Abstract
This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.
1. Introduction and Preliminaries
In [1], authors proposed a mathematical model governed by ordinary differential equations related to the interaction between a plant and an insect. We extend this model by interchanging ordinary differential equations into fractional-order differential equations. Arguing as in [1], we assume that the larval stage of an insect develops and grows at the expense of nonreproductive tissues of the plant; on the other hand, these insects at their adult stage convey pollen to or deposit pollen on the flowers of the same plant, so allowing fertilization. Moreover, we denote , , , and as densities of plant species, larval species of insect, adult species of insect, and flower species involving in pollination, respectively. The interaction among , , , and is described by the following system of nonlinear differential equations:Then, we recall the definition of Caputo fractional-order derivative [2] for any function , and it is given bywhere is the least positive integer satisfying and denotes integral operator of Riemann-Liouville type with order , and it is defined bywhere represents Euler’s Gamma function. Furthermore, the fractional-order counterpart of (5) is given by the following system:where is in the sense of the Caputo fractional derivative defined in (2), is plant intrinsic growth rate, is plant intraspecific self-regulation coefficient (also the inverse is its carrying capacity), denotes pollination rate, is called herbivory rate, is flower production rate, is flower decay rate, and are larva and adult mortality rates, is plant pollination efficiency ratio, denotes adult consumption efficiency ratio, and is called the maturation rate for brevity. Moreover, parameter represents a reproduction rate resulting from the pollination of other plants species. We now consider the fact that flowers last for a very short time as compared to the life cycles of plants and insects. This means that the variables , , and have slower dynamics, and, on the other hand, the variable has fast dynamics [3]. In case of steady states of plants and insects, one can find a steady state of flowers by putting right hand side of second equation of system (4) that is equal to zero, that is, ; then it follows that [1]. Putting in system (4), we obtain the following 3-dimensional fractional-order system:For lenient mathematical analysis, one can reduce the number of parameters in system (5) by using the following transformations: We have the following dimensionless system:where , , , , , and . Since population densities cannot be negative, the state space of system (7) is given by Due to efficient computational results, discrete dynamical systems are much better than related systems in differential equations. Particularly, in case of nonoverlapping generations, difference equations are more suitable to study the behavior of population models [4–8]. For more details on some interesting population models both in differential equations and in difference equations, we refer the interested reader to [9–12]. It is very interesting to investigate the parametric conditions for existence of Neimark-Sacker bifurcation and to discuss chaos control techniques due to emergence of Neimark-Sacker bifurcation for discrete-time population models. For some interesting results related to Neimark-Sacker bifurcation and chaos control of discrete-time population models, we refer the reader to [13–17].
Now we consider the counterpart of (7) with piecewise constant arguments as follows:with initial conditions , , . Furthermore, assume that ; then it follows that . So for , system (9) givesThe solution of (9) is given bywhere is the Riemann-Liouville integral operator of order which is defined in (3). From (3) and (11), it follows thatSimilarly, for , so that , we obtainThe solution of (13) is given byRepeating the above process -times, the solution of (9) for is given byNext taking and adopting the notation of difference equations, system (15) yieldsOur aim in this paper is to study the local asymptotic stability of equilibrium points of system (16). Moreover, Neimark-Sacker bifurcation for positive equilibrium of system (16) is also investigated. In order to control the chaos due to emergence of Neimark-Sacker bifurcation, pole-placement and hybrid control strategies are implemented on system (16). Similar methods of discretization for fractional-order systems are also used in [18–22].
2. Linearized Stability of System (16)
First, we consider possible steady-state (equilibrium point) of system (16), which can be obtained by solving the following system:Then it is easy to see that and are two solutions of system (17). It follows that and are two equilibria of system (16). Neglecting the trivial equilibrium, we are left withIt must be noted that the equilibrium point of system (16), that is, the solution of system (18), may not be unique, but we do not care how many. For biological reasons, we are only interested in positive solutions of (18). From system (18), we obtain where is one of the roots of the following quartic polynomial:such that We are looking for the unique positive equilibrium point of system (18); for this, we have the following Descartes’s rule of signs.
Lemma 1 (see [23]). Let be a polynomial function with real coefficients. Then the number of positive real roots of is either the same as the number of sign changes of or less than the number of sign changes of by a positive even integer. Moreover, if has only one variation in sign, then has exactly one positive real root.
Using Lemma 1, we have the following result for existence of unique positive real root of polynomial given in (20).
Lemma 2. Polynomial in (20) has unique positive real root if one of the following conditions hold: (i)(ii)
Due to above analysis, we have the following result about the existence and uniqueness of positive equilibrium point of system (16).
Lemma 3. Under the conditions of Lemma 2, system (16) has unique positive equilibrium point if the following condition holds: where is unique positive real root of polynomial (20).
Next, the Jacobian matrix for system (16) evaluated at is given by
Theorem 4. For system (16), the following statements hold true: (i)The trivial equilibrium point is unstable.(ii)The equilibrium point is locally asymptotically stable if and only if and
Proof. (i) The Jacobian matrix for system (16) evaluated at trivial equilibrium is given by Now it is easy to see that eigenvalues of Jacobian matrix are , , and . Since and implies that , then it follows that . Hence is unstable.
(ii) The Jacobian matrix for system (16) evaluated at trivial equilibrium is given by The eigenvalues of Jacobian matrix are given by and Now it is easy to see that if and only if and if and only if
Theorem 5. The unique positive equilibrium of system (16) is locally asymptotically stable if the following condition is satisfied: where
Proof. The Jacobian matrix for system (16) evaluated at unique positive equilibrium is given by The characteristic polynomial of Jacobian matrix evaluated at positive equilibrium is given bywhereNow applying the Jury condition [11], the unique positive equilibrium point is locally asymptotically stable if the following conditions are satisfied:
In order to study the Neimark-Sacker bifurcation in system (16), we need the following explicit criterion of Hopf bifurcation.
Lemma 6 (see [24]). Consider an -dimensional discrete dynamical system , where is bifurcation parameter. Let be a fixed point of and the characteristic polynomial for Jacobian matrix of -dimensional map is given bywhere , and is control parameter or another parameter to be determined. Let , be a sequence of determinants defined by , , where Moreover, the following conditions hold:, (H1) Eigenvalue assignment: , , , , , , when is even or odd, respectively.(H2)Transversality condition: .(H3) Nonresonance condition: , or resonance condition , where , and . Then Neimark-Sacker bifurcation occurs at .
The following result shows that system (16) undergoes Neimark-Sacker bifurcation if we take as bifurcation parameter.
Theorem 7. The unique positive equilibrium point of system (16) undergoes Neimark-Sacker bifurcation if the following conditions hold: where , , and are given in (32).
Proof. According to Lemma 6, for , we have in (32) the characteristic polynomial of system (16) evaluated at its unique positive equilibrium, then we obtain the following equalities and inequalities:
3. Chaos Control
Controlling chaos in discrete-time models is a topic of great interest for many researchers in recent times, and practical methods can be used in many fields such as biochemistry, cardiology, communications, physics laboratories, and turbulence [25]. Chaos control in discrete-time models can be obtained by using various methods. In this section, we concentrate on two procedures only, that is, pole-placement technique which is based on feedback control methodology and the other one is hybrid control based on feedback control strategy and parameter perturbation.
First, we study chaos controlling technique based on pole-placement methodology introduced by Romeiras et al. [26] (also see [27]), which may be treated as generalized OGY method studied for the first time by Ott et al. [28]. In order to apply pole-placement technique to system (16), we rewrite system (16) as follows:where is taken as control parameter. Moreover, is restricted to lie in some small interval with and denotes the nominal value belonging to chaotic region. We apply the stabilizing feedback control strategy in order to move the trajectory towards the desired orbit. Assume that is unstable equilibrium point of system (16) in chaotic region produced by Neimark-Sacker bifurcation; then system (39) can be approximated in the neighborhood of the unstable equilibrium point by the following linear map:where It is easy to see that system (39) is controllable provided that the following matrixhas rank . Next, we assume that , where ; then system (40) can be written asMoreover, equilibrium point is locally asymptotically stable if and only if all three eigenvalues of the matrix , say, , , and , lie in an open unit disk. These eigenvalues are known as regulator poles, and problem of placing these regulator poles at suitable position is known as pole-placement problem. Assume that rank of the matrix is ; therefore pole-placement problem has a unique solution. Next, we assume that is characteristic equation of and is characteristic equation of ; then unique solution of the pole-placement problem is given bywhere and
Next in order to control Neimark-Sacker bifurcation in system (16), we apply hybrid control feedback methodology [29, 30]. Assuming that system (16) undergoes Neimark-Sacker bifurcation at equilibrium point , then corresponding controlled system can be written aswhere and controlled strategy in (45) is a combination of both parameter perturbation and feedback control. Moreover, by suitable choice of controlled parameter , the Neimark-Sacker bifurcation of the equilibrium point of controlled system (45) can be advanced (delayed) or even completely eliminated. The Jacobian matrix of controlled system (45) evaluated at positive equilibrium is given byThen, positive equilibrium of the controlled system (45) is locally asymptotically stable if roots of the characteristic polynomial of (47) lie in an open unit disk.
4. Numerical Simulations
Example 8. First, we take , , , , , , , , , , , , , and in system (16); then system (16) undergoes Neimark-Sacker bifurcation when is taken as bifurcation parameter. In this case, at , the unique positive equilibrium point of system (16) is = (0.4784797606357651, 0.0996595753165892, 0.32260552713822044). The characteristic polynomial equation evaluated at this positive equilibrium is given byThe roots of (48) are and with . Moreover, we haveHence, according to Theorem 7, the conditions of Neimark-Sacker bifurcation are obtained near the positive equilibrium point = (0.4784797606357651, 0.0996595753165892, 0.32260552713822044) at the critical value of Neimark-Sacker bifurcation .
Furthermore, Figure 1 shows that all three populations undergo Neimark-Sacker bifurcation (see Figures 1(a), 1(b), and 1(c)) and corresponding maximum Lyapunov exponents (MLEs) are shown in Figure 1(d). These MLEs confirm the existence of the chaotic sets. In general, the positive Lyapunov exponent is considered to be one of the characteristics implying the existence of chaos.
(a) Bifurcation diagram for
(b) Bifurcation diagram for
(c) Bifurcation diagram for
(d) Maximum Lyapunov exponents
Example 9. Next we take , , , , , , , , , , , , , and in system (16); then system (16) undergoes Neimark-Sacker bifurcation when is taken as bifurcation parameter. In this case, at , the unique positive equilibrium point of system (16) is ) = (0.3691168623529498, 0.2335056090113552, 0.20752106517420343). The characteristic polynomial equation evaluated at this positive equilibrium is given byThe roots of (48) are and with . Moreover, we verify the conditions of Theorem 7 as follows:Hence, according to Theorem 7, the conditions of Neimark-Sacker bifurcation are obtained near the positive equilibrium point = (0.3691168623529498, 0.2335056090113552, 0.20752106517420343) at the critical value of Neimark-Sacker bifurcation .
Furthermore, Figure 2 shows that all three populations undergo Neimark-Sacker bifurcation (see Figures 2(a), 2(b), and 2(c)) and corresponding maximum Lyapunov exponents (MLEs) are shown in Figure 2(d).
Next, we fix the value of . In Figure 3, the plot of is shown in Figure 3(a), the plot of is shown in Figure 3(b), the plot of is shown in Figure 3(c), and phase portrait of system (16) is shown in Figure 3(d). It is clear from Figure 3 that system (16) has unique positive equilibrium point which is locally asymptotically stable.
Furthermore, the phase portraits of system (16) for different values of are shown in Figure 4.
(a) Bifurcation diagram for
(b) Bifurcation diagram for
(c) Bifurcation diagram for
(d) Maximum Lyapunov exponents
Example 10. We choose the parametric values , , , , , , , , , , and and initial conditions , , and in system (16); then system (16) has a unique positive equilibrium point = which is unstable and Figure 4(c) shows a chaotic attractor for system (16) for these parametric values. So, one can take in order to apply pole-placement technique for system (16). In this case, the corresponding controlled system is given bywhere , , , , , , , , , , , and . Furthermore, we haveThen it is easy to check that matrix is of rank . Thus system (52) is controllable and its Jacobian matrix at is given by The characteristic equation of is given byThen a necessary and sufficient condition for all roots of the equation (55) to lie inside the unit disk iswhere , , and . For simplicity, if we choose and , then the unique positive equilibrium point of the controlled system (52) is locally asymptotically stable if and only if . Next, we choose , and selecting as bifurcation parameter for the controlled system (52). Moreover, the bifurcation diagrams for the controlled system are shown in Figure 5. It can be noticed from Figure 5 that the unique positive equilibrium point of the controlled system (52) undergoes period-doubling bifurcation at ; then both periodic orbits inter the region in which these orbits undergo Neimark-Sacker bifurcation separately. Thus we succeeded to advance the Neimark-Sacker bifurcation by introducing the pole-placement control technique in this case.
Finally, choosing the parametric values , , , , , , , , , , and and initial conditions , , and in system (16), then system (16) undergoes Neimark-Sacker bifurcation for these parametric values. Then unique positive equilibrium point of system (16) is unstable and Figure 4(a) shows that closed invariant circle appears at enclosing this unstable equilibrium. For these parametric values, the controlled system (45) can be written aswhere , , , , , , , , , , , and . Then Jacobian matrix of system (57) evaluated at is given byThe characteristic polynomial of (58) is given byAccording to the Jury condition, the roots of (58) lie in the unit open disk if and only if . Moreover, for , the plots for the controlled system (57) are shown in Figure 6.
5. Concluding Remarks
We study the qualitative behavior of a 3-dimensional discrete-time fractional-order plant-herbivore model, and thus we obtain the mathematical results related to existence and uniqueness of positive steady state and conditions for local asymptotic stability of positive equilibrium. In order to confirm complexity in system (16), the existence of Neimark-Sacker bifurcation for positive equilibrium point is proved mathematically and numerical simulations are provided. Through numerical simulation, we show that system (16) undergoes Neimark-Sacker for wide range of bifurcation parameter . Neimark-Sacker bifurcation is successfully controlled with two different control strategies. From our numerical investigation, it is clear that pole-placement technique based on feedback control methodology restores the stability for wide range of parameters, whereas the hybrid control based on feedback control and parameter perturbation works more effectively. Although hybrid control strategy is basically formulated for control of period-doubling bifurcation, it can also stabilize the unstable steady state for maximum possible range of control parameter in case of Neimark-Sacker bifurcation as well. Moreover, through numerical simulations, it is observed that the original system (16) undergoes Neimark-Sacker bifurcation only and there is no chance of period-doubling bifurcation for this system. But an application of pole-placement control strategy gives an evidence of period-doubling bifurcation for corresponding controlled system.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.