TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 210846 10.1155/2013/210846 210846 Research Article Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation Garić-Demirović M. 1 Kulenović M. R. S. 2 Nurkanović M. 1 Iiduka H. Mohiuddine S. A. 1 Department of Mathematics University of Tuzla 75000 Tuzla Bosnia and Herzegovina untz.ba 2 Department of Mathematics University of Rhode Island Kingston RI 02881-0816 USA uri.edu 2013 4 12 2013 2013 31 08 2013 01 10 2013 2013 Copyright © 2013 M. Garić-Demirović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form xn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,, where the parameters   a,  b, and  c   are positive numbers and the initial conditions x-1 and x0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

1. Introduction

We investigate global behavior of the equation (1)xn+1=xn-12axn2+bxnxn-1+cxn-12=f(xn,xn-1), where the parameters  a,  b, and  c  are positive numbers and the initial conditions  x-1,  and  x0  are arbitrary nonnegative numbers such that  x-1+x0>0. Equation (1) is a special case of equations (2)xn+1=Axn2+Bxnxn-1+Cxn-12axn2+bxnxn-1+cxn-12,n=0,1,2,,(3)xn+1=Axn2+Bxnxn-1+Cxn-12+Dxn+Exn-1+Faxn2+bxnxn-1+cxn-12+dxn+exn-1+f,mmmmmmmmmmmmmmmmmmnln=0,1,2,. Some special cases of (3) have been considered in the series of papers . Some special second-order quadratic fractional difference equations have appeared in analysis of competitive and anticompetitive systems of linear fractional difference equations in the plane; see . Describing the global dynamics of (3) is a formidable task as this equation contains as a special case many equations with complicated dynamics, such as the linear fractional difference equation (4)xn+1=Dxn+Exn-1+Fdxn+exn-1+f,n=0,1,2, which dynamics was investigated in great detail in  and in many papers which solved some conjectures and open problems posed in . Equation (2) can be brought to the form (5)xn+1=A(xn/xn-1)2+B(xn/xn-1)+Ca(xn/xn-1)2+b(xn/xn-1)+c,n=0,1,2,, and one can take the advantage of this auxiliary equation to describe the dynamics of (2). This approach was used in [14, 12]. In this paper, we take a different approach based on the theory of monotone maps developed in [13, 14] and use it to describe precisely the basins of attraction of all attractors of this equation. The special case of (1) when  a=0  is the linear fractional difference equation whose global dynamics is described in . We show that (1) exhibits three types of global behavior characterized by the existence of a unique positive equilibrium solution and one or two minimal period-two solutions, one of which is locally stable and the other is a saddle point. The unique feature of (1) is the coexistence of an equilibrium solution and the minimal period-two solution both being locally asymptotically stable. This new phenomenon is caused by the presence of quadratic terms and did not exist in the case of (4).

Our results will be based on the following theorem for a general second-order difference equation (6)xn+1=f(xn,xn-1),n=0,1,2,; see .

Theorem 1.

Let  I  be a set of real numbers and let  f:I×II  be a function which is nonincreasing in the first variable and nondecreasing in the second variable. Then, for every solution  {xn}n=-1  of the equation (7)xn+1=f(xn,xn-1),x-1,x0I,n=0,1,2,, the subsequences  {x2n}n=0  and  {x2n-1}n=0  of even and odd terms of the solution do exactly one of the following:

eventually, they are both monotonically increasing;

eventually, they are both monotonically decreasing;

one of them is monotonically increasing and the other is monotonically decreasing.

The consequence of Theorem 1 is that every bounded solution of (7) converges to either equilibrium or period-two solution or to the point on the boundary, and the most important question becomes determining the basins of attraction of these solutions as well as the unbounded solutions. The answer to this question follows from an application of theory of monotone maps in the plane which will be presented in Section 2.

2. Preliminaries

We now give some basic notions about monotone maps in the plane.

Consider a partial ordering    on  2. Two points  x,y2  are said to be related if  xy  or  yx. Also, a strict inequality between points may be defined as  xy  if  xy  and  xy. A stronger inequality may be defined as  x=(x1,x2)y=(y1,y2)  if  xy  with  x1y1  and  x2y2.

A map T on a nonempty set 2 is a continuous function T:. The map T is monotone if xy implies that T(x)T(y) for all x,y, and it is strongly monotone on if xy implies that T(x)T(y) for all x,y. The map is strictly monotone on if xy implies that T(x)T(y) for all x,y. Clearly, being related is invariant under iteration of a strongly monotone map.

Throughout this paper, we will use the North-East ordering (NE) for which the positive cone is the first quadrant; that is, this partial ordering is defined by  (x1,y1)ne(x2,y2)  if  x1x2  and  y1y2  and the South-East (SE) ordering defined as  (x1,y1)se(x2,y2)  if  x1x2  and  y1y2.

A map T on a nonempty set 2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive.

If T is a differentiable map on a nonempty set , a sufficient condition for T to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points x has the following sign configuration: (8)  sign(JT(x))=[+--+], provided that is open and convex.

For x2, define Q(x) for =1,,4 to be the usual four quadrants based at x and numbered in a counterclockwise direction; for example, Q1(x)={yR2:  x1y1,  x2y2}. Basin of attraction of a fixed point (x-,y-) of a map T, denoted as ((x-,y-)), is defined as the set of all initial points (x0,y0) for which the sequence of iterates Tn((x0,y0)) converges to (x-,y-). Similarly, we define a basin of attraction of a periodic point of period p. The next five results, from [13, 14], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by Smith in [5, 16].

Theorem 2.

Let  T  be a competitive map on a rectangular region 2. Let x¯ be a fixed point of  T  such that  Δ:=int(Q1(x¯)Q3(x¯))  is nonempty (i.e.,  x¯  is not the NW or SE vertex of ), and  T  is strongly competitive on  Δ. Suppose that the following statements are true.

The map  T  has a  C1  extension to a neighborhood of  x¯.

The Jacobian JT(x¯) of T at x¯ has real eigenvalues λ, μ such that 0<|λ|<μ, where |λ|<1, and the eigenspace Eλ associated with λ is not a coordinate axis.

Then, there exists a curve 𝒞 through x¯ that is invariant and a subset of the basin of attraction of x¯, such that 𝒞 is tangential to the eigenspace Eλ at x¯, and 𝒞 is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of 𝒞 in the interior of are either fixed points or minimal period-two points. In the latter case, the set of endpoints of 𝒞 is a minimal period-two orbit of T.

We will see in Theorem 4 that the situation where the endpoints of 𝒞 are boundary points of is of interest. The following result gives a sufficient condition for this case.

Theorem 3.

For the curve 𝒞 of Theorem 2 to have endpoints in  , it is sufficient that at least one of the following conditions is satisfied.

The map  T  has no fixed points nor periodic points of minimal period-two in  Δ.

The map  T  has no fixed points in  Δ,  detJT(x¯)>0, and  T(x)=x¯  has no solutions  xΔ.

The map  T  has no points of minimal period-two in  Δ,  detJT(x¯)<0, and  T(x)=x¯  has no solutions  xΔ.

For maps that are strongly competitive near the fixed point, hypothesis (b) of Theorem 2 reduces just to  |λ|<1. This follows from a change of variables  that allows the Perron-Frobenius theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis.

The next result is useful for determining basins of attraction of fixed points of competitive maps.

Theorem 4.

(A) Assume the hypotheses of Theorem 2, and let 𝒞 be the curve whose existence is guaranteed by Theorem 2. If the endpoints of 𝒞 belong to , then 𝒞 separates into two connected components, namely, (9)𝒲-:={x𝒞:y𝒞with x sey},𝒲+:={x𝒞:y𝒞withysex}, such that the following statements are true.

𝒲-  is invariant, and  dist(Tn(x),Q2(x¯))0  as  n  for every  x𝒲-.

𝒲+  is invariant, and  dist(Tn(x),Q4(x¯))0  as  n  for every  x𝒲+.

(B) If, in addition to the hypotheses of part (A),  x¯  is an interior point of and  T  is  C2  and strongly competitive in a neighborhood of  x¯, then  T  has no periodic points in the boundary of  Q1(x¯)Q3(x¯)  except for  x¯, and the following statements are true.

For every  x𝒲-, there exists  n0such that  Tn(x)intQ2(x¯)  for  nn0.

For every  x𝒲+, there exists  n0such that  Tn(x)intQ4(x¯)  for  nn0.

If  T  is a map on a set and if  x¯  is a fixed point of  T, the stable set  𝒲s(x¯)  of  x¯  is the set  {x:Tn(x)x¯}  and unstable set  𝒲u(x¯)  of  x¯  is the set (10){x0=x,limn-xn=x¯}x:there  exists  {xn}n=-0s.t.  T(xn)=xn+1,x0=x,limn-xn=x¯}.

When  T  is noninvertible, the set  𝒲s(x¯)  may not be connected and made up of infinitely many curves or  𝒲u(x¯)  may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on , the sets  𝒲s(x¯)  and  𝒲u(x¯)  are the stable and unstable manifolds of  x¯.

Theorem 5.

In addition to the hypotheses of part (B) of Theorem 4, suppose that μ>1 and that the eigenspace Eμ associated with μ is not a coordinate axis. If the curve 𝒞 of Theorem 2 has endpoints in , then 𝒞 is the stable set 𝒲s(x¯) of x¯, and the unstable set 𝒲u(x¯) of x¯ is a curve in that is tangential to Eμ at x¯ and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of 𝒲u(x¯) in are fixed points of T.

Remark 6.

We say that  f(u,v)  is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative  D1f  negative and first partial derivative  D2f  positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of (7) follows from the fact that if  f  is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to (7) is a strictly competitive map on  I×I; see .

Set  xn-1=un  and  xn=vn  in (7) to obtain the equivalent system (11)un+1=vnvn+1=f(vn,un),n=0,1,. Let  T(u,v)=(v,f(v,u)). The second iterate  T2  is given by (12)T2(u,v)=(f(v,u),f(f(v,u),v)) and it is strictly competitive on  I×I  (see ).

Remark 7.

The characteristic equation of (7) at an equilibrium point  (x-,x-), (13)λ2-D1f(x-,x-)λ-D2f(x-,x-)=0, has two real roots  λ,μ  which satisfy  λ<0<μ  and  |λ|<μ, whenever  f  is strictly decreasing in first and increasing in second variable. Thus, the applicability of Theorems 25 depends on the nonexistence of minimal period-two solution.

There are several global attractivity results for (7). Some of these results give the sufficient conditions for all solutions to approach a unique equilibrium and they were used efficiently in . The next result is from .

Theorem 8 (see [<xref ref-type="bibr" rid="B6">17</xref>]).

Consider (7) where  f:I×II  is a continuous function and  f  is decreasing in the first argument and increasing in the second argument. Assume that  x¯  is a unique equilibrium point which is locally asymptotically stable and assume that  (φ,ψ)  and  (ψ,φ)  are minimal period-two solutions which are saddle points such that (14)(φ,ψ)se(x¯,x¯)se(ψ,φ). Then, the basin of attraction ((x¯,x¯)) of (x¯,x¯) is the region between the global stable sets 𝒲s((φ,ψ)) and 𝒲s((ψ,φ)). More precisely, (15)((x¯,x¯))={(x,y):yu,yl:yu<y<yl,c(x,yl)𝒲s((φ,ψ)),(x,yu)𝒲s((ψ,φ))}. The basins of attraction ((φ,ψ))=𝒲s((φ,ψ)) and ((ψ,φ))=𝒲s((ψ,φ)) are exactly the global stable sets of (φ,ψ) and (ψ,φ).

If  (x-1,x0)𝒲+((ψ,φ))  or  (x-1,x0)𝒲-((φ,ψ)), then  Tn((x-1,x0))  converges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the region  I×I.

3. Local Stability Analysis

Denote (16)f(u,v)=v2au2+buv+cv2, and notice that the function  f(u,v)  is decreasing in the first variable and increasing in the second variable. By Theorem 1, for every solutions  {xn}n=-1  of (1) the subsequences  {x2n}n=0  and  {x2n-1}n=0  are eventually monotonic.

It is clear that (1) has a unique positive equilibrium solution  x¯=1/(a+b+c)  and that a linearization of (1) is of the form (17)yn+1=syn+tyn-1, where (18)s=-t=fu(x¯,x¯)=(-v2(2au+bv)(au2+buv+cv2)2)(x¯,x¯)=-x¯2(2ax¯+bx¯)(ax¯2+bx¯2+cx¯2)2=-1x¯·2a+b(a+b+c)2=-2a+ba+b+c.

Lemma 9.

Equation (1) has a unique positive equilibrium solution  x¯=1/(a+b+c).

If  3a+b-c<0, then equilibrium solution  x¯  is locally asymptotically stable.

If  3a+b-c>0, then equilibrium solution  x¯  is a saddle point.

If  3a+b-c=0, then equilibrium solution  x¯  is nonhyperbolic (with eigenvalues  λ1=-1  and  λ2=1/2).

Proof.

By (17), a linearization of (1) is of the form (19)yn+1+2a+ba+b+cyn-2a+ba+b+cyn-1=0. Its characteristic equation is (20)λ2+2a+ba+b+cλ-2a+ba+b+c=0, with eigenvalues  λ±=(-(2a+b)±D)/2(a+b+c), where  D=(2a+b)(6a+5b+4c)  . It is clear that  λ-<0  and  λ+>0. Now, we prove that  λ+(0,1)  and (21)λ-{(-1,0)for  3a+b-c<0<-1for  3a+b-c>0=-1for  3a+b-c=0. Namely,

λ+<1-(2a+b)+D<2(a+b+c)D<4a+3b+2c4(a+b+c)2>0, which is always satisfied;

also, (22)λ-<-1-(2a+b)-D<-2(a+b+c)D>b+2c(2a+b)2>(a-c)2(a+b+c)(3a+b-c)>03a+b-c>0;

λ->-13a+b-c<0  and  λ-=-13a+b-c=0.

Also, if  3a+b-c=0, then  c=3a+b, and we have  λ+=1/2.

4. Periodic Solutions

In this section, we present results for the existence of minimal period-two solutions of (7).

Theorem 10.

(a) Equation (1) has the minimal period-two solution (23){,0,1c,0,1c,}, for all positive values of parameters  a,b, and  c.

(b) If 3a+b-c<0, then (1) has the minimal period-two solution: (24){,φ,ψ,φ,ψ,},(φψ,φ>0,ψ>0), where (25)φ=1-D2(c-a),ψ=1+D2(c-a),D=3a+b-cb-a-c>0.

Proof.

Suppose that there is a minimal period-two solution  {φ,ψ,φ,ψ,}  of (1), where  φ  and  ψ  are distinct nonnegative real numbers such that  φ2+ψ20. Then,  φ,ψ  satisfy (26)φ=φ2aψ2+bψφ+cφ2,ψ=ψ2aφ2+bφψ+cψ2, from which we obtain three cases: (27)φ=0,aφ2+bφψ+cψ2=ψ,(28)aψ2+bφψ+cφ2=φ,ψ=0,(29)aψ2+bφψ+cφ2=φ,(30)aφ2+bφψ+cψ2=ψ. Conclusion (a) follows from (27) and (28).

Subtracting (30) from (29), we have (31)(ψ-φ)[(a-c)(φ-ψ)+1]=0, that is, (32)φ+ψ=1c-a for  a<c. Substituting (32) in (29), we obtain (33)φ2-1c-aφ+a(a-b+c)(c-a)2=0, from which (34)φ±=1±D2(c-a). Equation (32) implies that (35)ψ±=1c-a-φ±=φ. It is clear that (36)D=3a+b-cb-a-c>0{(3a+b-c>0b-a-c>0)cc×(3a+b-c<0b-a-c<0)}(b-a>c3a+b<c). If  b-a>c, then (37)D=1+4ab-a-c>1(φ-<0,φ+>0)(ψ->0,ψ+<0), which is a contradiction.

If  3a+b<c, then (38)0<D=1+4ab-a-c<1(φ±>0,ψ±>0).

By using substitution (39)xn-1=un,xn=vn, equation (1) becomes the system of equations (40)un+1=vn,vn+1=un2avn2+bunvn+cun2.

The map  T  corresponding to the system (40) is of the form (41)T(uv)=(vg(u,v)), where  g(u,v)=u2/(av2+buv+cu2).  The second iteration of the map  T  is (42)T2(uv)=T(vg(u,v))=(g(u,v)g(v,g(u,v)))=(F(u,v)G(u,v)), where (43)F(u,v)=g(u,v),G(u,v)=v2aF2(u,v)+bvF(u,v)+cv2, and the map  T2  is competitive by Remark 7. The Jacobian matrix of the map  T2  is (44)JT2(uv)=(FuFvGuGv).

Notice that periodic solutions  (0,1/c),  (1/c,0),  (φ,ψ), and  (ψ,φ)  of (1) are equilibrium points of the map  T2.

Now, we have (45)Fu=2u(av2+buv+cu2)-u2(bv+2cu)(av2+buv+cu2)2=uv(2av+bu)(av2+buv+cu2)2,Fv=-u2(2av+bu)(av2+buv+cu2)2,Gu=-v2(aF2(u,v)+bvF(u,v)+cv2)2·(2aF(u,v)+bv)Fu,Gv=(Fv2v(aF2(u,v)+bvF(u,v)+cv2)-v2(2aF(u,v)(F/v)+bF(u,v)+bv(F/v)+2cv)Gv)×(aF2(u,v)+bvF(u,v)+cv2)-2=(Fvv(2aF(u,v)+bv)F(u,v)-(2aF(u,v)+bv)v2(F/v)Gv)×(aF2(u,v)+bvF(u,v)+cv2)-2.

Theorem 11.

(i) The minimal period-two points (46)(0,1c),(1c,0) are locally asymptotically stable.

(ii) If a<c and 3a+b-c<0, then the minimal period-two points (47)(φ,ψ),(ψ,φ)(φψ,φ>0,ψ>0), where  φ  and  ψ  satisfy (25), are saddle points.

Proof.

(i) Since  F(0,1/c)=0, for periodic point  (0,1/c), we have (48)Fu(01c)=Fv(01c)=Gu(01c)=Gv(01c)=0,JT2(01c)=(0000) with eigenvalues  λ1,2=0, which implies that  (0,1/c)  is locally asymptotically stable.

Similarly, since  F(1/c,0)=1/c, for periodic solution  (1/c,0), we have (49)JT2(1c0)=(0-bc00) with eigenvalues  λ1,2=0, which implies that  (0,1/c)  is locally asymptotically stable.

(ii) By (26), we have that (50)F(φ,ψ)=φ,(51)Fu(φψ)=ψ(2aψφ+b),(52)Fv(φψ)=-φ(2aψφ+b). By (50) and (51), we obtain (53)Gu(φψ)=-ψ2(aφ2+bψφ+cψ2)2·(2aφ+bψ)Fu(φψ)=-ψ(2aφ+bψ)(2aψφ+b); that is, (54)Gu(φψ)=-ψ2(2aφψ+b)(2aψφ+b). Similarly, we have (55)Gv(φψ)=ψ(2aφ+bψ)φ-(2aφ+bψ)ψ2(F/v)(φψ)(aφ2+bψφ+cψ2)2=ψ(2aφ+bψ)φ+(2aφ+bψ)ψ2(2aψ+bφ)ψ2, so that (56)Gv(φψ)=(2aφψ+b)φ+(2aφψ+b)(2aψφ+b)φψ.

Now, we obtain that Jacobian matrix of the map  T2  at the point  (φ,ψ)  is of the form (57)JT2(φψ)=(Fu(φψ)Fv(φψ)Gu(φψ)Gv(φψ)).

The corresponding characteristic equation is (58)λ2-pλ+q=0, where (59)p=Fu(φψ)+Gv(φψ)=ψ(2aψφ+b)+φ(2aφψ+bψ)+φψ(2aφψ+b)(2aψφ+b), that is, (60)p=m+n+mn, where (61)m=ψ(2aψφ+b),n=φ(2aφψ+b).

Notice that (62)Fu(φψ)=m,  Fv(φψ)=-φψm,Gu(φψ)=-ψφmn,  Gv(φψ)=n(1+m), so that (63)q=detJT2(φψ)=mn(1+m)-m2n=mn.

We need to show that (64)|p|>|1+q|p2-4q>0.

Consider that (65)p2-4q>0(m+n+mn)2-4mn>0m2+n2+m2n2+2m2n+2mn2>2mn, which is satisfied because  m2+n22mn.

Notice that (66)m=ψ(2aψφ+b)=2aψ2φ+bψ=(26)2φ(φ-bφψ-cφ2)+bψ=2-bψ-2cφn=φ(2aφψ+b)=2aφ2ψ+bφ=(26)2ψ(ψ-bφψ-cψ2)+bφ=2-bφ-2cφ.

This implies that (67)|p|>|1+q|p>1+qm+n>13-(b+c)(φ+ψ)>0(18)3-b+cc-a>03a+b-c<0, which is satisfied.

5. Global Results and Basins of Attraction

In this section, we present global dynamics results for (1).

Notice that  T(u,0)=(0,1/c),  u>0, and  T(0,v)=(v,0),  v>0.

Theorem 12.

If 3a+b-c<0, then (7) has a unique equilibrium point x¯, which is locally asymptotically stable, and has the minimal period-two solution (φ,ψ), (ψ,φ), which is a saddle point and has the minimal period-two solution (0,1/c), (1/c,0) which is locally asymptotically stable. The basin of attraction ((x¯,x¯)) of (x¯,x¯) is the region between the global stable sets 𝒲s((φ,ψ)) and 𝒲s((ψ,φ)). The basins of attraction ((φ,ψ))=𝒲s((φ,ψ)) and ((ψ,φ))=𝒲s((ψ,φ)) are exactly the global stable sets of (φ,ψ) and (ψ,φ). Furthermore, the basin of attraction of the minimal period-two solution (0,1/c), (1/c,0) is the union of the regions above 𝒲s((φ,ψ)) and below 𝒲s((ψ,φ)) in SE ordering; that is,

if  (x-1,x0)𝒲-((φ,ψ)), then  limnx2n=1/c  and  limnx2n+1=0;

if  (x-1,x0)𝒲+(ψ,φ), then  limnx2n=0  and  limnx2n+1=1/c.

Proof.

Using assumption  3a+b-c<0  (and its consequences:  c-a>0,  b-c-a<0) and (25), it is easy to check that  (0,1/c)se(φ,ψ)se(x¯,x¯)se(ψ,φ)se(1/c,0). It is easy to check that the equilibrium point  (x¯,x¯)  is locally asymptotically stable for the strictly competitive map  T2  as well. Equation (1) is equivalent to the system of difference equations (40) which can be decomposed into the system of the even-indexed and odd-indexed terms as follows: (68)u2n=v2n-1,u2n+1=v2n,v2n=v2n-12au2n-12+bu2n-1v2n-1+cv2n-12,v2n+1=v2n2au2n2+bu2nv2n+cv2n2.

The conclusion follows from Lemma 9 and from Theorems 10, 11, and 8 and using the facts that

if  (x-1,x0)𝒲-((φ,ψ)), then (69)(u2n,v2n)=T2n((u0,v0))(0,1c),(u2n+1,v2n+1)=T2n+1((u0,v0))(1c,0);

if  (u0,v0)𝒲+((x¯,x¯)), then (70)(u2n,v2n)=T2n((u0,v0))(1c,0),(u2n+1,v2n+1)=T2n+1((u0,v0))(0,1c).

It means that

if  (x-1,x0)𝒲-((x¯,x¯)), then  T2n((x-1,x0))(0,1/c)  and  T2n+1((x-1,x0))(1/c,0), that is, (71)limnx2n=1climnx2n+1=0;

if  (x-1,x0)𝒲+(ψ,φ), then  T2n+1((x-1,x0))(0,1/c)  and  T2n((x-1,x0))(1/c,0), that is, (72)limnx2n=0limnx2n+1=1c

(see Figure 1).

Global dynamics of (1) for  a=1,  b=1, and  c=5—an illustration of Theorem 12.

Theorem 13.

If 3a+b-c>0, then (1) has a unique equilibrium point x¯ which is saddle point and has the minimal period-two solution (0,1/c), (1/c,0) which is locally asymptotically stable. There exists a set 𝒞 which is an invariant subset of the basin of attraction of (x¯,x¯). The set 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates (0,0), where =[0,)×[0,), into two connected and invariant components 𝒲-((x¯,x¯)) and 𝒲+((x¯,x¯)) which satisfy that

if  (x-1,x0)𝒲-((x¯,x¯)), then  limnx2n=1/c  and  limnx2n+1=0;

if  (x-1,x0)𝒲+((x¯,x¯)), then  limnx2n=0  and  limnx2n+1=1/c.

Proof.

It is easy to check that  (x¯,x¯)  is a saddle point for the strictly competitive map  T2  as well. The existence of the set 𝒞 with stated properties follows from Lemma 9 and Theorems 2, 4, and 10. Therefore, using (68), we obtain that

if  (u0,v0)𝒲-((x¯,x¯)),then (73)(u2n,v2n)=T2n((u0,v0))(0,1c),(u2n+1,v2n+1)=T2n+1((u0,v0))(1c,0);

if  (u0,v0)𝒲+((x¯,x¯)), then (74)(u2n,v2n)=T2n((u0,v0))(1c,0),(u2n+1,v2n+1)=T2n+1((u0,v0))(0,1c).

Consequently,

if  (x-1,x0)𝒲-((x¯,x¯)), then  T2n((x-1,x0))(0,1/c)  and  T2n+1((x-1,x0))(1/c,0); that is, (75)limnx2n=1climnx2n+1=0;

if  (x-1,x0)𝒲+((x¯,x¯)), then  T2n+1((x-1,x0))(0,1/c)  and  T2n((x-1,x0))(1/c,0); that is, (76)limnx2n=0limnx2n+1=1c

(see Figure 2).

Global dynamics of (1) for  a=2,  b=2, and  c=1—an illustration of Theorem 13.

Theorem 14.

If 3a+b-c=0, then (1) has a unique equilibrium point x¯ which is nonhyperbolic and has two minimal period-two points (0,1/c), (1/c,0) which are locally asymptotically stable points. There exists a set 𝒞 which is an invariant subset of the basin of attraction of (x¯,x¯). The set 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates (0,0), where =[0,)×[0,), into two connected and invariant components 𝒲-((x¯,x¯)) and 𝒲+((x¯,x¯)) which satisfy that

if  (x-1,x0)𝒲-((x¯,x¯)), then  limnx2n=1/c  and  limnx2n+1=0;

if  (x-1,x0)𝒲+((x¯,x¯)),  limnx2n=0  and  limnx2n+1=1/c.

Proof.

In view of Lemma 9, the eigenvalues of the map  T  at the equilibrium point  (x¯,x¯)  are  λ1=-1  and  λ2=1/2, which means that  μ1=λ12=1  and  μ2=λ22=1/4  are the eigenvalues of the map  T2.  Using (51), (52), (54), and (56), we obtain (77)JT2(x¯,x¯)=(Q-Q-Q2(Q-1)(Q-2)), where  Q=(2a+b)/(a+b+c).  A straightforward calculation yields that the eigenvector corresponding to the eigenvalue  μ2=1/4  is of the form (78)v2=(2αα),α{0}.

We see that eigenvector  v2  is not parallel to coordinate axes. Therefore, all conditions of Theorem 2 are satisfied for the map  T2  with =(0,)×(0,). As a consequence of this and using (68), we have that

if  (u0,v0)𝒲-((x¯,x¯)),then  (u2n,v2n)(0,1/c)  and  (u2n+1,v2n+1)(1/c,0);

if  (u0,v0)𝒲+((x¯,x¯)), then  (u2n,v2n)(1/c,0)  and  (u2n+1,v2n+1)(0,1/c).

It means that

if  (x-1,x0)𝒲-((x¯,x¯)), then  T2n((x-1,x0))(0,1/c)  and  T2n+1((x-1,x0))(1/c,0); that is, (79)limnx2n=1climnx2n+1=0;

if  (x-1,x0)𝒲+((x¯,x¯)), then  T2n+1((x-1,x0))(0,1/c)  and  T2n((x-1,x0))(1/c,0); that is, (80)limnx2n=0limnx2n+1=1c

(see Figure 3).

Global dynamics of (1) for  a=1/9,  b=1/3, and  c=2/3—an illustration of Theorem 14.

Remark 15.

As one may notice from the figures all stable manifolds of either saddle point equilibrium points or saddle period-two solutions are asymptotic to the origin, which is the point where (1) is not defined. These manifolds cannot end in any other point on the axes since the union of axes without the origin is an invariant set. Thus, the limiting points of the global stable manifolds of either saddle point equilibrium points or saddle period-two solutions have endpoints at  (0,0)  and  (,).

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