Abstract
The local behavior with topological classifications, bifurcation analysis, chaos control, boundedness, and global attractivity of the discrete-time Kolmogorov model with piecewise-constant argument are investigated. It is explored that Kolmogorov model has trivial and two semitrival fixed points for all involved parameters, but it has an interior fixed point under definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrival, and interior fixed points. Further for the discrete Kolmogorov model, existence of periodic points is also investigated. It is also investigated the occurrence of bifurcations at interior fixed point and proved that at interior fixed point, there exists no bifurcation, except flip bifurcation by bifurcation theory. Next, feedback control method is utilized to stabilize chaos existing in discrete Kolmogorov model. Boundedness and global attractivity of the discrete Kolmogorov model are also investigated. Finally, obtained results are numerically verified.
1. Introduction
1.1. Motivation and Literature Review
There are many symbiotic interactions existing in nature between two or more species in an ecosystem. Mutualism is an example of such interaction where interacted species get benefit from each other. For example, termites eat cellulose of wood but cannot digest it, and flagellates reside in the termite’s gut decomposing the cellulose of food and thus providing nutrients to termites. On the other hand, termite’s gut provides food and shelter to flagellates. It is pointed out that in theoretical ecology, mutualist behavior of symbiosis or mutualism is very significant [1]. This field is not widely studied as the other fields of mathematical biology even for two species, although its importance is equal to the other competitive interactions such as host-parasitoid and prey-predator interactions. So, this topic of mutualism system seems interesting to study. For instance, May [2] suggested the two-species hybrid continuous-time Kolmogorov model represented by the following system of differential equations:where represents the greatest integer in , and parameters , and are positive numbers. Moreover, due to the effect of piecewise-constant argument, we have integrated both sides of the Kolmogorov model, which is depicted in (1). On interval , (1) can be written aswhere for , and resulted in the following solution:
It is important here to mention that discrete-time models directed by difference equations are more appropriate than the continuous ones in the case where populations have nonoverlapping generations, and also these models provide efficient computational results as compared to continuous models. Due to that reason in recent years, many scholars, researchers, and scientists have studied the dynamics of biological systems such as ratio-dependent predator-prey system, discrete hyperchaotic system, and host-parasitoid model. For instance, Bhattacharya and Saha [3] studied the dynamics characteristics of discrete Kolmogorov system. Cheng and Cao [4] explored dynamic characteristics of a discrete ratio-dependent prey-predator model. Jing and Yang [5] explored the dynamic characteristics of discrete prey-predator model. Kangalgil and Topsakal [6] explored dynamics characteristics of a discrete prey-predator system. Ran et al. [7] explored the Neimark-Sacker bifurcation of a stochastic discrete hyperchaotic system. Beddington et al. [8] explored dynamic characteristics in prey-predator models. Chen [9] studied the global dynamics and permanence of a discrete multispecies system. Lu and Zhang [10] investigated global attractivity and permanence of a discrete system with Holling type-II functional response. Fang and Chen [11] studied permanence of a discrete multispecies Lotka–Volterra model with delays. Fang et al. [12] investigated the dynamic characteristics of a discrete system. Jana and Samanta [13] studied prey-predator system in discrete-time scale using interval parameters.
1.2. Objective, Contributions, and Novelties
Motivated from aforementioned studies, the objective of the present work is to explore the global dynamics, bifurcations, and chaos in a discrete Kolmogorov model with piecewise-constant argument (3). More precisely, our main finding in this paper includes Topological classifications at fixed points of the discrete Kolmogorov model (3) by linear stability theory Exploration of periodic points of the discrete Kolmogorov model (3) Flip bifurcation analysis at interior fixed point by bifurcation theory. Investigation of chaos by feedback control method To explore boundedness and global attractivity of discrete Kolmogorov model Validation of obtained results numerically
1.3. Paper Structure
The rest of the paper is structured as follows: Section 2 relates with the investigation of topological classifications of discrete Kolmogorov model (3) at fixed points. In Section 3, periodic points of prime period-1 and period- of the discrete Kolmogorov model (3) are explored, whereas comprehensive analysis of the bifurcation at fixed point is explored in Section 4. In Section 5, chaos control is explored by feedback control method, whereas Section 6 is about the presentation of numerical simulations to validate obtained results. In Section 7, boundedness and global dynamics are explored. The conclusion and future work are given in Section 8.
2. Topological Classifications of Discrete Kolmogorov Model (3) at Fixed Points
Here, local dynamical properties with topological classifications at fixed points are explored in . For this, first it is easy to verify that discrete Kolmogorov model (3) has trivial fixed point , boundary fixed points , and , but if , then it has an interior fixed point . Additionally, the variational matrix at fixed point under the following map,whereis
Now, topological classifications at fixed points , and are explored for the completion of this section.
2.1. Topological Classifications at
The variational matrix at fixed point iswhose eigenvalues areThus, the topological classifications at are summarized as Table 1.
Remark 1. Fixed point of the discrete Kolmogorov model (3) is never sink, saddle, and nonhyperbolic.
2.2. Topological Classifications at
The variational matrix at fixed point iswhose eigenvalues are
So, the topological classifications at of the discrete Kolmogorov model (3) can be summarized as Table 2.
Remark 2. Fixed point of the discrete Kolmogorov model (3) is never sink, source, and nonhyperbolic.
2.3. Topological Classifications at
The variational matrix at fixed point iswhose eigenvalues are
Based on eigenvalues , we will summarize the topological classifications at of the discrete-time Kolmogorov model (3) as Table 3.
Remark 3. Fixed point of the discrete Kolmogorov model (3) is never sink, source, and nonhyperbolic.
2.4. Topological Classifications at
The variational matrix at fixed point iswith corresponding characteristic equation of the form,where
Finally, roots of (14) arewhere
Since , therefore it is important here to note that fixed point is never stable focus, unstable focus, and nonhyperbolic. So, we will summarize the topological classifications of discrete Kolmogorov model (3) at as follows.
3. Exploration of Periodic Points
In the following proposition, periodic points of prime period-1 of the discrete Kolmogorov model (3) are explored.
Proposition 1. Fixed points , and of the Kolmogorov model (3) are periodic points of prime period-1.
Proof. From (3), definingwhere and are defined in (5), after some computations, one getsFrom (19)–(22), one can summarize that , and of the Kolmogorov model (3) are periodic points of prime period-1.
In the following proposition, periodic points of period- of the discrete-time Kolmogorov model (3) are explored.
Proposition 2. Fixed points , and of the Kolmogorov model (3) are periodic points of period-.
Proof. After some straightforward computations, from (18), one getsFrom (23), one concludes the required statement. Similarly, one can show that fixed points , and of the discrete Kolmogorov model (3) are periodic points of period-.
4. Analysis of Bifurcation
It is easy to note that there exists no bifurcation at fixed points , and , but in the following subsection, we explore that at discrete, Kolmogorov model (3) undergoes only flip bifurcation by bifurcation theory [14–20].
4.1. Flip Bifurcation about
From Table 4, it is noted that equilibrium point of the discrete Kolmogorov model (3) is nonhyperbolic if . Therefore, eigenvalues of at nonhyperbolic condition are computed and one gets , but which conclude that at fixed point , Kolmogorov model (3) undergoes flip bifurcation if are located in the following set:
Hereafter, in the following, we will present comprehensive flip bifurcation analysis at of the Kolmogorov model (3).
Theorem 1. If , then the discrete Kolmogorov model (3) undergoes the flip bifurcation.
Proof. It is recalled from Table 4 that if , then is nonhyperbolic where . Moreover, at , one gets , where as , which gives the existence of flip bifurcation at by choosing as a bifurcation parameter. So, if varies in a neighborhood of , then model (3) takes the form,Now, it is noted that by using transformation,one can transform to where (25) takes the following form:whereNow, (27) takes the following form:wherebyNow, for (29), center manifold about is examined in a small neighborhood of , and therefore can be expressed as the following expression:Therefore, the computation yieldsThus, the map (29) restricting to iswhereIn order to show (34) undergoes flip bifurcation, it is required that must be nonzero, i.e.,After calculating, one getswhereIn view of (38) and (39), if , then as , discrete Kolmogorov model (3) undergoes a flip bifurcation. Additionally, period-2 points from are stable (respectively unstable), if .
5. Chaos Control
This section is purely dedicated for the exploration of chaos control in the sense of state feedback control method [21, 22]. After adding control force , discrete Kolmogorov model (3) takes the form,where . It is noted here that denotes feedback gains. Moreover, for the controlled system (40) is
If denotes the characteristic roots of at , then
Now, the answer of and determines the strains of marginal stability, and these prerequisites give that . If , then from (43), one gets
If , then from (42) and (43), one gets
Finally, if , then from (42) and (43), one gets
Therefore, from (44)–(46), lines , and in -plane give the triangular region, which further gives the fact that .
6. Numerical Simulations
We present numerical simulations to validate the corresponding obtained results. For instance, if , then from nonhyperbolic condition, one gets . From theoretical discussion, of the discrete Kolmogorov model (3) is a stable node if . So, if one chose the bifurcation value , then it is clear from Figure 1(a) that fixed point of Kolmogorov model (3) is a stable node. Moreover, for some more parametric values, if , then Figures 1(b)–1(d) also indicate that the fixed point of Kolmogorov model (3) is also a stable node. Further, if , then fixed point becomes unstable and meanwhile flip bifurcation occurs, i.e., if , then by mathematical computation from (37), one gets . Moreover, from (38), one gets , which indicates the fact that stable period-2 points bifurcate from , and hence flip bifurcation diagram along with maximum Lyapunov exponents are plotted and presented in Figure 2. Moreover, 3D flip bifurcation diagrams are presented in Figure 3. Finally, Figure 4 shows the complex dynamics with orbits of period-.
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Hereafter, we will prove the validity of obtained results in Section 5. For instance, if , then from (44)–(46), one gets
Hence, lines that are presented in (47)–(49) determine triangular region that gives (see Figure 5).
7. Boundedness and Global Attractivity of the Kolmogorov Model (3)
Boundedness with construction of invariant rectangle and global dynamics at interior fixed point of the discrete Kolmogorov model (3) are explored in this section.
7.1. Boundedness Along with Construction of Invariant Rectangle
Theorem 2. Every positive solution of the Kolmogorov model (3) is bounded if
Proof. If is positive solution of the discrete Kolmogorov model (3), thenFrom inequality of (51), one getsUtilizing (52) in the inequality of (51), one getswhich further gives the following solution:Now, from the inequality of (51), one getsUtilizing (55) in the inequality of (51), one getswhose solution isIf (50) holds, then from (54) and (57), one gets
Theorem 3. For the discrete Kolmogorov model (3), the invariant rectangle is .
Proof. Followed by induction.
7.2. Attractivity at
From (5), it is noted that is increasing in for some fixed . Also, is increasing in for fixed . This motivates the study of global attractivity at of discrete Kolmogorov model (3), as follows.
Theorem 4. of discrete Kolmogorov model (3) is the global attractor.
Proof. Since and satisfied the above monotonic characteristics and so ifis the solution of following system:In view of (3) and (59), one getsFrom (60), one hasFrom (61), one hasFrom (62), one getsSimilarly, from (63), one getsForm (64) and (65), one gets , which implies that . Finally, from (65), one gets . So, of discrete Kolmogorov model (3) is a global attractor by Theorem 1.16 of [23].
8. Conclusion and Future Work
The work is about the local dynamical characteristics at fixed points, existence of periodic points, boundedness, global attractivity, chaos control, and bifurcations of a discrete Kolmogorov model with piecewise-constant argument. By existing linear stability theory, it is proved that fixed point is never sink, saddle, and nonhyperbolic, but it is a source; is never sink, source, and nonhyperbolic, but it is a saddle; is never sink, source, and nonhyperbolic, but it is a saddle; and finally, is a stable node if , unstable node if , and nonhyperbolic if , but it is never stable focus, unstable focus, and nonhyperbolic. Further, it is proved that fixed points , and are periodic having prime period-1. Then, in order to understand under consideration model deeply, we also explored the occurrence of bifurcations at . It is proved that at , the model undergoes flip bifurcation if . Further, state feedback control method is utilized in order to stabilize chaos existing in discrete Kolmogorov model (3). Moreover, numerical simulations are also presented to validate obtained results. Finally, boundedness and global attractivity of the Kolmogorov model (3) at are also explored. The use of robust control to improve the control process and stabilization for the two-species hybrid Kolmogorov model are our next aim to study.
Appendix
A. Derivation of (3) from (2)
Dividing the equation of system (2) by and the equation of system (2) by , one gets
Now, putting in the equation of system (A.1) and in the equation of system (A.1), one gets
Moreover, multiplying the equation of system (A.2) by integrating factor and the equation of system (A.2) by integrating factor , one gets
Now, integrating from and , one gets
After some simplifications, the solution of (A.4) becomeswhich can also be written as
Data Availability
All the data utilized in this article have been included, and the sources from where they were adopted were cited accordingly.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.