Abstract
We present the bifurcation results for the difference equation where and are positive numbers and the initial conditions and are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.
1. Introduction and Preliminaries
In this paper, we consider the difference equationwhere the parameters and are positive numbers and the initial conditions and are nonnegative numbers.
Equation (1) can be considered as a nonlinear perturbation of the sigmoid Beverton–Holt difference equationwhich is a major mathematical model in population dynamics and is the simplest model that exhibits Allee’s effect, see [1, 2]. A related difference equation of the formwhere the initial conditions and are nonnegative numbers considered in [3]. Equation (1) is a square version of the well-known Pielou difference equationwhich is another major model in population dynamics [1, 4]. Equation (4) has the same global dynamics as more general difference equation
Both equations (4) and (5) exhibit transcritical bifurcation, where the zero equilibrium is globally asymptotically stable up to some critical value where positive equilibrium appears and assumes global asymptotic stability.
A perturbation of original Beverton–Holt equation is the difference equation studied in [5]where the parameters and initial conditions and are nonnegative numbers exhibiting Naimark–Sacker bifurcation and chaos. Another perturbation of original delayed Beverton–Holt equation studied in [6]where the parameters and initial conditions and are nonnegative numbers, exhibiting transcritical and period-doubling (flip) bifurcation.
A perturbation of delayed sigmoid Beverton–Holt equation (2) considered in [7] iswhere the parameters and are positive numbers and the initial conditions and are nonnegative numbers.
Equation (2) exhibits a global asymptotic stability of either zero or positive equilibrium solutions and exchange of stability or transcritical bifurcation. Equation (8), where all solutions converge to either an equilibrium or to one of the three period-two solutions, exhibits the transcritical and period-doubling bifurcations while equation (6) exhibits Neimark–Sacker bifurcation and possibly chaos, but not a period-doubling (flip) bifurcation. In view of Theorem 4.2 in [8], equation (1) cannot have period-two solution, so period-doubling (flip) bifurcation is impossible. Related models with similar dynamics were considered in [1, 2, 9]. If we search for a model that exhibits Allee’s effect, transcritical bifurcation, and Neimark–Sacker bifurcation to a periodic solution, then equation (1) is probably the simplest such model. The dynamical difference between equations (1) and (6) is that equation (6) cannot exhibit Allee’s effect and has at most two equilibrium solutions. The dynamical difference between equations (1) and (8) is that equation (1) exhibits transcritical and Neimark–Sacker bifurcation while equation (8) exhibits transcritical and period-doubling bifurcation, proving this difference is the main objective of this paper.
In this paper, we will show that local asymptotic stability of the zero equilibrium will also imply its global asymptotic stability. In the case of the positive equilibrium solution, we will show that global asymptotic stability holds in some subspaces of the parametric region of local asymptotic stability. The technique used in proving global asymptotic stability of the positive equilibrium solution is based on global attractivity results for maps with invariant boxes, see [10, 11]. Related rational difference equations which exhibit similar behavior were considered in [5]. Equation (1) is one of the possible perturbations of the sigmoid Beverton–Holt difference equation which can exhibit chaos and shows that models based on such perturbation of the sigmoid Beverton–Holt equation can be used to model any kind of dynamic behavior from Allee’s effect, flip bifurcations to Naimark–Sacker bifurcation, and chaos. Consequently, all these dynamic scenarios can be fitted by appropriate perturbation of the sigmoid Beverton–Holt equation. For instance, equation (8) exhibits period-doubling bifurcation in addition to the transcritical bifurcation, so depending on the dynamics of the observed data, one can choose one of the models (1) and (6)–(8) to fit appropriate coefficients. See [12] and references therein for examples of fitting parameters of models. Some most recent applications of Neimark–Sacker bifurcation for differential equations can be found in [13] and for difference equations in [14]. Some global asymptotic results for second order difference equations related to the results from [10], which will be used in the present paper, can be found in [15].
Now, for the sake of completeness, we give the basic facts about the Neimark–Sacker bifurcation.
The Neimark–Sacker bifurcation is the discrete counterpart to the Hopf bifurcation for a system of ordinary differential equations in two or more dimensions, see [16–18]. It occurs for such a discrete system depending on a parameter, for instance, along with a fixed point, the Jacobian matrix of which has a pair of complex conjugate eigenvalues. These eigenvalues and will cross the unit circle at transversally. When this occurs in the discrete setting a periodic solution, which will in general be of an unknown period, will appear and this solution will be locally stable. To represent this periodic solution, we use the Murakami computational approach, see [19], to identify an asymptotic formula for an invariant curve in the phase plane which is locally attracting.
The following result is referred to as the Neimark–Sacker bifurcation theorem, see [16–18].
Theorem 1 (Neimark–Sacker bifurcation). Letbe a map depending on real parameter satisfying the following conditions:(i) for near some fixed ;(ii) has two nonreal eigenvalues and for near with ;(iii) at (transversality condition);(iv) for (nonresonance condition).
Then, there is a smooth -dependent change of coordinate bringing into the formand there are smooth functions , , and so that in polar coordinates, the function is given by
If , then the Neimark–Sacker bifurcation at is supercritical and there exists a unique closed invariant curve , which is attracting, and bifurcates from for .
Consider a general map that has a fixed point at the origin with complex eigenvalues and satisfying and . Assume thatwhere is the Jacobian matrix of evaluated at the fixed point , and
Here, we denote and . We let and be the eigenvectors of associated with satisfyingand . Assume thatand
Letthen
The next result of Murakami [19] gives an approximate formula for the periodic solution.
Corollary 1. Assumeandwhereis a sufficient small parameter. Ifis a fixed point of, then the invariant curvefrom Theorem 1can be approximated bywhere
Here, “” represents the real parts of these complex numbers. The calculation of is given by [20].
The rest of the paper is organized as follows; Section 2 gives local and global stability analysis of the zero equilibrium and positive equilibrium solutions in some regions of parameters; Section 3 presents the computation of Neimark–Sacker bifurcation; Section 4 presents the approximations of stable, unstable, and center manifolds of the equilibrium solutions of equation (1); finally, Section 5 establishes that the rate of convergence of the solutions that converge to the zero equilibrium is quadratic while the rate of convergence of the solutions that converge to any positive equilibrium solution is linear.
2. Local and Global Stability
Equation (1) has always the zero equilibrium . The positive equilibrium solutions of equation (1) are the positive solutions of the equation , that is,whenandwhen
The linearized equation associated with equation (1) about the equilibrium point iswhere
Now, the following results hold.
Lemma 1. For the equilibrium pointof equation (1), the equilibrium is always locally asymptotically stable.
The proof of the lemma follows from the fact that linearized equation at is .
Lemma 2. Assume that (22) holds. The positive equilibriumof equation (1) satisfies the following:(i)If , the equilibrium point is locally asymptotically stable.(ii)If , the equilibrium point is a repeller.(iii)If , the equilibrium point is nonhyperbolic of stable type with eigenvalues 1 and .
Proof. (i)One can see that and The rest of the proof follows from Theorem 1.1 [10]. We notice that the linearized equation at any positive equilibrium is and the corresponding characteristic equation is(iii)In the case (iii), we have that the characteristic equation at the equilibrium point iswhich solutions are 1 and . In view of the condition , we have .
Lemma 3. Assume that (22) holds. The positive equilibriumis always a saddle point.
Proof. One can see thatandwhich implyThe rest of the proof follows from Theorem 1.1 [10].
First, we give the global asymptotic result for zero equilibrium.
Theorem 2. Assume that
Then, the zero equilibrium of equation (1) is globally asymptotically stable.
Proof. Every solution of equation (1) satisfiesand the functionis increasing in and decreasing in , with property that it has an invariant and attracting interval . Now, we will employ Theorem 1.13 from [10]. Consider the system of equationsand prove that . This system becomeswhich after eliminating becomesNow, the discriminant of first quadratic polynomial in (40) is in view of (19) and the discriminant of second quadratic polynomial in (40) is in view of (35). Finally, the discriminant of the third polynomial in (20) isand for is negative in view of (35). If , the third polynomial simply becomes a constant 1. Thus, the only solution of (40) is . The same holds for in view of symmetry of the considered system. So, , and by Theorem 1.13 in [10], the zero solution of equation (1) is globally asymptotically stable.
As Figure 1 shows the boundary of the basins of attraction of two locally asymptotically stable equilibrium solutions and seem to be the global stable manifold of the smaller equilibrium solution , which is a saddle point for all values of parameters. In Section 3, we will derive the asymptotic formulas for both stable and unstable manifolds based on the functional equations that the two manifolds satisfy. We will visually compare these manifolds with the image of the basin of attraction.
Now, we give some results about the basins of attraction of the positive equilibrium solutions. We will show that local asymptotic stability of a positive equilibrium will also imply its global asymptotic stability in a substantial subregion of the parametric space and within the basins of attraction of locally stable equilibrium solutions.
Lemma 4. Assume that (14) holds. If is nonzero solution of equation (1), then the following hold:(i)If , then . The solution is a decreasing sequence and so(ii)If , then . The solution is an increasing sequence and so(iii)If , then . The solution is a decreasing sequence and so it converges to one of the equilibrium solutions.
Proof. SetThen, the positive equilibrium solutions are solutions of the equilibrium equation and if and only if .(i)Now, we have By using induction, we can prove that the solution is a decreasing sequence, and since , it can only converge to the zero equilibrium.(ii)Now, we have By using induction, we can prove that the solution is an increasing and bounded sequence, and since , it can only converge to the larger positive equilibrium .(iii)The proof is identical to the proof of part (i) and it will be omitted.
Remark 1. An immediate consequence of Lemma 4 is that the set is a part of the basin of attraction of the zero equilibrium and is a part of the basin of attraction of the larger positive equilibrium , that is, . Based on our simulations, we formulate the following conjecture.
Conjecture 1. .
3. The Neimark–Sacker Bifurcation
In this section, we bring the system that corresponds to equation (1) to the normal form which can be used for the computation of the relevant coefficients of the Neimark–Sacker bifurcation.
If we make a change of variable , then the transformed equation is given by
Setand write equation (1) in the equivalent form:
Let be the corresponding map defined by
Then, has the unique fixed point and the Jacobian matrix of at is given by
The eigenvalues of are and where
One can prove that for , we obtain and
One can see that for and
Furthermore, we get
The eigenvectors corresponding to and are and , where
For , we getwhereand
Hence, for , system (49) is equivalent to
Define the basis of by .
Let
By using this, one can see thatand
By using , and , we have that
Next, we have that and , where
One can see that
Theorem 3. Let and
Then, there is a neighborhood of the equilibrium point and some such that forand , the -limit set of a solution of equation (1), with initial conditions , is the equilibrium point ifand belongs to a closed invariant curve encircling the equilibrium point if
Furthermore, and the invariant curve can be approximated bywhere
Proof. The proof follows from the above discussion and Theorem 1 and Corollary 1. See Figures 2 and 3 for a graphical illustration.
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(b)
4. The Invariant Manifolds
In this section, we derive the asymptotic formulas for the local stable and unstable manifolds for the equilibrium point and provide some numerical examples where we compare visually the local approximations of stable and unstable manifolds and center manifold, obtained by using Mathematica, with the boundaries of the basins of attraction obtained by using the software package Dynamica.
From Lemma 3, it follows that is a saddle point if . In order to apply the theorem for the stable and unstable manifolds, we make a change of variable . Then, the transformed equation is given by
Setand write equation (1) in the equivalent form:
Let be the corresponding map defined by
We expand and as a Taylor series about to write
Assume that the invariant manifold at is locally represented as the graph of a function such that . Then, from , and by using package Mathematica, we obtainandand
Then, the dynamics restricted to the invariant manifold are given locally by the equation
Note that the Jacobian matrix of at is given byand the eigenvalues of are .
Theorem 4. Assume that . Then, the equilibrium point of equation (1) is a saddle point if . The stable manifold and unstable manifold at are given byand
The equilibrium point is nonhyperbolic if , and it is semiasymptotically stable from the right.
Proof. The proof of first part of the theorem follows from the above discussion. If , then and we obtain , and . The rest of the proof follows from the fact that the dynamics of equation (1) are dynamics restricted to the center manifold which is given locally by equation (81). See Figures 4 and 5 for a graphical illustration.
Bifurcation diagram of equation (1) in certain range of parameters indicate chaos, see Figure 6.
(a)
(b)
(a)
(b)
(a)
(b)
5. Rate of Convergence
In this section, we will shortly discuss the rate of convergence of solutions of equation (1) that converge to the equilibrium solutions. We will show that the convergence toward the zero equilibrium is quadratic and toward any positive equilibrium is linear.
Assume that for some solutions of equation (1). Then, equation (1) impliesand sowhich shows that the convergence toward the zero equilibrium is quadratic.
Let for some solutions of equation (1). Then, we havewhereand
Setting , we see that the limiting equation is exactly the linearized equation (29). Now, in view of Poincaré-Perron theorem, we conclude thatwhere are roots of the characteristic equation (30), and [6, 8, 21]. Thus, in this case, the convergence is linear since . See Table 1 for numerical comparison of rates of convergence to zero equilibrium and positive equilibrium.
6. Conclusions
In this paper, we give the global dynamics of the difference equation where and are positive numbers and the initial conditions and are nonnegative numbers in a part of parametric space. We show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Moreover, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the saddle point and nonhyperbolic equilibrium solution. Finally, we give the rates of convergence toward all equilibrium solutions, proving that the convergence to the zero equilibrium solution is quadratic and convergence to the positive equilibrium solution is linear.
This difference equation is the simplest perturbation of sigmoid Beverton–Holt difference equation that exhibits Allee’s effect, transcritical bifurcation, and Neimark–Sacker bifurcation, but not a flip bifurcation.
Data Availability
No data were used for this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.