Research Article Dynamics of a Predator-Prey System with a Mate-Finding Allee Effect

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1 Abstract and Applied Analsis, Article ID , 14 pages Research Article Dnamics of a Predator-Pre Sstem with a Mate-Finding Allee Effect Ruiwen Wu and Xiuiang Liu School of Mathematical Sciences, South China Normal Universit, Guangzhou 51631, China Correspondence should be addressed to Xiuiang Liu; liu@scnu.edu.cn Received 26 Jul 214; Revised 11 September 214; Accepted 11 September 214; Published 2 October 214 Academic Editor: Weinian Zhang Copright 214 R. Wu and X. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. We consider a ratio-dependent predator-pre sstem with a mate-finding Allee effect on pre. The stabilit properties of the equilibria and a complete bifurcation analsis, including the eistence of asaddle-node, ahopf bifurcation, and, abogdanov-takens bifurcations, have been proved theoreticall and numericall. The blow-up method has been applied to investigate the structure of a neighborhood of the origin. Our mathematical results show the mate-finding Allee effect can reduce the compleit of sstem behaviors b making the complicated equilibrium less complicated, and it can be a destabilizing force as well, which makes the sstem has a high possibilit of being threatened with etinction in ecolog. 1. Introduction Just as pointed out b Berrman in [1] that thisdnamical relationship between predators and their pre has long been and will continue to be one of the dominant themes in both ecolog and mathematical ecolog due to its universal eistence and importance, both ecologists and mathematicians areinterestedinthednamicalanalsisofpredator-pre models. It is well known that the classical Gause tpe predator-pre sstem is written b dn dτ dp dτ =G(N) N f(n, P) P, =ef(n, P) mp, where N(τ) and P(τ) are the densities of pre and predator, respectivel, e represents the trophic efficienc or the conversion efficienc of predator, and the parameter m characterizes the predator natural mortalit rate. The functional response f(n, P), which quantifies the amount of pre consumed per predator per unit time and plas an important role in predator-pre dnamics, is conventionall modeled as predependent, where the pre consumption rate b an average predator is onl a function of pre densit alone; that is, f(n, P) = f(n). Different pre-dependent response tpes (1) (e.g., the mass-action approach in Lotka-Volterra model and Holling tpes I III) have been used to model the predatorpre interactions and get success in describing some ecological communities. When the spatial structure of one or both of the interacting populations is involved, it would be more plausible to take the predator-dependent functional form, where both predator and pre densities affect the response, for eample, Hassell and Varle function response [2] and Beddington-DeAngelis functional response functional [3]. In this paper, we are interested in one important form of predator-dependent functional response, called ratiodependent response; that is, the functional response depends on the term N/P and the corresponding Monod-Holling hperbolic form is f( N P )= an N+ahP, (2) where a is the maimum pre consumption rate and h is the predator handling time, which is proposed b Arditi and Ginzburg [4]andstudiedbmanauthors;see,foreample, Berezovskaa et al. [5, 6], Kuang et al. [7 1], and Zhang et al. [11, 12] in which we know this ratio-dependent functional response provides more reasonable eplanations and accurate predictions when communit-level situations of food chains and food webs are considered.

2 2 Abstract and Applied Analsis Another function G(N) in (1) is the per capita growth rate of pre population in the absence of predators. The classical viewofpopulardnamicsisthatthemajorecologicalforce at work is the release from the constraints of intraspecific competition when a population is small or at low densit. The fewer we are, the more we all have and the better welfare be there. In general, G(N) takes the form G (N) =r(1 N ), (3) K where r and K are the intrinsic growth rate and the environment carring capacit of pre, respectivel. However, the ecologist Allee gathered sufficient eperimental and observational data to conclude that the evolution of social structures was not onl driven b competition, but that cooperation was another, if not the most, fundamental principle in animal species [13]. The dnamical consequences of this importance ofanimalaggregationsdirectlledtowhatodumcalledin 1953 the Allee principle, now known as the Allee effect [14]. There are two conventional tpes of Allee effects: component Allee effect and demographic Allee effect. The positive relationship between an component of individual fitness and population size can be regarded as a component Allee effect, whileademographicalleeeffectislinkedtothelevelof overallindividualfitness[15]. Moreover, a demographic Allee effect is strong if there eists an Allee threshold, below which the per capita growth rate becomes negative. The classical and simplest tpes of demographic Allee effects are considered commonl, which takes the mathematical form of G (N) =r(n A) (1 N ), (4) K where A denotes the Allee threshold if A>.Muchworkhas been done on strong Allee effects; see, Hilker et al. [16, 17], Shi et al. [18, 19], and González-Olivares et al. [2, 21] andtheir citations. A wide range of mechanisms which ma result in Allee effects are considered and discovered, from fertilization efficienc in sessile organisms to pollen limitation in plants andtocooperativehuntinginanimals.sofar,oneofthemost important and probabl the most studied mechanisms for Allee effects is called made-finding; that is, individuals in a population fail to find a suitable mate at low densit, thus resulting in fewer reproductive outputs and eamples include Glanville fritillar butterfl [15], sheep ticks, and whales [15, 22], and such mechanism ma cause a mate-finding (component) Allee effect potentiall. Mathematicall, a mate-finding process can be modeled b a female mating rate which has positive dependence on densit, denoted b M(N). The hperbolic function M (N) = N N+θ, (5) where θ scales the mate-finding Allee effects, is widel used [15, 22 24]. In our paper, we aim to stud a predator-pre sstem which is subject to the ratio-dependent functional response and a mate-finding Allee effect, and we believe it will give people a better understanding of Allee effects in predatorpre sstems. More precisel, we consider the following sstem: dn dτ =bn N θ+n dn(1+n K ) anp N+ahP, dp dτ = eanp N+ahP mp, with the initial conditions N() >, P() >.Bintroducing the dimensionless variables given b t=dτ, =N/K, and = ahp/k, then the sstem (6)becomes d dt =g δ+ d dt = α 1 + β:=f 2 (, ), (1+) α + := F 1 (, ), where g = b/d, α = 1/dh, α 1 =ae/d, β=m/d,andδ= θ/k (relative strength of the mate-finding Allee effect). Now we have rescaled the sstem from (6) to(7) breducingthe number of parameters from seven to five. For the sake of consistenc, we make the same dimensionless transformation of g = b/d as that in the sstem studied b Pavlová etal.[25]. Note that b>d>,so (A)g=b/d>1. We assume that (A) holds in our paper and all the other parameters are positive; that is, δ, α, α 1,β>. We organize the paper as follows. In Section 2, weshow thesstemisdissipativeandthestabilitandbifurcationanalsis of complicated equilibrium E, predator-free equilibria, and positive equilibria are given. We discuss our findings and summarize our conclusions in Section 3, and numerical simulations are also carried out to support our findings. 2. Stabilit and Bifurcation Analsis 2.1. Dissipativetit. Note that /(+) is not defined at (, ), and we can redefine the derivative as follows: (6) (7) F 1 (, ) = F 2 (, ) =, (, ) = (, ). (8) Finall we define F:=( F 1 F 2 ). Clearl, with the etended definition, there globall eists a unique solution of sstem (7) for an given nonnegative initial condition. Obviousl, R 2 + = {(, ), } is an invariant set. The following lemma shows that sstem (7)is dissipative. Lemma 1. Let ((t), (t)) be a solution of sstem (7);then,one has lim sup t + ( (t) + α (t)) (g+β 1)2. (9) α 1 4β

3 Abstract and Applied Analsis (a) δ =.42, δ>δ 1 (b) δ =.4, δ=δ (c) δ =.32, δ<δ 1 Figure 1: Phase portraits of sstem (7)ifα 1 <βand for g = 1.44, δ 1 =.4, α =.4, α 1 =.1,andβ =.3. Proof. Let V(t) = (t)+(α/α 1 )(t). Differentiating V,onehas For the sake of simplicit in calculation, we focus on the following sstem which is equivalent to sstem (7): V (t) = g2 +δ 2 +(β 1) βv(t) < 2 +(g+β 1) βv(t) (g+β 1)2 4 βv(t). (1) d dt = 3 +(g 1 δ) 2 δ α2 +αδ, + d dt = α 1 2 +α 1 δ β(+δ). + (11) Thus, we have lim sup t + V(t) (g+β 1) 2 /4β and sstem (7) is dissipative. This completes the proof Complicated Equilibrium E (, ) and Its Stabilit. Since the Jacobian matri cannot be evaluated at E (, ) (noting that F 1 and F 2 are not differentiable at (, )), the classical stabilit anlsis methods are not applied. We appl the algorithm presented in [6] to investigate the structure of a neighborhood of E (, ) and show that sstem (7)hasastableE (, ) for all sstem parameters. Such result is obtained numericall in Figures 1, 2, 4,and6(a). Definition 2. AvectorfieldW(, ) = P(, )( / )+Q(, ) ( / ) is nondegenerate if it satisfies the following. (A1) Polnomials P n (, ) and Q n (, ) have no common factors of the form μ+],wherep(, ) = P n (, )+ P(, ), Q(, ) = Q n (, ) + Q(, ), and at least one of the constants μ, ] is nonzero. (A2) Polnomial L(, ) has no factors of the form (μ + ]) k,wherek > 1,andL(, ) = Q n (, ) P n (, ).

4 4 Abstract and Applied Analsis (a) A saddle E 1 and a stable E 2 (δ =.8) (b) A saddle E 1 and a stable E 2 surrounded b a homoclinic loop (δ =.926) (c) A saddle E 1 and an unstable E 2 (δ =.99) Figure 2: Phase portraits of sstem (7)ifα>α 1 and for g = 1.44, α =.8, α 1 =.4,andβ =.3. B the time scale change dt dt/(+),thesstem(11) takes the form Let d dt = 4 3 +(g 1 δ) 3 +(g 1 δ α) 2 δ 2 (α+1) δ := P, d dt =(α 1 β) 2 β 2 +(α 1 β)δ βδ 2 := Q. (12) P 2 (, ) = δ 2 (α+1) δ, andthenonehas P(,)=P 2 (, ) + P(,), P(,)=o(, 3 ), Q(,)=Q 2 (, ) + Q(,), Q(,)=o(, 3 ), L (, ) =Q 2 (, ) P 2 (, ) =δ[(α 1 β+1)+(α β+1)]. (14) It is eas to see that the vector field W(, ) of (12) isnondegenerate. After appling the blow-up transformations (, ) (, u) with u=, =, (15) Q 2 (, ) = (α 1 β)δ βδ 2, P(,)= (g 1 δ) 3 +(g 1 δ α) 2, Q(,)=(α 1 β) 2 β 2, (13) and the time change dt dt,onehas d dt =P 2 (1, u) +G 1 (, u), du dt =L 1 (u) +G 2 (, u), (16)

5 Abstract and Applied Analsis δ 2 δ Hopf magnitude δ h δ (a) Bifurcation structure (b) δ =.9724(δ = δ h ), the unstable limit ccle when a subcritical Hopf bifurcation occurs (c) δ =.9724(δ = δ h ), the trajector around the unstable limit ccle when a subcritical Hopf bifurcation occurs Figure 3: Bifurcation structure of E 2 for Hopf bifurcation and phase portraits of sstem (7)forg = 1.44, α =.8, α 1 =.4,andβ = (a) Inside the limit ccle (b) Outside the limit ccle Figure 4: An unstable limit ccle coeists with a stable E 2,forg = 1.44, δ =.95, α =.84, α 1 =.4,andβ =.3.

6 6 Abstract and Applied Analsis H SN H SN δ δ (a) Pre population scenario (b) Predator population scenario Figure 5: A saddle-node bifurcation of sstem (7) atδ =.1932 (markassn),forg = 1.44, α =.8, α 1 =.4, andβ=.3.ahopf bifurcation also occurs..45 Domain 1.8 α BT Limit point curve (green) Domain 2 Hopf curve (blue) Homoclinic curve (magenta) δ (a) Bifurcation structure (b) A cusp E Figure 6: Bifurcation structure of sstem (7)in(δ, α 1 ) with a Bogdanov-Takens bifurcation and phase portraits for g = 1.44, δ =.22859, α =.4, α 1 =.4,andβ =.3. where L 1 (u) =L(1, u) =δu[(α 1 β+1)+(α β+1)u], G 1 (, u) = P (, u), G 2 (, u) = ( Q (, u) P (, u) u) 2. (17) (1) If α β+1<and α 1 β+1<, O 1 (, u 1 ) is a single stable node. (2) If α β+1>and α 1 β+1>, O 1 (, u 1 ) is a single saddle. (3) If α β+1> and α 1 β+1 <, O 1 (, u 1 ) is a stable node and O 2 (, u 2 ) is a saddle. Thus, if (α β+1)(α 1 β+1) >, L 1 (u) has onl one nonnegative root u 1 =,andconsequentl,sstem(16)hasoneequilibrium O 1 (, u 1 ). L 1 (u) has two nonnegative roots: u 1 = and u 2 = (α 1 β+1)/(α β+1) if (α β+1)(α 1 β+1) <,and thus sstem (16)hastwoequilibriaO 1 (, u 1 ) and O 2 (, u 2 ) on u-ais. We state the following lemma. (4) If α β+1 < and α 1 β+1 >, O 1 (, u 1 ) is a saddle and O 2 (, u 2 ) is a stable node. Proof. We onl verif (3) since the other three results can be showninasimilarwa.itiseastoseethat Lemma 3. We have the following results. P 2 (1, u 1 )= δ<, L 1 (u 1)=δ(α 1 β+1),

7 Abstract and Applied Analsis 7 P 2 (1, u 2 )= δ(αα 1 αβ+α 1 ), α β+1 L 1 (u 2)= δ(α 1 β+1). (18) If α β+1>and α 1 β+1<,thatis,α 1 <β 1<α, then L 1 (u 1)<, L 1 (u 2)>,and P 2 (1, u 2 )= δ(αα 1 αβ+α 1 ) α β+1 < δ( α+α 1) α β+1 <. (19) Based on [6,Proposition3],wecanconcludethatthatO 1 is a stable node and O 2 is a saddle, respectivel. We obtain the following sstem in a similar wa b making transformations (, ) (V,)with and dt dt;then,we obtain V =, =, (2) dv dt =L 2 (V) +J 1 (V,), d dt =Q 2 (V,1) +J 2 (V,), where L 2 (V) = L(V,1) = δv [(α 1 β+1)v +(α β+1)], J 1 (V,)= ( P(V, ) Q(V, ) V) 2, J 2 (V,)= Q(V, ). (21) (22) Thus, if (α β + 1)(α 1 β+1)>,sstem(21) has onl one nonnegative equilibrium O 1 (,). The sstem (21) hastwo equilibria O 1 (, ) and O 2 (V 2,)on the V-ais if (α β+1)(α 1 β+1)<.however,o 2 corresponds to O 2;thus,wedonot need to stud it again. O 1 is a new equilibrium and it does not eist in the (, u)-plane. Note that L 2 () = δ(α + 1 β) and Q 2 (, 1) = βδ <. Again, based on [6,Proposition3],then comes the following conclusion. Lemma 4. We sa that (1) if α β+1<, O 1 (, ) is a saddle; (2) if α β+1>, O 1 (, ) is a stable node. According to Lemmas 3 and 4, itthenfollowsfrom[6] that the following result holds for the origin E (, ) of sstem (7), which is equivalent to sstem (11). Theorem 5. If (α β + 1)(α 1 β+1) =, then there is at least one attracting parabolic sector in the neighborhood of (, ) in thefirstquadrant,andnohperbolicorellipticsectorcanbe found; that is, the complicated equilibrium E (, ) is stable. Remark 6. If (α β + 1)(α 1 β+1)>, the neighborhood of E (, ) in the first quadrant has a parabolic (attracting) sector, but if (α β+1)(α 1 β+1) <,therearetwoparabolic (attracting) sectors; in fact, we can regard those two sectors as a whole attracting one. If (α β + 1)(α 1 β+1)=, O i, i=1,2arealsocomplicated equilibria and needed to appl the blow-up transformation again. Now we can see that the relative strength of mate-finding Allee effect δ greatl changes the structure of the neighborhood of E (, ) in our sstem (7), because P 2 (1, u 1 ) = δ and Q 2 (, 1) = βδ alwas sta negative and O 1 (, ) and O 1 (, ) cannot be unstable; thus, the diversit of phase portraits around E (, ) of ratiodependent pre-predator sstems decreases. More importantl, the sstem becomes fragile and even has a higher risk of etinction due to mate-finding Allee effect Eistence and Stabilit of Predator-Free Equilibria. In order to analze the predator-free equilibria of the sstem (7), we set Δ 1 =(g δ 1) 2 4δ. (23) We can notice that, for =, there eist two boundar equilibria E 1 =( 1,)and E 2 =( 2,)if <δ<g 1 and Δ 1 >;thatis, (B1) < δ < δ 1, where δ 1 := ( g 1) 2.Let i, i = 1,2 be the roots of the quadratic equation 2 +(1+δ g)+δ=;thatis, 1 = 1 2 (g δ 1 (g δ 1) 2 4δ), 2 = 1 2 (g δ 1+ (g δ 1) 2 4δ), (24) and the appear in pairs. Besides, there is no equilibrium on -ais. The Jacobian matri evaluated at E i, i=1,2is given b J i =( i ( i +δ) 2 (gδ ( i +δ) 2 ) α ), (25) α 1 β and one eigenvalue is λ 1 (E i )=( i /( i +δ) 2 )(gδ ( i +δ) 2 ), i=1,2,andthesecondeigenvalueisλ 2 (E i )=α 1 β, i=1,2. Note that the sign of λ 1 (E i ), i = 1, 2, depends on ψ i := gδ δ i, i=1,2. Using the epression of i, i=1,2,from(24), we can have ψ 1 = 1 2 (2 gδ δ g+1+ (g δ 1) 2 4δ), ψ 2 = 1 2 (2 gδ δ g+1 (g δ 1) 2 4δ). Under the restriction (B1),it follows that ψ 1 > δ( g δ 1)>, ψ 2 = 1 2 ( ( g δ) 2 +1 Δ 1 )< Δ 1 2 <. (26) (27)

8 8 Abstract and Applied Analsis Hence λ 1 (E 1 ) is alwas positive, while λ 1 (E 2 ) is negative. B using the facts above, we have the following theorem. Theorem 7. Assume (B1)holds.Then (1) if α 1 < β, E 1 is a saddle and E 2 is a stable node (Figure 1(c)), (2) if α 1 >β, E 1 is an unstable node and E 2 is a saddle (Figure 2). Remark 8. If δ=δ 1,sstem(7) hasauniquepredator-free boundar equilibrium point E =(,) (also known as the saddle-node equilibrium), where = 1 2 (g δ 1 1)= δ 1. (28) Figure 1(b) shows the instantaneous equilibrium E in the case of α 1 <βand the behaviors of trajectories are divided b the stable manifold W s (E ) of E (the green curve) Positive Equilibria and Their Stabilit and Bifurcation Results Eistence and Stabilit of Positive Equilibria. The interior (positive) equilibria can be evaluated b the intersections ofthezeroisoclines g +δ (1+) = α 1 + =β, α +, (29) in the first quadrant. The predator zero isocline, with the slope (α 1 β)/β, is a straight line passing through the origin, and it lies in the first quadrant if the following restriction holds (C1) α 1 >β. Besides, we can define σ:=1+α αβ. (3) α 1 Note that σ>1under the restriction (C1).Andfor 2 +(δ+ σ g)+δσ=,wealsoset Δ 2 =(g δ σ) 2 4δσ. (31) If <δ<g σand Δ 2 >,thatis, (B2) < δ < δ 2, where δ 2 := ( g σ) 2,therearetwopositiveequilibria E 1 = ( 1, 1 ) and E 2 = ( 2, 2 ),whichalsoappear simultaneousl, with the coordinates 1 = 1 2 (g δ σ (g δ σ) 2 4δσ), 2 = 1 2 (g δ σ+ (g δ σ) 2 4δσ), (32) i = α 1 β β i, i = 1,2. (33) Note that <δ 2 <δ 1,itsuggeststhatif(B2) holds, then (B1) holds; that is, if there eists an interior equilibrium in the first quadrant, then two predator-free equilibria will be detected. The Jacobian matri J i, i = 1,2 evaluated at an interior equilibrium is given b gδ i ( J i =( i +δ) 2 i + α 1 2 i ( i + i ) 2 α i i ( i + i ) 2 α 2 i det (J i )= α 1 2 i i [( i +δ) 2 gδ] ( i + i ) 2 ( i +δ) 2. ( i + i ) 2 α ), 1 i i ( i + i ) 2 Similarl, we have det(j 1 )<and det(j 2 )>.Moreover, (34) gδ tr (J 2 ) = 2 ( 2 +δ) β(α α 1)(α 1 β), (35) and it is interesting to find that tr(j 2 ) is an increasing function of δ. In fact, we can have d 2 dδ = 1 2 (1 + α 2 1 g δ+σ )<, (36) (g δ+σ) 2 4δσ and we can define 2 := d 2/dδ.Then,itfollowsthat since dtr (J 2 ) dδ = 2 [ gδ ( 2 +δ) 2 1] + 2 g ( 2 +δ) 3 [ 2 δ 2δ 2 ]>, δ 2δ 2 = δ+δ(1+ g δ+σ ) (g δ+σ) 2 4δσ = δ(g δ+σ) >. (g δ+σ) 2 4δσ Moreover, we can have the following conclusion. (37) (38) Theorem 9. Assume that (B2) and (C1) hold. Then one has the following. (1) E 1 is a saddle. (2) If α α 1, E 2 is stable. (3) If α>α 1, E 2 canbestableorunstabledependingonδ (Figure 2).

9 Abstract and Applied Analsis Hopf Bifurcation. Here assume that (B2) and (C1) hold, and we consider δ as the bifurcation parameter to discuss the Hopf bifurcation. It is eas to know that the Hopf bifurcation can occur onl at E 2 as E 1 is alwas a saddle point. Then we tr to show the eistence of δ=δ h such that tr(j 2 )= and det(j 2 ) >. From the epression of tr(j 2 ) and some numerical attempts, we find that other parameters g, α, α 1,andβ still have an important impact on the qualitative properties of the Hopf bifurcation though onl δ is considered as the bifurcation parameter. The following assumption is used to guarantee the esitence of the Hopf bifurcation with bifurcation parameter δ: (C2) α > α 1, (1 (β/α 1 ))(α β + (αβ/α 1 )) < g 1. Assumption (C2) implies that tr(j 2 ) δ= < and tr(j 2 ) δ=δ2 >, and we have known that tr(j 2 ) is a continuous increasing function of δ in [, + ); thus, there eists δ=δ h such that tr(j 2 )=. Furthermore, the transversalit condition (d/dδ) tr(j 2 ) δ=δh >is satisfied. Using the facts above, sstem (7) can undergo a Hopf bifurcation at E 2 for δ=δ h if (C2) holds. Now we tr to discuss the stabilit of the limit ccle of sstem (7)asaHopfbifurcationoccursbcomputingthefirst Lapunov coefficient l [26] atthee 2. Transformations u 1 = 2 and u 2 = 2 areusedtotranslatetheequilibrium E 2 ( 2, 2 ) of the sstem (7) to the origin. Then we get u 1 =a 1 u 1 +a 1 u 2 +a 2 u 2 1 +a 11u 1 u 2 +a 2 u 2 2 +a 3u 3 1 +a 21 u 2 1 u 2 +a 12 u 1 u 2 2 +a 3u 3 2 +R 1 (u 1,u 2 ), u 2 =b 1 u 1 +b 1 u 2 +b 2 u 2 1 +b 11u 1 u 2 +b 2 u 2 2 +b 3u 3 1 where +b 21 u 2 1 u 2 +b 12 u 1 u 2 2 +b 3u 3 2 +R 2 (u 1,u 2 ), a 1 = a 1 = a 2 = a 11 = a 3 = 2 gδ ( 2 +δ) 2 + α 2 2 ( ) 2 2, α 2 2 ( ) 2, gδ 2 2 ( 2 +δ) 3 + α2 ( ) 3 1, 2α 2 2 ( ) 3, a 2 = α 2 2 ( ) 3, gδ 2 2 ( 2 +δ) 4 α2 ( ) 4, (39) a 21 = 2α 2 2 α 2 2 ( ) 4, a 12 = 2α α2 ( ) 4, a 3 = α 2 2 ( ) 4, b 1 = b 2 = b 2 = α ( ) 2, b 1 = α ( ) 2, α ( ) 3, b 11 = 2α ( ) 3, α ( ) 3, b α 3 = ( ) 4, b 21 = 2α α ( ) 4, b 12 = 2α α α ( ) 4, b 3 = ( ) 4, (4) and R i (, ), i=1,2arepower series in the powers of u j 1 uk 2 satisfing j+k 4.TheLapunovnumberl (as defined in [26]) is given b 3π l= 2a 1 Δ 3/2 ([a 1 b 1 (a a 11b 2 +a 2 b 11 ) +a 1 a 1 (b a 2b 11 +a 11 b 2 ) +b 2 1 (a 11a 2 +2a 2 b 2 ) 2a 1 b 1 (b 2 2 a 2a 2 ) 2a 1 a 1 (a 2 2 b 2b 2 ) a 2 1 (2a 2b 2 +b 11 b 2 ) +(a 1 b 1 2a 2 1 )(b 11b 2 a 11 a 2 )] (a 2 1 +a 1b 1 )[3(b 1 b 3 a 1 a 3 ) +2a 1 (a 21 +b 12 ) +(b 1 a 12 a 1 b 21 )] ), (41) where Δ = a 1 b 1 a 1 b 1. Since the epression Lapunov number for l is ver comple, we fail to discuss the sign of l preciselthoughasubcriticalhopfbifurcationhasbeen found numericall with the Lapunov number l = > (Figure 3). Obviousl, the assumption (C2) is satisfied in Figure 3. Weneedtorequireδ<δ 1 (green line) and δ<δ 2 (blue line) to guarantee the eistence of E 2 ;thatis,(b1) and (B2) hold (Figure 3(a)). tr(j 2 ) is a continuous increasing function of δ. If < δ < δ h,sstem(7) hasastablee 2 (Figure 2(a)). As δ increases, a homoclinic loop is created b joining the stable and unstable manifolds of the saddle E 1 at some δ ( δ <δ h )withastablee 2 inside (Figure 2(b)). Then as δ getslarger,anunstablelimitccleappearsand coeists with a stable E 2 (Figure 4). However, the limit ccle begins to shrink with the increasing δ. Theshrinking unstable limit ccle (Figures3(b) and 3(c)) disappearsasδ passes through δ h (δ h is also the root of equation tr(j 2 )=, δ h =.9724). Thenδ h <δ<δ 2, E 2 becomes unstable (Figure 2(c)).

10 1 Abstract and Applied Analsis Saddle-Node Bifurcation. We know that when g, δ, α, α 1,andβ satisf (C1) and the following equation δ ( g σ) 2 =, (42) where σ = 1+α αβ/α 1, then there is onl one interior equilibrium E of sstem (7), whose coordinates are given b = 1 2 (g δ σ )= δ σ, = α 1 β 2β (g δ σ )= α 1 β β δ σ. And the Jacobian matri evaluated at E is (43) α α ( J =( + ) 2 2 ( + ) 2 α 1 2 ( + ) 2 α ). (44) 1 ( + ) 2 Net we choose the relative strength of mate-finding Allee effect δ as the bifurcation parameter; then, we have the following theorem. Theorem 1. Assume (C1) holds. For δ = δ and α 1 = α, a saddle-node bifurcation occurs at the unique positive equilibrium E of sstem (7). Proof. First, we have det(j )=; therefore, J has an eigenvalue λ =,andifα 1 =α,thenλ =is simple. Let W 1 and W 2 be the eigenvectors corresponding to the λ =for J and J T,respectivel.Thenweobtain 1 W 1 = ( α 1 β ):=( w 11), w β 12 (45) 1 W 2 = ( α β ). α 1 (α 1 β ) From the epressions for W 1 and W 2,weget W T 2 F g δ ((, );δ )= 2 ( +δ ) 2 <( =), where W T 2 D2 F(W 1,W 1 )= 2 <( =), +δ F δ = F δ, D 2 F(W 1,W 1 )= 2 F 2 w 11w F w 11w F w 12w F 2 w 12w 12. (46) (47) Thus b Sotomaor s theorem [26], sstem (7) undergoes a saddle-node bifurcation at E as δ passes δ = δ if α 1 = α. A numerical eample has been carried out to demonstrate a saddle-node bifurcation of sstem (7)(Figure 5). The green curves represent the smaller unstable equilibria. The stabilit of the larger equilibria changes from stable (blue curves) to unstable (black curves) as δ passes through δ h (δ h =.9724) Bogdanov-Takens Bifurcation. Let us focus on E again. After knowing that a saddle-node bifurcation can occur at E if δ=δ and α 1 =α,wegoontoconsiderthecaseofδ= δ 2, det(j )=,andtr(j )=. If det(j )=and tr(j )=, then the Jacobian matri of E has λ 1 = λ 2 =.When det(j )=,itiseasilshownthatifα 1 =α then tr(j )=, and hence we choose the relative mate-finding strength δ and the predator growing abilit α 1 as two bifurcation parameters. Then the following statement holds. Theorem 11. Let (C1) hold. The sstem (7) undergoes a Bogdanov-Takens bifurcation around the equilibrium point E when δ=δ and α 1 =α. Proof. Following the ideas in [27, 28], we consider the neighborhood of (δ,α 1 );thatis,δ=δ +r 1 and α 1 =α 1 + r 2,wherer i, i = 1,2,aresufficientsmall,andsstem(7) becomes d dt = g 2 δ +r 1 + d (1+) α 1 +, d dt = (α 1 +r 2 ) β +. (48) We make z 1 =, z 2 = for translating E =(, ) to the origin (, ),andweobtain g z 1 =[ 2 ( +δ ) 2 r 1 +ξ 1 (r 1 )] +[ 2g δ ( +δ ) 3 r 1 +ξ 2 (r 1 )+ α 1 ( + ) 2 ]z 1 α 1 2 ( + ) 2 z 2 +[ g δ (2 δ ) ( +δ ) 4 r 1 +ξ 3 (r 1 ) 2α 1 ( + ) 3 z α 1z ( + ) 3 z2 2 +T 1 (z 1,z 2 ):=Q 1 (z 1,z 2,r 1,r 2 ), + α 1 2 ( + ) 3 ]z 2 1 +δ

11 Abstract and Applied Analsis 11 z 2 = α 1 2 r + 2 +[ ( + ) 2 + +[ α 2 1 ( + ) 2 + r 2 ( + ) 2 ]z 2 [ α ( + ) 3 + ( + ) 3 r 2]z r 2 ( + ) 2 ]z 1 +[ 2α 1 ( + ) ( + ) 3 r 2]z 1 z 2 [ α ( + ) 3 + ( + ) 3 r 2]z 2 2 +T 2 (z 1,z 2 ):=Q 2 (z 1,z 2,r 1,r 2 ), (49) where ξ i, i = 1, 2, 3 are polnomial functions of r 1 with r p 1 satisfing p 2,andT 1 (z 1,z 2 ) and T 2 (z 1,z 2 ) are C functions z j 1 zk 2 satisfing j+k 3. Thenweappltheaffinetransformation V 1 =z 1, (BT) 2 α 1 ( J (,,) =( + ) 2 α 1 ( + ) 2 α 1 2 ( + ) 2 α 1 ( + ) 2 ) =θ 2 2, (52) (BT1) [ ( /( +δ )) + ξ 1 (r 1 )+ η 4 (r 1,r 2 )] r1 =,r 2 = = / ( +δ )<, (BT2) [ (α 1 2 /(( +δ )( + ) 2 ))+ η 3 (r 1,r 2 )] r1 =,r 2 = = α 1 2 /(( +δ )( + ) 2 )<, (BT3) the map (( z 1 z 2 ),( r 1 r 2 )) (( Q 1 Q 2 ),tr ( Q i z i ),det ( Q i z i )), is regular at i=1,2 (53) ( z 1)=( z 2 ), (r 1)=( ). (54) r 2 According to [27, 28], the sstem (7) undergoes a Bogdanov- Takens bifurcation around the equilibrium point E when δ= δ and α 1 =α.thiscompletestheproof. V 2 =[ 2g δ ( +δ ) 3 r 1 +ξ 2 (r 1 )+ α 1 ( + ) 2 ]z 1 to obtain α 1 2 ( + ) 2 z 2, V 1 = V 2 +[ + ξ +δ 1 (r 1 )] V ξ 2 (r 1 ) V 1 V α 1 2 V T 1 (V 1, V 2,r 1,r 2 ), V 2 = η 1 (r 1,r 2 ) V 1 + η 2 (r 1,r 2 ) V 2 α +[ 1 2 ( +δ )( + ) 2 + η 3 (r 1,r 2 )] V 2 1 (5) (51) Remark 12. Furthermore, the quantit ω [27, 28] which determines the structure of the bifurcation is given b α ω=sign [ 1 3 ( +δ ) 2 ]=1. 2 (55) ( + ) The bifurcation diagram of sstem (7) in(δ, α 1 ) is presented in Figure 6(a). There are no positive equilibria of sstem (7) in Domain 1. In the domain bounded b Limit point ccle (green) and Hopf curve (blue), there eist a saddle E 1 and an unstable E 2. In the domain bounded b Hopf curve (blue) and Homoclinic curve (dot and magenta), there is a saddle E 1 and a stable E 2 surrounded b an unstable limit ccle. In Domain 2, sstem (7) has a saddle E 1 and a stable E 2.AtBTpoint,E istheuniquepositiveequilibriumofsstem (7)inthefirstquadrant,andthelocaldnamicsnearE is characterized b two trajectories (green curves, Figure 6(b)), which meet at E, but with opposite time orientation, and also behave like a cusp-like configuration. + η 4 (r 1,r 2 ) V 1 V 2 +[ 1 + η 5 (r 1,r 2 )] V T 2 (V 1, V 2,r 1,r 2 ), where ξ i (r 1 ), i = 1, 2, 3 and η i (r 1,r 2 ), i = 1, 2, 3, 4, 5 are polnomial functions with r p 1 satisfing p 1and rp 1 rq 2 satisfing p+q 1,respectivel. T i, i = 1,2,areC functions in their variables with V j 1 Vk 2 satisfing j+k 3. Furthermore, if r 1 =r 2 =,wecanseethat 3. Discussion In this paper, we have discussed a ratio-dependent predatorpre model with a (component) mate-finding Allee effect on pre. Allee effects and ratio-dependent functional response are ver popular among theoretical ecologists as the can displa realistic and complicated dnamical behaviors, which provide better eplanations for the ecological observations. And the hperbolic function that we concerned for matefinding Allee effects is considered as the most appropriate analtical form from ecological point of view.

12 12 Abstract and Applied Analsis (a) δ =.66 (b) δ= Figure 7: The same ratio-dependent predator-pre sstem with and without a mate-finding Allee effect for g = 1.3, α =.2, α 1 =.8, and β = The Less Complicated Equilibrium E (, ). We know that ratio-dependent predator-pre sstems with the logistic growth of pre ehibit ver complicated dnamic behaviors around the origin [5, 6, 8, 9]; for instance, the origin behaves like a stable node and an unstable saddle at the same time [5]. However, such dnamic properties can not be observed in our sstem, because E (, ) is alwas asmptoticall stable, which implies that an trajector, starting from a certain neighborhood of E (, ), goestowardstheoriginforall sstem parameters. In other words, the complicated equilibrium E (, ) becomes less complicated because of the matefinding Allee effect on pre, which provides insights into pest control b introducing a mate-finding Allee effect to reduce the compleit around the origin. We have constructed diagrams with and without a matefinding Allee effect in the same ratio-dependent predatorpre sstem for better understanding and comparison. It is eas to see that the origin can be a unstable node and a saddle at the same time in a ratio-dependent predator-pre sstem where the mate-finding Allee effect is absent in Figure 7(b). It is interesting to note that, in both cases, we set α < α 1. Generall, the predator growing abilit α 1 is assumed to be less than the consumptionabilitα in predator-pre sstems. Hence α<α 1 describes the case that predators can repl on other resources, but the pre is still the limiting factor [5, 8]. So we can see obviousl how the mate-finding Allee effect in the pre population influences the dnamics of the ratiodependent predator-pre sstems: it increases the etinction risk of both pre and predators, even in the case that pre population is not the onl resources of the predator population The Impacts of δ. In order to understand the influence of therelativestrengthofthemate-findingalleeeffectδ on the ratio-dependent pre-predator sstem, we consider δ as the bifurcation parameter and find that sstem (7) can ehibit a saddle-node, a subcritical Hopf, and a co-2 Bogdanov-Takens bifurcations with the other bifurcation parameter α 1. It is eas to see that if δ>δ 1,sstem(7) hasnointerior equilibria or other boundar equilibria ecept E (, );then, E (, ) is globall asmptoticall stable. In other words, if the mate-finding Allee effect on the pre population is ver strong, an trajector converges to E as t tends to infinit; that is, both the pre and predators become etinct regardless of their initial densit, because the pre population has great difficult in finding mates at low densit and suffers a heav loss from predation at the same time; thus, such situation leads to a continued decline in predator population which is illustratedin Figure 1(a). Furthermore, different from general predator-pre sstems with logistic growth on pre population [5, 6]orthosesubjecttothedemographicAlleeeffectson pre [29], the eistence of predator-free boundar equilibria becomes conditional in pre-predator sstems where the pre population suffers from a mate-finding Allee effect, and if such equilibria eist, the appear simultaneousl with an unstable smaller equilibrium E 1. If < δ < δ 2, α 1 > β, there are two predator-free equilibria (an unstable node E 1 and a saddle E 2 whose stable manifolds lie on -ais) and two positive equilibria: E 1 (alwas a saddle) and E 2 of sstem (7)simultaneousl.In some cases, E 2 changes from stable to unstable as the matefinding Allee effect strength gets stronger. It is interesting that, following from Theorem 5, there do not eist an global asmptoticall stable positive equilibria for an g, δ, α, α 1, and β, which means that a mate-finding Allee effect on pre increases the risk of etinction for sstem (7). In particular, there can eist a stable E 2 surrounded b an unstable limit ccle for some δ; then, the other trajectories outside the unstable limit ccle are attracted to E (, ) ecept the stable manifolds of E 1, which indicates that the mate-finding Allee effect on pre population ma destabilize the sstem. Though sstem (7) ma eperience unstable oscillation for some certain δ, it still needs etra assumption (C2) for other sstem parameters apart from those basic assumptions (A) and (C1), which indicates actuall that the impacts δ are limit and sstem (7) itself is highl sensitive to its initial

13 Abstract and Applied Analsis 13 condition and present states. Harder to encounter a receptive mate at low densit leads to fewer reproductive output directl, however, we take measures to mitigate the negative effects of mate-finding Allee effects b changing other sstem parameters. We also leave the co-3 bifurcation problem for future discussion. Another important and interesting issue that requires further eploration is a more sophisticated proof of the eistence and stabilit of the limit ccles. It has been show n that, with different mathematical forms of the Allee effect in a predator-pre sstem, the number of limit ccles changes [2]. Recentl, uniqueness of limit ccle in a Gause-tpe predator-pre sstem with an Allee effect has been proved b Olivares et al. [21]. We believe their work on the form of demographic Allee effects will be helpful for future studies on those with component Allee effects. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors are ver thankful to Dr. Ludĕk Berec for his nice suggestions and the two anonmous reviewers for their careful reading and thoughtful comments. The work of is supported in part b the NSF of Guangdong Province (S ). References [1] A. A. Berrman, The origins and evolution of predator-pre theor, Ecolog,vol.73,no.5,pp ,1992. [2]M.P.HassellandG.C.Varle, Newinductivepopulation model for insect parasites and its bearing on biological control, Nature, vol. 223, no. 5211, pp , [3] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficienc, Animal Ecolog,vol.44,pp ,1975. [4] R. Arditi and L. R. Ginzburg, Coupling in predator-pre dnamics: ratio-dependence, JournalofTheoreticalBiolog, vol. 139, no. 3, pp , [5] F. Berezovskaa, G. Karev, and R. 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14 14 Abstract and Applied Analsis [29] M. Sen, M. Banerjee, and A. Morozov, Bifurcation analsis of a ratio-dependent pre-predator model with the Allee effect, Ecological Compleit,vol.11,pp.12 27,212.

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