m)]dy−
Z
N
2
Q
2
(
1−x
y
−c)dy
=
Z
N
Q
cmdy>0
dž)gñ,lüØ‚7
P
S
4;•35,y..
œ/1 : 0 <
d
ec(1−m)
<σ<1.
3ù«œ/e,²ï:E
+
2
´¢²ï:,²ï:E
−
2
´J²ï:,–²ï:E
p
Ø•3.
½n3.4.0 <
d
ec(1−m)
<σ<1ž,E
+
2
ÛìC-½.
y².²ï:E
+
2
•¢,´fXÚ(2.6)ÛÜìC-½(:.ÏL½n(3.2)Ú(3.3)y²L§,·
‚•?Û;,˜>w,Ò¬÷XwÄãABle•þ£Ä.ùžÿŠâ½n(3.1),vk
u«•G
1
½G
2
4•‚.¿…,qd½n(3.2)Ú(3.3),·‚•Ø•3•¹Ü©wÄ‚½Œ7w
ÄãAB4•‚.Ïd,l«•G
2
m©;,‡o†r•E
+
2
,‡o E þwÄ‚,÷Xù^‚le
à:Aþà:B,,•ªr•E
+
2
(Xã3¤ «).Ïd,¤k;,•ªòª•u²ï:E
+
2
,¤
±²ï:E
+
2
ÛìC-½.ùÒ¤y².
Figure3.E
−
2
isgloballyasymptoticallystableinthe
system(2.3)(σ=0.8,e=0.8,c=0.5,d=0.1,m=0.5)
ã3.E
+
2
3XÚ(2.3)¥ÛìC-½"Ù¥σ=0.8§
e=0.8§c=0.5§d=0.1§m=0.5
œ/2 : 0 <σ<1 <
d
ec
.
3ù«œ/e,²ï:E
−
1
´¢²ï:,²ï:E
+
2
´J²ï:,–²ï:E
p
Ø•3.
DOI:10.12677/pm.2023.134096911nØêÆ
Ûû
½n3.5.0 <σ<1 <
d
ec
ž,E
−
1
ÛìC-½.
y².²ï:E
−
1
•¢,´f XÚ(2.7)ÛÜìC-½(:.ÏL½n(3.2)Ú(3.3)y²L§,·
‚•?Û;,˜>w,Ò¬÷XwÄãABle•þ£Ä.ùžÿŠâ½n(3.1),vk
u«•G
1
½G
2
4•‚.¿…,qd½n(3.2)Ú(3.3),·‚•Ø•3•¹Ü©wÄ‚½Œ7w
ÄãAB4•‚.Ïd,l«•G
1
m©;,‡o†r•E
−
2
,‡oEþwÄ‚,÷Xù^‚lþ
ýweà:B,,•ªr•E
−
1
(Xã4¤«).Ïd,¤k;,•ªòª•u²ï:E
−
1
,¤±
²ï:E
−
1
ÛìC-½.ùÒ¤y².
Figure4.E
−
1
isgloballyasymptoticallystableinthe
system(2.3)(σ=0.9,e=0.5,c=0.4§d=0.8,m=0.6)
ã4.E
−
1
3XÚ(2.3)¥ÛìC-½"Ù¥σ=0.9§
e=0.5§c=0.4§d=0.8§m=0.6
œ/3 : 0 <σ<
d
ec
<1.
3ù«œ/e,²ï:E
−
2
´¢²ï:,²ï:E
+
2
´J²ï:,–²ï:E
p
Ø•3.
½n3.6.0 <σ<
d
ec
<1ž,E
−
2
ÛìC-½.
y².²ï:E
−
2
•¢,´f XÚ(2.7)ÛÜìC-½(:.ÏL½n(3.2)Ú(3.3)y²L§,·
‚•?Û;,˜>w,Ò¬÷XwÄãABle•þ£Ä.ùžÿŠâ½n(3.1),vk
u«•G
1
½G
2
4•‚.¿…,qd½n(3.2)Ú(3.3),·‚•Ø•3•¹Ü©wÄ‚½Œ7w
ÄãAB4•‚.Ïd,l«•G
1
m©;,‡o†r•E
−
2
,‡oEþwÄ‚,÷ X ù^ ‚ l e
ýwþà:B,,•ªr•E
−
2
(Xã5¤«).Ïd,¤k;,•ªòª•u²ï:E
−
2
,¤±
²ï:E
−
2
ÛìC-½.ùÒ¤y².
œ/4 : 0 <
d
ec
<σ<
d
ec(1−m)
.
3ù«œ/e,²ï:E
+
2
´J²ï:,²ï:E
−
2
´J²ï:,–²ï:E
p
•3.
½n3.7.0 <
d
ec
<σ<
d
ec(1−m)
ž,E
p
ÛìC-½.
y².lG
2
m©;,lmCE
−
2
ž,§‚Eƒ†‚,÷Xƒ†‚wĽ?\«•G
1
,
lG
1
m©;,•þCE
+
2
ž,§‚Eƒ†‚,÷ Xƒ†‚wĽ?\«•G
2
,ü«;,3
DOI:10.12677/pm.2023.134096912nØêÆ
Ûû
Figure5.E
−
2
isgloballyasymptoticallystableinthe
system(2.3)(σ=0.8,e=0.8,c=0.6,d=0.4,m=0.2)
ã5.E
−
2
3XÚ(2.3)¥ÛìC-½"Ù¥σ=0.8§
e=0.8§c=0.6§d=0.4§m=0.2
ƒ†‚þ-Ež,Ñyäk–²ïwã(Xã6¤«),^½n(3.1),(3.2)Ú(3.3)üØ4•‚
•3,qÏ•E
P
´ÛÜìC-½,Œ±éN´/íÑ–²ïE
P
´ÛìC-½.
Figure6.E
p
isgloballyasymptoticallystableinthe
system(2.3)(σ=0.7,e=0.8,c=0.6,d=0.2,m=0.5)
ã6.E
p
3XÚ(2.3)¥ÛìC-½"Ù¥σ=0.7§
e=0.8§c=0.6§d=0.2§m=0.5
œ/5 : σ=
d
ec
>0.
3ù«œ/e,–²ï:E
p
,¢²ï:E
−
2
ÚŒ„mò:T
1
-Ü,C¤˜‡>.²ï:
E
1
=

d
ec
,
ec−d
ec
2

.
½n3.8.σ=
d
ec
>0ž,E
1
ÛìC-½.
DOI:10.12677/pm.2023.134096913nØêÆ
Ûû
y².e0<
d
ec
=σ,ÏL½n(3.3)y²L§,·‚•Ø•3Œ7Σ
s
4;,J²ï:E
+
2
ÚE
−
2
©O3G
1
ÚG
2
¥´ÛìC-½,KäkЊ);‚‡o†?\Σ
s
,‡okBLΣ,
,3k•žmSˆΣ
s
,‡o;‚3w•ABþlþ•e£Ä,•ªª•>.²ï:E
1
(Xã
7¤«).=E
1
´ÛìC-½¢²ï:,y..
Figure7.E
1
isgloballyasymptoticallystableinthe
system(2.3)(σ=0.75,e=0.5,c=0.8,d=0.3,m=0.7)
ã7.E
1
3XÚ(2.3)¥ÛìC-½"Ù¥σ=0.75§
e=0.5§c=0.8§d=0.3§m=0.7
œ/6 : σ=
d
ec(1−m)
>0.
3ù«œ/e,–²ï:E
p
,¢²ï:E
+
2
ÚŒ„mò:T
2
-Ü,C¤˜‡>.²ï:
E
2
=

x
+
2
,y
+
2

=

d
ec(1−m)
,
ec(1−m)−d
ec
2
(1−m)
2

.
Figure8.E
2
isgloballyasymptoticallystableinthe
system(2.3)(σ=0.625,e=0.8,c=0.8,d=0.2,m=0.5)
ã8.E
2
3XÚ(2.3)¥ÛìC-½"Ù¥σ=0.625§
e=0.8§c=0.8§d=0.2§m=0.5
DOI:10.12677/pm.2023.134096914nØêÆ
Ûû
½n3.9.0 <
d
ec(1−m)
= σž,E
2
ÛìC-½.
y².e0<
d
ec(1−m)
=σ,ÏL½n(3.3)y² L§,·‚•Ø• 3Œ7Σ
s
4;,J²ï:E
−
2
ÚE
+
2
©O3G
2
ÚG
1
¥´ÛìC-½,KäkЊ);‚‡o†?\Σ
s
,‡okBLΣ,
,3k•žmSˆΣ
s
,‡o;‚3w•ABþlþ•e£Ä,•ªª•>.²ï:E
2
(Xã8
¤«).=E
2
´ÛìC-½¢²ï:,y..
4.(Ø
©•Ä·KŠüÑ,ïÄäk'~Ô;J¤HollingI.Ó - .
ÛÄåÆ.|^Lyapunov¼ê{Ú‚úª,ïÄˆa²ï:ÛìC-½5.ïÄuy:
XJ0<
d
ec(1−m)
<σ<1ž,K¢²ï:E
+
2
´ÛìC-½;XJ0<σ<1<
d
ec
ž,K¢²ï
:E
−
1
ÛìC-½; 0 <σ<
d
ec
<1 ž, K¢²ï:E
−
2
ÛìC-½; 0 <
d
ec
<σ<
d
ec(1−m)
ž,
–²ï:E
p
ÛìC-½;eσ=
d
ec
>0,3ù«œ/e,–²ï:E
p
,¢²ï:E
−
2
ÚŒ„mò
:T
1
-Ü,-ܤ˜‡>.²ï:E
1
,E
1
ÛìC-½;e0<
d
ec(1−m)
=σ,3ù«œ/e,–²ï
:E
p
,¢²ï:E
+
2
ÚŒ„mò:T
2
-Ü,-ܤ˜‡>.²ï:E
2
,E
2
ÛìC-½.
ÏL±þ(ØŒ•;J¤éÔk˜½oŠ^,ïᘽêþ;J¤éoÄÔ«+
õ5ék7‡.©•ïÄ;J¤éHollingI.Ó - .K•,ù´{ü˜a.,
3ƒïÄ¥Œ ±•Ä•\E,œ¹,'X•Ä;J¤é1aõU‡A¼êÓ - 
.K•,ŒU¬ؘ(Ø.
ë•©z
[1]Eduardo,G.O.andRodrigo,R.J.(2003)DynamicConsequencesofPreyRefugesinaSimple
ModelSystem:MorePrey,FewerPredatorsandEnhancedStability.EcologicalModelling,
166,135-146.https://doi.org/10.1016/S0304-3800(03)00131-5
[2]Mcnair,J.N.(1987)StabilityEffectsofPreyRefugeswithEntry-ExitDynamics.Journalof
TheoreticalBiology,125,449-464.https://doi.org/10.1016/S0022-5193(87)80213-8
[3]Mcnair,J.N.(1986)TheEffects ofRefugesonPredator-Prey Interactions:A Reconsideration.
TheoreticalPopulationBiology,29,38-63.https://doi.org/10.1016/0040-5809(86)90004-3
[4]Ko,W. andRyu, K.(2006) Qualitative Analysisof aPredator-Prey Modelwith HollingType
IIFunctionalResponseIncorporatingaPreyRefuge.JournalofDifferentialEquations,231,
534-550.https://doi.org/10.1016/j.jde.2006.08.001
[5]Sih,A.(1987)PreyRefugesandPredator-PreyStability.TheoreticalPopulationBiology,31,
1-12.https://doi.org/10.1016/0040-5809(87)90019-0
[6]Wang,Y.andWang,J.Z.(2012)InfluenceofPreyRefugeonPredator-PreyDynamics.Non-
linearDynamics,67,191-201.https://doi.org/10.1007/s11071-011-9971-z
DOI:10.12677/pm.2023.134096915nØêÆ
Ûû
[7]Chen,L.andChen,F.(2010)QualitativeAnalysisofaPredator-PreyModelwithHolling
Type IIFunctionalResponseIncorporating aConstant Prey Refuge. NonlinearAnalysis:Real
WorldApplications,11,246-252.https://doi.org/10.1016/j.nonrwa.2008.10.056
[8]Ma,Z.H.,Li,W.L.andZhao,Y.(2009)EffectsofPreyRefugesonaPredator-PreyModel
withaClassofFunctionalResponses:TheRoleofRefuges.MathematicalBiosciences,218,
73-79.https://doi.org/10.1016/j.mbs.2008.12.008
[9]MaynardSmith,J.(1974)ModelsinEcology.CambridgeUniversityPress,Cambridge.
[10]Gause,G.F.,Smaragdova,N.P.andWitt,A.A.(1936)FurtherStudiesofInteractionbetween
PredatorsandPrey.JournalofAnimalEcology,5,1-18.https://doi.org/10.2307/1087
[11]Filippov,A.F.(1988)DifferentialEquationswithDiscontinuousRighthandSides.Springer,
Dordrecht.https://doi.org/10.1007/978-94-015-7793-9
DOI:10.12677/pm.2023.134096916nØêÆ

版权所有：汉斯出版社 (Hans Publishers) Copyright © 2023 Hans Publishers Inc. All rights reserved.