DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2015/847360 847360 Research Article Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect Brett A. Kulenović M. R. S. Papashinopoulos Garyfalos Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816 USA uri.edu 2015 1212015 2015 26 08 2014 29 10 2014 1212015 2015 Copyright © 2015 A. Brett and M. R. S. Kulenović. 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 consider the following system of difference equations: xn+1=xn2/B1xn2+C1yn2, yn+1=yn2/A2+B2xn2+C2yn2,  n=0, 1, ,   where B1, C1, A2, B2, C2 are positive constants and x0, y00 are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at (0,0), which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at (0,0) and thus describe the global dynamics of this system. Since the singular point at (0,0) always possesses a basin of attraction this system exhibits Allee’s effect.

1. Introduction

The following difference equation is known as the Beverton-Holt model: (1)xn+1=axn1+xn,n=0,1,, where a>0 is the rate of change (growth or decay) and xn is the size of the population at the nth generation.

This model was introduced by Beverton and Holt in 1957. It depicts density dependent recruitment of a population with limited resources which are not shared equally. The model assumes that the per capita number of offspring is inversely proportional to a linearly increasing function of the number of adults.

The Beverton-Holt model is well studied and understood and exhibits the following properties.

Equation (1) has two equilibrium points 0 and a-1 when a>1.

All solutions of (1) are monotonic (increasing or decreasing) sequences.

If a1, then the zero equilibrium is a global attractor; that is, limnxn=0, for all x00.

If a>1, then the equilibrium point a-1 is a global attractor; that is, limnxn=a-1, for all x0>0.

Both equilibrium points are globally asymptotically stable in the corresponding regions of parameters a1 and a>1; that is, they are global attractors with the property that small changes of initial condition x0 result in small changes of the corresponding solution {xn}.

All these properties can be derived from the explicit form of the solution of (1): (2)xn=11/a-1+1/x0-1/a-11/anifa1,xn=1n+1/x0,ifa=1. See .

The following difference equation, (3)xn+1=axn21+xn2,n=0,1,, was introduced by Thomson  as a depensatory generalization of the Beverton-Holt stock-recruitment relationship used to develop a set of constraints designed to safeguard against overfishing; see  for further references. In view of the sigmoid shape of the function f(u)=au2/(1+u2)  (3) is called the Sigmoid Beverton-Holt model. A very important feature of the Sigmoid Beverton-Holt model is that it exhibits the Allee effect; that is, zero equilibrium has a substantial basin of attraction, as we can see from the following results.

Equation (3) has a unique zero equilibrium when a<2.

Equation (3) has a zero equilibrium and the positive equilibrium x¯=1/2, when a=2.

There exist a zero equilibrium and two positive equilibria, x¯- and x¯+, when a>2.

All solutions of (3) are monotonic (increasing or decreasing) sequences.

If a<2, then the equilibrium point 0 is a global attractor; that is, limnxn=0.

If a=2, then the equilibrium point 0 is a global attractor, with the basin of attraction B(0)=(0,x¯) and x¯=1/2 is a nonhyperbolic equilibrium point with the basin of attraction B(x¯)=[x¯,).

If a>2, then zero equilibrium and x¯+ are locally asymptotically stable, while x¯- is repeller and the basins of attraction of the equilibrium points are given as (4)B0=x0:0x0<x¯-,Bx¯+=x0:x¯-<x0<.

In other words, the smaller positive equilibrium serves as the boundary between two basins of attraction. The zero equilibrium has the basin of attraction B(0) and the model exhibits the Allee effect.

The equilibrium points 0 and x¯+ are globally asymptotically stable in the corresponding basins of attractions B(0) and B(x¯+).

The two dimensional analogue of (1) is the uncoupled system (5)xn+1=axn1+xn,yn+1=byn1+yn,h1n=0,1,, where a,b are positive parameters. The dynamics of system (5) can be derived from dynamics of each equation. Therefore, this system has an explicit solution given by (2).

Two species can interact in several different ways through competition, cooperation, or host-parasitoid interactions. For each of these interactions, we obtain variations of system (5) all of which may require different mathematical analysis.

One such variation that exhibits competitive interaction is the following model, known as the Leslie-Gower model, which was considered in Cushing et al. : (6)xn+1=axn1+xn+c1yn,yn+1=byn1+c2xn+yn,hhhhh11n=0,1,, where all parameters are positive and the initial conditions are nonnegative. The global dynamics of system (6) was completed in . Several variations of system (6) where the competition of two species was modeled by linear fractional difference equations were considered in . An interesting fact is that none of these models exhibited the Allee effect.

The two dimensional analogue of system (3) is the following uncoupled system: (7)xn+1=axn21+xn2,yn+1=byn21+yn2,1hn=0,1,, where a,b are positive parameters. The dynamics of system (7) can be derived from the dynamics of each equation in the system. Since each equation in system (7) has three possible dynamic scenarios, then system (7) possesses nine dynamic scenarios.

A variation of system (7) that exhibits competitive interactions is the system (8)xn+1=xn2B1xn2+C1yn2,yn+1=yn2A2+B2xn2+C2yn2,hhhhhhhhhh1n=0,1,, where B1,C1,A2,B2,C2>0. This system will be considered in the remainder of this paper. We will show that system (8) has similar but more complex dynamics than system (7). We will see that like system (7) the coupled system (8) may possess 1, 3, 5, or 7 equilibrium points in the hyperbolic case and 2, 4, or 6 equilibrium points in the nonhyperbolic case. In each of these cases we will show that the Allee effect is present, although (0,0) is outside of the domain of definition of system (8). We will precisely describe the basins of attraction of all equilibrium points and the singular point (0,0). We will show that the boundaries of the basins of attraction of the equilibrium points are the global stable manifolds of the saddle or the nonhyperbolic equilibrium points. See [10, 11, 1318] for related results and  for dynamics of competitive system with a singular point at the origin. The biological interpretation of a related system is given in [20, 21] and similar system is treated in . The specific feature of our results is that no equilibrium point in the interior of the first quadrant is computable and so our analysis is based on geometric analysis of the equilibrium curves.

2. Preliminaries

Our proofs use some recent general results for competitive systems of difference equations of the form: (9)xn+1=fxn,yn,yn+1=gxn,yn, where f and g are continuous functions and f(x,y) is nondecreasing in x and nonincreasing in y and g(x,y) is nonincreasing in x and nondecreasing in y in some domain A.

Competitive systems of the form (9) were studied by many authors in [6, 7, 9, 13, 14, 2337] and others.

Here we give some basic notions about monotonic maps in the plane.

We define a partial order se on R2 (so-called South-East ordering) so that the positive cone is the fourth quadrant; that is, this partial order is defined by (10)x1y1sex2y2x1x2y1y2.

Similarly, we define North-East ordering as (11)x1y1nex2y2x1x2y1y2.

A map F is called competitive if it is nondecreasing with respect to se, that is, if the following holds: (12)x1y1x2y2Fx1y1Fx2y2.

For each v=(v1,v2)R+2, define Qi(v) for i=1,,4 to be the usual four quadrants based on v and numbered in a counterclockwise direction; for example, Q1(v)={(x,y)R+2:v1x,v2y}.

For SR+2 let S denote the interior of S.

The following definition is from .

Definition 1.

Let R be a nonempty subset of R2. A competitive map T:RR is said to satisfy condition (O+) if for every x, y in R, T(x)neT(y) implies xney, and T is said to satisfy condition (O-) if for every x, y in R, T(x)neT(y) implies ynex.

The following theorem was proved by de Mottoni and Schiaffino  for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps .

Theorem 2.

Let R be a nonempty subset of R2. If T is a competitive map for which (O+) holds, then for all xR, {Tn(x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T. If instead (O-) holds, then for all xR, {T2n} is eventually componentwise monotone. If the orbit of x has compact closure in R, then its omega limit set is either a period-two orbit or a fixed point.

It is well known that a stable period-two orbit and a stable fixed point may coexist; see Hess .

The following result is from , with the domain of the map specialized to be the cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O-).

Theorem 3.

Let RR2 be the cartesian product of two intervals in R. Let T:RR be a C competitive map. If T is injective and detJT(x)>0 for all xR then T satisfies (O+). If T is injective and detJT(x)<0 for all xR then T satisfies (O-).

Theorems 2 and 3 are quite applicable as we have shown in , in the case of competitive systems in the plane consisting of rational equations.

The following result is from , which generalizes the corresponding result for hyperbolic case from . Related results have been obtained by Smith in .

Theorem 4.

Let R be a rectangular subset of R2 and let T be a competitive map on R. Let x¯R be a fixed point of T such that (Q1(x¯)Q3(x¯))R has nonempty interior (i.e., x¯ is not the NW or SE vertex of R).

Suppose that the following statements are true.

The map T is strongly competitive on int((Q1(x¯)Q3(x¯))R).

T is C2 on a relative neighborhood of x¯.

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

Either λ0 and (13)Txx¯,TxxxintQ1x¯Q3x¯R,

or λ<0 and (14)T2xxxintQ1x¯Q3x¯R.

Then there exists a curve C in R such that

C is invariant and a subset of Ws(x¯);

the endpoints of C lie on R;

x¯C;

C is the graph of a strictly increasing continuous function of the first variable;

C is differentiable at x¯ if x¯int(R) or one sided differentiable if x¯R, and in all cases C is tangential to Eλ at x¯;

C separates R into two connected components, namely, (15)W-=xR:yCwithxy,W+=xR:yCwithyx;

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+.

The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess (see [7, 39]) and is helpful for determining the basins of attraction of the equilibrium points.

Corollary 5.

If the nonnegative cone of is a generalized quadrant in Rn, and if T has no fixed points in the ordered interval I(u1,u2) other than u1 and u2, then the interior of I(u1,u2) is either a subset of the basin of attraction of u1 or a subset of the basin of attraction of u2.

The next results give the existence and uniqueness of invariant curves emanating from a nonhyperbolic point of unstable type, that is, a nonhyperbolic point where second eigenvalue is outside interval [-1,1]. Similar result for a nonhyperbolic point of stable type, that is, a nonhyperbolic point where second eigenvalue is in the interval (-1,1), follows from Theorem 4. See Kulenović and Merino, Invariant Curves of Planar Competitive and Cooperative Maps.

Theorem 6.

Let R=(a1,a2)×(b1,b2) and let T:RR be a strongly competitive map with a unique fixed point x¯R, such that T is continuously differentiable in a neighborhood of x¯. Assume further that at the point x¯ the map T has associated characteristic values μ and ν satisfying 1<μ and -μ<ν<μ.

Then there exist curves C1, C2 in R and there exist p1,p2R with p1sex¯sep2 such that

for l=1,2, Cl is invariant, north-east strongly linearly ordered, such that x¯Cl and ClQ3(x¯)Q1(x¯); the endpoints ql, rl of Cl, where qlnerl, belong to the boundary of R. For l,j{1,2} with lj, Cl is a subset of the closure of one of the components of RCj. Both C1 and C2 are tangential at x¯ to the eigenspace associated with ν;

for l=1,2, let Bl be the component of RCl whose closure contains pl. Then Bl is invariant. Also, for xB1, Tn(x) accumulates on Q2(p1)R, and for xB2, Tn(x) accumulates on Q4(p2)R.

Let D1=Q1(x¯)R(B1B2) and D2=Q3(x¯)R(B1B2).

Then D1D2 is invariant.

Corollary 7.

Let a map T with fixed point x¯ be as in Theorem 6. Let D1, D2 be the sets as in Theorem 6. If T satisfies (O+), then for l=1,2, Dl is invariant, and for every xDl, the iterates Tn(x) converge to x¯ or to a point of R. If T satisfies (O-), then T(D1)D2 and T(D2)D1. For every xD1D2, the iterates Tn(x) either converge to x¯ or converge to a period-two point or to a point of R.

3. Local Stability of Equilibrium Points

First we present the local stability analysis of the equilibrium points. It is interesting that the local stability analysis is the more difficult part of our analysis.

The equilibrium points of system (8) satisfy the following system of equations: (16)x¯=x¯2B1x¯2+C1y¯2,y¯=y¯2A2+B2x¯2+C2y¯2,n=0,1,.

All solutions of system (16) with at least one zero component are given as Ex¯x¯,0 where x¯=1/B1,  Ey¯0,y¯ where y¯=1/2C2, and Ey¯±0,y¯± where y¯±=(1±1-4C2A2)/2C2. The equilibrium point Ey¯0,y¯ exists when 1=4C2A2, and Ey¯±0,y¯± exists when 1>4C2A2.

The equilibrium points with strictly positive coordinates satisfy the following system of equations: (17)B1x2+C1y2-x=0,A2+B2x2+C2y2-y=0.

From (17) we have that all real solutions of the system (17) belong to the positive quadrant, since B1x2+C1y2=x>0 and A2+B2x2+C2y2=y>0. By eliminating y from (17) we obtain (18)x4B2C1-B1C22+2C2x3B2C1-B1C2+x22A2B2C12+B1C1-2A2C1C2+C22+C1x2A2C2-1+A22C12=0.

The next result gives the necessary and sufficient conditions for (18) and so system (16) to have between zero and 4 solutions. As we show in Section 4.2 the global dynamics depends on the number of the equilibrium points with positive coordinates.

Lemma 8.

Let (19)Δ3=16A22B14C121-4A2C22-4B13C14A2C2-1×32A23B2C12-8A22C22+6A2C2-1+B12256A24B22C14+128A23B2C22C12-8A23B2C12+C23+16A224B2C12C2+C24+C22+2B2B1C14A2-64A22B2C2C12+4A23B2C12+4C23-13C22+9C2+B2256A23B22C14+B2C1216A2C29-8A2C2-27+4C234A2C2-1,Δ2=-2B13C12A2C2-14A2C2-1+B1232A22B2C2C12-4A23B2C12+C23+C22-4B2B1C1A24A2B2C12+C22-C2-B2B2C129-8A2C2+2C23,Δ1=4A2B1C1C2-2C12A2B2C1+B1+C22.

Assume that B2C1B1C2. Then the following holds.

If Δ3>0, Δ2>0, and Δ1>0, then (18) has four simple real roots.

If Δ3>0 and Δ20(Δ2>0Δ10), then (18) has no real roots.

If Δ3<0, then (18) has two simple real roots.

If Δ3=0 and Δ2<0, then (18) has one real double root.

If Δ3=0 and Δ2>0, then (18) has two real simple roots and one real double root.

If Δ3=0, Δ2=0, and Δ1>0, then (18) has two real double roots.

If Δ3=0, Δ2=0, and Δ1<0, then (18) has no real roots.

If Δ3=0, Δ2=0, and Δ1=0, then (18) has one real root of multiplicity four.

Proof.

The discrimination matrix  of f(x)=Ax4+Bx3+Cx2+Dx+E and f(x) is given by (20)Discrf,f=ABCDE00004A3B2CD0000ABCDE00004A3B2CD0000ABCDE00004A3B2CD0000ABCDE00004A3B2CD. Let Dk denote the determinant of the submatrix of Discr(f~,f~), formed by the first 2k rows and the first 2k columns, for k=1,2,3,4 where (21)f~x=x4B2C1-B1C22+2C2x3B2C1-B1C2+x22A2B2C12+B1C1-2A2C1C2+C22+C1x2A2C2-1+A22C12. So, by straightforward calculation one can see that (22)D1=4B2C1-B1C24,D2=4Δ1B2C1-B1C26,D3=4Δ2C12B2C1-B1C26,D4=Δ3C14B2C1-B1C26. The rest of the proof follows in view of Theorem 1 in .

Geometrically solutions of system (17) are intersections of two ellipses that satisfy the equations (23)x-1/2B121/4B12+y21/4B1C1=1,x21/4B2C2-A2/B2+y-1/2C221/4C22-A2/C2=1, with respective vertices 1/2B1,0 and 0,1/2C2. See Figure 1.

The equilibrium curves of system (8).

Consequently when 1>4C2A2, in addition to the three equilibrium points on the axes, system (8) may have 1,2,3, or 4 positive equilibrium points. We will refer to these equilibrium points as ESW(x¯,y¯) (southwest), ESE(x¯,y¯) (southeast), ENW(x¯,y¯) (northwest), and ENE(x¯,y¯) (northeast) where (24)ENWseENEseESE,ESWneENW.

When a positive equilibrium point is nonhyperbolic we will refer to it as EN(x¯,y¯).

The map associated with system (8) has the form: (25)Txy=x2B1x2+C1y2y2A2+B2x2+C2y2.

The Jacobian matrix of T is (26)JTx,y=2C1xy2B1x2+C1y22-2C1x2yB1x2+C1y22-2B2xy2A2+B2x2+C2y222A2y+2B2x2yA2+B2x2+C2y22, and the Jacobian matrix of T evaluated at an equilibrium E(x¯,y¯) with positive coordinates has the following form: (27)JT(x¯,y¯)=2C1y¯2x¯-2C1y¯-2B2x¯2A2+2B22x¯y¯. The determinant and trace of (27) are (28)detJTx¯,y¯=4A2C1y¯x¯,trJT(x¯,y¯)=2C1y¯2x¯+2A2+2B2x¯2y¯.

It is worth noting that detJT(x¯,y¯) and trJT(x¯,y¯) of (27) are both positive.

Using the equilibrium condition (17), we may rewrite the determinant and trace in the more useful form: (29)detJTx¯,y¯=4x¯y¯B1C2-4y¯C2-4x¯B1-4x¯y¯B2C1+4,trJTx¯,y¯=4-2y¯C2-2x¯B1.

The characteristic equation of the matrix (27) is (30)λ2-trJTx¯,y¯λ+detJTx¯,y¯=0,

whose solutions are the eigenvalues (31)λ=trJTx¯,y¯-trJTx¯,y¯2-4detJTx¯,y¯2,μ=trJTx¯,y¯+trJTx¯,y¯2-4detJTx¯,y¯2.

The corresponding eigenvectors of (31) are (32)Eλ=12xB2xB1-yC2+xB1-yC22+4B2C1xy,1,Eμ=-12xB2yC2-xB1+xB1-yC22+4B2C1xy,1.

We will now consider two lemmas that will be used to prove the local stability character of the positive equilibrium points of system (8). The nonzero coordinates (x¯,y¯) of all equilibrium points will subsequently be designated with the subscripts: r (repeller), a (attractor), s, s1, s2 (saddlepoint), ns (nonhyperbolic of the stable type), and nu (nonhyperbolic of the unstable type).

Lemma 9.

The following conditions hold for the coordinates of the positive equilibrium points, Ex¯,y¯, of system (8).

For ESWx¯r,y¯r and ENx¯nu,y¯nu, (33)x¯<12B1,y¯<12C2.

For ENWx¯s1,y¯s1, (34)x¯<12B1,y¯>12C2.

For ENEx¯a,y¯a, ENEx¯s,y¯s, and ENx¯ns,y¯ns, (35)x¯>12B1,y¯>12C2.

For ESEx¯s2,y¯s2, (36)x¯>12B1,y¯<12C2.

Proof.

This is clear from geometry. See Figure 2.

Local stability.

Lemma 10.

The following conditions hold for the coordinates of the positive equilibrium points, Ex¯,y¯, of System (8).

For ESWx¯r,y¯r and ENWx¯s1,y¯s1, (37)4x¯y¯B1C2-4B2C1x¯y¯+1>2y¯C2+2x¯B1.

For ENEx¯a,y¯a, ENEx¯s,y¯s, and ESEx¯s2,y¯s2, (38)4x¯y¯B1C2-4B2C1x¯y¯+1<2y¯C2+2x¯B1.

For ENx¯ns,y¯ns and ENx¯nu,y¯nu, (39)4x¯y¯B1C2-4B2C1x¯y¯+1=2y¯C2+2x¯B1.

Proof.

(i) Let mE1 be the slope of the tangent line to ellipse E1 at Ex¯,y¯=ESWx¯r,y¯r and let mE2 be the slope of the tangent line to ellipse E2 at Ex¯,y¯=ESWx¯r,y¯r. It is clear from geometry that (40)mE1>mE2>0.

See Figure 2. It follows that (41)dydxE1x¯,y¯>dxdyE2x¯,y¯>0, and in turn (42)1-2B1x¯2C1y¯>2B2x¯1-2C2y¯>0.

Therefore (43)4x¯y¯B1C2-4B2C1x¯y¯+1>2y¯C2+2x¯B1.

The proofs for the remaining case in (i) and all cases in (ii) and (iii) are similar and will be omitted.

Theorem 11.

The following conditions hold for the equilibrium points Ex¯,y¯ of system (8):

Ex¯x¯a,0 is a locally asymptotically stable;

Ey¯0,y¯ns is nonhyperbolic of the stable type;

Ey¯+0,y¯+a is locally asymptotically stable and Ey¯-0,y¯-s is a saddle point;

ESW(x¯r,y¯r) is a repeller;

ENW(x¯s1,y¯s1), ESE(xs2¯,y¯s2), and ENE(x¯s,y¯s) are saddle points;

ENE(x¯a,y¯a) is locally asymptotically stable;

EN(x¯ns,y¯ns) is nonhyperbolic of the stable type;

EN(x¯nu,y¯nu) is nonhyperbolic of the unstable type.

Proof.

(i) The eigenvalues of (26), evaluated at Ex¯x¯a,0, are λ=0 and μ=0.

(ii) The eigenvalues of (26), evaluated at Ey¯0,y¯ns, are λ=0 and μ=1 when 1=4C2A2.

(iii) The eigenvalues of (26), evaluated at Ey¯+0,y¯+a and Ey¯-0,y¯-s, respectively, are λ=0 and μ±=2A2/y¯± when 1>4C2A2.

Note that when 1>4C2A2, (44)y¯+=1+1-4C2A22C2>12C2>2A2.

Therefore μ+=2A2/y¯+<1.

Note that when 1>4C2A2, 1-4A2C2>1-4A2C2. Therefore (45)μ-=2A2y¯-=4A2C21-1-4A2C2>1-1-4A2C21-1-4A2C2=1.

In both cases, the conclusion follows.

(iv) We need to show that trJT(x¯,y¯)<1+detJT(x¯,y¯) and detJT(x¯,y¯)>1 when E(x¯,y¯)=ESW(x¯r,y¯r). Since trJT(x¯,y¯) and detJT(x¯,y¯) are both positive, our conditions become trJT(x¯,y¯)<1+detJT(x¯,y¯) and detJT(x¯,y¯)>1. We will first show that detJT(x¯,y¯)>1. By (37) we have (46)detJTx¯y¯-1=4x¯y¯B1C2-4x¯y¯B2C1-4y¯C2-4x¯B1+4-1>2y¯C2+2x¯B1-1-4y¯C2-4x¯B1+4-1=1-2y¯C2+1-2x¯B1.

By (33) we have 1-2y¯C2+1-2x¯B1>0.

Therefore detJT(x¯,y¯)>1. We will next show that trJT(x¯,y¯)<1+detJT(x¯,y¯).

By (37) we have (47)1+detJTx¯,y¯-trJTx¯,y¯=1+4x¯y¯B1C2-4y¯C2-4x¯B1-4x¯y¯B2C1+4-4-2y¯C2-2x¯B1=4x¯y¯B1C2-4x¯y¯B2C1+1-2y¯C2-2x¯B1>2y¯C2+2x¯B1-2y¯C2-2x¯B1=0.

Therefore trJT(x¯,y¯)<1+detJT(x¯,y¯).

(v) We need to show that trJ(x¯,y¯)>1+detJT(x¯,y¯) when E(x¯,y¯)=ENW(x¯s1,y¯s1). Since trJT(x¯,y¯) and detJT(x¯,y¯) are both positive, our condition becomes trJT(x¯,y¯)>1+detJT(x¯,y¯). By (37) we have (48)trJTx¯,y¯-1+detJTx¯,y¯=4-2y¯C2-2x¯B1-1+4x¯y¯B1C2-4y¯C2-4x¯B1-4x¯y¯B2C1+4=2x¯B1+2y¯C2-4x¯y¯B1C2+4x¯y¯B2C1-1>4x¯y¯B1C2-4B2C1x¯y¯+1-4x¯y¯B1C2+4x¯y¯B2C1-1. Therefore trJT(x¯,y¯)>1+detJT(x¯,y¯). The proofs that ESE(x¯s2,y¯s2) and ENE(x¯s,y¯s) are saddle points are similar and will be omitted.

(vi) We need to show that trJT(x¯,y¯)<1+detJT(x¯,y¯) and detJT(x¯,y¯)<1 when E(x¯,y¯)=ENE(x¯a,y¯a). Since trJT(x¯,y¯) and detJT(x¯,y¯) are both positive, our conditions become trJT(x¯,y¯)<1+detJT(x¯,y¯) and detJT(x¯,y¯)<1. We will first show that detJT(x¯,y¯)<1. By (38) we have (49)detJT(x¯,y¯)-1=4x¯y¯B1C2-4y¯C2-4x¯B1-4x¯y¯B2C1+4-1=4x¯y¯B1C2-4x¯y¯B2C1-4y¯C2-4x¯B1+3<2y¯C2+2x¯B1-1-4y¯C2-4x¯B1+3=1-2y¯C2+1-2x¯B1. By (35) we have 1-2y¯C2+1-2x¯B1<0.

Therefore detJT(x¯,y¯)<1. We will next show that trJT(x¯,y¯)<1+detJT(x¯,y¯). By (38) we have (50)1+detJTx¯,y¯-trJTx¯,y¯=1+4x¯y¯B1C2-4y¯C2-4x¯B1-4x¯y¯B2C1+4-4-2y¯C2-2x¯B1=4x¯y¯B1C2-4x¯y¯B2C1+1-2y¯C2-2x¯B1>2y¯C2+2x¯B1-2y¯C2-2x¯B1.

Therefore trJT(x¯,y¯)<1+detJT(x¯,y¯).

(vii) By (29) and (31) we have (51)λ=-44xyB1C2-4yC2-4xB1-4xyB2C1+44-2yC2-2xB121/24-2yC2-2xB1-4-2yC2-2xB12-44xyB1C2-4yC2-4xB1-4xyB2C1+44-2yC2-2xB121/2×2-1,μ=-44xyB1C2-4yC2-4xB1-4xyB2C1+44-2yC2-2xB121/24-2yC2-2xB1+-44xyB1C2-4yC2-4xB1-4xyB2C1+44-2yC2-2xB12-44xyB1C2-4yC2-4xB1-4xyB2C1+44-2yC2-2xB121/24-2yC2-2xB1×2-1.

By (39), we have λ=3-2y¯C2-2x¯B1 and μ=1. By (35), we have λ<1. The conclusion follows.

(viii) The proof of (viii) is similar to the proof of (vii) and will be omitted.

4. Global Results

In this section we combine the results from Sections 2 and 3 to prove the global results for system (8). First, we present the behavior of the solutions of system (8) on coordinate axes and then we prove that the map T which corresponds to system (8) is injective and that it satisfies (O+).

4.1. Convergence of Solutions on the Coordinate Axes: Injectivity and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M450"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>O</mml:mi><mml:mo mathvariant="bold">+</mml:mo><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

When yn=0, system (8) becomes (52)xn+1=1B1,yn+1=0,n=0,1,.

When xn=0, system (8) becomes (53)xn+1=0,yn+1=yn2A2+C2yn2,n=0,1,.

It follows from (52) and (53) that solutions of system (8) with initial conditions on the x-axis remain on the x-axis and solutions of system (8) with initial conditions on the y-axis remain on the y-axis.

Theorem 12.

The following conditions hold for solutions (xn,yn) of system (8) with initial conditions on the x or y-axis.

Ex¯x¯a,0 is a superattractor of all solutions (xn,yn) of system (8) with initial conditions on the x-axis.

When no equilibrium points exist on the y axis, if x0=0, then limn(xn,yn)=(0,0).

When Ey¯0,y¯ns exists,

if x0=0 and y0>y¯ns, then limn(xn,yn)=(0,y¯ns);

if x0=0 and 0<y0<y¯ns, then limn(xn,yn)=(0,0).

When Ey¯+0,y¯+a and Ey¯-0,y¯-s exist,

if x0=0 and y0>y¯+a, then limn(xn,yn)=(0,y¯+a);

if x0=0 and y¯-s<y0<y¯+a, then limn(xn,yn)=(0,y¯+a);

if x0=0 and 0<y0<y¯-s, then limn(xn,yn)=(0,0).

Proof.

(i) When y0=0, it follows directly from (52) that (xn,yn)=(x¯a,0) for n>1.

(ii) In this case 1<4A2C2. By (53) it can be shown that (54)yn+1-yn=-ynC2yn-1/2C22+A2-1/4C2A2+C2yn2.

By (54), when 1<4A2C2, it is clear that {yn} is a stricly decreasing sequence, and so is convergent. It follows that {yn} converges to 0.

(iii) In this case, 1=4A2C2, and we may rewrite (54) as (55)yn+1-yn=-ynC2yn-y¯ns2A2+C2yn2.

By (55) it is clear that {yn} is a stricly decreasing sequence, and so is convergent. It follows that {yn} converges to y¯ns when y0>y¯ns, and {yn} converges to 0 when 0<y0<y¯ns.

(iv) In this case, 1>4A2C2. By (53), it can be shown that (56)yn+1-yn=-C2ynyn-y¯+ayn-y¯-sA2+C2yn2.

By (56), it is clear that {yn} is a stricly decreasing sequence (and so is convergent) when y0>y¯+a and when 0<y0<y¯-s, and a strictly increasing sequence (and so is convergent) when y¯-s<y0<y¯+a. It follows that {yn} converges to y¯+a when y0>y¯+a and when y¯-s<y0<y¯+a and converges to 0 when 0<y0<y¯-s.

Theorem 13.

The map T which corresponds to system (8) is injective.

Proof.

Indeed, (57)Tx1y1=Tx2y2x12B1x12+C1y12y12A2+B2x12+C2y12=x22B1x22+C1y22y22A2+B2x22+C2y22 which is equivalent to (58)y22x12=y12x22,y1=y2. This immediatly implies x1=x2.

Theorem 14.

The map T which corresponds to system (8) satisfies (O+). All solutions of system (8) converge to either an equilibrium point or to (0,0).

Proof.

Assume that (59)Tx1y1neTx2y2x12B1x12+C1y12y12A2+B2x12+C2y12nex22B1x22+C1y22y22A2+B2x22+C2y22.

The last inequality is equivalent to (60)y22x12y12x22,y1y2. Suppose x2<x1. Then y12x22<y22x12, which contradicts (60). Consequently x1x2 and so x1y1nex2y2.

Thus we conclude that all solutions of system (8) are eventually monotonic for all values of parameters. Furthermore it is clear that all solutions are bounded. Indeed every solution of (8) satisfies (61)xn1B1,yn1C2.

Consequently, all solutions of system (8) converge to an equilibrium point or to (0,0).

4.2. Global Dynamics

In this section we show that there are seven dynamic scenarios for global dynamics of system (8). See Figures 3 and 4 for geometric interpretations of these scenarios.

Global stability.

Global stability.

Theorem 15.

Assume that 1<4A2C2. Then system (8) has one equilibrium point Ex¯ which is locally asymptotically stable. The singular point E0(0,0) is global attractor of all points on y-axis and every point on x-axis is attracted to Ex¯. Furthermore, every point in the interior of the first quadrant is attracted to E0 or Ex¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. In view of Theorem 12, every solution that starts on the y-axis converges to 0 in a decreasing manner and every solution that starts on x-axis is equal to Ex¯ in a single step. Let (x0,y0) be an arbitrary initial point in the interior of the first quadrant. Then (0,y0)se(x0,y0)se(x0,0) and T(0,y0)seT(x0,y0)seT(x0,0)=Ex¯ and so Tn(0,y0)seTn(x0,y0)seTn(x0,0)=Ex¯. In view of Theorems 12 and 14  Tn(x0,y0)Ex¯ or Tn(x0,y0)E0 as n.

Theorem 16.

Assume that 1=4A2C2. Then system (8) has two equilibrium points, Ex¯ which is locally asymptotically stable and Ey¯ which is nonhyperbolic of the stable type. The singular point E0 is global attractor of all points on the y-axis, which start below Ey¯. Furthermore, every point in the interior of the first quadrant below Ws(Ey¯) is attracted to E0(0,0) or Ex¯ and every point in the first quadrant which starts above Ws(Ey¯) is attracted to Ey¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. In view of Theorem 12, every solution that starts on the y-axis below Ey¯ converges to 0 in a decreasing manner and every solution that starts on the x-axis is equal to Ex¯ in a single step. In addition, every solution that starts on the y-axis above Ey¯ converges to Ey¯ in a decreasing way. Let (x0,y0) be an arbitrary initial point in the interior of the first quadrant below Ws(Ey¯). Then (0,y0)se(x0,y0)se(x0,0) which implies T(0,y0)seT(x0,y0)seT(x0,0)=Ex¯ and so Tn(0,y0)seTn(x0,y0)seTn(x0,0)=Ex¯. If y0>y¯ then Tn(x0,y0) will eventually enter the ordered interval I(Ey¯,Ex¯)={(x,y):0<xx¯,0<yy¯}. In view of Theorems 12 and 14, Tn(x0,y0)Ex¯ or Tn(x0,y0)E0 as n.

Now, let (x0,y0) be an arbitrary initial point in the interior of the first quadrant above Ws(Ey¯). Then (0,y0)se(x0,y0)se(x0,yW), where (x0,yW)Ws(Ey¯). This implies T(0,y0)seT(x0,y0)seT(x0,yW) and so Tn(0,y0)seTn(x0,y0)seTn(x0,yW). Since Tn(0,y0)Ey¯, T(x0,yW)Ey¯ as n, we conclude that Tn(x0,y0)Ey¯ as n.

Theorem 17.

Assume that 1>4A2C2 and system (8) has three equilibrium points, Ex¯ and Ey¯+ which are locally asymptotically stable and Ey¯- which is a saddle point. The singular point E0(0,0) is global attractor of all points on y-axis, which start below Ey¯-. The basins of attraction of two equilibrium points are given as (62)BEy¯+=x0,y0:pointsaboveWsEy¯-,BEy¯-=WsEy¯-, where Ws(Ey¯-) denotes the global stable manifold guaranteed by Theorem 4. Furthermore, every initial point below Ws(Ey¯-) is attracted to E0(0,0) or Ex¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. The existence of the global stable manifold is guaranteed by Theorem 4 in view of Theorem 13.

By Theorem 12, every solution that starts on the y-axis below Ey¯- converges to E0 in a decreasing manner and every solution that starts on the x-axis is equal to Ex¯ in a single step. In addition, every solution that starts on the y-axis above Ey¯- converges to Ey¯+ in a monotonic way.

Let (x0,y0) be an arbitrary initial point in the interior of the first quadrant below Ws(Ey¯-). Then (x0,yW)se(x0,y0)se(x0,0) which implies T(x0,yW)seT(x0,y0)seT(x0,0)=Ex¯ and so Tn(x0,yW)seTn(x0,y0)seTn(x0,0)=Ex¯. Since Tn(x0,yW)Ey¯- as n, we conclude that Tn(x0,y0) eventually enters the ordered interval I(Ey¯-,Ex¯)={(x,y):0<xx¯,0<yy¯-}, in which case it converges to Ex¯ or E0(0,0).

Finally, let (x0,y0) be an arbitrary initial point in the interior of the first quadrant above Ws(Ey¯-). Then (0,y0)se(x0,y0)se(x0,yW), where (x0,yW)Ws(Ey¯-). Thus Tn(0,y0)seTn(x0,y0)seTn(x0,yW), which, by Tn(x0,yW)Ey¯- as n, implies that Tn(x0,y0) eventually lands on the part of y-axis above Ey¯- and so it converges to Ey¯+.

Theorem 18.

Assume that 1>4A2C2 and system (8) has four equilibrium points, Ex¯ and Ey¯+ which are locally asymptotically stable, Ey¯- which is a saddle point, and EN which is nonhyperbolic of the unstable type. The singular point E0(0,0) is global attractor of all points on the y-axis, which start below Ey¯-. The basins of attraction of three of the equilibrium points are given as (63)x0,y0:pointsbelowClsuchthatx0xNBEx¯,BEy¯+=x0,y0:pointsaboveWsEy¯-Cu,BEN=x0,y0:pointsbetweenClandCu,BEy¯-=WsEy¯-, where Ws(Ey¯-) denotes the global stable manifold guaranteed by Theorem 4 and Cl,Cu are continuous nondecreasing curves emanating from EN, whose existence and properties are guaranteed by Corollary 7. Furthermore, every initial point below Ws(Ey¯-) is attracted to E0(0,0) or Ex¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. The existence of the global stable manifold is guaranteed by Theorems 4 and 13.

By Theorem 12, every solution that starts on the y-axis below Ey¯- converges to E0 in a decreasing manner and every solution that starts on the x-axis is equal to Ex¯ in a single step. In addition, every solution that starts on y-axis above Ey¯- converges to Ey¯+ in a monotonic way.

Let (x0,y0) be an arbitrary initial point in the interior of the first quadrant below Ws(Ey¯-)Cl. Assume that x0x¯N. Then (x0,yW)se(x0,y0)se(x0,0) and so T(x0,yW)seT(x0,y0)seT(x0,0)=Ex¯, where (x0,yW)Cl and so Tn(x0,yW)seTn(x0,y0)seTn(x0,0)=Ex¯. Since Tn(x0,yW)EN and Tn(x0,0)Ex¯ as n, we conclude that Tn(x0,y0) eventually enters the ordered interval I(EN,Ex¯), in which case, in view of Corollary 5, it converges to Ex¯.

Next, assume that 0<x0<x¯N. Then (x0,yW)se(x0,y0)se(x0,0), where (x0,yW)Ws(Ey¯-) and so T(x0,yW)seT(x0,y0)seT(x0,0)=Ex¯ and so Tn(x0,yW)seTn(x0,y0)seTn(x0,0)=Ex¯. Since Tn(x0,yW)Ey¯- and Tn(x0,0)Ex¯ as n, we conclude that Tn(x0,y0) eventually enters the ordered interval I(Ey¯-,Ex¯), in which case, by Theorems 12 and 14, Tn(x0,y0)Ex¯ or Tn(x0,y0)E0 as n.

Now, let (x0,y0) be an arbitrary initial point in the interior of the first quadrant above Ws(Ey¯-)Cu. Assume that x0>x¯N. Then (0,y0)se(x0,y0)se(x0,yW). Assume that (x0,yW)Cu. Thus Tn(0,y0)seTn(x0,y0)seTn(x0,yW), which by Tn(0,y0)Ey¯+ and Tn(x0,yW)EN as n implies that Tn(x0,y0) eventually enters the ordered interval I(Ey¯+,EN), in which case, in view of Corollary 5, it converges to Ey¯+.

Next, assume that 0<x0x¯N. Then (0,y0)se(x0,y0)se(x0,yW), where (x0,yW)Ws(Ey¯-) and so Tn(0,y0)seTn(x0,y0)seTn(x0,yW). Since Tn(x0,yW)Ey¯- and Tn(0,y0)Ey¯+ as n, we conclude that Tn(x0,y0) converges to Ey¯+.

Finally, let (x0,y0) be an arbitrary initial point between Cl and Cu. Then Tn(x0,y0) stays between Cl and Cu for all n and in view of Corollary 7 it must converge to EN.

Conjecture 19.

Based on our numerical simulations we believe that Cl=Cu in Theorem 18.

Theorem 20.

Assume that 1>4A2C2 and system (8) has five equilibrium points, Ex¯, Ey¯+ which are locally asymptotically stable, Ey¯- and ENW (resp., ESE) which are saddle points, and ESW which is a repeller. The singular point E0(0,0) is global attractor of all points on the y-axis, which start below Ey¯-. The basins of attraction of four of the equilibrium points are given as (64)x0,y0:pointsbelowWsENWBEx¯,BEy¯+=x0,y0:pointsaboveWsEy¯-WsENW,BENW=WsENW,BEy¯-=WsEy¯-, where Ws(Ey¯-) and Ws(ENW) denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below Ws(Ey¯-) is attracted to E0 or Ey¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. We present the proof in the case of the equilibrium point ENW. The proof in the case of the equilibrium point ESE is similar.

The existence of the global stable manifold is guaranteed by Theorems 4 and 13.

By Theorem 12, every solution that starts on the y-axis below Ey¯- converges to E0 in a decreasing manner and every solution that starts on the x-axis is equal to Ex¯ in a single step. In addition, every solution that starts on the y-axis above Ey¯- converges to Ey¯+ in a monotonic way.

Let (x0,y0) be an arbitrary initial point in the interior of the first quadrant below Ws(Ey¯-)Ws(ENW). Assume that x0>x¯SW. Then (x0,yW)se(x0,y0)se(x0,0) which implies T(x0,yW)seT(x0,y0)seT(x0,0)=Ex¯, where (x0,yW)Ws(ENW) and so Tn(x0,yW)seTn(x0,y0)seTn(x0,0)=Ex¯. Since Tn(x0,yW)ENW and Tn(x0,0)Ex¯ as n, we conclude that Tn(x0,y0) eventually enters the ordered interval I(ENW,Ex¯), in which case, in view of Corollary 5, it converges to Ex¯.

Next, assume that 0<x0x¯SW. Then (x0,yW)se(x0,y0)se(x0,0), where (x0,yW)Ws(Ey¯-). Thus T(x0,yW)seT(x0,y0)seT(x0,0)=Ex¯ and so Tn(x0,yW)seTn(x0,y0)seTn(x0,0)=Ex¯. Since Tn(x0,yW)Ey¯- and Tn(x0,0)Ex¯ as n, we conclude that Tn(x0,y0) eventually enters the interior of the ordered interval I(Ey¯-,Ex¯), in which case, it converges to E0 or Ex¯.

Now, let (x0,y0) be an arbitrary initial point in the interior of the first quadrant above Ws(Ey¯-)Ws(ENW). Assume x0>x¯SW. Then (0,y0)se(x0,y0)se(x0,yW), where (x0,yW)Ws(ENW) and so Tn(0,y0)seTn(x0,y0)seTn(x0,yW). Since Tn(0,y0)Ey¯+ and Tn(x0,yW)ENW as n, then Tn(x0,y0) eventually enters the ordered interval I(Ey¯+,ENW), in which case, in view of Corollary 5, it converges to Ey¯+.

Next, assume that 0<x0x¯SW. Then (0,y0)se(x0,y0)se(x0,yW), where (x0,yW)Ws(Ey¯-) and so Tn(0,y0)seTn(x0,y0)seTn(x0,yW). Since Tn(x0,yW)Ey¯- and Tn(0,y0)Ey¯+ as n, we conclude that Tn(x0,y0) converges to Ey¯+.

Theorem 21.

Assume that 1>4A2C2 and system (8) has six equilibrium points, Ex¯, Ey¯+ which are locally asymptotically stable, Ey¯- and ENE (resp., ESE or ENW) which are saddle points, ESW which is a repeller, and EN which is nonhyperbolic of the stable type. The singular point E0(0,0) is global attractor of all points on the y-axis, which start below Ey¯-. The basins of attraction of five of the equilibrium points are given as (65)x0,y0:pointsbelowWsENBEx¯,BEy¯+=x0,y0:pointsaboveWsEy¯-WsENE,BEN=x0,y0:regionboundedbyWsENandWsENE,BEy¯-=WsEy¯-,BENE=WsENE, where Ws(Ey¯-), Ws(EN), and Ws(ENE) denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below Ws(Ey¯-) is attracted to E0 or Ex¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. We present the proof in the case of the equilibrium point ENE. The proof in the case of the equilibrium points ESE and ENW is similar.

The existence of the global stable manifolds are guaranteed by Theorems 4 and 13.

The proofs of the basins of attractions B(Ex¯), B(Ey¯+) are the same as the proofs for the corresponding basins of attraction in Theorem 20, so we will only give the proof for B(EN). Indeed, B(EN) is an invariant set and Tn(B(EN)) is a subset of the interior of the ordered interval I(ENE,EN) for n large. In view of Corollary 5 the interior of the ordered interval I(ENE,EN) is attracted to EN.

Theorem 22.

Assume that 1>4A2C2 and system (8) has seven equilibrium points, Ex¯, Ey¯+, ENE which are locally asymptotically stable, Ey¯-,ESE,ENW which are saddle points, and ESW which is a repeller. The singular point E0(0,0) is global attractor of all points on y-axis, which start below Ey¯-. The basins of attraction of six of the equilibrium points are given as (66)x0,y0:pointsbelowWsESEBEx¯,BEy¯+=x0,y0:pointsaboveWsEy¯-WsENW,BENE=x0,y0:regionboundedbyWsESEandWsENW,BEy¯-=WsEy¯-,BESE=WsESE,BENW=WsENW, where Ws(Ey¯-), Ws(ENW), and Ws(ESE) denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below Ws(Ey¯-) is attracted to E0 or Ex¯.

Proof.

Local stability of all equilibrium points follows from Theorem 11. Proofs of the basins of attractions B(Ex¯), B(Ey¯+) are the same as the proofs for corresponding basins of attraction in Theorem 20. So we only give the proof for B(ENE). Indeed, B(ENE) is an invariant set and Tn(B(ENE)) is a subset of the interior of the ordered interval I(ENW,ESE) for n large. In view of Corollary 5 the interior of the ordered interval (ENW,ESE) is attracted to ENE.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Kulenovic M. R. S. Ladas G. Dynamics of Second Order Rational Difference Equations 2001 Boca Raton, Fla, USA Chapman & Hall/CRC Kulenovic M. R. S. Merino O. Disrete Dynamical Systems and Difference Equations with Mathematica 2002 Boca Raton, Fla, USA Chapman & Hall/CRC Press Thieme H. R. Mathematics in Population Biology 2003 Princeton, NJ, USA Princeton University Press Princeton Series in Theoretical and Computational Biology Thomson G. G. Smith S. J. Hunt J. J. Rivard D. A proposal for a threshold stock size and maximum fishing mortality rate Risk Evaluation and Biological Reference Points for Fisheries Management 1993 120 Canadian Special Publication of Fisheries and Aquatic Sciences 303 320 Harry A. J. Kent C. M. Kocic V. L. Global behavior of solutions of a periodically forced Sigmoid Beverton-Holt model Journal of Biological Dynamics 2012 6 2 212 234 10.1080/17513758.2011.552738 MR2897874 2-s2.0-84868336011 Cushing J. M. Levarge S. Chitnis N. Henson S. M. Some discrete competition models and the competitive exclusion principle Journal of Difference Equations and Applications 2004 10 13–15 1139 1151 10.1080/10236190410001652739 MR2100718 2-s2.0-8744244403 Kulenović M. R. S. Merino O. Global bifurcation for discrete competitive systems in the plane Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences 2009 12 1 133 149 10.3934/dcdsb.2009.12.133 MR2505666 2-s2.0-69249198864 Clark D. Kulenović M. R. S. A coupled system of rational difference equations Computers & Mathematics with Applications 2002 43 6-7 849 867 10.1016/S0898-1221(01)00326-1 MR1884331 2-s2.0-0036498930 Clark D. Kulenović M. R. Selgrade J. F. Global asymptotic behavior of a two-dimensional difference equation modelling competition Nonlinear Analysis. Theory, Methods & Applications 2003 52 7 1765 1776 10.1016/S0362-546X(02)00294-8 MR1956175 2-s2.0-0037333658 Kalabušić S. Kulenović M. R. S. Pilav E. Dynamics of a two-dimensional system of rational difference equations of Leslie-Gower type Advances in Difference Equations 2011 2011, article 29 MR2830974 10.1186/1687-1847-2011-29 Kalabušić S. Kulenović M. R. Pilav E. Multiple attractors for a competitive system of rational difference equations in the plane Abstract and Applied Analysis 2011 2011 35 295308 10.1155/2011/295308 MR2854930 2-s2.0-84855555043 Kulenović M. R. S. Merino O. Nurkanović M. Global dynamics of certain competitive system in the plane Journal of Difference Equations and Applications 2012 18 12 1951 1966 10.1080/10236198.2011.605357 MR3001343 Kulenović M. R. S. Nurkanović M. Asymptotic behavior of a system of linear fractional difference equations Journal of Inequalities and Applications 2005 2 127 143 MR2173357 10.1155/JIA.2005.127 Kulenović M. R. S. Nurkanović M. Asymptotic behavior of a competitive system of linear fractional difference equations Advances in Difference Equations 2006 2006 13 19756 MR2255151 10.1155/ADE/2006/19756 Basu S. Merino O. On the global behavior of solutions to a planar system of difference equations Communications on Applied Nonlinear Analysis 2009 16 1 89 101 ZBL1176.39014 2-s2.0-61749089471 MR2494303 Brett A. Kulenović M. R. S. Basins of attraction of equilibrium points of monotone difference equations Sarajevo Journal of Mathematics 2009 5 2 211 233 MR2567754 Burgić D. Kalabuvić S. Kulenović M. R. S. Nonhyperbolic dynamics for competitive systems in the plane and global period-doubling bifurcations Advances in Dynamical Systems and Applications 2008 3 2 229 249 MR2548022 Kulenović M. R. S. Merino O. Invariant manifolds for competitive discrete systems in the plane International Journal of Bifurcation and Chaos 2010 20 8 2471 2486 10.1142/S0218127410027118 MR2738710 2-s2.0-78549242035 Kang Y. Smith H. L. Global dynamics of a discrete two-species lottery-Ricker competition model Journal of Biological Dynamics 2012 6 2 358 376 10.1080/17513758.2011.586064 MR2897879 2-s2.0-84868311119 Livadiotis G. Elaydi S. General Allee effect in two-species population biology Journal of Biological Dynamics 2012 6 2 959 973 10.1080/17513758.2012.700075 2-s2.0-84868307063 Livadiotis G. Assas L. Elaydi S. Kwessi E. Ribble D. Competition models with Allee effects Journal of Difference Equations and Applications 2014 20 8 1127 1151 10.1080/10236198.2014.897341 MR3216890 ZBL1291.39042 2-s2.0-84899469608 Chow Y. Jang S. R. Multiple attractors in a Leslie-Gower competition system with Allee effects Journal of Difference Equations and Applications 2014 20 2 169 187 10.1080/10236198.2013.815166 MR3173541 ZBL06259237 2-s2.0-84889264195 Franke J. E. Yakubu A.-A. Mutual exclusion versus coexistence for discrete competitive systems Journal of Mathematical Biology 1991 30 2 161 168 10.1007/BF00160333 MR1138846 ZBL0735.92023 Franke J. E. Yakubu A.-A. Global attractors in competitive systems Nonlinear Analysis: Theory, Methods & Applications 1991 16 2 111 129 10.1016/0362-546X(91)90163-U MR1090785 2-s2.0-0005729536 Franke J. E. Yakubu A.-A. Geometry of exclusion principles in discrete systems Journal of Mathematical Analysis and Applications 1992 168 2 385 400 10.1016/0022-247X(92)90167-C MR1175998 ZBL0778.93012 2-s2.0-38249007755 Hassell M. P. Comins H. N. Discrete time models for two-species competition Theoretical Population Biology 1976 9 2 202 221 10.1016/0040-5809(76)90045-9 MR0421737 2-s2.0-0016947226 Hirsch M. Smith H. Monotone dynamical systems Handbook of Differential Equations: Ordinary Differential Equations 2005 2 Amsterdam, The Netherlands Elsevier B. V. 239 357 Hirsch M. W. Smith H. Monotone maps: a review Journal of Difference Equations and Applications 2005 11 4-5 379 398 10.1080/10236190412331335445 MR2151682 ZBL1080.37016 2-s2.0-22544480015 Jiang H. Rogers T. D. The discrete dynamics of symmetric competition in the plane Journal of Mathematical Biology 1987 25 6 573 596 10.1007/BF00275495 MR918829 ZBL0668.92011 2-s2.0-0023487191 Krawcewicz W. Rogers T. D. Perfect harmony: the discrete dynamics of cooperation Journal of Mathematical Biology 1990 28 4 383 410 10.1007/BF00178325 MR1057045 ZBL0729.92022 Kulenović M. R. S. Merino O. Competitive-exclusion versus competitive-coexistence for systems in the plane Discrete and Continuous Dynamical Systems, Series B 2006 6 5 1141 1156 2-s2.0-33748570775 10.3934/dcdsb.2006.6.1141 MR2224875 Kulenović M. R. S. Nurkanović M. Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation Journal of Difference Equations and Applications 2003 9 1 149 159 10.1080/1023619031000060981 MR1958309 May R. M. Stability and Complexity in Model Ecosystems 2001 Princeton, NJ, USA Princeton University Press Smith H. L. Periodic competitive differential equations and the discrete dynamics of competitive maps Journal of Differential Equations 1986 64 2 165 194 10.1016/0022-0396(86)90086-0 MR851910 ZBL0596.34013 2-s2.0-38249038749 Smith H. L. Planar competitive and cooperative difference equations Journal of Difference Equations and Applications 1998 3 5-6 335 357 10.1080/10236199708808108 MR1618123 ZBL0907.39004 Yakubu A.-A. The effects of planting and harvesting on endangered species in discrete competitive systems Mathematical Biosciences 1995 126 1 1 20 10.1016/0025-5564(94)00033-V MR1317925 ZBL0820.92024 2-s2.0-0028985509 Yakubu A.-A. A discrete competitive system with planting Journal of Difference Equations and Applications 1998 4 213 214 10.1080/10236199808808137 de Mottoni P. Schiaffino A. Competition systems with periodic coefficients: a geometric approach Journal of Mathematical Biology 1981 11 3 319 335 10.1007/BF00276900 MR613766 ZBL0474.92015 2-s2.0-0019495964 Hess P. Periodic-Parabolic Boundary Value Problems and Positivity 1991 247 Harlow, UK Longman Scientific & Technical Pitman Research Notes in Mathematics Series MR1100011 Camouzis E. Kulenović M. R. S. Ladas G. Merino O. Rational systems in the plane Journal of Difference Equations and Applications 2009 15 3 303 323 MR2498776 10.1080/10236190802125264 2-s2.0-61449200503 Yang L. Hou X. Zeng Z. A complete discrimination system for polynomials Science in China. Series E. Technological Sciences 1996 39 6 628 646 MR1438754 ZBL0866.68104