Nonautonomous periodic systems with Allee effects

Transcription

1 Journal of Difference Equations and Applications Vol 00, No 00, February 2009, 1 16 Nonautonomous periodic systems with Allee effects Rafael Luís a, Saber Elaydi b and Henrique Oliveira c a Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, Technical University of Lisbon, Portual; b Department of Mathematics, Trinity University, San Antonio, Texas, USA; c Department of Mathematics, Instituto Superior Técnico, Technical University of Lisbon, Portual Received) A new class of maps called unimodal Allee maps are introduced Such maps arise in the study of population dynamics in which the population oes extinct if its size falls below a threshold value A unimodal Allee map is thus a unimodal map with tree fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bier positive fixed point, called the carryin capacity In this paper the properties and stability of the three fixed points are studied in the settin of nonautonomous periodic dynamical systems or difference equations Finally we investiate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps Keywords: Unimodal Allee maps, Threshold point, Carryin capacity, Composition map, Stability, Bifurcation 1 Introduction The Allee effect is a phenomena in population dynamics attributed to the bioloist Wander Claude Allee [1] Allee proposed that the per capita birth rate declines at low density or population sizes In the lanuaes of dynamical systems or difference equations, a map representin the Allee effect must have tree fixed points, an asymptotically stable zero fixed point, a small unstable fixed point, called the threshold point, and a bier positive fixed point, called the carryin capacity, that is asymptotically stable at least for smaller values of the parameters Recently, there has been a sure in research activities on models with Allee effect and a publication of a book dedicated solely to this phenomenon [5] Some of the relevant work may be found in Yakubu [15, 16], Jan [21], Li, Son, and Wan [12], Elaydi and Sacker [17], Allen, Faan, Honas, and Faerholm [2], Luís, Elaydi and Oliveira [13], Schreiber [19], Dennis [6] and Cushin [7] Our main interest in this paper is to study nonautonomous periodic difference equations/discrete dynamical systems in which the maps of the system are unimodal Allee maps Such systems model population with fluctuatin habitat and they are commonly called periodically forced systems Correspondin author rafaelluis@netmadeiracom selaydi@trinityedu holiv@mathistutlpt This work is part of the first author s PhD dissertation ISSN: print/issn online c 2009 Taylor & Francis DOI: / YYxxxxxxxx

2 2 Rafael Luís, Saber Elaydi and Henrique Oliveira 2 Preliminaries Consider the set F = {f 0, f 1,, f p 1 } of continuous maps on I = [0, b], where b The set F enerates the nonautonomous p periodic difference equation x n+1 = f n x n ), n Z +, 1) where Z + := {0, 1, 2, 3, } and f n+p = f n, n Z + Thouh the nonautonomous periodic difference equation 1) does not enerate a discrete dynamical system [9], one may speak about the nonautonomous p periodic dynamical system F One of the most effective way of convertin the nonautonomous difference equation 1) into a enuine discrete dynamical system is the construction of the associated skew-product system as described a recent paper by Elaydi and Sacker [10] It is noteworthy to mention that this idea was oriinally used to study nonautonomous differential equations by Sacker and Sell [18] However, since the focus here will be on the case when F consists of two maps, we will not utilize the skew-product construction as it is more appropriate for more complicated settin We now present few basic definitions that will be used in the sequel Definition 21 A point x is a fixed point of Eq 1) or the systems F if f n x ) = x for all n Z + In other words, x is a fixed point of all the maps in F Definition 22 Let C r = {x 0, x 1,, x r 1 } be a ordered set in I Then C r is called an periodic r cycle orbit) if f i+nr)modp x i ) = x i+1)modr, 0 i r 1 To this end we have talked about eneral continuous maps on an interval The focus in this paper will be on special types of map that we call unimodal Allee maps A definition of these maps now follows Definition 23 Let I = [0, b] R + A continuous function f : I I is called an Allee map if the followin hold: f 0) = 0, and there are positive points A f and K f such that f x) < x for x 0, A f ) K f, b) and f x) > x for x A f, K f ) If, in addition, the map is unimodal, then it is called an unimodal Allee map Explicitly, we require the followin: fb) = 0 when b is finite or lim f x) = 0 otherwise x + There exists a unique critical point C f of f, where f x) is strictly increasin on [0, C f ) and strictly decreasin on C f, b) or C f, + ) when b = + ) Fi 1 depicts a prototype of unimodal Allee maps From definition 23, it follows that A f and K f are positive fixed points We call the smaller positive fixed point the threshold point A f of the map f, and the reater positive fixed point K f the carryin capacity of the map f It is easy to verify that x = 0 is an attractin fixed point and [0, A f ) Af, B f 0), where B f 0) is the basin of attraction of zero and A Af f = f 1 A f ), ie f = A f, with A f > K f Note that the threshold point A f is always repellin while the carryin

3 f x K f Journal of Difference Equations and Applications 3 A f 0 A f K f A f x Fiure 1 An instance of one unimodal Allee map f capacity K f may or not stable Next we define a unimodal Allee map f by two maps, a left map f l and a riht map f r Thus we have f x) = { fl x) if 0 x A f f r x) if A f < x b 2) It follows that f 0) = f l 0) = f r b) = 0 or lim x f r x) = 0) Since f is continuous in R +, it follows that f A f ) = f l A f ) = f r A f ) = A f and f K f ) = f r K f ) = K f To facilitate our study we introduce two zones, the threshold zone and the carryin capacity zone Definition 24 1) The square that contains the oriin and the points A f, 0), 0, A f ) and A f, A f ) will be called the threshold zone Af 2) The rectanle that contains the points A f, A f ), A f, f C f )),, A f Af and, f C f ) will be called the carryin capacity zone Consider now the nonautonomous periodic system W = {f, } where f and are unimodal Allee maps with f x) > x) for all x on 0, b) We note that under this hypothesis, we have 0 < A f < A < K < K f Henceforth we assume that the riht end point b of I is fixed for all the unimodal Allee maps The composition map f may be written as follows f = { f l x)) if 0 x A f r x)) if A < x b 3) The first branch of 3) may be written as { fl f l x)) = l x)) if 0 1 x) A f 0 x A f r l x)) if A f 1 x) < b 0 x A

4 4 Rafael Luís, Saber Elaydi and Henrique Oliveira hence where A f or equivalently { fl l x)) if 0 x A f f l x)) = f r l x)) if A f < x A, 4) represents the left preimae of A f under the map, ie, A f = A f, ) ) f l l A f = f r l A f = f A f ) = A f The second branch of 3) may be written as where A + f or equivalently f r X)) = { fr r x)) if A < x < A + f f l r x)) if A + f x b, 5) represents the riht preimae of A f under the map, ie, A + f = A f, From 4) and 5) we obtain Similarly ) ) f r r A + f = f l r A + f = f A f ) = A f f l l x)) if 0 x A f f r f = l x)) if A i < x A f r r x)) if A < x < A + 6) f f l l x)) if A + f x b l f l x)) if 0 x A f f = l f r x)) if A f < x < A r f r x)) if A x A + l f r x)) if A + < x b 7) where A and A + represents the left and the riht preimaes of A under the map f, respectively, that is, f A ) = f A + = A In other words we have and Fiure 2 summarises above remarks l fr A )) = r fr A )) = A ) = A r fr A + )) = l fr A + )) = A ) = A Lemma 25 Let f, W If C f > A, where C f is the unique critical point of f and A is the threshold point of, then f and, both, are homeomorphisms on [0, A ] 3 Threshold points of the composition map In this section we prove the existence of the fixed points, called threshold points, of the composition map In addition we establish an order relation between these

5 Journal of Difference Equations and Applications 5 I II f f A f f A f A f A f A A f A f III IV A f f f A f f A f A A A A Fiure 2 Parts I and II depicts the left and the riht preimaes of A f under while parts III and IV depicts the left and the riht preimae of A under the map f fixed points From here to the rest of the paper we assume that A f and A are the threshold points of the unimodal Allee maps f and, respectively We also assume that A f and A are, respectively, the first preimae of A f by the map and the first preimae of A by the map f Theorem 31 Let f and be two unimodal Allee maps such that f x) > x) for all x on 0, b) Then both f and f, have threshold points, that we denote by A f and A f, respectively Moreover A f < A f < A and A f < A f < A Proof Assume that f and satisfy the conditions of the hypothesis of the theorem First let us to prove the existence of A f and A f We know that l A f ) < A f and on [0, A f ] f is increasin This implies that f l l A f )) < f l A f ) = A f On the other hand f r l A )) = f r A ) > l A ) = A Hence the function f x) x chanes sin on A f, A ) Then there exits x A f, A ) such that f x) = x, ie A f < A f < A In the same way we prove that A f < A f < A [ ] To proof that A f A f, A ) first we will proof that A f / A f, A f Let [ ] x A f, A f We know that f A f ) < A f and f A f = A f < A f When if x A f, A f ) we have that l x) < A f and so f l l x)) < A f < x Therefore [ ] A f, A f Now let x A f, A ) By one side we have f A ) = f A ) > A > x and on A f / the other side f A f ) = A f < x, consequently, there exits y A f, A ) such that f y) = y, that is, A f A f, A ) Followin the same reasonin we prove A f < A f < A Next we establish an order relation between these two threshold points of f

6 6 Rafael Luís, Saber Elaydi and Henrique Oliveira and f, respectively Theorem 32 Let f and be two unimodal Allee maps such that f x) > x) for all x on 0, b) Suppose that in the threshold reion, ie, on J = [0, A ], these two maps are convex, f is increasin and f x) > x), x J Suppose also that Then A A f Moreover A f < A f f A ) + A ) f A f ) A ) 8) Proof By hypothesis we have A A f = ε > 0 and x J = [0, A ], f x) x) = δ x) > 0 such that δ x) is increasin We need to prove that the first preimae of A f and A, both, satisfy the relation A A f or f 1 A ) 1 A f ), that is equivalent By the Taylor s series we know that A f 1 A f ) 9) 1 A f ) = 1 A ε) = 1 A ) 1 A ) ε + O ε 2) = A ε A ) + O ε 2) Substitutin the previous relation in 9) we et A f A ε A ) + O ε 2)) and aain by Taylor s series that is A = f A ) f A ) A ) ε + O ε 2) f A ) A ) ε f A ) A + O ε 2) Once that f A ) A = f A ) A ) = δ we et So relation 9) is equivalent to relation 10) f is convex and therefore f A ) A ) ε δ + O ε 2) δ 10) f A f ) < fa) faf ) A A f = f M) < f A ), where M ]A f, A [ So f M) = fa ) A +A A f ε, and therefore f M) = δ+ε ε By hypothesis we have f A ) + A ) f A f ) A ), that is equivalent to f A ) A + 1 f ) A f ) But f A f ) < f M) so f A ) A ) + 1 f M) = δ+ε ε Multiplyin by ε both of the members of the last relation we et f A ) A ) ε δ

7 Journal of Difference Equations and Applications 7 that is equivalent to relation 10), and therefore this part of the theorem is proved From Theorem 31 and by the fact that A A f it follows A f < A f Hypothesis 8) requires that f and stay sufficiently far apart to avoid the collapse of the interval where the threshold points of f and f belons 4 The carryin capacity of the composition map In this section we study the existence, the location and the properties of the carryin capacity of the composition map Note that if f and are two unimodal Allee maps such that f x) > x) for all x on 0, b), then the critical points of f are the solutions of the equation f x)) x) = 0 This implies that C is a critical point of both and f The equation f x)) = 0 has a solution if and only if the equation x) = C f has a solution Thus either 1 C f ) = or 1 C f ) consists of two points one on the left side of C and the other on the riht side of C Let us to represent the points by C f resp C+ f ), the critical point of the composition map f on the left resp on the riht) side of C So if C f and C+ f exists then the composition map f has four intervals of monotonicity otherwise f [ has two ] intervals of monotonnicity) Explicitly, [ ] f is strictly increasin on 0, C f C, C + f and is strictly decreasin on [ C f, C ] [C + f, b ] The same analysis can be made for the map f Note that the threshold point of the composition map f resp f) belons always to the first interval where the composition map is increasin Recall from the previous sections that K f and K are the carryin capacities of f and, respectively, and A + f resp A+ ) the riht positive preimae of A f resp A ) under the map resp f) We also follows the notation about the critical points of the composition map that we described above Theorem 41 Let f and be two unimodal Allee maps such that f x) > x) for all x on 0, b) Then both f and f, have carryin capacities, that we denote by K f and K f, respectively Moreover K < K f < A + f and A < K f < A + Proof It is clear that 0 < A f < A < K < K f, A + f > K f and A < A < K < A + We can see that f K ) = f K ) > K ) = K and f A + f = f A f ) = A f < A + f Therefore the map f x) x chanes sin on K, A + f ) Hence there exists x K, A + f ) such that f x) = x Note that C + f < A+ f To see this fact suppose by contradiction that C+ f A+ f or equivalently 1 C f ) 1 A f ) We know that C + f, A+ f > C and is decreasin on C, b) Consequently, applyin in both sides of the last inequality we et C f A f that is impossible Similarly we prove C + f < A+ Since C + f < A+ f the carryin capacity of f, K f, is the reater root of f x) = x on K, A + f ) We also can see that f A + ) = A ) < A + and f A ) = A > A So the map f x) x chanes sin on A, A + ) and therefore there exists K f A, A + ) such that f K f ) = K f since C + f < A+ Remark 1 Let f and be two unimodal Allee maps such that f x) > x) for

8 8 Rafael Luís, Saber Elaydi and Henrique Oliveira all x on 0, b) If f C ) > C resp f C f ) > C f ) then the map f resp f) has exactly two positives fixed points, the threshold point and the carryin capacity Corollary 42 Let f and be two unimodal Allee maps such that f x) > x) for all x on 0, b) If C, C f > K f then f x) > f x), x [K, K f ] Moreover, K < K f < K f < K f Proof If C, C f > K f we have that f and are increasin on [K, K f ] The composition of increasin maps is an increasin map The interval [K, K f ] is invariant under composition because f K ) > K, f K f ) < K f and f K f ) < K f, f K ) > K So the map f x) x resp fx) x) chanes sin on [K, K f ] We know that f K ) > K and therefore f K ) < f K ) = f K ) x) < x, x > K ) On other hand we know that K f ) < K f so f K f ) > K f ) = f K f ) f x) > x, x ]A f, K f [) Consequently, f x) > f x), x [K, K f ] Once that f C ) < C resp fc f ) < C f ) from remark 1 it follows that K f resp K f ) is the unique fixed point of f resp f) on [K, K f ] The relation order between K f and K f is an immediate consequence of the relation order between the composition maps Corollary 43 Let f and be two unimodal Allee maps such that f x) > x) for all x on 0, b) Then followin statements holds true 1) if C f > K f and C > K then K f, K f K, K f ) 2) if C f < K f then we have a) K f K, K f ) if f K f ) < K f and f C + f < C + f in the case that C + f doesn t exists we have K f K, K f ) if f K f ) < K f ); b) K f K f, A + f ) if f K f ) > K f ; c) K f A, K ) if f K ) < K and f C + f < C + f ; d) K f K, K f ) if f K ) > K and f C + f < C + f ; e) K f C + f, A+ ) if f C + f > C + f 3) if C f > K f and C < K then we have a) K f K, K f ) b) The situation of K f is similar to 2)c, 2)d and 2)e From the previous corollary it is possible, in certain cases, to establish an order relation between the two carryin capacities K f and K f of the composition maps f and f In particular we are interested in an order when such fixed points are between the carryin capacities of the individual maps The next result provides this information, respectively Theorem 44 Let f and be two unimodal Allee maps such that f x) > ) x) for all x on 0, b) Suppose that C f < K f, C < K, f K f ) < K f, f C + f < C + f, f K ) > K and f C + f > C + f Let y J = [k, k f ] and suppose that y) > K f, y J, where K f is the left preimae of K f by the map f Then f y) < f y), y J Moreover, K < K f < K f < K f Proof Let K f be the left preimae of K f by the map f, ie, f K f = K f Then A f < K f < K f Note that is decreasin on J = [k, k f ], y) < y, y J and f y) > y, y J From the hypothesis we have y) > K f and therefore f y) > K f, y J

9 Journal of Difference Equations and Applications 9 On the other hand f y) > K f > y > K, and then f y) < f y) < K, y J Consequently, f y) < f y), y J From the hypothesis of the theorem and remark 1 the theorem is established 5 Stability The first objective in this section is to formulate in a more precise form definitions of stability in the settins of eneral periodic difference equations of the form x n+1 = f n x n ), f n+p = f n, n Z + 11) Equivalently, one may speak about the stability in the settins of the nonautonomous periodic dynamical systems F = {f 0, f 1,, f p 1 } Stability notions for periodic difference equations have been investiated by many authors Includin, to cite few, AlSharawi and Anelos [3], AlSharawi, Anelos, Elaydi and Rakesh [4], Henson [11], Yakubu [15], Oliveira and D Aniello [14], and Selrado and Roberds [20] It is our hope that our definitions will standarize the notion and terminoloy in the area of nonautonomous systems Definition 51 Φ i = f i 1 f 1 f 0 is called the composition operator with order i associated with Eq 11) and Φ i = f 0 f 1 f i 1 is called the reverse composition operator with order i associated with Eq 11) Definition 52 Let x be a fixed point of F, ie, x is a fixed point of all the members of the set F Then 1) x is stable if iven ε > 0, there exits δ > 0 such that x 0 x < δ implies Φ i x 0 ) x < ε, i 1 Otherwise, x is unstable 2) x is attractin if there exists η > 0 such that x 0 x < η implies lim n Φ n x 0 ) = x 3) x is asymptotically stable if it is both stable and attractin 4) x is lobally asymptotically stable if it is asymptotically stable and η = Many authors use the notion of attractivity of nonhyperbolic fixed point instead of our eneral definition The followin result provides us the connection between these two notions Lemma 53 Suppose that F is a set of continuously differentiable maps at x If Φ i x ) < 1, i 1 then x is an asymptotically stable fixed point of F Proof From the hypothesis we have for all i 1 Φ i x ) = f i 1x )f i 2x )f 1 x ) f 0 x ) M < 1 Hence there exists an open interval J = x ε, x +ε) such that Φ i y) M < 1, y J By the mean value theorem we know that Φ 1 x 0 ) x = Φ 1y) x 0 x M x 0 x, for any x 0 I and y between x 0 and x The last inequality implies that Φ 1 x 0 ) I

10 10 Rafael Luís, Saber Elaydi and Henrique Oliveira since M < 1 Usin the same arument we et Φ 2 x 0 ) x M Φ 1 x 0 ) x M 2 x 0 x By mathematical induction, we can prove that Φ n x 0 ) x M n x 0 x, n 1 12) Assumin δ = ε/2, for any ε > 0, from x 0 x < δ follows that Φ n x 0 ) x M n ε/2 < ε, n 1 since M < 1 and consequently x is stable Moreover, lim Φ n x 0 ) = x and thus x is attractin n In particular, if F is a set formed by unimodal Allee maps we have that x = 0 is a fixed point of Φ i x), for all i 1 Since this fixed point is asymptotically stable for each map we have Φ i 0) < 1 and thus from the previous lemma x = 0 is an asymptotically stable fixed point of F Now let us focus our attention on the stability of a periodic cycle for a periodic nonautonomous difference equation Our definition of stability now follows Definition 54 Let C r = {x 0, x 1,, x r 1 } be a r periodic cycle of equation 11) where f n+p = f n, n Z +, p > 1 and either r divides p or r is a multiple of p 1) C r is stable if iven ε > 0 there exists δ > 0 such that x 0 x 0 < δ implies Φ i x 0 ) Φ i x 0 ) < ε, i 1 Otherwise, C r is said unstable 2) C r is attractin if there exits η > 0 such that x 0 x 0 < η implies lim n Φ rn+i x 0 ) = x i, 0 i r 1 3) C r is asymptotically stable if it is both stable and attractin 4) C r is lobally asymptotically stable if it is asymptotically stable and η = An immediate consequence of this definition now follows Lemma 55 Suppose that F = {f 0, f 1,, f p 1 } is a set of continuously differentiable maps at x 0 If Φ p x 0 ) < 1 then the r periodic cycle C r of Eq 12) is asymptotically stable, where f n+p = f n, n Z +, p > 1 and either r divides p or r is a multiple of p Proof The proof is divided into two parts 1) Consider the first case where r divides p, that is, mr = p Suppose that Φ p x 0 ) < 1, ie f 0x 0 )f 1x 1 )f r 1x r 1 )f rx 0 )f 2r 1x r 1 )f 2rx 0 )f mr 1x r 1 ) M < 1 Followin the same arument that we the used in the prove of lemma 53 we et Φ m r x 0 ) x 0 M mr x 0 x 0 13) This implies that lim m Φm r x 0 ) = x 0 since M < 1 By continuity the composition of continuous maps is a continuous map) the followin statement yields Φ i Φ m r x 0 ) = Φ m r+ix 0 ) m Φ i x 0 ) = x i and thus C r is attractin Note that Φ p+i = Φ mr+i = Φ i Φ m r

11 Journal of Difference Equations and Applications 11 To see the stability of C r let us to take δ < ε/2, for any iven ε > 0 Since x 0 C r we have x 0 = Φ r x 0 ) = Φ rm x 0 ) and thus from 13) we have Φ rm x 0 ) Φ rm x 0 ) < ε Consequently, Φ i x 0 ) Φ i x 0 ) < ε, 1 2) In the second case let us to assume that r = mp The dynamics of the points in the r cycle is x 1 = Φ 1 x 0 ), x 2 = Φ 2 x 0 ),, x p = Φ p x 0 ),, x 2p = Φ 2 px 0 ),, x 0 = Φ m p x 0 ) Followin the same arument of the previous case we show that The rest of the prove is similar Φ nm p x 0 ) x 0 < M nm x 0 x 0 14) In the particular case, when F is a periodic set formed by unimodal Allee maps such that f i < f i+1, i {0, 1,, p}, the threshold point A Φp of Φ p is unstable since the map Φ p x) is increasin on [0, A Φp ] and Φ p x) < x, x 0, A Φp ) The same happens for Φ i on [0, A Φ i ] Remark 1 The above theorems cover the hyperbolic case when Φ x ) 1 When Φ x ) = 1 or 1, the critical point is called neutral A complete analysis of these nonhyperbolic cases may be found in Elaydi s book Discrete Chaos [8] 6 Bifurcation The study of various notions of bifurcation in the settin of nonautonomous systems is still in its infancy stae The main contribution in this area are the papers by Henson [11], AlSharawi and Anelo [3], and Oliveira and D Aniello [14] Our main oal here is to ive precise and complete definitions and notions for the various bifurcation notions in the settin of nonautonomous systems Thouh our focus here will be on 2 periodic systems, the ideas presented can be easily extended to the eneral periodic case We start our exposition presentin the followin theorem due by Henson [11] where the idea is to perturb the parameters Theorem 61 Suppose that F α, x) : R R R is non-linear in x, one to one in α, C 2 in both x and α, and for a specific real number α, the autonomous difference equation x n+1 = F α, x n ) has an attractin r periodic cycle {x 0, x 1,, x r 1 }, with a minimal period r Then for sufficiently small ɛ > 0, if α 0, α 1,, α p 1 α ɛ, α+ɛ) then the p periodic difference equation x n+1 = F α n, x n ), α n+p = α n has t attractin periodic solutions of minimal period s, where t = cdp, r) reatest common divisor of p and r), and s = lcmp, r) least common multiple of p and r) Proof See [11] Consider the 2 periodic system F = {f 0, f 1 }, f 0 f 1, where both maps arise from a one-parameter family of maps f α in which f 0 = f α0 and f 1 = f α1 Let C r = {x 0, x 1,, x r 1 } be an r periodic cycle of f α In the followin table we summarize the ideas presents in theorem 61 for sufficiently small ɛ and α 0, α 1 α ɛ, α + ɛ)

12 12 Rafael Luís, Saber Elaydi and Henrique Oliveira Fiure 3 Sequence of the periodic points {x 0, x 1,, x r 1 } in the fibers and f r cd r, 2) lcm r, 2) Conclusion F has one GAS 2 periodic cycle F has two AS 2 periodic cycle F has two AS 4 periodic cycle 2m 2 2m F has two AS 2m periodic cycle Let r = 2m, m 1 With Φ 2 = f, one may write the orbit of C r as see fiure 3) Ox 0 ) = { x 0, x 0 ), Φ 2 x 0 ), Φ 2 x 0 ), Φ 2 2x 0 ),, Φ m 1 2 x 0 ), Φ m 1 2 x 0 ) } 15) or equivalently Ox 1 ) = { m 1 f Φ 2 x 1 ), x 1, fx 1 ), Φ 2 x 1 ), f Φ } m 2 m 1 2 x 1 ),, f Φ 2 x 1 ), Φ 2 x 1 ) 16) where Φ 2 = f Hence the order of the composition is irrelevant to the dynamics of the system In the sequel, we assume that the maps f and arise from a one-parameter family of maps such that f = f α and = β with β = qα for some real number q > 0 Thus one may write, without loss of enerality, our system as F = {f α, α } The dynamics of F depends very much on the parameter and the qualitative structure of the dynamical system chanes as the parameter chanes These qualitative chanes in the dynamics of the system are called bifurcation and the parameter values at which they occur are called bifurcation points For autonomous systems or sinle maps the bifurcation analysis may be found in Elaydi [8] In a one-dimensional systems enerated by a one-parameter family of maps f α, bifurcation at a fixed point x occurs when f x α, x ) = 1 or 1 at a bifurcation point α The farmer case leads to a saddle-node bifurcation, while the latter case

13 Journal of Difference Equations and Applications 13 leads to a period-doublin bifurcation Our objective here is to extend this analysis to 2 periodic difference equations or F = {f α, α } To simplify the notation we write Φ instead of Φ 2, and we write Φα, x) instead of Φx) Let C r = {x 0, x 1,, x r 1 } be a r periodic cycle of F and suppose that 2m = r, Φ = f and Φ = f Then Φ m x 2i ) = x 2i)modr and Φ m x 2i+1 ) = x 2i+1)modr, 1 i m In eneral we have Φ nm x 2i ) = x 2i)modr and Φ nm x 2i+1 ) = x 2i+1)modr, n 1 Assumin Φm x α, x 0) = 1 at a bifurcation point α by the chain rule, we have or Φ x α, x 2m 2) Φ x α, x 2m 4) Φ x α, x 2) Φ x α, x 0) = 1 f αx 2m 1 ) αx 2m 2 )f αx 2m 3 ) αx 2m 4 )f αx 3 ) αx 2 )f αx 1 ) αx 0 ) = 1 17) Applyin α on both sides of the identity Φ m α, x 0 ) = x 0, yields Φ m α, x 1 ) = x 1 Differentiatin both sides of this equation we et or equivalently Φ x α, x 2m 1) Φ x α, x 2m 3) Φ x α, x 3) Φ x α, x 1) = 1 αx 0 )f αx 2m 1 ) αx 2m 2 )f αx 2m 3 ) αx 4 )f αx 3 ) αx 2 )f αx 1 ) = 1 18) Hence Eq 17) is equivalent to Eq 18) More enerally the followin the relation yields Φ m x α, x 2j) = Φ m x α, x 2j 1), j {0, 1,, m 1} 19) The next result ives conditions for the saddle-node bifurcation Theorem 62 Saddle-node Bifurcation Let C r = {x 0, x 1,, x r 1 } be a r periodic cycle of F Suppose that both 2 Φ x and 2 Φ 2 exist and are continuous in a neihbourhood of a periodic orbit such that Φm x α, x 0) = 1 for the periodic point x 0 Assume 2 also that A = Φm α α, x 0) 0 and B = 2 Φ m x 2 α, x 0) 0 Then there exists an interval J around the periodic orbit and a C 2 map α = hx), where h : J R such that hx 0 ) = α, and Φ m x, hx)) = x Moreover, if AB < 0, the periodic points exists for α > α, and, if AB > 0, the periodic points exists for α < α Proof The proof is similar to the proof of theorem 25 in [8, pp 86] and will be omitted When Φm x α, x 0) = 1 but Φm α α, x) = 0, two types of bifurcation appear The first is called transcritical bifurcation which appears when 2 Φ m x α, x 2 0 ) 0 and the second called pitchfork bifurcation which appears when 2 Φ m x α, x 2 0 ) = 0 see

14 14 Rafael Luís, Saber Elaydi and Henrique Oliveira table 21 in [8, pp 90] In [14], the authors studied the pitchfork bifurcation for 2 periodic systems in which the maps have neative Schwarzian derivative In the next result we characterize period-doublin bifurcation Theorem 63 Period-Doublin Bifurcation Let C r = {x 0, x 1,, x r 1 } be a r periodic cycle of F Assume that both 2 Φ x 2 and Φ α exist and are continuous in a neihbourhood of a periodic orbit, Φm x α, x 0) = 1 for the periodic point x 0 and 2 Φ 2m α x α, x 0) 0 Then, there exists an interval J around the periodic orbit and a function h : J R such that Φ m x, hx)) x but Φ 2m x, hx)) = x Proof The proof is similar to the proof of the theorem 27 in [8, pp 89] and will be omitted Note that if W is a periodic set formed by unimodal Allee maps, neither the zero fixed point nor the threshold point can contribute to bifurcation, since the former is always asymptotically stable and the latter is always unstable Hence bifurcation may only occur at the carryin capacity of W Now we are oin to apply the above results to study the bifurcation of the system W = {f 0, f 1 }, where f i x) = a i x 2 1 x), i = 0, 1, x [0, 1] and a i > 0 For an individual map x) = ax 2 1 x), the dynamics is interestin but predictable For a < 4 we have a lobally asymptotically stable zero fixed point and no other fixed point A new unstable fixed point born at a = 4 after which x) becomes a unimodal map with an Allee effect Henceforth, we assume that a > 4 To determine the two main types of bifurcation, we solve the equations { x0 = f 1 f 0 x 0 )) f 1 F 0 x 0 )) f 0 x 0) = 1 20) and { x0 = f 1 f 0 x 0 )) f 1 f 0 x 0 )) f 0 x 0) = 1 21) Usin the command resultant in Mathematica or Maple Software, we eliminate the variable x 0 in both systems Eq 20 yields a 0 a a 2 0a a 3 0a a 0 a a 2 0a a 3 0a a 0 a a 2 0a a 3 0a a 4 0a a 3 0a a 4 0a 4 1 = 0 while Eq 21 yields a 0 a a 2 0a a 3 0a a 0 a a 2 0a a 3 0a a 0 a a 2 0a a 3 0a a 4 0a a 3 0a a 4 0a 4 1 = 0 For each one of these two equations we invoke the implicit function theorem to plot, in the a 0, a 1 ) plane, the bifurcation curves see fiure 4) The black curves are the solutions of the first equation, while the rey curves are the solutions of the second equation For values of a 0 and a 1 in reion A, the fixed point x = 0 of W is lobally asymptotically stable For values of a 0, a 1 4, 527), approximately, in the reion B, we have one lobally asymptotically stable 2 periodic cycle Moreover, in the same reion B and for values

15 REFERENCES D 8 C 1 E a B C 2 A a 0 Fiure 4 Bifurcations curves for the 2 periodic nonautonomous difference equation with Allee effects x n+1 = a nx 2 n1 x n), a n+2 = a n and x n+2 = x n in the a 0, a 1 ) plane a 0, a 1 527, ), approximately, system W has two asymptotically stable 2 periodic cycles Computation shows that the hypotheses of theorem 62 are satisfied when a 0 and a 1 belon to a curve between A and B Consequently, this curve is the Saddle-node curve The black cusp is the pitchfork bifurcation curve In reion D, system W has two asymptotically stable 2 periodic cycles The rey curves are where the period-doublin occurs Therefore, for certain values of the parameters a 0 and a 1 in reions C 1, C 2, and E, the system W has two asymptotically stable 4 periodic cycles One rose from the period-doublin and the other from the pitchfork bifurcation References [1] W C Allee, The Social Life of Animals, William Heinemann, London 1938 [2] L Allen, J Faan, G Honas, and H Faerholm, Population extinction in discrete-time stochastic population models with an Allee effect, Journal of Difference Equations and Applications, Vol 11, No ), pp [3] Z AlSharawi and J Anelos, On the periodic loistic equation, Appl Math Comput, Vol 180, No 1, pp , 2006 [4] Z AlSharawi, J Anelos, S Elaydi and L Rakesh, An extension of Sharkovsky s theorem to periodic difference equations, J Math Anal Appl, J Math Anal Appl, Vol 316, , 2006 [5] F Courchamp, L Berec, and J Gascoine, Allee effects in Ecoloy and Conservation, Oxford University Press, 2008 [6] B Dennis, Allee effects: population rowth, critical density, and the chance of extinction, Nat Res Model, 3, pp , 1989 [7] J M Cushin, Oscillations in ae-structured population models with an Allee effect, Journal of Computational and Applied Mathematics 52, 71-80, 1994 [8] S Elaydi, Discrete Chaos: With Applications in Science and Enineerin, Chapman and Hall/CRC, Second Edition, 2008 [9] S Elaydi, and R Sacker, Skew-product dynamical systems: Applications to difference equations, Proceedins of the Second Annual Celebration of Mathematics, United Arab Emirates, 2005 [10] S Elaydi, and R Sacker, Global stability of periodic orbits of nonautonomous difference equations and population bioloy, Journal of Differential Equations, Vol 208, pp )

16 16 REFERENCES [11] S Henson, Multiple attractors an resonance in periodically forced population models, Physica D 140, pp 33-49, 2000 [12] J Li, B Son and X Wan, An extended discrete Ricker population model with Allee effects, Journal of Difference Equations and Applications, Vol 13, No ), pp [13] R Luís, SElaydi and H Oliveira, An economic model with Allee effect, Journal of Difference Equations and Applications, to appear [14] H Oliveira and E D Aniello, Pitchfork bifurcation for non autonomous interval maps, Journal of Difference Equations and Applications, to appear [15] A Yakubu, Multiple Attractors in Juvenile-adult Sinle Species Models, J Differ Equ Appl, Vol 9, No ), pp [16] A Yakubu, Allee effects in a discrete-time SIS epidemic model with infected newborns, J Differ Equ Appl Vol 13, No ), pp [17] R Sacker and S Elaydi, Population models with Allee effects: A new model, in proress, 2008 [18] R Sacker and J Sell, Liftin properties in skew-product flows with applications to Differential Equations, AMS Memoirs, Vol 11, No 190, 1977 [19] S Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Populations Bioloy, ), [20] J Selrade and H Roberds, On the structure of attractors for discrete periodically forced systems with applications to populations models, Physica D, 158, 69-82, 2001 [21] R Sophia and J Jan, Allee effects in a discrete-time host-parasitoid model, Journal of Difference Equations and Applications, Vol 12, No ),