Research Article A New Approach to Global Stability of ...

Research ArticleA New Approach to Global Stability of DiscreteLotka-Volterra Predator-Prey Models

Young-Hee Kim1 and Sangmok Choo2

1Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea2Department of Mathematics University of Ulsan Ulsan 680-749 Republic of Korea

Correspondence should be addressed to Sangmok Choo smchooulsanackr

Received 24 March 2015 Accepted 20 May 2015

Academic Editor Ryusuke Kon

Copyright copy 2015 Y-H Kim and S Choo This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An Euler difference scheme for a three-dimensional predator-prey model is considered and we introduce a new approach to showthe global stability of the scheme For this purpose we partition the three-dimensional space and calculate the sign of the ratechange of population of species in each partitioned region Our method is independent of dimension and then can be applicableto other dimensional discrete models Numerical examples are presented to verify the results in this paper

1 Introduction

Vito Volterra proposed a differential equation model toexplain the observed increase in predator fish and corre-sponding decrease in prey fish in 1926 At the same timethe equations studied byVolterra were derived independentlyby Alfred Lotka (1925) to describe a chemical reactionMany predator-prey models have been studied and a classicpredator-prey model is given by

119889119909

119889119905= 119909 (1199031 minus 11988611119909minus 11988612119910)

119889119910

119889119905= 119910 (minus1199032 + 11988621119909minus 11988622119910)

(1)

where 119903119894gt 0 119886119894119895gt 0 and 119909 and 119910 denote the population sizes

of the prey and predator respectivelyIt is recognized that the rate of prey capture per predator

cannot increase indefinitely as the number of prey increasesInstead the rate of prey capture is saturated when the popu-lation of prey is relatively large Then such nonlinear func-tional responses are employed to describe the phenomenaof predation including the Holling types [1ndash5] Beddington-DeAngelis type [6ndash8] Crowley-Martin type [9ndash11] and Ivlevtype of functional response [12ndash14]

In particular similar phenomena are observed in theinteractions in chemical reactions andmolecular eventswhenone species is abundant Thus linear response functionMichaelis-Menten kinetics and Hill function are related toHolling types I II and III respectively Holling type IV is alsocalled the Monod-Haldane function [4 5]

The functional responses have been applied to predator-prey models to express the Allee effect [15ndash19] whichdescribes a positive relation between population density andthe per capita growth rate

Most of researches on the predator-prey models assumethat the distribution of the predators and prey is homo-geneous which leads to ordinary differential equationsHowever both predators and prey have the natural tendencyto diffuse so that there have beenmodels to take into accountthe inhomogeneous distribution of the predators and prey[20ndash22]

On the other hand population is inevitably affected byenvironmental noise in natureTherefore many authors havetaken stochastic perturbation into deterministic models [23ndash25]

There are a number of works investigating continuoustime predator-prey models but relatively few theoreticalpapers are published on their discretized models [26 27]As far as we know there is no theoretical research on the

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 674027 11 pageshttpdxdoiorg1011552015674027

2 Discrete Dynamics in Nature and Society

global stability of the discrete-time models of type (1) withmore than two species except for [28] The author in [28]introduced a method to present global stability for the casethat all species coexist at a unique equilibrium Then a newapproach needs to be developed for the other cases

For explaining our new approach we consider a modelwith one prey and two predators

119889119909119894

119889119905= 119909119894(120590119894119903119894+ sum

1le119895le119894minus1119886119894119895119909119895minus sum

119894le119895le3119886119894119895119909119895) (2)

where 119894 = 1 2 3 1205901 = 1 1205902 = 1205903 = minus1 and119903311988631

lt119903111988611

lt119903211988621

(3)

119903111988612

lt119903311988632

(4)

1198862111988623

lt1198863111988633

(5)

Here 1199091 denotes the population number of the prey 1199092 and1199093 denote the population numbers of the predators Letting

119891119894(119909

1 119909

2 119909

3) = 120590119894119903119894+ sum

1le119895le119894minus1119886119894119895119909119895minus sum

119894le119895le3119886119894119895119909119895 (6)

the Euler difference scheme for (2) is as follows

Δ119909119894

119896

Δ119905= 119909119894

119896119891119894(119909

1119896 119909

2119896 119909

3119896) 119896 = 0 1 (7)

where 1199091198940 gt 0 (119894 = 1 2 3) Δ119909119894119896= 119909119894

119896+1 minus 119909119894

119896 and Δ119905 is a step

sizeThe three conditions (3)ndash(5) mean that the two planes

1198911(119909 119910 119911) = 0 and 1198913(119909 119910 119911) = 0 intersect in the first octantand the plane 1198912(119909 119910 119911) = 0 is not intersected with either1198911(119909 119910 119911) = 0 or 1198913(119909 119910 119911) = 0 in the first octant Forexample let (119903

119894 1198861198941 1198861198942 1198861198943) (1 le 119894 le 3) be

(2 1 2 2)

(3 1 1 1)

(2 2 1 1)

(8)

respectively Then the three conditions (3)ndash(5) are satisfiedand Figure 1 shows the three planes and regions with theglobally stable point of scheme (7) The global stability of thepoint will be shown in Section 3

Using Theorem 41 in [28] we have the positivity andboundedness of the solutions of scheme (7) Letting 120594

119894(1 le

119894 le 3) satisfy

120594119894lt1 + 120590119894119903119894Δ119905 minus sum

119894+1le119895le119899 119886119894119895120594119895Δ119905

2119886119894119894Δ119905

120590119894119903119894+ sum1le119895le119894minus1 119886119894119895120594119895

119886119894119894

lt 120594119894

(120590119894119903119894+ sum

1le119895le119894minus1119886119894119895120594119895)Δ119905 lt 1

(9)

we have that for all 119896

(1199091119896 119909

2119896 119909

3119896) isin (0 1205941) times (0 1205942) times (0 1205943) (10)

For small values of Δ119905 there exist infinitely many 120594119894(1 le 119894 le

3) satisfying (9)Consider the five regions

I = (119909 119910 119911) isinS | 1198911 (119909 119910 119911) ge 0 1198912 (119909 119910 119911)

lt 0 1198913 (119909 119910 119911) le 0

II = (119909 119910 119911) isinS | 1198911 (119909 119910 119911) ge 0 1198912 (119909 119910 119911)

lt 0 1198913 (119909 119910 119911) ge 0

III = (119909 119910 119911) isinS | 1198911 (119909 119910 119911) le 0 1198912 (119909 119910 119911)

lt 0 1198913 (119909 119910 119911) le 0

IV = (119909 119910 119911) isinS | 1198911 (119909 119910 119911) le 0 1198912 (119909 119910 119911)

le 0 1198913 (119909 119910 119911) ge 0

V = (119909 119910 119911) isinS | 1198911 (119909 119910 119911) lt 0 1198912 (119909 119910 119911)

ge 0 1198913 (119909 119910 119911) gt 0

(11)

where S = (119909 119910 119911) isin R3| 119909 gt 0 119910 gt 0 119911 gt 0 cap

prod1le119894le3(0 120594119894) (see Figure 1)For convenience we denote the set I by

I (+ minus minus)

or (+ minus minus) (12)

and then

119875 isin I (+ minus minus)

iff 1198911 (119875) ge 0 1198912 (119875) lt 0 1198913 (119875) le 0(13)

We adapt similar notations for the other regions II to VNote that region I has the property

sup 119909 | (119909 119910 119911) isin I (+ minus minus) le119903111988611

(14)

since1198911(119909 119910 119911) = 1199031minus11988611119909minus11988612119910minus11988613119911 ge 0 for all (119909 119910 119911) isinI(+ minus minus)

2 Method

In this section we present theorems that describe ourapproach The theorems will be used to obtain the globalstability of scheme (7) in the next section

Assume that Δ119905 satisfies

Δ119905 1205941 (11988611 + 119886131198862111988623

)+120594211988622 +1205943 (11988633 + 119886321198861311988612

)

lt 1(15)

Δ119905

1 minus 119886331205943Δ11990511988631120594

21 lt

119886311199032 minus 11988621119903311988621

(16)

Discrete Dynamics in Nature and Society 3

01

23

0

1

2

0

1

2

xy

z

(a)

0 1 2 3 40

1

x

y

III

III

IV

V

f1 = 0

f2 = 0

f3 = 0

(b)

0 1 2 3 40

1

x

z

f1 = 0

f2 = 0

f3 = 0

(c)

Figure 1 Three planes and regions with the globally stable point (a) The three planes 119891119894(119909 119910 119911) = 0 (1 le 119894 le 3) in (0 4) times (0 3) times (0 2)

The magenta circle denotes the globally stable point (65 0 25) of scheme (7) (b) Five regions projected on the 119909119910-plane (c) Five regionsprojected on the 119909119911-plane

Let119875 = (1198751 1198752 1198753) and119879 = (1198791 1198792 1198793) be the vector functiondefined on S by

119879119894 (119875) = 119875119894 1+Δ119905119891119894 (119875) 119894 = 1 2 3 (17)

Then scheme (7) can be written as

(1199091119896+1 119909

2119896+1 119909

3119896+1) = 119879 (119909

1119896 119909

2119896 119909

3119896) (18)

4 Discrete Dynamics in Nature and Society

Note that the map 119879 has the three fixed points with allnonnegative components (0 0 0) (119903111988611 0 0) and 120599 =

(1205991 0 1205993) satisfying

1198911 (120599) = 1198913 (120599) = 0 (19)

Then the fixed point 120599 with 1205991 = (119903111988633 + 119903311988613)(1198861111988633 +1198863111988613) and 1205993 = (119903111988631 minus 119903311988611)(1198861311988631 + 1198863311988611) is locallystable as follows

Lemma 1 Let 120599 be the fixed point of the map 119879 satisfying(19) Assume that Δ119905 satisfies (15) Then the fixed point 120599 isasymptotically stable

Proof Applying the linearization method to the discretesystem of (7) at 120599 the matrix of the linearized system has thethree eigenvalues

1+1198912 (120599) Δ119905

1minus 12(120599111988611 + 120599311988633) Δ119905

plusmn12(120599111988611 minus 120599311988633)

2minus 4120599112059931198861311988631

12Δ119905

(20)

Substituting 1205991 = (119903111988633 + 119903311988613)(1198861111988633 + 1198863111988613)minus1 to

1198912 (120599) = minus 1199032 + 119886211205991 minus 119886231205993 lt minus 1199032 + 119886211205991 (21)

and using (3) we obtain

1198912 (120599) lt 0 (22)

In addition (15) gives that the other two eigenvalues havemagnitudes less than 1 Hence the spectral radius of thematrix is less than 1 which completes the proof

The two fixed points (0 0 0) and (119903111988611 0 0) are unstablesince the matrices of the linearized system at (0 0 0) and(119903111988611 0 0) have the eigenvalues 1 + 1198911(0 0 0)Δ119905 and 1 +1198913(119903111988611 0 0)Δ119905 with 1198911(0 0 0) gt 0 and 1198913(119903111988611 0 0) gt0 respectively Then we only consider the fixed point 120599 =

(1205991 0 1205993) of the map 119879 to show global stability in the nextsection

Our methodology to obtain the global stability is basedon the approach to determine regions among regions I to V inwhich119879(119875) is contained by calculating the sign of119891

119894(119879(119875)) for

119875 in each region Then we use the sign symbol 119875 isin (+ lowast lowast)

as follows119875 isin (+ lowast lowast) if and only if 1198911(119875) ge 0 and the signs of

119891119894(119875) (119894 = 2 3) are unknownOther sign symbols are similarly defined

Theorem 2 Suppose that Δ119905 satisfies (9) and (15) Let 119904 isin

+ minus

119868119891119875 isin (119904 minus 119904 minus 119904) 119905ℎ119890119899 119905ℎ119890 119904119894119892119899 1199001198911198911 (119879 (119875)) 119894119904 119904 (23)

Proof Let 119875 = (1198751 1198752 1198753) Using the definition of1198911 we have

1198911 (119879 (119875)) = 1199031 minus 119886111198791 (119875) minus 119886121198792 (119875) minus 119886131198793 (119875)

= 1199031 minus sum

1le119894le31198861119894119875119894 1+Δ119905119891119894 (119875)

= 1199031 minus sum

1le119894le31198861119894119875119894 minus sum

1le119894le3Δ1199051198861119894119875119894119891119894 (119875)

= 1198911 (119875119894) minus sum

1le119894le3Δ1199051198861119894119875119894119891119894 (119875)

= (1minusΔ119905119886111198751) 1198911 (119875)

minusΔ119905 1198861211987521198912 (119875) + 1198861311987531198913 (119875)

(24)

Note that if 1198912(119909 119910 119911) = 0 for nonnegative 119909 119910 and 119911 then

119909 =111988621

(1199032 + 11988622119910+ 11988623119911) (25)

with which the nonnegativity of 119909 119910 and 119911 gives

1198911 (119909 119910 119911) = 1199031 minus 11988611119909minus 11988612119910minus 11988613119911

= 1199031 minus 11988611111988621

(1199032 + 11988622119910+ 11988623119911) minus 11988612119910

minus 11988613119911 le 1199031 minus 11988611111988621

1199032

= 11988611 (119903111988611

minus119903211988621

) lt 0

(26)

and similarly

1198913 (119909 119910 119911) gt 0 (27)

due to (3) Hence there exist no nonnegative numbers 119909 119910and 119911 such that 1198911(119909 119910 119911) = 1198912(119909 119910 119911) = 1198913(119909 119910 119911) = 0Since 119875 isin (119904 minus119904 minus119904) and it is impossible that 1198911(119875) = 1198912(119875) =1198913(119875) = 0 (24) gives that the sign of 1198911(119879(119875)) is 119904

UsingTheorem 2 we have the following

If 119875 isin I (+ minus minus)

then 119879 (119875) notin III cup IV cup V (minus lowast lowast) (28)

If 119875 isin V (minus + +) then 119879 (119875) notin I (+ minus minus) (29)

Remark 3 Suppose that Δ119905 satisfies (9) and (15) Let 119904 isin

+ minus Then the definitions of 1198912 and 1198913 yield

1198912 (119879 (119875)) = (1minusΔ119905119886221198752) 1198912 (119875)

+Δ119905 1198862111987511198911 (119875) minus 1198862311987531198913 (119875) (30)

1198913 (119879 (119875)) = (1minusΔ119905119886331198753) 1198913 (119875)

+Δ119905 1198863111987511198911 (119875) + 1198863211987521198912 (119875) (31)

and hence we can also obtain the property as inTheorem 2

Discrete Dynamics in Nature and Society 5

(a) If 119875 isin (119904 119904 minus119904) then the sign of 1198912(119879(119875)) is 119904(b) If 119875 isin (119904 119904 119904) then the sign of 1198913(119879(119875)) is 119904

Using (a) and (b) we can obtain the following

If 119875 isin IV (minus minus +) then 119879 (119875) notin V (minus + +) (32)

If 119875 isin III (minus minus minus)

then 119879 (119875) notin II cup IV cup V (lowast lowast +) (33)

It follows from (28) and (29) that every point119875 in a regioncannot move by the map 119879 to regions with three differentsigns In the case of regions with two different signs it is alsoimpossible by the following theorem

Theorem 4 Suppose that Δ119905 satisfies (9) (15) and (16)

(a) If 119875 isin 119868119868(+ minus +) then 119879(119875) notin 119881(minus + +)(b) If 119875 isin 119881(minus + +) then 119879(119875) notin 119868119868(+ minus +)(c) If 119875 isin 119868119868(+ minus +) then 119879(119875) notin 119868119868119868(minus minus minus)(d) If 119875 isin 119868119868119868(minus minus minus) then 119879(119875) notin 119868119868(+ minus +)(e) If 119875 isin 119868119868119868(minus minus minus) then 119879(119875) notin 119881(minus + +)(f) If 119875 isin 119881(minus + +) then 119879(119875) notin 119868119868119868(minus minus minus)

Proof (a) Suppose that119879(119875) isin V(minus + +)Then (24) and (30)with 119875 isin II(+ minus +) give

(1minusΔ119905119886111198751) 1198911 (119875) +Δ11990511988612119875210038161003816100381610038161198912 (119875)

1003816100381610038161003816

lt Δ1199051198861311987531198913 (119875) (34)

1198862311987531198913 (119875) le 1198862111987511198911 (119875) (35)

respectively Combining (34) and (35) we obtain

(1minusΔ119905119886111198751) 1198911 (119875) lt Δ1199051198861311988621119875111988623

1198911 (119875) (36)

Since 1198911(119875) ge 0 inequality (36) is a contradiction to (15)Therefore the proof of (a) is completed

(b) Suppose that 119879(119875) isin II(+ minus +) Then the inclusion of119875 isin V(minus + +) gives a contradiction

0 le 1198911 (119879 (119875))

= (1minusΔ119905119886111198751) 1198911 (119875)

minusΔ119905 1198861211987521198912 (119875) + 1198861311987531198913 (119875) lt 0

(37)

(c) Suppose that 119879(119875) isin III(minus minus minus) Then (24) and (31)with 119875 isin II(+ minus +) give

(1minusΔ119905119886111198751) 1198911 (119875) +Δ11990511988612119875210038161003816100381610038161198912 (119875)

1003816100381610038161003816

le Δ1199051198861311987531198913 (119875)

(1minusΔ119905119886331198753) 1198913 (119875) le Δ11990511988632119875210038161003816100381610038161198912 (119875)

1003816100381610038161003816

(38)

which yield

(1minusΔ119905119886331198753) 1198913 (119875) le Δ1199051198863211988613119875311988612

1198913 (119875) (39)

Hence if 1198913(119875) gt 0 or 1198913(119875) = 0 then

(1minusΔ119905119886331198753) lt Δ1199051198863211988613119875311988612

or 119891119894 (119875) = 0 (1 le 119894 le 3)

(40)

respectively these are contradictions due to (15) and thefact that there is no solution of the system of equations119891119894(119909 119910 119911) = 0 (119894 = 1 2 3)(d) and (e) are proved by (33)(f) Suppose that 119879(119875) isin III(minus minus minus) Then (31) with 119875 isin

V(minus + +) gives

0 lt 1198913 (119875) leΔ119905

1 minus Δ119905119886331198753119886311198751

10038161003816100381610038161198911 (119875)1003816100381610038161003816 (41)

It follows from 1198912(119875) = minus1199032 + 119886211198751 minus 119886221198752 minus 119886231198753 ge 0 (5)and (3) that

1198913 (119875) ge minus 1199033 + 11988631111988621

(1199032 + 119886221198752 + 119886231198753) + 119886321198752

minus 119886331198753

gt111988621

minus119886211199033 + 119886311199032 + (1198863111988623 minus 1198862111988633) 1198753

gt111988621

(minus119886211199033 + 119886311199032)

(42)

Since minus119886211199033 + 119886311199032 gt 0 by (3) inequality (41) with (42) is acontradiction to (16) which completes the proof

Theorem 5 Suppose that Δ119905 satisfies (9) If 119875 isin 119881 then1198791198981(119875) notin 119881 for some1198981

Proof Suppose that 119879119898(119875) isin V(minus + +) for all119898 Then thereexist constants 119909

lowast 119910lowast and 119911

lowastsuch that

119909lowast= lim119896rarrinfin

1199091119896ge 0

119910lowast= lim119896rarrinfin

1199092119896gt 0

119911lowast= lim119896rarrinfin

1199093119896gt 0

(43)

and so

1198912 (119909lowast 119910lowast 119911lowast) = 1198913 (119909lowast 119910lowast 119911lowast) = 0 (44)

This is a contradiction since the system of (44) gives that

0 lt (1198862211988631 + 1198863211988621) 119910lowast

= minus (119886311199032 minus 119886211199033) minus (1198862311988631 minus 1198863311988621) 119911lowast lt 0(45)

due to both (3) and (4) with 119911lowastgt 0

The results we obtained are summarized in Table 1Finally using Table 1 we can obtain the following theo-

rem

6 Discrete Dynamics in Nature and Society

Table 1The regions containing119879(119875)The symbols I to V denote theregions defined in (11) The region I in the second column denotes119875 isin I the symbol times at (119875 119879(119875)) = (I III) means 119879(119875) notin III andthen the two circles ∘ together at both (I I) and (I II) denote that119879(119875) isin Icup II by (28)The symbolM denotes that 119875 isinV and1198791198981 (119875) notinV for some1198981 byTheorem 5

119879(119875) Equation andtheoremI II III IV V

119875

I ∘ ∘ times times times Equation (28)

II ∘ ∘ times ∘ timesTheorem 4(c)

and (a)III ∘ times ∘ times times Equation (33)IV ∘ ∘ ∘ ∘ times Equation (32)

V times times times ∘ M

Equation (29)Theorem 4(b)and (f) andTheorem 5

Theorem 6 Suppose that Δ119905 satisfies (9) (15) and (16) If 119875 isin119868 cup 119868119868 cup 119868119868119868 cup 119868119881 cup 119881 then for all sufficiently large119898

119879119898(119875) isin 119868 cup 119868119868 cup 119868119868119868 cup 119868119881 (lowast minus lowast) (46)

Remark 7 Table 1 and Theorem 6 are obtained for the casethat only two planes 119891

119894(119909 119910 119911) = 0 (119894 = 1 3) intersect in the

first octant We can also apply the approach to the other casesand then have a table and a theorem similar to Table 1 andTheorem 6

3 Global Stability

In this section we show that the fixed point 120599 = (1205991 0 1205993) ofthe map 119879 satisfying (19) is globally stable Let Δ119905 satisfy that

1198861198941 (1205941 + 1205991) + 11988611989421205942 + 1198861198943 (1205943 + 1205993) Δ119905 lt

12

119894 = 1 3(47)

Δ119905 ltmin 1198863111988611 1198861311988633

211988631 (11988612 + 1198601205991119872) + 211988613 (11988632 + 1198601205993119872) (48)

where 119860 = 3sum1le119895le3(1198861119895 + 1198863119895) and119872 = 6max1le119894119895le3119886119894119895

Theorem 8 Let 120599 be the point defined in (19) Suppose that Δ119905satisfies (9) (15) (16) (47) and (48) Assume that the initialpoint (11990910 119909

20 119909

30) of the Euler scheme (7) satisfies

(11990910 119909

20 119909

30) isin prod

1le119894le3(0 120594119894) (49)

Then

lim119896rarrinfin

(1199091119896 119909

2119896 119909

3119896) = 120599 (50)

Proof Theorem 6 gives that for all 119896

(1199091

119896 1199092

119896 1199093

119896) isin Icup IIcup IIIcup IV (lowast minus lowast) (51)

and then (10) gives that for a nonnegative constant 119910lowast

119910lowast= lim119896rarrinfin

1199092119896 (52)

Now we claim that 119910lowast= 0 suppose on the contrary that

119910lowastgt 0 (53)

Applying both (52) and (53) to (7) we have

lim119896rarrinfin

1198912 (1199091119896 119909

2119896 119909

3119896) = 0 (54)

which implies that there exists a constant 1205980 such that for allsufficiently large 119896

0 lt 1205980

lt 11988621 max 119903211988621

minus119903111988611

119903211988621

minus119903311988631

(55)

1198912 (1199091119896 119909

2119896 119909

3119896) gt minus 1205980 (56)

Then (56) and (55) give 1199091119896gt (111988621)(1199032 minus 1205980) gt 119903111988611 and so

(14) gives

(1199091

119896 1199092

119896 1199093

119896) notin I (+ minus minus) (57)

Using (56) we have

1199091119896gt

111988621

(1199032 minus 1205980 + 119886221199092119896+ 11988623119909

3119896) (58)

and so it follows from (42) and (55) that for all sufficientlylarge 119896

1198913 (1199091119896 119909

2119896 119909

3119896) gt

111988621

11988631 (1199032 minus 1205980) minus 119886211199033 gt 0 (59)

which gives

(1199091

119896 1199092

119896 1199093

119896) notin III (minus minus minus) (60)

Hence (51) with (57) and (60) gives that for all sufficientlylarge 119896

(1199091

119896 1199092

119896 1199093

119896) isin IIcup IV (lowast minus +) (61)

Both (61) and (53) yield that there exist the two limits

119909lowast= lim119896rarrinfin

1199091119896

119911lowast= lim119896rarrinfin

1199093119896gt 0

(62)

and then

1198912 (119909lowast 119910lowast 119911lowast) = 1198913 (119909lowast 119910lowast 119911lowast) = 0 (63)

This is a contradiction due to (45) and finally we obtain theclaim

lim119896rarrinfin

1199092119896= 0 (64)

Discrete Dynamics in Nature and Society 7

Consider the function 119881119896defined by

119881119896= 11988631 (119909

1119896minus 1205991 ln119909

1119896) + 11988613 (119909

3119896minus 1205993 ln119909

3119896) (65)

Letting 119886119894119895= 119886119894119895Δ119905 and

1205991119896 = 1205991 minus1199091119896

1205993119896 = 1205993 minus1199093119896

(66)

scheme (7) and the fixed point (19) yield

Δ1199091119896= 119909

1119896(119886111205991119896 minus 11988612119909

2119896+ 119886131205993119896)

Δ1199093119896= 119909

3119896(minus119886311205991119896 + 11988632119909

2119896+ 119886331205993119896)

(67)

Then the mean value theorem gives that for some 120572 120573 with0 lt 120572 120573 lt 1

119881119896+1 minus119881119896 = 11988631Δ119909

1119896(1minus 1205991

Δ ln1199091119896

Δ1199091119896

)

+11988613Δ1199093119896(1minus 1205993

Δ ln1199093119896

Δ1199093119896

)

= 11988631 (119886111205991119896 minus 119886121199092119896+ 119886131205993119896)

sdot (1199091119896minus 1205991

1199091119896

120572Δ1199091119896+ 1199091119896

)

+11988613 (minus119886311205991119896 + 119886321199092119896+ 119886331205993119896)

sdot (1199093119896minus 1205993

1199093119896

120573Δ1199093119896+ 1199093119896

)

(68)

Note that

1199091119896

120572Δ1199091119896+ 1199091119896

= 1minus120572 (119886111205991119896 minus 11988612119909

2119896+ 119886131205993119896) Δ119905

120572 (119886111205991119896 minus 119886121199092119896+ 119886131205993119896) Δ119905 + 1

equiv 1minus (11986211205991119896 +11986221199092119896+11986231205993119896) Δ119905

1199093119896

120573Δ1199093119896+ 1199093119896

= 1

minus120573 (minus119886311205991119896 + 11988632119909

2119896+ 119886331205993119896) Δ119905

120573 (minus119886311205991119896 + 119886321199092119896+ 119886331205993119896) Δ119905 + 1

equiv 1minus (11986241205991119896 +11986251199092119896+11986261205993119896) Δ119905

(69)

where (47) gives

max1le119894le6

10038161003816100381610038161198621198941003816100381610038161003816 lt 2 max

1le119894119895le3119886119894119895 (70)

Now suppose on the contrary that (1199091119896 119909

2119896 119909

3119896) does not

converge to (1205991 0 1205993)Since 120599 is asymptotically stable by Lemma 1 the supposi-

tion with lim119896rarrinfin

1199092119896= 0 implies that

100381610038161003816100381612059911198961003816100381610038161003816 +10038161003816100381610038161205993119896

1003816100381610038161003816 has a positive lower bound (71)

Again using lim119896rarrinfin

1199092119896= 0 with (71) we can have that for

all sufficiently large 119896100381610038161003816100381610038161199092119896

10038161003816100381610038161003816lt Δ119905 (

100381610038161003816100381612059911198961003816100381610038161003816 +10038161003816100381610038161205993119896

1003816100381610038161003816) (72)

Then (68) becomes

119881119896+1 minus119881119896 le minusΔ119905 (1198863111988611 minus1198627Δ119905) 120599

21119896

minusΔ119905 (1198861311988633 minus1198628Δ119905) 12059923119896

(73)

where

max 1198627 1198628 lt 211988631 (11988612 +11986012059911003816100381610038161003816119862123

1003816100381610038161003816)

+ 211988613 (11988632 +11986012059931003816100381610038161003816119862456

1003816100381610038161003816)

(74)

with 119860 = 3sum1le119895le3(1198861119895 + 1198863119895) and |119862119894119895119896| = |119862119894| + |119862

119895| + |119862

119896|

Hence (73) with (48) becomes

119881119896+1 minus119881119896 le minus1198629 (120599

21119896 + 120599

23119896) Δ119905 (75)

for a positive constant 1198629 so that there exists a positiveconstantC such that for all sufficiently large 119896

119881119896+1 minus119881k le minusCΔ119905 (76)

and then

119881119896le 1198810 minus 119896CΔ119905 (77)

Hence we have

lim119896rarrinfin

119881119896= lim119896rarrinfin

(1198810 minus 119896CΔ119905) = minusinfin (78)

which is a contradiction due to (10)

Remark 9 The global stability in Theorem 8 is obtained forthe discrete predator-prey model with one prey and twopredators by using both Theorem 6 and the Lyapunov typefunction 119881

119896 At first we show that one predator is extinct

by Theorem 6 and then we can apply the function 119881119896which

was used in the lower dimensional case the two-dimensionaldiscrete model with one prey and one predatorTherefore ourapproach can utilize the methods used in lower dimensionalmodels

On the other hand in the case that119903111988611

lt119903311988631

lt119903211988621

119903111988612

lt119903311988632

1198862111988623

lt1198863111988633

(79)

region II is empty and similarly we can obtain the globalstability of the fixed point (119903111988611 0 0) of the map 119879 by usingTheorems 2ndash5without applying a Lyapunov type function like119881119896

8 Discrete Dynamics in Nature and Society

0

2

4

0

1

20

1

V

IVII

I

III

x 2

x3

x1

(a)

0 1 2 3 40

1

I II

III IV

V

x1

x2

f1 = 0

f2 = 0

f3 = 0

(b)

0 1 2 3 40

1

x1

x3

f1 = 0

f2 = 0

f3 = 0

(c)

Figure 2 Trajectories for five different initial points denoted by the black circlesThe initial points are (05 0375 03562) (115 02125 01281)(0125 15 15) (15 0125 02813) and (4 05 025) which are contained in regions I II III IV and V defined in (11) respectivelyThe browndotted lines mean the trajectories which converge to 120599 = (65 0 25) denoted by the magenta circle The three planes 119891

119894= 0 are depicted in

Figure 1

Discrete Dynamics in Nature and Society 9

0 10 20 30 40 50

I

II

III

IV

V

Time

Region

(a)

II

(b)

III

(c)

IV

(d)

V

(e)

Figure 3 The region containing (1199091119896 119909

2119896 119909

3119896) at time 119905 = 119896Δ119905 for the five trajectories in Figure 2 (a) The initial point (11990910 119909

20 119909

30) is (05 0375

03562) denoted by the black circle and located at (0 I) The green segments denote the regions containing (1199091119896 119909

2119896 119909

3119896) The region having 119875

is connected to the region having 119879(119875) by the brown line The other figures (b) (c) (d) and (e) are similarly obtained by using the differentinitial points in Figure 2

10 Discrete Dynamics in Nature and Society

4 Numerical Examples

In this section we consider the Euler difference scheme for(2)

Δ1199091119896= 119909

1119896(2minus1199091

119896minus 21199092119896minus 21199093119896) Δ119905

Δ1199092119896= 119909

2119896(minus3+1199091

119896minus119909

2119896minus119909

3119896) Δ119905

Δ1199093119896= 119909

3119896(minus2+ 21199091

119896+119909

2119896minus119909

3119896) Δ119905

(80)

where Δ119905 = 0001 and 119896 = 0 1 Applying Theorem 41 in[28] to (80) we can obtain the positivity and boundedness ofthe solutions For example

(1199091119896 119909

2119896 119909

3119896) isin (0 4) times (0 2) times (0 8) forall119896 (81)

The fixed point 120599 in (19) becomes 120599 = (65 0 25) which isdenoted by themagenta circle in Figures 1 and 2 and the valueΔ119905 = 0001 satisfies all conditions (9) (15) (16) (47) and(48) Consequently 120599 = (65 0 25) is globally stable whichis demonstrated in Figure 2 for five different initial points(119909

10 119909

20 119909

30) contained in regions I to V respectively

In order to verify the results in Table 1 we mark theregions containing 119879(119875) for 119875 located in the five trajectoriesin Figure 2 and present the result in Figure 3 which demon-strates that the regions containing 119879(119875) follow the rule inTable 1 For example we can see in Figure 3 that 119879(119875) cannotbe contained in V for every 119875 isin Icup IIcup IIIcup IV and if 119875 isin IIthen 119879(119875) can be contained only in II cup IV

5 Conclusion

In this paper we have developed a new approach to obtainthe global stability of the fixed point of a discrete predator-prey system with one prey and two predators The main ideaof our approach is to describe how to trace the trajectoriesIn this process we calculate the sign of the rate change ofpopulation of species so that we call our method the signmethod Although we have applied our sign method for thethree-dimensional discrete model the sign method can beutilized for two-dimensional and other higher dimensionaldiscrete models

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the 2015 Research Fund ofUniversity of Ulsan

References

[1] J Alebraheem and Y Abu-Hasan ldquoPersistence of predators ina two predators-one prey model with non-periodic solutionrdquoApplied Mathematical Sciences vol 6 no 19 pp 943ndash956 2012

[2] S B Hsu and T W Huang ldquoGlobal stability for a class ofpredator-prey systemsrdquo SIAM Journal on Applied Mathematicsvol 55 no 3 pp 763ndash783 1995

[3] J Huang S Ruan and J Song ldquoBifurcations in a predator-prey system of Leslie type with generalized Holling type IIIfunctional responserdquo Journal of Differential Equations vol 257no 6 pp 1721ndash1752 2014

[4] Y Li and D Xiao ldquoBifurcations of a predator-prey system ofHolling and Leslie typesrdquo Chaos Solitons amp Fractals vol 34 no2 pp 606ndash620 2007

[5] S Ruan and D Xiao ldquoGlobal analysis in a predator-prey systemwith nonmonotonic functional responserdquo SIAM Journal onApplied Mathematics vol 61 no 4 pp 1445ndash1472 2000

[6] H K Baek and D S Kim ldquoDynamics of a predator-preysystem with mixed functional responsesrdquo Journal of AppliedMathematics vol 2014 Article ID 536019 10 pages 2014

[7] S Liu and E Beretta ldquoA stage-structured predator-prey modelof Beddington-DeAngelis typerdquo SIAM Journal onAppliedMath-ematics vol 66 no 4 pp 1101ndash1129 2006

[8] S Shulin and G Cuihua ldquoDynamics of a Beddington-DeAngelis type predator-prey model with impulsive effectrdquoJournal of Mathematics vol 2013 Article ID 826857 11 pages2013

[9] H Xiang X-Y Meng H-F Huo and Q-Y Yin ldquoStability in apredator-prey model with Crowley-Martin function and stagestructure for preyrdquo Applied Mathematics and Computation vol232 pp 810ndash819 2014

[10] X Y Zhou and J G Cui ldquoGlobal stability of the viral dynam-ics with Crowley-Martin functional responserdquo Bulletin of theKorean Mathematical Society vol 48 no 3 pp 555ndash574 2011

[11] P H Crowley and E K Martin ldquoFunctional responses andinterference within and between year classes of a dragony popu-lationrdquo Journal of the North American Benthological Society vol8 no 3 pp 211ndash221 1989

[12] X Wang and H Ma ldquoA Lyapunov function and global stabilityfor a class of predator-prey modelsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 218785 8 pages 2012

[13] H B Xiao ldquoGlobal analysis of Ivlevrsquos type predator-preydynamic systemsrdquo Applied Mathematics and Mechanics vol 28no 4 pp 419ndash427 2007

[14] J Sugie ldquoTwo-parameter bifurcation in a predator-prey systemof Ivlev typerdquo Journal of Mathematical Analysis and Applica-tions vol 217 no 2 pp 349ndash371 1998

[15] C Celik andODuman ldquoAllee effect in a discrete-time predator-prey systemrdquoChaos Solitons amp Fractals vol 40 no 4 pp 1956ndash1962 2009

[16] S R J Jang ldquoAllee effects in a discrete-time host-parasitoidmodelrdquo Journal of Difference Equations and Applications vol 12no 2 pp 165ndash181 2006

[17] P C Tabares J D Ferreira and V Sree Hari Rao ldquoWeak Alleeeffect in a predator-prey system involving distributed delaysrdquoComputational amp Applied Mathematics vol 30 no 3 pp 675ndash699 2011

[18] R Cui J Shi and B Wu ldquoStrong Allee effect in a diffusivepredator-prey system with a protection zonerdquo Journal of Differ-ential Equations vol 256 no 1 pp 108ndash129 2014

[19] J D Flores and E Gonzalez-Olivares ldquoDynamics of a predator-prey model with Allee effect on prey and ratio-dependentfunctional responserdquo Ecological Complexity vol 18 pp 59ndash662014

Discrete Dynamics in Nature and Society 11

[20] N Shigesada K Kawasaki and E Teramoto ldquoSpatial segrega-tion of interacting speciesrdquo Journal of Theoretical Biology vol79 no 1 pp 83ndash99 1979

[21] G Zhang and X Wang ldquoEffect of diffusion and cross-diffusionin a predator-prey model with a transmissible disease in thepredator speciesrdquo Abstract and Applied Analysis vol 2014Article ID 167856 12 pages 2014

[22] Z Xie ldquoCross-diffusion induced Turing instability for a threespecies food chainmodelrdquo Journal ofMathematical Analysis andApplications vol 388 no 1 pp 539ndash547 2012

[23] L Yang and S Zhong ldquoGlobal stability of a stage-structuredpredator-prey model with stochastic perturbationrdquo DiscreteDynamics in Nature and Society vol 2014 Article ID 512817 8pages 2014

[24] T C Gard ldquoPersistence in stochastic food webmodelsrdquo Bulletinof Mathematical Biology vol 46 no 3 pp 357ndash370 1984

[25] M Vasilova ldquoAsymptotic behavior of a stochastic Gilpin-Ayalapredator-prey system with time-dependent delayrdquo Mathemati-cal and Computer Modelling vol 57 no 3-4 pp 764ndash781 2013

[26] Q Din ldquoDynamics of a discrete Lotka-Volterra modelrdquoAdvances in Difference Equations vol 2013 article 95 13 pages2013

[27] T Wu ldquoDynamic behaviors of a discrete two species predator-prey system incorporating harvestingrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 429076 12 pages 2012

[28] S M Choo ldquoGlobal stability in n-dimensional discrete Lotka-Volterra predator-prey modelsrdquo Advances in Difference Equa-tions vol 2014 no 1 article 11 17 pages 2014

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