Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect

where B 1 , C 1 ,A 2 , B 2 , C 2 are positive constants and x 0 , y 0 ≥ 0 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.


Introduction
The following difference equation is known as the Beverton-Holt model: where  > 0 is the rate of change (growth or decay) and   is the size of the population at the th 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.that is, lim  → ∞   = 0, for all  0 ≥ 0. (d) If  > 1, then the equilibrium point  − 1 is a global attractor; that is, lim  → ∞   =  − 1, for all  0 > 0.
(e) Both equilibrium points are globally asymptotically stable in the corresponding regions of parameters  ≤ 1 and  > 1; that is, they are global attractors with the property that small changes of initial condition  0 result in small changes of the corresponding solution {  }.
All these properties can be derived from the explicit form of the solution of (1): See [1][2][3].
The following difference equation, ,  = 0, 1, . . ., was introduced by Thomson [4] as a depensatory generalization of the Beverton-Holt stock-recruitment relationship used to develop a set of constraints designed to safeguard against overfishing; see [5] for further references.In view of the sigmoid shape of the function () =  2 /(1 +  2 ) (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.
(c) There exist a zero equilibrium and two positive equilibria,  − and  + , when  > 2.
(d) All solutions of (3) are monotonic (increasing or decreasing) sequences.
(g) If  > 2, then zero equilibrium and  + are locally asymptotically stable, while  − is repeller and the basins of attraction of the equilibrium points are given as In other words, the smaller positive equilibrium serves as the boundary between two basins of attraction.The zero equilibrium has the basin of attraction (0) and the model exhibits the Allee effect.
(h) The equilibrium points 0 and  + are globally asymptotically stable in the corresponding basins of attractions (0) and ( + ).
The two dimensional analogue of (1) is the uncoupled system where ,  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]: where all parameters are positive and the initial conditions are nonnegative.The global dynamics of system (6) was completed in [7].Several variations of system (6) where the competition of two species was modeled by linear fractional difference equations were considered in [8][9][10][11][12][13][14].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: where ,  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 where  1 ,  1 ,  2 ,  2 ,  2 > 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,[13][14][15][16][17][18] for related results and [19] 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 [22].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.

Preliminaries
Our proofs use some recent general results for competitive systems of difference equations of the form: where  and  are continuous functions and (, ) is nondecreasing in  and nonincreasing in  and (, ) is nonincreasing in  and nondecreasing in  in some domain .
Here we give some basic notions about monotonic maps in the plane.
We define a partial order ⪯ se on R 2 (so-called South-East ordering) so that the positive cone is the fourth quadrant; that is, this partial order is defined by Similarly, we define North-East ordering as A map  is called competitive if it is nondecreasing with respect to ⪯ se , that is, if the following holds: For each k for  = 1, . . ., 4 to be the usual four quadrants based on V and numbered in a counterclockwise direction; for example, For  ⊂  2 + let  ∘ denote the interior of .The following definition is from [35].Definition 1.Let  be a nonempty subset of R 2 .A competitive map  :  →  is said to satisfy condition (+) if for every ,  in , () ⪯ ne () implies  ⪯ ne , and  is said to satisfy condition (−) if for every ,  in , () ⪯ ne () implies  ⪯ ne .
The following theorem was proved by de Mottoni and Schiaffino [38] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations.Smith generalized the proof to competitive and cooperative maps [34].
Theorem 2. Let  be a nonempty subset of R 2 .If  is a competitive map for which (+) holds, then for all  ∈ , {  ()} is eventually componentwise monotone.If the orbit of  has compact closure, then it converges to a fixed point of .If instead (−) holds, then for all  ∈ , { 2 } is eventually componentwise monotone.If the orbit of  has compact closure in , 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 [39].
The following result is from [35], with the domain of the map specialized to be the cartesian product of intervals of real numbers.It gives a sufficient condition for conditions (+) and (−).
Theorem 3. Let  ⊂ R 2 be the cartesian product of two intervals in R. Let  :  →  be a   competitive map.If  is injective and det   () > 0 for all  ∈  then  satisfies (+).If  is injective and det   () < 0 for all  ∈  then  satisfies (−).
Theorems 2 and 3 are quite applicable as we have shown in [40], in the case of competitive systems in the plane consisting of rational equations.
The following result is from [18], which generalizes the corresponding result for hyperbolic case from [7].Related results have been obtained by Smith in [34].
Theorem 4. Let R be a rectangular subset of R 2 and let  be a competitive map on R. Let  ∈ R be a fixed point of  such that (Q 1 () ∪ Q 3 ()) ∩ R has nonempty interior (i.e.,  is not the NW or SE vertex of R).
Suppose that the following statements are true.
(a) The map  is strongly competitive on int((Q (b)  is  2 on a relative neighborhood of .
(c) The Jacobian matrix of  at  has real eigenvalues ,  such that || < , where  is stable and the eigenspace   associated with  is not a coordinate axis.
(d) Either  ≥ 0 and or  < 0 and Then there exists a curve C in R such that (i) C is invariant and a subset of W  (); (ii) the endpoints of C lie on R; (iii)  ∈ C; (iv) C is the graph of a strictly increasing continuous function of the first variable; 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 R  , and if  has no fixed points in the ordered interval ( 1 ,  2 ) other than  1 and  2 , then the interior of ( 1 ,  2 ) is either a subset of the basin of attraction of  1 or a subset of the basin of attraction of  2 .
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 = ( 1 ,  2 ) × ( 1 ,  2 ) and let  : R → R be a strongly competitive map with a unique fixed point x ∈ R, such that  is continuously differentiable in a neighborhood of x.Assume further that at the point x the map  has associated characteristic values  and ] satisfying 1 <  and − < ] < .
Then there exist curves C 1 , C 2 in R and there exist p 1 , p 2 ∈ R with p 1 ≪ se x ≪ se p 2 such that the endpoints q ℓ , r ℓ of C ℓ , where q ℓ ⪯ ne r ℓ , belong to the boundary of R. For ℓ,  ∈ {1, 2} with ℓ ̸ = , C ℓ is a subset of the closure of one of the components of R\C  .Both C 1 and C 2 are tangential at x to the eigenspace associated with ]; Then D 1 ∪ D 2 is invariant.
Corollary 7. Let a map  with fixed point x be as in Theorem 6.Let D 1 , D 2 be the sets as in Theorem 6.If  satisfies ( + ), then for ℓ = 1, 2, D ℓ is invariant, and for every x ∈ D ℓ , the iterates   (x) converge to x or to a point of R.If  satisfies ( − ), then (D 1 ) ⊂ D 2 and (D 2 ) ⊂ D 1 .For every x ∈ D 1 ∪D 2 , the iterates   (x) either converge to x or converge to a period-two point or to a point of R.

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 with strictly positive coordinates satisfy the following system of equations: From (17) we have that all real solutions of the system (17) belong to the positive quadrant, since 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.
Proof.The discrimination matrix [41] of () =  4 +  3 +  2 +  +  and   () is given by Let   denote the determinant of the submatrix of Discr( f, f ), formed by the first 2 rows and the first 2 columns, for  = 1, 2, 3, 4 where So, by straightforward calculation one can see that The rest of the proof follows in view of Theorem 1 in [41].
The map associated with system (8) has the form: ) . ( The Jacobian matrix of  is and the Jacobian matrix of  evaluated at an equilibrium (, ) with positive coordinates has the following form: The determinant and trace of ( 27) are It is worth noting that det   (, ) and tr   (, ) of ( 27) are both positive.
Using the equilibrium condition (17), we may rewrite the determinant and trace in the more useful form: The characteristic equation of the matrix ( 27) is whose solutions are the eigenvalues The corresponding eigenvectors of (31) are 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 (, ) of all equilibrium points will subsequently be designated with the subscripts:  (repeller),  (attractor), ,  1 ,  2 (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, (, ), of system (8).
Lemma 10.The following conditions hold for the coordinates of the positive equilibrium points, (, ), of System (8).
(ii) For  NE (  ,   ),  NE (  ,   ), and (iii) For   ( ns ,  ns ) and   ( nu ,  nu ), Proof.(i) Let  1 be the slope of the tangent line to ellipse  1 at (, ) =  SW (  ,   ) and let  2 be the slope of the tangent line to ellipse  2 at (, ) =  SW (  ,   ).It is clear from geometry that See Figure 2. It follows that and in turn Therefore The proofs for the remaining case in (i) and all cases in (ii) and (iii) are similar and will be omitted.
(viii) The proof of (viii) is similar to the proof of (vii) and will be omitted.

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  which corresponds to system ( 8) is injective and that it satisfies (+).

Convergence of Solutions on the Coordinate Axes: Injectivity and (𝑂+).
When   = 0, system (8) becomes When   = 0, system (8) becomes It follows from ( 52) and (53) that solutions of system (8) with initial conditions on the -axis remain on the -axis and solutions of system (8) with initial conditions on the -axis remain on the -axis.
Theorem 12.The following conditions hold for solutions {(  ,   )} of system (8) with initial conditions on the  or -axis.
(iii) In this case, 1 = 4 2  2 , and we may rewrite (54) as By (55) it is clear that {  } is a stricly decreasing sequence, and so is convergent.It follows that {  } converges to  ns when  0 >  ns , and {  } converges to 0 when 0 <  0 <  ns .
Theorem 13.The map  which corresponds to system (8) is injective. Proof.Indeed, which is equivalent to This immediatly implies  1 =  2 .
Proof.Assume that ) . ( The last inequality is equivalent to , which contradicts (60).Consequently  1 ≤  2 and so ( 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 Consequently, all solutions of system (8) converge to an equilibrium point or to (0, 0).

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.
Theorem 15.Assume that 1 < 4 2  2 .Then system (8) has one equilibrium point   which is locally asymptotically stable.The singular point  0 (0, 0) is global attractor of all points on axis and every point on -axis is attracted to   .Furthermore, every point in the interior of the first quadrant is attracted to  0 or   .
Theorem 16.Assume that 1 = 4 2  2 .Then system (8) has two equilibrium points,   which is locally asymptotically stable and   which is nonhyperbolic of the stable type.The singular point  0 is global attractor of all points on the -axis, which start below   .Furthermore, every point in the interior of the first quadrant below W  (  ) is attracted to  0 (0, 0) or     and every point in the first quadrant which starts above W s (  ) is attracted to   .
Theorem 17. Assume that 1 > 4 2  2 and system (8) has three equilibrium points,   and   + which are locally asymptotically where W  (  − ) and W  ( NW ) denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below W  (  − ) is attracted to  0 or   .
Proof.Local stability of all equilibrium points follows from Theorem 11.We present the proof in the case of the equilibrium point  NW .The proof in the case of the equilibrium point  SE 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 -axis below   − converges to  0 in a decreasing manner and every solution that starts on the -axis is equal to   in a single step.In addition, every solution that starts on the -axis above   − converges to   + in a monotonic way.
Theorem 21.Assume that 1 > 4 2  2 and system (8)   Proof.Local stability of all equilibrium points follows from Theorem 11.We present the proof in the case of the equilibrium point  NE .The proof in the case of the equilibrium points  SE and  NW is similar.The existence of the global stable manifolds are guaranteed by Theorems 4 and 13.
The proofs of the basins of attractions (  ), (  + ) are the same as the proofs for the corresponding basins of attraction in Theorem 20, so we will only give the proof for (  ).Indeed, (  ) is an invariant set and   ((  )) is a subset of the interior of the ordered interval ( NE ,   ) for  large.In view of Corollary 5 the interior of the ordered interval ( NE ,   ) is attracted to   .Theorem 22. Assume that 1 > 4 2  2 and system (8) (a) Equation (1) has two equilibrium points 0 and  − 1 when  > 1.(b) All solutions of (1) are monotonic (increasing or decreasing) sequences.(c) If  ≤ 1, then the zero equilibrium is a global attractor;

One positive equilibrium point Two positive equilibrium points-case 1 Two positive equilibrium points-case 2
Three positive equilibrium points-case 1 Three positive equilibrium points-case 2Three positive equilibrium points-case 3