Abstract
We consider the following system of difference equations: where , , , , are positive constants and are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at , which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at and thus describe the global dynamics of this system. Since the singular point at 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: where 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.(a)Equation (1) has two equilibrium points and when .(b)All solutions of (1) are monotonic (increasing or decreasing) sequences.(c)If , then the zero equilibrium is a global attractor; that is, , for all .(d)If , then the equilibrium point is a global attractor; that is, , for all .(e)Both equilibrium points are globally asymptotically stable in the corresponding regions of parameters and ; that is, they are global attractors with the property that small changes of initial condition result in small changes of the corresponding solution .
All these properties can be derived from the explicit form of the solution of (1): See [1–3].
The following difference equation, 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 (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.(a)Equation (3) has a unique zero equilibrium when .(b)Equation (3) has a zero equilibrium and the positive equilibrium , when .(c) There exist a zero equilibrium and two positive equilibria, and , when .(d) All solutions of (3) are monotonic (increasing or decreasing) sequences.(e) If , then the equilibrium point is a global attractor; that is, .(f)If , then the equilibrium point is a global attractor, with the basin of attraction and is a nonhyperbolic equilibrium point with the basin of attraction .(g)If , 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 and the model exhibits the Allee effect.(h) The equilibrium points and are globally asymptotically stable in the corresponding basins of attractions 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 [8–14]. 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 . 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 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 . 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, 13–18] 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.
2. 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 .
Competitive systems of the form (9) were studied by many authors in [6, 7, 9, 13, 14, 23–37] and others.
Here we give some basic notions about monotonic maps in the plane.
We define a partial order on (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 , that is, if the following holds:
For each , define for to be the usual four quadrants based on and numbered in a counterclockwise direction; for example, .
For let denote the interior of .
The following definition is from [35].
Definition 1. Let be a nonempty subset of . A competitive map is said to satisfy condition () if for every , in , implies , and is said to satisfy condition () if for every , in , implies .
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 . 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 , 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 be the cartesian product of two intervals in . Let be a competitive map. If is injective and for all then satisfies (). If is injective and 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 be a rectangular subset of and let be a competitive map on . Let be a fixed point of such that has nonempty interior (i.e., is not the NW or SE vertex of ).
Suppose that the following statements are true. (a)The map is strongly competitive on .(b) is 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 and
or and
Then there exists a curve in such that (i) is invariant and a subset of ;(ii) the endpoints of lie on ;(iii) ;(iv) is the graph of a strictly increasing continuous function of the first variable;(v) is differentiable at if or one sided differentiable if , and in all cases is tangential to at ;(vi) separates into two connected components, namely,
(vii) is invariant, and as for every ;(viii) is invariant, and as for every .
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 , and if has no fixed points in the ordered interval other than and , then the interior of is either a subset of the basin of attraction of or a subset of the basin of attraction of .
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 . Similar result for a nonhyperbolic point of stable type, that is, a nonhyperbolic point where second eigenvalue is in the interval , follows from Theorem 4. See Kulenović and Merino, Invariant Curves of Planar Competitive and Cooperative Maps.
Theorem 6. Let and let be a strongly competitive map with a unique fixed point , such that is continuously differentiable in a neighborhood of . Assume further that at the point the map has associated characteristic values and satisfying and .
Then there exist curves , in and there exist with such that (i)for , is invariant, north-east strongly linearly ordered, such that and ; the endpoints , of , where , belong to the boundary of . For with , is a subset of the closure of one of the components of . Both and are tangential at to the eigenspace associated with ;(ii)for , let be the component of whose closure contains . Then is invariant. Also, for , accumulates on , and for , accumulates on .(iii)Let and .
Then is invariant.
Corollary 7. Let a map with fixed point be as in Theorem 6. Let , be the sets as in Theorem 6. If satisfies (), then for , is invariant, and for every , the iterates converge to or to a point of . If satisfies (), then and . For every , the iterates either converge to or converge to a period-two point or to a point of .
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:
All solutions of system (16) with at least one zero component are given as where , where , and where . The equilibrium point exists when , and exists when .
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 and . By eliminating from (17) we obtain
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
Assume that . Then the following holds. (a)If , , and , then (18) has four simple real roots.(b)If and , then (18) has no real roots.(c)If , then (18) has two simple real roots.(d)If and , then (18) has one real double root.(e)If and , then (18) has two real simple roots and one real double root.(f)If , , and , then (18) has two real double roots.(g)If , , and , then (18) has no real roots.(h)If , , and , then (18) has one real root of multiplicity four.
Proof. The discrimination matrix [41] of and is given by Let denote the determinant of the submatrix of , formed by the first rows and the first columns, for where So, by straightforward calculation one can see that The rest of the proof follows in view of Theorem 1 in [41].
Geometrically solutions of system (17) are intersections of two ellipses that satisfy the equations with respective vertices and . See Figure 1.
Consequently when , in addition to the three equilibrium points on the axes, system (8) may have , or positive equilibrium points. We will refer to these equilibrium points as (southwest), (southeast), (northwest), and (northeast) where
When a positive equilibrium point is nonhyperbolic we will refer to it as .
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 and 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), , , (saddlepoint), (nonhyperbolic of the stable type), and (nonhyperbolic of the unstable type).
Lemma 9. The following conditions hold for the coordinates of the positive equilibrium points, , of system (8).(i)For and , (ii)For , (iii)For , , and , (iv)For ,
Proof. This is clear from geometry. See Figure 2.
Lemma 10. The following conditions hold for the coordinates of the positive equilibrium points, , of System (8).(i)For and , (ii)For , , and , (iii)For and ,
Proof. (i) Let be the slope of the tangent line to ellipse at and let be the slope of the tangent line to ellipse at . 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.
Theorem 11. The following conditions hold for the equilibrium points of system (8):(i) is a locally asymptotically stable;(ii) is nonhyperbolic of the stable type;(iii) is locally asymptotically stable and is a saddle point;(iv) is a repeller;(v), , and are saddle points;(vi) is locally asymptotically stable;(vii) is nonhyperbolic of the stable type;(viii) is nonhyperbolic of the unstable type.
Proof. (i) The eigenvalues of (26), evaluated at , are and .
(ii) The eigenvalues of (26), evaluated at , are and when .
(iii) The eigenvalues of (26), evaluated at and , respectively, are and when .(a)Note that when ,
Therefore .(b)Note that when , . Therefore
In both cases, the conclusion follows.
(iv) We need to show that and when . Since and are both positive, our conditions become and . We will first show that . By (37) we have
By (33) we have .
Therefore . We will next show that .
By (37) we have
Therefore .
(v) We need to show that when . Since and are both positive, our condition becomes . By (37) we have
Therefore . The proofs that and are saddle points are similar and will be omitted.
(vi) We need to show that and when . Since and are both positive, our conditions become and . We will first show that . By (38) we have
By (35) we have .
Therefore . We will next show that . By (38) we have
Therefore .
(vii) By (29) and (31) we have
By (39), we have and . By (35), we have . 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 which corresponds to system (8) is injective and that it satisfies .
4.1. Convergence of Solutions on the Coordinate Axes: Injectivity and
When , system (8) becomes
When , 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.(i) is a superattractor of all solutions of system (8) with initial conditions on the -axis.(ii)When no equilibrium points exist on the axis, if , then .(iii)When exists,(a)if and , then ;(b)if and , then .(iv)When and exist,(a)if and , then ;(b)if and , then ;(c)if and , then .
Proof. (i) When , it follows directly from (52) that for .
(ii) In this case . By (53) it can be shown that
By (54), when , it is clear that is a stricly decreasing sequence, and so is convergent. It follows that converges to .
(iii) In this case, , 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 when , and converges to when .
(iv) In this case, . By (53), it can be shown that
By (56), it is clear that is a stricly decreasing sequence (and so is convergent) when and when , and a strictly increasing sequence (and so is convergent) when . It follows that converges to when and when and converges to when .
Theorem 13. The map which corresponds to system (8) is injective.
Proof. Indeed, which is equivalent to This immediatly implies .
Theorem 14. The map which corresponds to system (8) satisfies . All solutions of system (8) converge to either an equilibrium point or to .
Proof. Assume that
The last inequality is equivalent to
Suppose . Then , which contradicts (60). Consequently 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 .
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.
Theorem 15. Assume that . Then system (8) has one equilibrium point which is locally asymptotically stable. The singular point 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 or .
Proof. Local stability of all equilibrium points follows from Theorem 11. In view of Theorem 12, every solution that starts on the -axis converges to in a decreasing manner and every solution that starts on -axis is equal to in a single step. Let be an arbitrary initial point in the interior of the first quadrant. Then and and so . In view of Theorems 12 and 14 or as .
Theorem 16. Assume that . Then system (8) has two equilibrium points, which is locally asymptotically stable and which is nonhyperbolic of the stable type. The singular point is global attractor of all points on the -axis, which start below . Furthermore, every point in the interior of the first quadrant below is attracted to or and every point in the first quadrant which starts above is attracted to .
Proof. Local stability of all equilibrium points follows from Theorem 11. In view of Theorem 12, every solution that starts on the -axis below converges to 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 decreasing way. Let be an arbitrary initial point in the interior of the first quadrant below . Then which implies and so . If then will eventually enter the ordered interval . In view of Theorems 12 and 14, or as .
Now, let be an arbitrary initial point in the interior of the first quadrant above . Then , where . This implies and so . Since , as , we conclude that as .
Theorem 17. Assume that and system (8) has three equilibrium points, and which are locally asymptotically stable and which is a saddle point. The singular point is global attractor of all points on -axis, which start below . The basins of attraction of two equilibrium points are given as where denotes the global stable manifold guaranteed by Theorem 4. Furthermore, every initial point below is attracted to or .
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 -axis below converges to 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.
Let be an arbitrary initial point in the interior of the first quadrant below . Then which implies and so . Since as , we conclude that eventually enters the ordered interval , in which case it converges to or .
Finally, let be an arbitrary initial point in the interior of the first quadrant above . Then , where . Thus , which, by as , implies that eventually lands on the part of -axis above and so it converges to .
Theorem 18. Assume that and system (8) has four equilibrium points, and which are locally asymptotically stable, which is a saddle point, and which is nonhyperbolic of the unstable type. The singular point is global attractor of all points on the -axis, which start below . The basins of attraction of three of the equilibrium points are given as where denotes the global stable manifold guaranteed by Theorem 4 and are continuous nondecreasing curves emanating from , whose existence and properties are guaranteed by Corollary 7. Furthermore, every initial point below is attracted to or .
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 -axis below converges to 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 -axis above converges to in a monotonic way.
Let be an arbitrary initial point in the interior of the first quadrant below . Assume that . Then and so , where and so . Since and as , we conclude that eventually enters the ordered interval , in which case, in view of Corollary 5, it converges to .
Next, assume that . Then , where and so and so . Since and as , we conclude that eventually enters the ordered interval , in which case, by Theorems 12 and 14, or as .
Now, let be an arbitrary initial point in the interior of the first quadrant above . Assume that . Then . Assume that . Thus , which by and as implies that eventually enters the ordered interval , in which case, in view of Corollary 5, it converges to .
Next, assume that . Then , where and so . Since and as , we conclude that converges to .
Finally, let be an arbitrary initial point between and . Then stays between and for all and in view of Corollary 7 it must converge to .
Conjecture 19. Based on our numerical simulations we believe that in Theorem 18.
Theorem 20. Assume that and system (8) has five equilibrium points, , which are locally asymptotically stable, and (resp., ) which are saddle points, and which is a repeller. The singular point is global attractor of all points on the -axis, which start below . The basins of attraction of four of the equilibrium points are given as where and denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below is attracted to or .
Proof. Local stability of all equilibrium points follows from Theorem 11. We present the proof in the case of the equilibrium point . The proof in the case of the equilibrium point 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 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.
Let be an arbitrary initial point in the interior of the first quadrant below . Assume that . Then which implies , where and so . Since and as , we conclude that eventually enters the ordered interval , in which case, in view of Corollary 5, it converges to .
Next, assume that . Then , where . Thus and so . Since and as , we conclude that eventually enters the interior of the ordered interval , in which case, it converges to or .
Now, let be an arbitrary initial point in the interior of the first quadrant above . Assume . Then , where and so . Since and as , then eventually enters the ordered interval , in which case, in view of Corollary 5, it converges to .
Next, assume that . Then , where and so . Since and as , we conclude that converges to .
Theorem 21. Assume that and system (8) has six equilibrium points, , which are locally asymptotically stable, and (resp., or ) which are saddle points, which is a repeller, and which is nonhyperbolic of the stable type. The singular point is global attractor of all points on the -axis, which start below . The basins of attraction of five of the equilibrium points are given as where , , and denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below is attracted to or .
Proof. Local stability of all equilibrium points follows from Theorem 11. We present the proof in the case of the equilibrium point . The proof in the case of the equilibrium points and 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 for large. In view of Corollary 5 the interior of the ordered interval is attracted to .
Theorem 22. Assume that and system (8) has seven equilibrium points, , , which are locally asymptotically stable, which are saddle points, and which is a repeller. The singular point is global attractor of all points on -axis, which start below . The basins of attraction of six of the equilibrium points are given as where , , and denote the global stable manifolds whose existence is guaranteed by Theorem 4. Furthermore, every initial point below is attracted to or .
Proof. Local stability of all equilibrium points follows from Theorem 11. Proofs of the basins of attractions , are the same as the proofs for corresponding basins of attraction in Theorem 20. So we only give the proof for . Indeed, is an invariant set and is a subset of the interior of the ordered interval for large. In view of Corollary 5 the interior of the ordered interval is attracted to .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.