Bifurcation analysis and chaos of a discrete-time Kolmogorov model

In this paper, we explore local dynamical characteristics with different topological classifications at fixed points, bifurcations and chaos in the discrete Kolmogorov model. More precisely, we investigate the existence of trivial, boundary and interior fixed points of the discrete Kolmogorov model by algebraic techniques. We prove that for all involved parameters, the discrete Kolmogorov model has trivial and two boundary fixed points, and the interior fixed point under specific parametric condition. Further we explore the local dynamics with topological classifications at fixed points and existence of periodic points of the discrete Kolmogorov model simultaneously. We also explore the occurrence of bifurcation at fixed points and prove that at boundary points there exists no flip bifurcation but it occurs at the interior fixed point. Moreover, we utilize feedback control method to stabilize chaos appears in the Kolmogorov model. Finally, we present numerical simulations to verify corresponding theoretical results and also reveal some new dynamics.


Introduction
In mathematics, diffeomorphism is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. In nature this property can be examined in different states of ecosystem. In ecosystem, different components interact with each other and also with the environment such as structure of trophic, relation among diversity production and flow of energy emerging from the variations of components that are interacted with each other [1]. These interactions affect the time taken by the ecosystem in its growth and therefore describing in cross scale-lay out solves the basic issues of services related to ecosystem [2][3][4]. An ecosystem develops over time in a well competitive adaptive manner. During its development competition exists between different components both at inter-specific and intra-specific levels, and is the basic class of dynamics [5,6].
In natural frameworks, time can be considered as a continuous function for many models [7][8][9]. In population's model, a permanent arrangement of deaths and births can be seen but in numerous life situations this standard is not appropriate (e.g. fish's genesis plan). In discrete type of population, it is not appropriate to take time as standard of an unbroken function or continuous function [10]. Biological systems are the main components of ecosystem. In recent years, number of scientists examined different aspects of a biological system, for example a predator-prey model, host parasite model, food chain model, hyperchaotic system and many others [11][12][13][14][15][16][17][18][19][20][21][22][23][24]. These are all related to a discrete type of population and numerous authors analysed the stability, bifurcation, chaotic behaviour and global dynamics along with biological interpretation of obtained results [25][26][27][28][29][30][31][32][33]. In a biological system, mutualism is a beneficial relationship exists between two equivalent species. Mutualism exists in large number and this can also be modified. The Lotka-Volterra model provided an opportunity for several scientific models to analyse mutualistic relationships [34][35][36][37]. For instance, May [38] suggested the two-species continuous-time Kolmogorov model represented by the following system of differential equations: where x(t) and y(t) denote the densities of species, and the parameters r 1 , r 2 , α 1 , α 2 , β 1 and β 2 are the positive numbers. It is important here to mention that the discrete-time models governed by difference equations are more appropriate than the continuous ones in the case where populations have non-overlapping generations, and discrete models can also provide efficient computational results for numerical simulations. So, in this study we will explore the bifurcation analysis and chaos of the discrete model corresponding to (1). By applying the forward Euler scheme, (1) becomes of the following form: The subsequent section purely dedicated for the dynamical characteristics at fixed points of the discrete Kolmogorov model (2). In Section 3, periodic points of period-1, 2, 3, . . . , n of the discrete Kolmogorov model (2) are explored, whereas comprehensive analysis of the bifurcation at fixed points 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 verify obtained results. The summary of present study is given in Section 7.

Dynamical characteristics at fixed points of discrete Kolmogorov model (2)
The present section is purely devoted for the exploration of dynamical characteristics at equilibrium points of the discrete Kolmogorov model (2) in R 2 + = {(x, y) : x, y ≥ 0}. For this, first we explore the existence of equilibrium points along with variational matrix.

Exploration of fixed points along with variational matrix
The summarized results regarding the exploration of fixed points can be stated as a following Lemma. (2) has at most four fixed points. More specifically,

Exploration of topological classifications at fixed points P, Q and R
The local dynamical characteristics with different topological classifications at fixed points P, Q and R of the discrete Kolmogorov model (2) are summarized in Table 1.

Exploration of topological classifications at fixed point S
The variational matrix at fixed point S is with corresponding characteristic equation is of the form Finally, roots of (9) become where Since > 0, therefore it is important here to note that fixed point S is never stable focus, unstable focus and non-hyperbolic. So we will summarize the behaviour of the discrete Kolmogorov model (2) at S in Table 2.
where f 1 and f 2 are defined in (7). Now after some computation from (13), one gets . . .

Bifurcation analysis of the discrete-time Kolmogorov model (2) at Q, R and S
The full possible bifurcation analysis of the discrete-time Kolmogorov model (2) at Q, R and S by bifurcation theory [40,41] is given in this section. Our investigation in this section reveals that at boundary fixed points Q, R there exist no flip bifurcation but flip bifurcation occurs at interior fixed point S and no other bifurcation occurs at it. So, for the completeness of this section first we will give definition of flip bifurcation as follows: Definition 4.1: Bifurcation related to the existence of λ 1 = −1 is known as flip bifurcation.

Bifurcation analysis at Q
From obtained results which are depicted in Table 1, (2) may undergo flip bifurcation if involved parameters (h, α 1 , α 2 , r 1 , r 2 , β 1 , β 2 ) are located in the following set: The following theorem shows that at Q no flip bifurca-
Proof: It is noted that discrete Kolmogorov model (2) is invariant with respect to x = 0 and hence for exploring mentioned bifurcation one restrict it to the line x = 0, where (2) takes the form Now from (19), one denote the map From (20), the computation yields and where h * = 2 r 2 and y * = β 2 . From (23), one can conclude that discrete Kolmogorov model (2) cannot undergo flip bifurcation if (h, α 1 , α 2 , r 1 , r 2 , β 1 , β 2 ) ∈ F| Q .

Bifurcation analysis at R
From summarized results, which are depicted in Table 1, (2) may undergo flip bifurcation if involved parameters (h, α 1 , α 2 , r 1 , r 2 , β 1 , β 2 ) are located in the following set: But by computation, the following theorem shows that it cannot occur if (h, α 1 , α 2 , r 1 , Proof: Since discrete Kolmogorov model (2) is invariant with respect to y = 0 and hence for exploring the flip bifurcation one restrict it to the line y = 0, where (2) takes the form Now from (25), one denote the map From (26), the computation yields and where h * = 2 r 1 and y * = β 1 . From (29), one can conclude that discrete Kolmogorov model (2) cannot undergo flip bifurcation if (h, α 1 , α 2 , r 1 , r 2 , β 1 , β 2 ) ∈ F| R .
Proof: From Table 2, it is recalled that S is non- . Moreover at parametric condition h = which gives the existence of flip bifurcation at S by choosing h as a bifurcation parameter. So if h varies in a neighbourhood of h * then model (2) takes the following form: Now using the following transformation in order to transform S into P = O(0, 0) In view of (32), (31) takes the following form: where Now using the following translation: Hereafter for the map (36), centre manifold M c P about P is explored in a neighbourhood of , and therefore M c P can be expressed as a following expression: The computation yields h 0 = 0, , Finally the map (36) is restricted to M c P as follows: where In order to ensure flip bifurcation for map (40), the discriminatory quantities should be non-zero [40,41] After calculating, one gets where and Now in view of (45) and (46), from (44) if one gets Q 2 = 0 as (h, α 1 , α 2 , r 1 , r 2 , β 1 , β 2 ) ∈ F| S then discrete Kolmogorov model (2) undergo the flip bifurcation. Additionally period-2 points from S are stable (respectively unstable) if Q 2 > 0 (respectively Q 2 < 0).

Chaos control
This section is purely devoted for the exploration of chaos control in the sense of state feedback control method [10,32,[42][43][44][45]. For the completeness of this section, first we will give the definition of marginal stability.

Numerical simulations
We present numerical simulations to validate the obtained results in this section. For this, first we will give the definition of Lyapunov exponent.
Further state feedback control method is utilized to stabilize chaos existing in discrete Kolmogorov model (2). Finally, corresponding theoretical results have been verified numerically. This research can provide a framework for theoretical basis and help for the research in different aspects of biology specifically in the field of ecology.