Stability of a Discrete-Time Predator-Prey System with Allee Effect

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1 Nonlinea Analsis an Diffeential Equations, Vol. 4, 6, no. 5, 5-33 HIKARI Lt, Stabilit of a Discete-Time Peato-Pe Sstem with Allee Effect Ming Zhao an Yunfei Du LMIB-School of Mathematics an Sstems Science Beihang Univesit, Beijing 9, China Copight c 6 Ming Zhao an Yunfei Du. This aticle is istibute une the Ceative Commons Attibution License, which pemits unesticte use, istibution, an epouction in an meium, povie the oiginal wok is popel cite. Abstact The stabilit of a iscete-time peato-pe sstem with an without Allee effect is investigate in this pape. B analzing both sstems, we fist obtain local stabilit conitions of the fie points without the Allee effect an then ehibit the impact of the Allee effect on stabilit when it is impose on pe population. The theoetical analsis an numeical simulations show that the Allee effect has a estabilizing foce on the local stabilit of the positive fie point of this sstem. Mathematics Subject Classification: 37N5, 65P4 Kewos: Peato-pe sstem; Stabilit; Allee effect Intouction Stuing the stabilit of population namic has been an active eseach aea fo a long time. Most namic population moels ae escibe b the iffeential an iffeence equations. Actuall iscete-time moels escibe ae moe easonable than the continuous-time moels when populations have non-ovelapping geneations. Moeove, using iscete-time moels can also povie moe efficient computational moels fo numeical simulations an these esults eveal fa iche namical behavios of the iscete-time moels compae to the continuous ones [4]. Allee effect is a phenomenon in population namics attibute to the biologist Allee[]. It escibes a positive coelation between population ensit

2 6 Ming Zhao an Yunfei Du (size) an the pe capita population gowth ate in small populations. Allee effect has a significant impact on population namics an pesistence, which can be case b ifficulties in fining mate an peato avoiance o efense []. Thee has been a enewe inteest in Allee effect ecentl an some of the elevant wok ma be foun in [5, 6, 7, 8, 9]. Howeve, a few papes have aesse the Allee effect with focus on the stabilit of iscete-time peato-pe sstems. Fo eample, the stabilit of a iscete-time peato-pe sstem was stuie in [6] an [9], in which the showe the stabilizing effect of Allee effect. The main aim of this pape is to investigate the stabilit of a iscete-time peato-pe sstem with an without Allee effect. We consie the following sstem in [3]: { n+ = n ( n ) b n n n+ = n n () whee an epesent population ensities of a pe an a peato, espectivel, an, b, ae positive paametes. Stabilit analsis of sstem Let u = an v = b, then the sstem () is convete to { un+ = u n ( u n ) u n v n. v n+ = u n v n. Fo simplicit, we will still use an instea of u an v, thus the sstem () can be ewitten as: { n+ = n ( n ) n n. () n+ = n n. Below we consie the above moifie sstem (). It is clea that the fie points of sstem () satisf the following equations: { ( ) =. =. B simple computation, we get that the sstem () has one etinction fie point (, ), one eclusion fie point (, ), an one coeistence fie point (, ) = (, ). Thus (, ) is the unique positive fie point of sstem (). Now we stu the stabilit of these fie points. Fo the fie point (, ), the coesponing chaacteistic equation is λ λ = an its oots ae λ = an λ =. Hence, (, ) is asmptoticall stable when < < an it is unstable when >.

3 Stabilit of a iscete-time peato-pe sstem 7 Fo the eclusion fie point (, ) fo >. Lineaizing the sstem () about (, ), we have the following coefficient mati: ( J = ( ) Cleal, J has eigenvalues λ =, λ = ( ). λ i < (i =, ) hols iff ). < < 3 an < <. Below we focus on the positive fie point (, ) = (, ) fo > ( > ). The Jacobian evaluate at the positive fie point (, ) is given b ( ) J =, an the chaacteistic equation of the Jacobian mati J can be witten as: P (λ) = λ (tj )λ + etj = λ ( ) λ + =. (3) Accoing to the Ju conitions[4], in oe to fin the asmptoticall stable egion of (, ), we nee to fin the egion that satisfies the following conitions: P () >, P ( ) > an etj <. Since P () =, P ( ) = 3 3+, etj = ( ), then fom the elations P () >, P ( ) > an etj <, we obtain < 3, < < o 3 < < 9, < <. Theefoe, we can summaize the above analsis in the following esult. Theoem.. Fo the peato-pe sstem (), the following statements ae tue: () (, ) is asmptoticall stable if < < ; () (, ) is asmptoticall stable if < < 3 an < < ; (3) (, ) is asmptoticall stable if an onl if one of the following conitions hols: (i) < 3 an < < ; (ii) 3 < < 9 an 3 < <. 3+

4 8 Ming Zhao an Yunfei Du 3 Allee effect on pe population In this section we consie the peato-pe sstem () is subject to an Allee effect on pe population an analze the following sstem: { n+ = n ( n ) n ε+ n n n, (4) n+ = n n, 4 whee we take n ε+ n as the Allee effect function an ε as the Allee constant satisfing the assumption < ε <. Afte a simple calculation, we have the sstem () has one etinction fie point (, ), two eclusion fie points ( ( ) 4ε, ) an ( + ( ) 4ε, ), an one coeistence (positive) fie point ( ε, ε) = (, ε). (+ε) Fo the fie point (, ), the coesponing chaacteistic equation is λ = an its oots ae λ = λ = that means (, ) is asmptoticall stable. Fo the eclusion fie point ( ( ) 4ε, ) fo > an < ε < ( ). Lineaizing the sstem (4) about ( ( ) 4ε, ), we have the coesponing Jacobian Mati as follows: A J = ( (+ε)+ ( ) 4ε) ( ) 4ε ( ( ) 4ε) 4 whee A = ( ε + 8 ε 3 ε + 6 ε ) + ( 3 + 5ε + ε) ( ) 4ε. Obviousl, J has chaacteistic oots λ = A, λ ( (+ε)+ ( ) 4ε) = ( ( ) 4ε), an λ > fo > an < ε < ( ). Hence, the fie point ( ( ) 4ε, ) is unstable. 4 Fo the eclusion fie point ( + ( ) 4ε, ) fo > an < ε < ( ). Lineaizing the sstem (4) about ( + ( ) 4ε, ), we have the following coefficient mati: J = B ((+ε) + ( ) 4ε) + ( ) 4ε ( + ( ) 4ε) whee B = ( ε+8 ε 3 ε+6 ε ) ( 3 + 5ε+ ε) ( ) B 4ε. J has chaacteistic oots λ = ((+ε) + ( ), 4ε) λ = ( ( ) 4ε). λ i < (i =, ) hols if an onl if { B < (( + ε) + ( ) 4ε), < ( ) 4ε < + (5).,,

5 Stabilit of a iscete-time peato-pe sstem 9 Une assumption < ε <, the peato-pe sstem (4) has unique positive fie point ( ε, ε) = (, ε). It is clea that the point (+ε) ε is smalle than, which implies peato ensit at the fie positive is ecease ue to Allee effect. The Jacobian evaluate at the positive fie point ( ε, ε) is given b ) J ε = ( ε+ ε(+)+ 3 ε (+ε) + + ε +ε an the chaacteistic equation of the Jacobian mati J ε can be witten as:, P ε (λ) =λ (tj ε )λ + etj ε =λ + ε(4 + ) + 3 ε + ( ε) λ + ( 3ε + ε) ( + ε) ( + ε) =. Then we have an Pε () = ε, ( + ε) P ε ( ) = ε 5ε + 3 ε ε ( + ε), Consequentl, P ε () > iff P ε ( ) > iff etj ε < iff etj ε = ( 3ε + ε) ( + ε). + + ε <, ε 5ε + 3 ε ε >, + + ε + 3ε ε + 3 ε >. Thus, we get the following esult. Theoem 3.. Fo the peato-pe sstem (4), the following statements ae tue: () (, ) is asmptoticall stable; () ( ( ) 4ε, ) is unstable;

6 3 Ming Zhao an Yunfei Du (3) ( + ( ) 4ε, ) is asmptoticall stable if >, < ε < ( ) an 4 the conitions (5) ae satisfie. (4) (, ε) is asmptoticall stable if an onl if all of the following (+ε) conitions hol: (i) + + ε < ; (ii) ε 5ε + 3 ε ε > ; (iii) + + ε + 3ε ε + 3 ε >. Remak. If we choose the Allee constant ε =, then sstem (4) euces to sstem () immeiatel. Howeve, when < ε <, we will see that the asmptotic stabilit of the fie point ( ε, ε) is weake than that of (, ) in numeical simulations (see Section 4). Futhemoe, fo some fie paametes, an ε satisfing the conitions in Theoem., we will see that (, ) is asmptoticall stable while ( ε, ε) is unstable in section 4. 4 Numeical simulations In this section, we pesent the gaphs of the solutions aoun the positive fie point fo the peato-pe sstems with an without Allee effect (sstem () an (4), espectivel) an show the impact of the Allee effect on the obits of the solutions. The phase potaits of sstems () an (4) fo = 3.5 with initial value (.,.5) ae ispose in Fig., which cleal epicts the obits of peato an pe ensities. We take =.8 in Fig.(a) an (b) while =. in Fig.(c) an (). Hee (a) an (c) show the obits of pe an peato ensities in sstem (), howeve, (b) an () coespon to sstem (4) that is subject to the Allee effect b taking the same paametes as in Fig.(a) an (c). Fom Fig.(a) an (b). we obseve that when the pe population is subject to an Allee effect, the local stabilit of the fie eceases an obit of the solution appoimates to the coesponing fie point much slowe. Futhemoe, Fig.(c) an () pesents that the coesponing fie points move fom stabilit to instabilit une the Allee effect. (a) (b)

7 3 Stabilit of a iscete-time peato-pe sstem (c) () Fig. : The obits of pe an peato ensities with an without Allee effect fo = 3.5 an the initial value is (.,.5). The gaph in (a) (esp. (c)) inicates the solution of sstem () with =.8 (esp. =.), howeve, the gaph in (b) (esp. ()) coespons to sstem (4) when the pe population is subject to the Allee effect with =.8 (esp. =.) an ε = a a (a) (b) a (c) a () Fig. : Bifucation iagams of pe an peato ensities in sstems () an (4) fo the paamete values = 3.5, ε =.5 an the initial value (.,.5). (a) an (c) ae given b sstem () while (b) an () coespon to sstem (4). Fig. shows the bifucation iagams of pe an peato ensities of sstems () an (4) with the initial conitions (.,.5) as above an the

8 3 Ming Zhao an Yunfei Du paamete values = 3.5, ε =.5. Fig.(b) an () show the bifucations of pe an peato ensities of sstem (4), espectivel, when the pe population is subject to the Allee effect; howeve, the othe gaphs coespon to the bifucations in the peato-pe sstem (). The show how the bifucation iagams ae change une the Allee effect. It can be seen fom the gaphs that the Allee effect has a estabilizing effect fo this case. 5 Conclusions Allee effect plas an impotant ole on the stabilit analsis of fie points of a population namics sstem. In this pape, we stuie the estabilizing effect of Allee effect on pe population in a iscete-time peato-pe sstem. B numeical simulations, we have shown the impact of the Allee effect (on pe population) on the stabilit of the positive fie point fo sstem (). The fie point coul be change fom stable to unstable o othewise, will take much longe time to each the stable state when it is stable. Refeences [] W.C. Allee, The Social Life of Animals, William Heinemann, Loon, [] F. Couchamp, L. Beec, J. Gascoigne, Allee Effects in Ecolog an Consevation, Ofo Univesit Pess, 8. [3] M. Danca, S. Coeanu, B. Bakó, Detaile Analsis of a Nonlinea Pe-peato Moel, Jounal of Biological Phsics, 3 (997), -. [4] J.D. Mua, Mathematical Biolog, Spinge-Velag, New Yok, [5] I. Scheuing, Allee effect inceases the namics stabilit of populations, J. Theo. Biol., 99 (999), [6] C. Úelik, O. Duman, Allee effect in a iscete-time peato-pe sstem, Chaos, Solitons an Factals, 4 (9), [7] S.N. Elai, R.J. Sacke, Population moels with Allee effect: A new moel, Jounal of Biological Dnamics, 4 (),

9 Stabilit of a iscete-time peato-pe sstem 33 [8] Ünal Ufuktepe, Allee effects in population namics, Biomath Communications, (4), -9. [9] W.X. Wang, Y.Bo. Zhang, C.Z. Liu, Analsis of a iscete-time peatope sstem with Allee effect, Ecological Compleit, 8 (), Receive: Mach 7, 6; Publishe: Apil, 6

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