# Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

Authors: V. Hadžiabdić; M. R. S. Kulenović; E. Pilav
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We investigate global dynamics of the following systems of difference equationsxn+1=xn/A1+B1xn+C1yn, yn+1=yn2/A2+B2xn+C2yn2, n=0,1,…, where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.


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## Body

## 1. Introduction

In this paper we study the global dynamics of the following rational system of difference equations:(1)xn+1=xnA1+B1xn+C1yn,yn+1=yn2A2+B2xn+C2yn2,n=0,1,…,where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers.System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane; see [1]. System (1) can be used as a mathematical model for competition in population dynamics. System (1) is related to Leslie-Gower competition model(2)xn+1=xnA1+B1xn+C1yn,yn+1=ynA2+B2xn+C2yn,n=0,1,…,where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers, considered in [2]. System (2) globally exhibits three dynamic scenarios in five parametric regions which are competitive exclusion, competitive coexistence, and existence of an infinite number of equilibrium solutions; see [1–3]. System (2) does not exhibit the Allee effect, which is desirable from modeling point of view. The simplest variation of system (2) which exhibits the Allee effect is probably system(3)xn+1=xn2A1+B1xn2+C1yn,yn+1=yn2A2+B2xn+C2yn2,n=0,1,…,where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers, considered in [4]. System (3) has between 1 and 9 equilibrium points and exhibits nine dynamics scenarios part of each is the Allee effect. In the case of the dynamic scenario with nine equilibrium points system (3) exhibits both competitive exclusion and competitive coexistence as well as the Allee effect. Another system with quadratic terms is(4)xn+1=xn2B1xn2+C1yn2,yn+1=yn2A2+B2xn2+C2yn2,n=0,1,…,where the parameters A2, B1, B2, C1, and C2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers such that x0+y0>0, considered in [5]. System (4) exhibits seven scenarios part of each is singular Allee’s effect, which means that the origin as the singular point of this system still has some basin of attraction. First systematic study for a system with quadratic terms was performed in [6] for system (5)xn+1=xnA1+yn2,yn+1=ynA2+xn2,n=0,1,…which exhibits nine dynamic scenarios and whose dynamics is very similar to the corresponding system without quadratic terms considered in [7].In general, it seems that an introduction of quadratic terms in equations of the Leslie-Gower model (2) generates the Allee effect. We will test this hypothesis in this paper by introducing the quadratic terms only in the second equation. System (1) can be considered as the competitive version of the decoupled system(6)xn+1=xnA1+B1xn,yn+1=yn2A2+C2yn2,n=0,1,…,where the parameters A1, A2, B1, and C2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers, whose dynamics can be directly obtained from two separate equations. Unlike system (2) which has five regions of parameters with distinct local behavior system (1) has eighteen regions of parameters with distinct local behavior, which is caused by the geometry of the problem, that is, by the geometry of equilibrium curves. More precisely, the equilibrium curves of system (2) are lines while the equilibrium curves of system (1) are a line and a parabola. In the case when A1>1, all equilibrium points are hyperbolic and all solutions are attracted to the three equilibrium points on the y-axis and we can describe this situation as competitive exclusion case. When A1=1, the equilibrium point E1 is nonhyperbolic and dynamics is analogous to the case when A1>1. In both cases the Allee effect is present. When A1<1, there exist 11 regions of parameters with different global dynamics. In nine of these regions the global dynamics is in competitive exclusion case, which means that all solutions converge to one of the equilibrium points on the axes and in only two situations we have competitive coexistence case, which means that the interior equilibrium points have substantial basin of attraction. In all 11 cases, the zero equilibrium has some basin of attraction which is a part of y-axis so we can say that in these cases system (1) exhibits weak Allee’s effect. Figure 3 gives the bifurcation diagram showing the transition from different global dynamics situations when A1<1, since the cases A1≥1 are simple and do not need graphical interpretation.The paper is organized as follows. Section2 contains some necessary results on competitive systems in the plane. Section 3 provides some basic information about the number of equilibrium points. Section 4 contains local stability analysis of all equilibrium solutions. Section 5 contains some global results on injectivity of the map associated with system (1). Section 6 gives global dynamics of system (1) in all regions of the parameters.


## 2. Preliminaries

A first-order system of difference equations(7)xn+1=fxn,yn,yn+1=gxn,yn,n=0,1,…,where S⊂R2, (f,g):S→S, f, g are continuous functions iscompetitive if f(x,y) is nondecreasing in x and nonincreasing in y, and g(x,y) is nonincreasing in x and nondecreasing in y. If both f and g are nondecreasing in x and y, system (7) iscooperative. Competitive and cooperative maps are defined similarly.Strongly competitive systems of difference equations or strongly competitive maps are those for which the functions f and g are coordinate-wise strictly monotone.Competitive and cooperative systems have been investigated by many authors; see [1–3, 7–16]. Special attention to discrete competitive and cooperative systems in the plane was given in [1–3, 16, 17]. One of the reasons for paying special attention to two-dimensional discrete competitive and cooperative systems is their applicability and the fact that many examples of mathematical models in biology and economy which involve competition or cooperation are models which involve two species. Another reason is that the theory of two-dimensional discrete competitive and cooperative systems is very well developed, unlike such theory for three-dimensional and higher systems. Part of the reason for this situation is de Mottoni-Schiaffino theorem given below, which provides relatively simple scenarios for possible behavior of many two-dimensional discrete competitive and cooperative systems. However, this does not mean that one can not encounter chaos in such systems as has been shown by Smith; see [16].Ifv=(u,v)∈R2, we denote with Ql(v), l∈{1,2,3,4}, the four quadrants in R2 relative to v, that is, Q1(v)={(x,y)∈R2:x≥u,y≥v}, Q2(v)={(x,y)∈R2:x≤u,y≥v}, and so on. Define theSouth-East partial order ⪯se on R2 by (x,y)⪯se(s,t) if and only if x≤s and y≥t. Similarly, we define theNorth-East partial order ⪯ne on R2 by (x,y)⪯ne(s,t) if and only if x≤s and y≤t. For A⊂R2 and x∈R2, define thedistance fromxtoA as dist(x,A)≔inf⁡{x-y:y∈A}. By intA we denote the interior of a set A.It is easy to show that a mapF is competitive if it is nondecreasing with respect to the South-East partial order, that is, if the following holds:(8)x1y1⪯sex2y2⟹Fx1y1⪯seFx2y2.For standard definitions of attracting fixed point, saddle point, stable manifold, and related notions see [10].We now state three results for competitive maps in the plane. The following definition is from [16].Definition 1.
LetS be a nonempty subset of R2. A competitive map T:S→S is said to satisfy condition (O+) if for every x, y in S, T(x)⪯neT(y) implies x⪯ney, and T is said to satisfy condition (O-) if for every x, y in S, T(x)⪯neT(y) implies y⪯nex.The following theorem was proved by de Mottoni-Schiaffino [17] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [13, 14].Theorem 2.
LetS be a nonempty subset of R2. If T is a competitive map for which (O+) holds then for all x∈S, {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 x∈S, {T2n(x)} is eventually componentwise monotone. If the orbit of x has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point.The following result is from [16], 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.
LetR⊂R2 be the Cartesian product of two intervals in R. Let T:R→R be a C1 competitive map. If T is injective and det⁡JT(x)>0 for all x∈R then T satisfies (O+). If T is injective and det⁡JT(x)<0 for all x∈R then T satisfies (O-).The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess (see [18]) and is helpful for determining the basins of attraction of the equilibrium points.Corollary 4.
If the nonnegative cone of⪯ is a generalized quadrant in Rn, and if T has no fixed points in u1,u2 other than u1 and u2, then the interior of u1,u2 is either a subset of the basin of attraction of u1 or a subset of the basin of attraction of u2.Next result is well known global attractivity result which holds in partially ordered Banach spaces as well; see [18].Theorem 5.
LetT be a monotone map on a closed and bounded rectangular region R⊂R2. Suppose that T has a unique fixed point e¯ in R. Then e¯ is a global attractor of T on R.The following theorems were proved by Kulenović and Merino [1] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or nonhyperbolic) is by absolute value smaller than 1 while the other has an arbitrary value. These results are useful for determining basins of attraction of fixed points of competitive maps.Theorem 6.
LetT be a competitive map on a rectangular region R⊂R2. Let x¯∈R be a fixed point of T such that Δ≔R∩int(Q1(x¯)∪Q3(x¯)) is nonempty (i.e., x¯ is not the NW or SE vertex of R), and T is strongly competitive on Δ. Suppose that the following statements are true:(a)
The mapT has a C1 extension to a neighborhood ofx¯.(b)
The JacobianJT(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⊂R 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 R 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.The situation where the endpoints ofC are boundary points of R is of interest. The following result gives a sufficient condition for this case.Theorem 7.
For the curveC of Theorem 6 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied(i)
The mapT has no fixed points or periodic points of minimal period-two in Δ.(ii)
The mapT has no fixed points in Δ, det⁡JT(x¯)>0, and T(x)=x¯ has no solutions x∈Δ.(iii)
The mapT has no points of minimal period-two in Δ, det⁡JT(x¯)<0, and T(x)=x¯ has no solutions x∈Δ.The next result is useful for determining basins of attraction of fixed points of competitive maps.Theorem 8.
(A) Assume the hypotheses of Theorem6, and let C be the curve whose existence is guaranteed by Theorem 6. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely,(9)W-≔x∈R∖C:∃y∈Cwithx⪯sey,W+≔x∈R∖C:∃y∈Cwithy⪯sex,such that the following statements are true:(i)
W- is invariant, and dist(Tn(x),Q2(x¯))→0 as n→∞ for every x∈W-.(ii)
W+ is invariant, and dist(Tn(x),Q4(x¯))→0 as n→∞ for every x∈W+.
(B) If, in addition to the hypotheses of part (A),x¯ is an interior point of R 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:(iii)
For everyx∈W- there exists n0∈N such that Tn(x)∈intQ2(x¯) for n≥n0.(iv)
For everyx∈W+ there exists n0∈N such that Tn(x)∈intQ4(x¯) for n≥n0.IfT is a map on a set R and if x¯ is a fixed point of T, thestable setWs(x¯) of x¯ is the set {x∈R:Tn(x)→x¯} andunstable setWu(x¯) of x¯ is the set (10)x∈R:thereexistsxnn=-∞0⊂Rs.t.Txn=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 R, the sets Ws(x¯) and Wu(x¯) are the stable and unstable manifolds of x¯.Theorem 9.
In addition to the hypotheses of part (B) of Theorem8, suppose that μ>1 and that the eigenspace Eμ associated with μ is not a coordinate axis. If the curve C of Theorem 6 has endpoints in ∂R, then C is the stable set Ws(x¯) of x¯, and the unstable set Wu(x¯) of x¯ is a curve in R 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 R are fixed points of T.


## 3. Number of Equilibria

In this section we give some basic facts which are used later. LetT be the map associated with system (1) given by(11)Tx,y=fx,y,gx,y=xA1+B1x+C1y,y2A2+B2x+C2y2.Let R=R+2. The equilibrium points (x¯,y¯) of system (1) satisfy equations(12)x¯A1+B1x¯+C1y¯=x¯,y¯2A2+B2x¯+C2y¯2=y¯.For x¯=0 we have (13)y¯=y¯2-A2y¯-C2y¯3from which we obtain three equilibrium points (14)E1=0,0,E2=0,1-Δ12C2,E3=0,1+Δ12C2,where Δ1=1-4A2C2.Assume thatx¯≠0. Then, from the first equation of system (12) we have (15)y¯=-A1+B1x¯-1C1.By substituting this into the second equation we obtain (16)A1+B1x¯-1=0or(17)g~x≡B12C2x2+xB12A1-1C2+C1+B2C12+A1-12C2+A1-1C1+A2C12=0,from which we obtain the other three equilibrium points (18)E4=1-A1B1,0,E5=-2A1B1C2-B2C12-B1C1+2B1C2+C1Δ22B12C2,B1+B2C1-Δ22B1C2,E6=-2A1B1C2-B2C12-B1C1+2B1C2-C1Δ22B12C2,B1+B2C1+Δ22B1C2,where (19)Δ2=B1+B2C12-4B1A2B1-A1-1B2C2.Lemma 10.
The following hold:(i)
The equilibrium pointsE2 and E3 exist if and only if Δ1≥0 and E2=E3 if and only if Δ1=0.(ii)
The equilibrium pointE4 exists if and only if A1≤1 and E4=E1 if and only if A1=1.(iii)
Assume thatΔ2≥0. The equilibrium point E5 exists if and only if A1<1 and (20)C1≤1-A1B11-A1B2+2A2B1,C2≤B2C1+B124B1A2B1+1-A1B2or(21)C1>1-A1B11-A1B2+2A2B1,C2≤C11-A1-A2C11-A12.(iv)
Assume thatΔ2≥0. The equilibrium point E6 exists if and only if A1<1 and (22)C1≤1-A1B11-A1B2+2A2B1,C11-A1-A2C11-A12≤C2≤B2C1+B124B1A2B1+1-A1B2.Proof.
The proof of the statements (i) and (ii) is trivial and we skip it. Now we prove the statement (iii). In view of Descartes’ rule of signs we obtain that (17) has no positive solutions if A1≥1. Now, we suppose that A1<1. One can see that y¯5>0 for all values of parameters. We consider two cases: (1)
Assume that(23)-2A1B1C2-B2C12-B1C1+2B1C2≥0,which is equivalent to(24)C2≥C1B2C1+B121-A1B1.Since(25)Δ2≥0⟺C2≤B2C1+B124B1A2B1+1-A1B2we have thatx¯5≥0 if and only if (26)C1B2C1+B121-A1B1≤C2≤B2C1+B124B1A2B1+1-A1B2,(27)B2C1+B124B1A2B1-A1-1B2-C1B2C1+B121-A1B1=-B2C1+B11-A1B2C1+B12A2C1+A1-141-A1B1A2B1+1-A1B2≥0which is equivalent to(28)C1≤1-A1B12A2B1+1-A1B2.From (27) and (28) it follows x¯5≥0 if and only if (29)C1≤1-A1B12A2B1+1-A1B2,C1B2C1+B121-A1B1≤C2≤B2C1+B124B1A2B1+1-A1B2.(2)
Assume that(30)-2A1B1C2-B2C12-B1C1+2B1C2<0which is equivalent to(31)C2<C1B2C1+B121-A1B1.Thenx¯5≥0 if and only if (32)Δ2C12--B12A1-1C2+C1-B2C122=-4B12C2A1-12C2+C1A2C1+A1-1≥0,which is equivalent to(33)C2≤C1-A2C1-A1+11-A12,(34)Δ2≥0⟺C2≤B2C1+B124B1A2B1+1-A1B2.Since(35)B2C1+B124B1A2B1-A1-1B2-C1-A2C1-A1+11-A12=1-A1B2C1+B1-2A2C1-A1+1241-A12B1A2B1+1-A1B2≥0then from (33) and Δ2≥0 we have(36)C2≤C11-A2C1-A11-A12.Since(37)C11-A2C1-A11-A12-C1B2C1+B121-A1B1=-C11-A1B2C1+B12A2C1+A1-12A1-12B1we have that (31) and (36) are equivalent to(38)C1>1-A1B12A2B1+1-A1B2,C2≤C11-A2C1-A11-A12or(39)C1≤1-A1B12A2B1+1-A1B2,C2<C1B2C1+B121-A1B1.Now, the proof of the statement (iii) follows from (28), (38), and (39). The proof of the statement (iv) is similar and we skip it.We now introduce the following notation for regions in parameter space(C1,C2) (see Figure 1):(40)R24=C1,C2:A1<1,Δ1>0,C1<1-A1B11-A1B2+2A2B1,C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2,R22=C1,C2:A1<1,Δ1>0,C2<C11-A1-A2C11-A12,R21=C1,C2:A1<1,Δ1>0,C2>B2C1+B124B1A2B1+1-A1B2,R25=C1,C2:A1<1,Δ1>0,1-A12A2>C1>1-A1B11-A1B2+2A2B1,C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2,R23=C1,C2:A1<1,Δ1>0,C1>1-A12A2,C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2,R1=C1,C2:A1<1,Δ1<0,C31=C1,C2:A1<1,Δ1>0,C1<1-A1B11-A1B2+2A2B1,C2=B2C1+B124B1A2B1+1-A1B2,C32=C1,C2:A1<1,Δ1>0,1-A12A2>C1>1-A1B11-A1B2+2A2B1,C2=B2C1+B124B1A2B1+1-A1B2,C21=C1,C2:A1<1,Δ1>0,C1<1-A1B11-A1B2+2A2B1,C2=C11-A1-A2C11-A12,C22=C1,C2:A1<1,Δ1>0,1-A12A2>C1>1-A1B11-A1B2+2A2B1,C2=C11-A1-A2C11-A12,C23=C1,C2:A1<1,Δ1>0,C1>1-A12A2,C2=C11-A1-A2C11-A12,C11=C1,C2:A1<1,Δ1=0,C1>1-A12A2,C12=C1,C2:A1<1,Δ1=0,C1<1-A12A2,P1=C1,C2:A1<1,C1=1-A12A2,C2=14A2,P2=C1,C2:A1<1,C1=1-A1B11-A1B2+2A2B1,C2=B1A2B1-A1-1B2A1-1B2-2A2B12,R31=C1,C2:A1=1,Δ1<0,R32=C1,C2:A1=1,Δ1>0,C3=C1,C2:A1=1,Δ1=0,R41=C1,C2:A1>1,Δ1<0,R42=C1,C2:A1>1,Δ1>0,C4=C1,C2:A1>1,Δ1=0.Figure 1
Parameter regions in the(C1,C2)-plane. The curves C21, C22, and C23 are defined as part of the parabola C2=C11-A1-A2C1/1-A12 and the curves C32 and C31 are defined as part of the parabola C2=B2C1+B12/4B1A2B1+1-A1B2.Figure1 gives a graphical representation of above sets. The following result gives a complete classification for the number of equilibrium solutions of system (1).Proposition 11.
LetA1, A2, B1, B2, C1, and C2 be positive real numbers. Then, the number of positive equilibrium solutions of system (1) with parameters A1, A2, B1, B2, C1, and C2 can be from 1 to 6. The different cases are given in Table 1.Table 1
The criteria for the existence of the equilibrium points.
Case
Equilibria
Region
The criteria for the existence
(i)
E
1
,
E
2
,
E
3
,
E
4
,
E
5
,
E
6
R
24
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1,  C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2
(ii)
E
1
,
E
2
,
E
3
,
E
4
,
E
5
=
E
6
C31
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1,  C2=B2C1+B124B1A2B1+1-A1B2
(iii)
E
1
,
E
2
,
E
3
,
E
4
,
E
5
R
22
A
1
<
1, Δ1>0, C2<C11-A1-A2C11-A12
(iv)
E
1
,
E
2
,
E
3
,
E
4
R21∪R25∪C32
A
1
<
1, Δ1>0, C2>B2C1+B124B1A2B1+1-A1B2 or A1<1, Δ1>0, 1-A12A2>C1>1-A1B11-A1B2+2A2B1,  C11-A1-A2C11-A12<C2≤B2C1+B124B1A2B1+1-A1B2
(v)
E1,E2,E3,E4
R
23
A
1
<
1, Δ1>0, C1>1-A12A2,  C11-A1-A2C11-A12<C2≤B2C1+B124B1A2B1+1-A1B2
(vi)
E
1
,
E
2
,
E
3
=
E
6
,
E
4
,
E
5
C21
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1, C2=C11-A1-A2C11-A12
(vii)
E1,E2,E3=E5=E6,E4
P2
A
1
<
1, Δ1>0, C1=1-A1B11-A1B2+2A2B1, C2=C11-A1-A2C11-A12
(viii)
E
1
,
E
2
,
E
3
=
E
5
,
E
4
C22
A
1
<
1, Δ1>0, 1-A12A2>C1>1-A1B11-A1B2+2A2B1, C2=C11-A1-A2C11-A12
(ix)
E1,E3,E2=E5,E4
C23
A
1
<
1, Δ1>0, C1>1-A12A2, C2=C11-A1-A2C11-A12
(x)
E
1
,
E
2
=
E
3
=
E
5
,
E
4
P1
A
1
<
1, Δ1=0, C1=1-A12A2, C2=C11-A1-A2C11-A12
(xi)
E1,E2=E3,E4
C11∪C12
A
1
<
1, Δ1=0, C1≠1-A12A2
(xii)
E1,E4
R1
A
1
<
1, Δ1<0
(xiii)
E1=E4,E2,E3
R
32
A
1
=
1, Δ1>0
(xiv)
E1=E4,E2=E3
C3
A
1
=
1, Δ1=0
(xv)
E1=E4
R
31
A
1
=
1, Δ1<0
(xvi)
E
1
,
E
2
,
E
3
R42
A
1
>
1, Δ1>0
(xvii)
E1,E2=E3
C
4
A
1
>
1, Δ1=0
(xviii)
E1
R
41
A
1
>
1, Δ1<0Proof.
The proof follows from Lemma10.


## 4. Linearized Stability Analysis

The Jacobian matrix of the mapT has the form(41)JT=A1+yC1A1+xB1+yC12-xC1A1+xB1+yC12-y2B2A2+xB2+y2C222yA2+xB2A2+xB2+y2C22.The determinant of (41) at the equilibrium point is given by(42)det⁡JTx¯,y¯=y¯2A1A2+B2x¯+y¯2A2+B2x¯C1A1+B1x¯+C1y¯2A2+B2x¯+C2y¯22and the trace of (41) at the equilibrium point is given by (43)trJTx¯,y¯=A1+y¯C1A1+B1x¯+C1y¯2+2y¯A2+B2x¯A2+B2x¯+C2y¯22.The characteristic equation has the form (44)λ2-λA1+y¯C1A1+B1x¯+C1y¯2+2y¯A2+B2x¯A2+B2x¯+C2y¯22+y¯2A1A2+B2x¯+y¯2A2+B2x¯C1A1+B1x¯+C1y¯2A2+B2x¯+C2y¯22=0.Lemma 12.
The following statements hold:(a)
E1 is locally asymptotically stable if A1>1.(b)
E1 is a saddle point if A1<1.(c)
E1 is a nonhyperbolic equilibrium point if A1=1.Proof.
We have that, for the equilibrium pointE1, trJT(E1)=1/A1 and det⁡JT(E1)=0. The characteristic equation of (50) at E1 has the form λ2-1/A1λ=0, from which the proof follows.Lemma 13.
The following statements hold:(a)
E4 is locally asymptotically stable if A1<1.(b)
E4 is a nonhyperbolic equilibrium point if A1=1.Proof.
We have that, for the equilibrium pointE4, trJT(E4)=A1 and det⁡JT(E4)=0. The characteristic equation of (50) at E4 has the form λ2-A1λ=0, from which the proof follows.The equilibrium pointsE5 and E6 are intersection points of the curves (45)xfy=1-A1-C1yB1,xgy=y-A2-C2y2B2.Let x~(y)=xf(y)-xg(y) for y∈[1-1-4A2C2/2C2,1+1-4A2C2/2C2].Lemma 14.
LetT=(f,g) be the map defined by (11). Then fx′(E5)<1, fx′(E6)<1, gy′(E6)<1. Let (46)f~y=B1C2y2-B2C1+B1y+A2B1+B21-A1.Then, y¯5 and y¯6 are zeros of f~(y) and signx~(y)=sign(f~(y)) for y∈[1-1-4A2C2/2C2,1+1-4A2C2/2C2].Proof.
The first derivative ofxf(y¯6) is given by (47)xf′y¯i=fy′Ei1-fx′Ei=-C1B1<0,i=5,6.Since fy′(Ei)<0, i=5,6, we get fx′(Ei)<1, i=5,6. Similarly, one can see that (48)xg′y¯6=1-gy′E6gx′E6=-B2C1+B12-4B1C2A2B1-A1-1B2+B2C1B1B2<0.Since gx′(E6)<0, we get gy′(E6)<1. Further, (49)x~y=1-A1-C1yB1-y-A2-C2y2B2=B1C2y2-B2C1+B1y+A2B1-A1B2+B2B1B2=f~yB1B2,from which the proof follows.Lemma 15.
LetT be the map associated with system (1) and(50)JTx¯i,y¯i=aibicidibe the Jacobian matrix of  T at fixed point Ei(i=5,6). Then the Jacobian matrix (50) has real and distinct eigenvalues λ1 and λ2 such that 0<λ1≤λ2. Furthermore, the following hold:(51)signx~′y¯6=sign1-λ26,λ26<1,(52)signx~′y¯5=sign1-λ251-λ25.Proof.
Implicit differentiation of the equations definingCf and Cg at Ei gives (53)xf′y¯i=fy′Ei1-fx′Ei,xg′y¯i=1-gy′Eigx′Ei.Characteristic equations associated with the Jacobian matrix of T at Ei are given by (54)pλ=λ2-fx′Ei+gy′Eiλ+fx′Eigy′Ei-fy′Eigx′Ei=λ2-ai+diλ+aidi-bici.Since the map T is competitive, then the eigenvalues of the Jacobian matrix of the map T, at the equilibrium Ei, are real and distinct and furthermore 0<λ1(i)<λ2(i). By (53), we have (55)x~′y¯i=xf′y¯i-xg′y¯i=fy′Ei1-fx′Ei-1-gy′Eigx′Ei=bi1-ai-1-dici=-1+ai+di-aidi-bicici1-ai=-p1ci1-ai=1-λ1i1-λ2iciai-1.In view of Lemma 14 and from trJT(E6)=a6+d6=fx′(E6)+gy′(E6)=λ1(6)+λ2(6)<2 we get λ1(6)<1. The map T is competitive, which implies c6=gx′(E6)<0. In view of Lemma 14 we get a6=fx′(E6)<1 from which it follows (51). Similarly, from c5=gx′(E5)<0 and a5=fx′(E5)<1 we obtain (52).The following lemma describes the local stability of the equilibrium pointsE5 and E6.Lemma 16.
Assume thatA1<1 and Δ2≥0. Then the following hold: (i)
IfΔ2>0 and E6 exists then it is locally asymptotically stable.(ii)
IfΔ2>0 and E5 exists then it is a saddle point.(iii)
IfΔ2=0 then E5=E6. Furthermore, if E5=E6 exists then it is nonhyperbolic equilibrium point. The eigenvalues of JT(E5=E6) are given by λ1=1 and λ2<1.Proof.
Assuming thatΔ2>0, then y¯5 and y¯6 are zeros of multiplicity one of (56)f~y=B1C2y2-B2C1+B1y+A2B1+B21-A1and y¯6>y¯5>0. From this we have f~(y)>0 for y∈(0,y¯5)∪(y¯6,+∞) and f~(y)<0 for y∈(y¯5,y¯5).
By Lemmas  6 and 7 from [19] the equilibrium curves xf(y) and xg(y) intersect transversally at E5 and E6, that is, x~′(y¯i)≠0i=5,6. By this and Lemma 14 and by continuity of function x~(y) there exists a neighborhood Uy¯i(i) of  y¯i such that x~′(y)>0 for y∈Uy¯6 and x~′(y)<0 for y∈Uy¯5(5). This implies that x~′(y¯6)>0 and x~′(y¯5)<0. By Lemma 15 we have that E6 is locally asymptotically stable and E5 is a saddle point whenever equilibrium points E5 and E6 exist.
Assume thatΔ2=0. Then y¯5=y¯6 is zero of f~(y) of multiplicity two. In view of Lemmas  6 and 7 from [19] we have that x~′(y¯5)=0. The rest of the proof follows from the proof of Lemma 15.Lemma 17.
Assume thatΔ1≥0. The following statements are true: (a)
E2 is a saddle point if Δ1>0, and A1≥1 or A1<1 and C1>1-A11+1-4A2C2/2A2.(b)
E2 is a repeller if Δ1>0, A1<1, and C1<1-A11+1-4A2C2/2A2.(c)
E2 is a nonhyperbolic equilibrium point if Δ1=0 or(57)Δ1>0,A1<1,C1=1-A11+1-4A2C22A2.IfΔ1=0 then the eigenvalues of JT(E2) are given by (58)λ1=1,λ2=12A2C1+A1with corresponding eigenvectors(59)v1=0,1T,v2=2A2C1+A1-1B22A2C1+A1,1T.If (57) holds then the eigenvalues of JT(E2) are given by (60)λ1=1,λ2=1+1-4A2C2>1with corresponding eigenvectors(61)v1=1-4A2C2B2,1T,v2=0,1T.Proof.
One can see that(62)1+det⁡JTE2-trJTE2=21-A11-4A2C2C2+C11-4A2C2-1-4A2C22A1C2+C11-1-4A2C2,1-det⁡JTE2=C11-1-4A2C2+2C2A1-1-1-4A2C22A1C2+C11-1-4A2C2.(a)
Sincedet⁡JT(E2)>0 and trJT(E2)>0, the equilibrium E2 is a saddle point if and only if 1+det⁡JT(E2)-trJT(E2)<0. If A1≥1 it is obvious that 1+det⁡JT(E2)-trJT(E2)<0. Assume that A1<1. Then 1+det⁡JT(E2)-trJT(E2)<0 if and only if (63)C1>21-A1C21-4A2C24A2C2-1+1-4A2C2=1-A11-4A2C2+12A2,from which the proof of the statement follows.(b)
Sincedet⁡JT(E2)>0 and trJT(E2)>0, the equilibrium E2 is repeller if and only if 1+det⁡JT(E2)-trJT(E2)>0 and 1-det⁡JT(E2)<0. The proof of the statement follows from the facts (64)1-det⁡JTE2<0⟺C1<2C21-4A2C2-A1+11-1-4A2C2,21-A1C21-4A2C24A2C2-1+1-4A2C2-2C21-4A2C2-A1+11-1-4A2C2=--4A2C2+1-4A2C2+12A2<0.(c)
Sincedet⁡JT(E2)>0 and trJT(E2)>0, the equilibrium E2 is nonhyperbolic if and only if 1+det⁡JT(E2)-trJT(E2)=0 or det⁡JT(E2)=1 and trJT(E2)≤2. From the proof of the statements (a) and (b) if (65)Δ1=0orΔ1>0,A1<1,C1=1-A11-4A2C2+12A2we obtain1+det⁡JT(E2)-trJT(E2)=0. Now, assume that Δ1>0 and (66)det⁡JTE2=1⟺C1=2C21-4A2C2-A1+11-1-4A2C2.This impliestrJT(E2)-2=1-4A2C2/1-4A2C2+1>0. The rest of the proof follows from the fact that if Δ1=0 then (67)JTE2=1A1+2A2C1001and if (57) holds then (68)JTE2=1001-4A2C2+1.Lemma 18.
Assume thatΔ1≥0. The following statements are true: (a)
E3 is locally asymptotically stable if Δ1>0, and A1≥1 or A1<1 and C1>1-A11-1-4A2C2/2A2.(b)
E3 is a saddle point if Δ1>0, and A1<1, C1<1-A11-1-4A2C2/2A2.(c)
E3 is a nonhyperbolic equilibrium point if Δ1=0 or (69)Δ1>0,A1<1,C1=1-A11-1-4A2C22A2.
IfΔ1=0 then the eigenvalues of JT(E3) are given by (70)λ1=1,λ2=12A2C1+A1
with corresponding eigenvectors(71)v1=0,1T,v2=2A2C1+A1-1B22A2C1+A1,1T.
If (57) holds then the eigenvalues of JT(E3) are given by (72)λ1=1,λ2=1-1-4A2C2<1
with corresponding eigenvectors(73)v1=-1-4A2C2B2,1Tv2=0,1T.Proof.
Since the proof of this lemma is similar to the proof of Lemma17, it is omitted.We summarize results about local stability in the following theorem.Theorem 19.
LetA1, A2, B1, B2, C1, and C2 be positive real numbers. Then, local stability of the equilibrium points for different parameter regions is given by Table 2.Table 2
The local stability of the equilibrium points.
Case
The local stability
Parameter region
(i)
Saddle:E1,E3,E5; LAS: E4,E6; repeller: E2
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1,   C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2
(ii)
Saddle:E1,E3; LAS: E4; nonhyperbolic: E5=E6; repeller: E2
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1,  C2=B2C1+B124B1A2B1+1-A1B2
(iii)
Saddle:E1,E5; LAS: E3,E4; repeller: E2
A
1
<
1, Δ1>0, C2<C11-A1-A2C11-A12
(iv)
Saddle:E1,E3; LAS: E4; repeller: E2
A
1
<
1, Δ1>0, C2>B2C1+B124B1A2B1+1-A1B2 or A1<1, Δ1>0, 1-A12A2>C1>1-A1B11-A1B2+2A2B1,  C11-A1-A2C11-A12<C2≤B2C1+B124B1A2B1+1-A1B2
(v)
Saddle:E1,E2; LAS: E3,E4
A
1
<
1, Δ1>0, C1>1-A12A2,  C11-A1-A2C11-A12<C2≤B2C1+B124B1A2B1+1-A1B2
(vi)
Saddle:E1,E5; LAS: E4; repeller: E2  Nonhyperbolic: E3=E6;
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1,  C2=C11-A1-A2C11-A12
(vii)
Saddle:E1; LAS: E4; repeller: E2 Nonhyperbolic: E3=E5=E6;
A
1
<
1, Δ1>0, C1=1-A1B11-A1B2+2A2B1,  C2=C11-A1-A2C11-A12
(viii)
Saddle:E1; LAS: E4; repeller: E2  Nonhyperbolic: E3=E5
A
1
<
1, Δ1>0, 1-A12A2>C1>1-A1B11-A1B2+2A2B1,  C2=C11-A1-A2C11-A12
(ix)
Saddle:E1; LAS: E3,E4; nonhyperbolic: E2=E5
A
1
<
1, Δ1>0, C1>1-A12A2, C2=C11-A1-A2C11-A12
(x)
Saddle:E1; LAS: E4; nonhyperbolic: E2=E3=E5
A
1
<
1, Δ1=0, C1=1-A12A2,  C2=C11-A1-A2C11-A12
(xi)
Saddle:E1; LAS: E4; nonhyperbolic: E2=E3
A
1
<
1, Δ1=0, C1≠1-A12A2
(xii)
Saddle:E1; LAS: E4
A
1
<
1, Δ1<0
(xiii)
Saddle:E2; LAS: E3; nonhyperbolic: E1=E4
A
1
=
1, Δ1>0
(xiv)
Nonhyperbolic:E1=E4, E2=E3
A
1
=
1, Δ1=0
(xv)
Nonhyperbolic:E1=E4
A
1
=
1, Δ1<0
(xvi)
Saddle:E1,E3; LAS: E2;
A
1
>
1, Δ1>0
(xvii)
LAS:E1; nonhyperbolic: E2=E3
A
1
>
1, Δ1=0
(xviii)
LAS:E1
A
1
>
1, Δ1<0Proof.
The proof follows from Theorem8 and Lemmas 17 and 18.Figure2 illustrates visually local stability of all equilibrium points of system (1).Figure 2
Equilibria in different parameter regions for the number of equilibria of system (1) when A1, A2, B1, and B2 are fixed positive real numbers, as given by Proposition 11. Each circle in the parameters (C1,C2)-plane indicates the existence of an isolated equilibrium point of system (1) in the nonnegative quadrant of the xy-plane. Local stability character of equilibria as given in Theorem 19 is indicated as follows: •, locally asymptotically stable equilibrium; ⊖, saddle; ◯, repelling equilibrium point: two-colored circle, semistable nonhyperbolic equilibrium.Figure 3
Parameter regions in terms of parametersC1 and C2 and corresponding dynamic scenarios for system (1) if A1<1.


## 5. Injectivity and Convergence to Equilibrium Points

In this section we prove some global properties of the mapT such as injectivity and (O+) property and give global behavior on the coordinate axes.Lemma 20.
The mapT is injective.Proof.
Assume thatT(x1,y1)=T(x2,y2). Then, we have (74)A1x1-A1x2-C1x2y1+C1x1y2A1+B1x1+C1y1A1+B1x2+C1y2,A2y12+B2x2y12-A2y22-B2x1y22A2+B2x1+C2y12A2+B2x2+C2y22=0,0.
Equation (74) is equivalent to(75)A1x1-A1x2-C1x2y1+C1x1y2=0,(76)A2y12+B2x2y12-A2y22-B2x1y22=0.Equation (75) implies(77)x1=A1x2+C1x2y1A1+C1y2.By substituting this into (76) we obtain(78)y1-y2A1A2y1+A1B2x2y1+A1A2y2+A1B2x2y2+A2C1y1y2+B2C1x2y1y2+A2C1y22A1+C1y2=0,from which it follows that y1=y2. From (77) we have x1=x2, which complete the proof.The global behavior ofT on the coordinate axes is described with the following result.Lemma 21.
The following statements hold:(i)
IfΔ1≥0 and (xn,yn)=Tn(0,y0) then xn=0 and (xn,yn)→E3 for y0∈1-1-4A2C2/2C2,∞ and (xn,yn)→E1for y0∈0,1+1-4A2C2/2C2.(ii)
IfΔ1<0 and (xn,yn)=Tn(0,y0) then xn=0 and (xn,yn)→E1 for y0∈0,∞.(iii)
IfA1<1 and (xn′,yn′)=Tn(x0,0) then yn′=0 and (xn′,yn′)→E4 for x0∈0,∞.(iv)
IfA1≥1 and (xn′,yn′)=Tn(x0,0) then yn′=0 and (xn′,yn′)→E1 for x0∈0,∞.(v)
T([0,∞)×[0,∞))⊂[0,1/B1]×[0,1/C2].Proof.
(i)
From (11) it is easy to see that if x0=0 then xn=0 for n>0. Since (79)T0,y0-0,y0=0,-y0C2y02-y0+A2A2+C2y02
we obtain that(80)0,y0⪯seT0,y0fory0∈0,1-1-4A2C22C2∪1+1-4A2C22C2,∞,T0,y0⪯se0,y0fory0∈1-1-4A2C22C2,1+1-4A2C22C2.
Takey0>1+1-4A2C2/2C2. Then Tn(0,y0)⪯seTn+1(0,y0)⪯seE3. Since Tn(0,y0)⪯seE3≺seE2≺seE1 we obtain Tn(0,y0)→E3 as n→∞. Similarly, if y0∈1-1-4A2C2/2C2,1+1-4A2C2/2C2 then E3⪯seTn+1(0,y0)⪯seTn(0,y0)⪯seE2≺seE1 which implies Tn(0,y0)→E3 as n→∞. If y0<1-1-4A2C2/2C2 then E3≺seE2≺seTn(0,y0)⪯seTn+1(0,y0)⪯seE1 which implies Tn(0,y0)→E1 as n→∞.
(ii)
IfΔ1<0 then T has only equilibrium E1 on y-axis and T(0,y0)⪯se(0,y0) for all y0≥0. Since T is monotone map we get Tn(0,y0)⪯seTn+1(0,y0)⪯seE1 which implies Tn(0,y0)→E1 as n→∞ from which the proof follows.
(iii)
The proof of the statements (iii) and (iv) is similar to the proof of statements (i) and (ii) and follows from the fact that(81)Tx0,0-x0,0=-x0A1+B1x0-1A1+B1x0,0
and will be omitted.
(v)
The proof follows from the facts thatf(x,y)≤1/B1 and g(x,y)≤1/C2.Lemma 22.
LetM(t)≡t,1-A1-B1t/C1. Then M(x¯6)=E6, M(x¯5)=E5, and M(x¯4)=E4 and the following hold: (i)
IfΔ2≥0 then M(t)⪯seT(M(t)) for t∈(0,x¯6)∪(x¯5,x¯4) and T(M(t))⪯seM(t) for t∈(x¯6,x¯5).(ii)
IfΔ2<0 then M(t)⪯seT(M(t)) for t∈(0,x¯4) and T(M(t))⪯seM(t) for t∈(x¯6,x¯5).Proof.
The proof follows from the fact(82)TMt-Mt=0,A1+B1t-1g~xC13A2+B2t+C2C1A1+B1t-12,where g~(x) is given by (17) and g~(x¯5)=g~(x¯6)=0 and x¯4=(1-A1)/B1.Theorem 23.
Every solution of system (1) converges to an equilibrium point.Proof.
The mapT associated with the system is injective. Relation (42) implies that determinant of Jacobian (41) is positive for all x∈[0,∞)×[0,∞). By using Lemma 20 we have that condition (O+) of Theorem 3 is satisfied for the map T (T is competitive). Theorem 2 implies that Tn(x) is eventually componentwise monotone for all x∈[0,∞)2. The statement (v) of Lemma 21 implies that every solution enters in compact set [0,1/B1]×[0,1/C2], from which the proof follows.Remark 24.
In view of Theorem23 the main objective in determining the global dynamics of system (1) is to characterize the basins of attractions of all equilibrium points. As we will see in Theorem 25 the boundaries of these basins of attractions will be the global stable manifolds of the saddle or nonhyperbolic equilibrium points, whose existence is guaranteed by Theorems 7, 8, and 9.


## 6. Global Behavior

In this section we give results which precisely describe global dynamics of system (1) including precise characterization of basins of attraction of different equilibrium points. The main result of this paper is the following.Theorem 25.
The global behavior of system (1) is given by Table 3. See Figure 3 for visual illustration of dynamic scenarios.Table 3
The global behavior of system (1).
Case
Parameter region
Global behavior
(i)
R
24
:
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1, C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2
There exist six equilibrium pointsE1, E2, E3, E4, E5, and E6, where E4 and E5 are locally asymptotically stable, E3, E4, and E6 are saddle points, and E2 is repeller. The stable manifold Ws(E6) of the saddle point E6 is an increasing separatrix with endpoint at E2, and solutions with initial point above the Ws(E6) converge to E5, while solutions with initial point below the Ws(E6) converge to E4. All orbits that start on Ws(E6) are attracted to E6. The basins of attraction of E1 and E3 are, respectively, B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2) and B(E3)=(0,y):y>(1-1-4A2C2)/(2A2)
(ii)
C
31
:
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1, C2=B2C1+B124B1A2B1+1-A1B2
There exist five equilibrium pointsE1, E2, E3, E4, and E5=E6, where E4 is locally asymptotically stable, E1 and E3 are saddle points, E2 is repeller, and E5=E6 is nonhyperbolic. There exists a continuous increasing curve C with endpoint at E2, which is a subset of the basin of attraction of E5=E6. All solutions with initial point above C converge to E5=E6, while solutions with initial point below C converge to E4. The basins of attraction of E1 and E3 are, respectively, B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2) and B(E3)=(0,y):y>(1-1-4A2C2)/(2A2)
(iii)
R
22
:
A
1
<
1, Δ1>0, C2<C11-A1-A2C11-A12
There exist five equilibrium pointsE1, E2, E3, E4, and E5, where E3 and E4 are locally asymptotically stable, E1 and E5 are saddle points, and E2 is repeller. The stable manifold Ws(E5) of the saddle point E5 is an increasing separatrix with endpoint at E2, and solutions with initial point above Ws(E5) converge to E3, while solutions with initial point below Ws(E5) converge to E4. All orbits that start on Ws(E5) are attracted to E5. The basin of attraction of E1 is B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2)
(iv)
R
21
∪
R
25
∪
C
32
:
A
1
<
1, Δ1>0, C2>B2C1+B124B1A2B1+1-A1B2 or A1<1, Δ1>0, 1-A12A2>C1>1-A1B11-A1B2+2A2B1, C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2
There exist four equilibrium pointsE1, E2, E3, and E4, where E4 is globally asymptotically stable, E1 and E3 are saddle points, and E2 is a repeller. The basins of attraction of E1, E3, and E4 are, respectively, B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2), B(E3)=(0,y):y>(1-1-4A2C2)/(2A2), and B(E4)=(0,∞)×[0,∞).
(v)
R
23
:
A
1
<
1, Δ1>0, C1>1-A1B11-A1B2+2A2B1, C11-A1-A2C11-A12<C2<B2C1+B124B1A2B1+1-A1B2
There exist four equilibrium pointsE1, E2, E3, and E4, where E3 and E4 are locally asymptotically stable and E1 and E2 are saddle points. The stable manifold Ws(E2) of the saddle point E2 is an increasing separatrix, and solutions with initial point above Ws(E1) converge to E3, while solutions with initial point below Ws(E2) converge to E4. All orbits that start on Ws(E2) are attracted to E2. The basin of attraction of E1 is given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2).
(vi)
C
21
:
A
1
<
1, Δ1>0, C1<1-A1B11-A1B2+2A2B1, C2=C11-A1-A2C11-A12
There exist five equilibrium pointsE1, E2, E3=E6, E4, and E5, where E4 is locally asymptotically stable, E1 and E5 are saddle points, E3=E6 is nonhyperbolic, and E2 is a repeller. The stable manifold Ws(E5) of the saddle point E5 is an increasing separatrix with endpoint at E2, and solutions with initial point above Ws(E5) converge to E3=E6, while solutions with initial point below Ws(E5) converge to E4. All orbits that start on Ws(E5) are attracted to E5. The basin of attraction of E1 is given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2).
(vii)
P
2
:
A
1
<
1, Δ1>0, C1=1-A1B11-A1B2+2A2B1, C2=C11-A1-A2C11-A12
There exist four equilibrium pointsE1, E2, E3=E5=E6, and E4, where E4 is locally asymptotically stable, E1 is a saddle point, E3=E5=E6 is nonhyperbolic, and E2 is a repeller. There exists a strictly increasing curve C with endpoint at E3=E5=E6, which is the subset of the basin of attraction of E3=E5=E6. All solutions with initial point above C converge to E3=E5=E6, while solutions with initial point below C converge to E4. The basin of attraction of E1 is given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2).
(viii)
C
22
:
A
1
<
1, Δ1>0, C2=C11-A1-A2C11-A12, 1-A12A2>C1>1-A1B11-A1B2+2A2B1
There exist five equilibrium pointsE1, E2, E3=E5, and E4, where E4 is locally asymptotically stable, E1 is a saddle point, E3=E5 is nonhyperbolic, and E2 is a repeller. The basins of attraction of E1 and E3 are given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2) and B(E3)=(0,y):y>(1-1-4A2C2)/(2A2) and the basin of attraction of E4 is B(E4)=(x,y):x>0,y≥0.
(ix)
C
23
:
A
1
<
1, Δ1>0, C2=C11-A1-A2C11-A12, 1-A12A2<C1
There exist four equilibrium pointsE1, E3, E2=E5, and E4, where E3 and E4 are locally asymptotically stable, E1 is a saddle point, and E2=E5 is nonhyperbolic. There exist continuous increasing curves C1 and C2 with endpoint at E2=E5, which are the subsets of the basin of attraction of E2=E5. Further, all solutions with initial point above C2 converge to E3 and all solutions with initial point above C1 and below C2 converge to E2, while solutions with initial point below C1 converge to E4. The basin of attraction of E1 is B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2).
(x)
A
1
<
1, Δ1=0, C1≠1-A12A2
There exist three equilibrium pointsE1, E2=E3, and E4, where E4 is locally asymptotically stable, E1 is a saddle point, and E2=E3 is nonhyperbolic. There exists a continuous increasing curve C with endpoint at E2=E3 which is a subset of the basin of attraction of E2=E3. All solutions with initial point above C converge to E2=E3, while solutions with initial point below C converge to E4. The basin of attraction of E1 is given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2).
(xi)
A
1
<
1, Δ1<0
There exist two equilibrium pointsE1, which is a saddle point, and E4, which is globally asymptotically stable, where B(E4)=(0,∞)×[0,∞) and B(E1)=(0,y):y≥0.
(xii)
A
1
=
1, Δ1>0
There exist three equilibrium pointsE1=E4, E2, and E3, where E1=E4 is nonhyperbolic, E3 is locally asymptotically stable, and E2 is a saddle point. The stable manifold Ws(E2) of the saddle point E2 is an increasing separatrix with endpoint at E2, and solutions with initial point above Ws(E2) converge to E3, while solutions with initial point below Ws(E2) converge to E4. All orbits that start on Ws(E2) are attracted to E2. The basin of attraction of E1 is given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2).
(xiii)
A
1
=
1, Δ1=0
There exist two equilibrium pointsE1=E4 and E2=E3 which are nonhyperbolic. There exists a continuous increasing curve C with endpoint at E2=E3 which is a subset of the basin of attraction of E2=E3. All solutions with initial point above C converge to E2=E3, while solutions with initial point below C converge to E1=E4.
(xiv)
A
1
=
1, Δ1<0
There exists one equilibrium pointE1=E4 which is nonhyperbolic and global attractor. The basin of attraction of E1 is B(E1)=[0,∞)×[0,∞)
(xv)
A
1
>
1, Δ1>0
There exist three equilibrium pointsE1, E2, and E3, where E1 and E3 are locally asymptotically stable and E2 is a saddle point. The stable manifold Ws(E2) of the saddle point E2 is an increasing separatrix with endpoint at E2, and solutions with initial point above Ws(E2) converge to E3, while solutions with initial point below Ws(E2) converge to E1. All orbits that start on Ws(E2) are attracted to E2.
(xvi)
A
1
>
1, Δ1<0
There exists one equilibrium pointE1 which is globally asymptotically stable. The basin of attraction of E1 is B(E1)=[0,∞)×[0,∞)
(xvii)
A
1
>
1, Δ1=0
There exist two equilibrium pointsE1 and E2=E3, where E1 is locally asymptotically stable and E2=E3 is nonhyperbolic. There exists a continuous increasing curve C with endpoint at E2=E3 which is a subset of the basin of attraction of E2=E3. All solutions with initial point above C converge to E2=E3, while solutions with initial point below C converge to E1.Proof.
We will prove statements (i)–(x) listed in the second column of Table3 in the given order. The proof of other statements is similar. Let R=[0,∞)×[0,∞).(i)
Suppose(C1,C2)∈R24. By Proposition 11, in Q1(0,0) there exist six equilibria E1, E2, E3, E4, E5, and E6. By Theorem 19 equilibria E4 and E5 are locally asymptotically stable; E3, E4, and E5 are the saddle points and E2 is repeller. In view of (41) the map T is competitive on R and strongly competitive on int(R). It follows from the Perron-Frobenius Theorem and a change of variables [16] that, at each point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that if the map is strongly competitive then no eigenvector is aligned with a coordinate axis. Hence, all conditions of Theorems 7, 8, and 9 are satisfied, which yields the existence of the global stable manifold Ws(E5), with endpoint at point E2, which is graph of an increasing function. Let W-={x,y∣(x,y)⪯se(x~0,y~0)forsome(x~0,y~0)∈Ws(E5)} and W+={(x,y)∣(x~1,y~1)⪯se(x,y)forsome(x~1,y~1)∈Ws(E5)}. By Lemma 21 and uniqueness of the global stable manifold we have Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)} and Ws(E3)=(0,y):y>(1-1-4A2C2)/(2A2). Take (x0,y0)∈W-∩int(Q1(0,0)). By Theorem 8 and Lemma 21 we have that there exists n0>0 such that Tn(x0,y0)∈int(Q4(E3)∩Q2(E5)), n>n0. In view of Corollary 4  [[E1,E6]]⊆B(E6) and [[E6,E5]]⊆B(E6). Since T is competitive, this implies int(Q4(E3)∩Q2(E5))=[[E1,E5]]⊆B(E6). Take (x0,y0)∈W+∩int(Q1(0,0)). By Theorem 8 and Lemma 21 we have that there exists n0′>0 such that Tn(x0,y0)∈int(Q4(E5)∩Q2(E4)), n>n0′. Since T is competitive, in view of Corollary 4  int(Q4(E5)∩Q2(E4))=[[E5,E4]]⊆B(E4). This completes the proof of statement (i).(ii)
Suppose(C1,C2)∈C31. By Proposition 11, in Q1(0,0) there exist five equilibria E1, E2, E3, E4, and E5=E6. By Theorem 19  E1 and E3 are the saddle points, E2 is repeller, and E5=E6 is nonhyperbolic. Similarly as in the proof of the statement (i), all conditions of Theorems 7, 8, and 9 are satisfied, which yields the existence of the invariant curve C with one endpoint at E2 and which is passing through E5=E6, and it is graph of an increasing function. Let W-={(x,y)∣(x,y)⪯se(x~0,y~0)forsome(x~0,y~0)∈C} and W+={(x,y)∣(x~1,y~1)⪯se(x,y)forsome(x~1,y~1)∈C}. By Lemma 21 and uniqueness of the global stable manifold we have Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)} and Ws(E3)={(0,y):y>(1-1-4A2C2)/(2A2)}. Take (x0,y0)∈W-∩int(Q1(0,0)). By Theorem 8 we have that there exists n0>0 such that Tn(x0,y0)∈int(Q4(E3)∩Q2(E5=E6)), n>n0. In view of Corollary 4  int(Q4(E3)∩Q2(E5=E6))=[[E1,E5=E6]]⊆B(E5=E6). Take (x0,y0)∈W+∩int(Q1(0,0)). By Theorem 8 and Lemma 21 we have that there exists n0′>0 such that Tn(x0,y0)∈int(Q4(E5)∩Q2(E4)), n>n0′. Since T is competitive, in view of Corollary 4  int(Q4(E5)∩Q2(E4))=[[E5,E4]]⊆B(E4). This completes the proof of statement (ii)(iii)
The proof is similar to the proof of case (i) and we skip it.(iv)
Suppose(C1,C2)∈R21∪R25∪C32. By Proposition 11, in Q1(0,0) there exist four equilibria E1, E2, E3, and E4. By Theorem 19  E1 and E3 are saddle points; E2 is repeller; E4 is locally asymptotically stable. By Lemma 21 and uniqueness of the global stable manifold we have B(E1)=Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)} and B(E3)=Ws(E3)={(0,y):y>(1-1-4A2C2)/(2A2)}. Since, by Theorem 23, every solution of system (1) converges to an equilibrium point, we have that B(E4)=(0,∞)×[0,∞).(v)
Suppose(C1,C2)∈R23. By Proposition 11, in Q1(0,0) there exist four equilibrium points E1, E2, E3, and E4. By Theorem 19  E1 and E2 are the saddle points; E3 and E4 are locally asymptotically stable. By Lemma 21 and uniqueness of the global stable manifold we have B(E1)=Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)}, and {(0,y):y>(1-1-4A2C2)/(2A2)}⊆B(E3) and {(x,0):x>0}⊆B(E4). Similarly as in the proof of case (i), all conditions of Theorems 7, 8, and 9 are satisfied, which yields the existence of the global stable manifold Ws(E2), which is a graph of an increasing function. The rest of the proof follows from the facts that W-={(x,y)∣(x,y)⪯se(x~0,y~0)forsome(x~0,y~0)∈Ws(E2)} and W+={(x,y)∣(x~1,y~1)⪯se(x,y)forsome(x~1,y~1)∈Ws(E2)} are invariant sets, E4∈W-, E3∈W+, uniqueness of the global stable manifold Ws(E2) and Theorem 23.(vi)
The proof is similar to the proof of case (i) and we skip it.(vii)
The proof is the same as the proof of case (viii) and we skip it.(viii)
Suppose(C1,C2)∈C22. By Proposition 11, in Q1(0,0) there exist four equilibrium points E1, E3=E5, E2, and E4. By Theorem 19 we have that E4 is locally asymptotically stable; E1 is a saddle point; E3=E5 is nonhyperbolic; and E2 is repeller. By Lemma 21 and uniqueness of the global stable manifold we have B(E1)=Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)}. In view of Lemma 18, for E3=E5 we have that the eigenspace E2λ associated with λ2 is a coordinate axis, so we can not use Theorem 6. By Lemma 21 we obtain {(0,y):y>(1-1-4A2C2)/(2A2)}⊆B(E3=E5). Similarly as in the proof of Theorem 8 (see [1] for more details) one can prove that for every x∈int(Q1(E3)) there exists n0∈N such that Tn(x)∈int(Q4(E3)) for n≥n0. By Lemmas 21 and 22 for (x0,y0)∈int(Q4(E3)), there exists t0′ and t0′′ such that E3⪯seM(t0′)⪯se(x0,y0)⪯se(t0′′,0). Since E3⪯seTn(M(t0′))⪯seTn+1(M(t0′))⪯E4 we have that Tn(M(t0′))→E4. Since Tn(t0′′,0)→E4 we obtain Tn(x0,y0)→E4. This implies int(Q4(E3))⊆B(E4), which completes the proof.(ix)
Suppose(C1,C2)∈C23. By Proposition 11, there exist four equilibrium points E1, E2=E5, E3, and E4. By Theorem 19 we have that E3 and E4 are locally asymptotically stable; E1 is a saddle point; E2=E5 is nonhyperbolic. By Lemma 21 and uniqueness of the global stable manifold we have B(E1)=Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)}. In view of Lemma 18, for E3, we have that λ1=1 and λ2>1, so we can not use Theorem 6. By Lemmas 21 and 22 for (x0,y0)∈int(Q4(E3)), there exists t0′ and t0′′ such that E3⪯seM(t0′)⪯se(x0,y0)⪯se(t0′′,0). Since E3⪯seTn(M(t0′))⪯seTn+1(M(t0′))⪯E4 we have that Tn(M(t0′))→E4. Since Tn(t0′′,0)→E4 we obtain Tn(x0,y0)→E4. This implies int(Q4(E3))⊆B(E4). Let C1 denote the boundary of B(E4) considered as a subset of int(Q1(E2)) and let C2 denote the boundary of B(E3) considered as a subset of int(Q1(E2)). It is easy to see by using Lemmas 21 and 22 that E2∈C1∩C2. Since T(int(R))⊂int(R), following the proof of Claims  1 and 2 [20], one can see that T(Ci)⊆Ci and Tn(x0,y0)→E2∈Ci for (x0,y0)∈Ci for i=1,2. Further, Ci are graphs of the continuous strictly increasing functions. If (x0,y0) is point above the curve C1 and below the curve C2 then there exists (x0′,y0′)∈C1 and (x0′′,y0′′)∈C2 such that (x0′′,y0′′)⪯se(x0,y0)⪯se(x0′,y0′). Since Tn(x0′′,y0′′)⪯seTn(x0,y0)⪯seTn(x0′,y0′) and Tn(x0′′,y0′′)→E2 and Tn(x0′,y0′)→E2 as n→∞ we have Tn(x0,y0)→E2 as n→∞.(x)
Suppose(C1,C2)∈C23. By Proposition 11, in Q1(0,0) there exist three equilibrium points E1, E2=E3, and E4. By Theorem 19  E2=E3 is nonhyperbolic, E1 is a saddle point, and E4 is locally asymptotically stable. By Lemma 21 and uniqueness of the global stable manifold we have B(E1)=Ws(E1)={(0,y):0≤y<(1-1-4A2C2)/(2A2)}. In view of Lemma 18 we have that λ1=1 and λ2<1 if C1>(1-A1)/(2A2) and λ2>1 if C1<(1-A1)/(2A2), so we can use Theorem 6 if C1>(1-A1)/(2A2). In this case there exists strictly increasing curve C with endpoint at E2=E3. The rest of the proof follows from Theorems 7, 8, and 9 and Lemma 21. Now, we assume that C1<(1-A1)/(2A2). By Lemmas 21 and 22 for (x0,y0)∈int(Q4(E3)), there exists t0′ and t0′′ such that E3⪯seM(t0′)⪯se(x0,y0)⪯se(t0′′,0). Since E3⪯seTn(M(t0′))⪯seTn+1(M(t0′))⪯E4 we have that Tn(M(t0′))→E4. Since Tn(t0′′,0)→E4 we obtain Tn(x0,y0)→E4. This implies int(Q4(E3))⊆B(E4). Let C denote the boundary of B(E4) considered as a subset of Q1(E3). It is easy to see by using Lemmas 21 and 22 that E3∈C. Since T(int(R))⊂int(R), following the proof of Claims  1 and 2 [20], one can see that T(C)⊆C. Further, C is graph of strictly increasing function. By Theorem 23 if (x0,y0)∈C then Tn(x0,y0)→E3∈C. If (x0,y0) is point above the curve C then there exists (x0′,y0′) and y0′′>y¯3 such that (0,y0′′)⪯se(x0,y)⪯se(x0′,y0′). Since Tn(0,y0′′)⪯seTn(x0,y0)⪯seTn(x0′,y0′), Tn(0,y0′′)→E3, and Tn(x0′,y0′)→E3 as n→∞ we have Tn(x0,y0)→E3 as n→∞.Based on a series of numerical simulations we propose the following conjectures.Conjecture 26.
Suppose that all assumptions of the statement (ix) of Theorem25 are satisfied; then the following holds: (83)BE2=C1=C2.Conjecture 27.
Assume thatC1,C2∈P1. There exist three equilibrium points E1, E3=E2=E5, and E4, where E4 is locally asymptotically stable; E1 is a saddle point; and E3=E2=E5 is nonhyperbolic. The basins of attraction of E1 and E3=E2=E5 are given by B(E1)=(0,y):0≤y<(1-1-4A2C2)/(2A2) and B(E3)=(0,y):y>(1-1-4A2C2)/(2A2) and the basin of attraction of E4 is B(E4)=(x,y):x>0,y≥0.



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*Source: 3104512-2017-08-02.xml*