Transcritical Bifurcation and Flip Bifurcation of a New Discrete Ratio-Dependent Predator-Prey System

After a discrete two-species predator-prey system with ratio-dependent functional response is topologically and equivalently reduced, some new dynamical properties for the new discrete system are formulated. The one is for the existence and local stability for all equilibria of this new system. Although the corresponding results for the equilibrium E3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_3$$\end{document} have been presented in a known literature, our results are more complete. The other is, what’s more important and difficult, to derive some sufficient conditions for the transcritical bifurcation and period-doubling bifurcation of this system at the equilibria E1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}, E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_2$$\end{document} and E3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_3$$\end{document} to occur, which are completely new. Numerical simulations are performed to not only illustrate the theoretical results obtained but also find new dynamics—chaos occuring. Our results sufficiently display that this system is very sensitive to its parameters. Namely, the perturbations of different parameters in this system will produce different bifurcations.


Introduction and Preliminaries
In recent years, the discrete dynamical models described by the difference equations have been extensively studied. Compared with continuous dynamical systems, discrete dynamical systems possess some obvious advantages as follows: to reduce the system (1.1) to the following dimensionless form x k+1 = x k e β(1−x m k −μ 2 y n k ) , y k+1 = y k e α(1−μ 1 x m k −y n k ) , (1.2) where μ 1 = ξ 1 K 1 , μ 2 = ξ 2 K 2 . Obviously, the system (1.2) is topologically equivalent to the system (1.1) but looks more simpler than the system (1.1). Anyway, considering there are still so many parameters in the system (1.2), the calculations for considering it will be very complicated, so we study its case of β = 1 in this paper, i.e.
x k+1 = x k e 1−x m k −μ 2 y n k , y k+1 = y k e α(1−μ 1 x m k −y n k ) , (1.3) where α, μ 1 , μ 2 > 0, m, n ∈ N + . The main purpose of this paper is to consider the bifurcation problems of the new discrete system (1.3) at its equilibria E 1 , E 2 and E 3 , namely, for their transcritical bifurcation and flip bifurcation, which are more important and difficult. Moreover, these problems have not been considered in any literature yet, hence, our results for these bifurcation problems are completely new. Besides, the stability for all of its equilibrium points is also considered, where our results for the fixed point E 3 is more complete than the known ones in the previous paper [1], and the results for the equilibrium points E 0 , E 1 and E 2 have not been studied in any known literature yet, and so are also totally new.
The paper is organized as follows. In Sect. 1, as preliminaries, a key lemma is given. In Sect. 2, we study the existence and local stability of all nonnegative fixed points of the system (1.3). In Sect. 3, we derive the sufficient conditions for the transcritical bifurcation and period-doubling bifurcation of the system (1.3) to occur. In Sect. 4, numerical simulations are performed to illustrate the obtained theoretical results and reveal some new dynamical properties of the system (1.3).
Before analyzing the fixed points of the system (1.3), we introduce a key lemma ( [18,19]), which will be useful in the sequel.

Existence and Stability of Fixed Points
In this section, we first consider the existence of fixed points and then analyze the local stability of each fixed point of the system (1.3). The fixed points of the system (1.3) satisfy Considering the biological meanings of the system (1.3), one only takes into account its nonnegative fixed points. Thereout, one can easily derive that the system (1.3) has and only has four nonnegative fixed points E 0 = (0, 0), E 1 = (1, 0), E 2 = (0, 1) and E 3 = (x 0 , y 0 ) for 0 < μ 1 , μ 2 < 1 or μ 1 , μ 2 > 1, where The Jacobian matrix of the system (1.3) at anyone fixed point E(x, y) takes the following form The characteristic polynomial of Jacobian matrix J (E) is Now, we derive the results of the stability of fixed points E 0 , E 1 , E 2 and E 3 in the following Theorems 2.1-2.4 respectively.

Theorem 2.2
The following statements about the fixed point E 1 = (1, 0) of the system (1.3) are true.
The proofs for Theorems 2.1-2.2 are easy and omitted here.

Theorem 2.3
The following results are valid for the fixed point E 2 .
Proof The Jacobian matrix of the system (1.3) at E 3 can be simplified as The characteristic polynomial of Jacobian matrix J (E 3 ) reads as By calculating we find Hence, it is impossible for F(−1) > 0 and q < 1 to hold simultaneously for m(1 − μ 2 ) > 2. Therefore, if one of the conditions 1.(a), (b) and (c) is met, then F(−1) > 0, q < 1. According to Lemma 1.1 (i.1), E 3 is a sink.

Bifurcation Analysis
In this section, we are in a position to use the center manifold theorem and bifurcation theory to analyze in detail the local bifurcation problems of the system (1.3) at the fixed points E 1 , E 2 and E 3 , respectively. For related work, refer to [20][21][22][23][24][25].

For Fixed Point E 1 = (1, 0)
Obviously, regardless of what values the parameters take in the system (1.3), the fixed point E 1 always exists. One can see from Theorem 2.2 that the fixed point E 1 is nonhyperbolic when m = 2 or μ 1 = 1. As soon as the parameter m or μ 1 goes through corresponding critical values, the dimensional numbers for the stable manifold and the unstable manifold of the fixed point E 1 vary. So, a bifurcation probably occurs for each case. By some computations, we find that the Taylor expansions of the system (1.3) at the fixed point E 1 for n = 1 and n ≥ 2 are different, hence, we respectively consider the two cases when discussing its bifurcation problem. Accordingly, the considered parameter space is divided into the following four cases: Case I. m = 2, n ∈ N + , μ 1 = 1, μ 2 > 0, α > 0; Case II. m = 2, n ∈ N + , μ 1 = 1, μ 2 > 0, α > 0; Case III. m = 2, n ∈ N + , μ 1 = 1, μ 2 > 0, α > 0; For the sake of convenience of statement in the following, now define First consider the Case I where the parameters (m, n, μ 1 , μ 2 , α) ∈ 1 . Then, 1 is further divided into 1 1 ∪ 2 1 , where When the parameters (m, n, μ 1 , μ 2 , α) ∈ 2 1 , although λ 1 = 1 and |λ 2 | = 1, some computations display that the conditions for a bifurcation to occur do not hold, so, we only discuss the case of (m, n, μ 1 , μ 2 , α) ∈ 1 1 . Then the following result is obtained. Proof In order to show the detailed process, we proceed according to the following steps.
Giving a small perturbation μ * 1 of the parameter μ 1 around μ 0 1 , i.e., μ * Expanding (3.3) as a Taylor series at (u k , v k , μ * k ) = (0, 0, 0) up to terms of order 3 produces the following model One can derive the three eigenvalues of the matrix A = Taking transformation Assume on the center manifold It is easy to derive So, the system (3.5) restricted to the center manifold is given by It is not difficult to calculate According to (21.1.43)-(21.1.46) in [29, p507], all conditions hold for a transcritical bifurcation to occur, hence, the system (1.3) undergoes a transcritical bifurcation at the fixed point E 1 . The proof is over.
Next one studies Case II where the parameters (m, n, μ 1 , μ 2 , α) ∈ 2 . The following result can be derived. Proof One adopts the following steps to show the detailed process. In order to verify this conclusion, let us divide 2 into 1 2 ∪ 2 2 , where First, we consider the parameters (m, n, Giving a small perturbation m * of the parameter m, i.e., m * = m − m 0 , with 0 < |m * | 1, the system (3.6) is perturbed into (3.8) Expand the system (3.8) as a Taylor series at Through the fowllowing transformation Suppose on this center manifold It is easy to derive m 20 = m 11 = m 02 = 0. Therefore, the center manifold equation is given by Thereout, one has Therefore, the following results are derived According to (21.2.17)-(21.2.22) in [18, p516], all conditions are satisfied for the occurrence of period-doubling bifurcation. Besides, So, the period-two orbit bifurcated from E 1 lies on the right of m 0 = 2. Of course, one can also compute the following two quantities, which respectively are the transversal condition and non-degenerate condition for judging the occurrence and stability of a period-doubling bifurcation (see [26][27][28][29]) α 2 > 0 means that the period-two orbit bifurcated from E 1 is stable. Next, we consider the parameters (m, n, μ 1 , μ 2 , α) ∈ 2 2 . Take X k = x k − 1, Y k = y k − 0, then the system (1.3) is reduced to Giving a small perturbation m * of the parameter m around the critical value m 0 with 0 < |m * | 1, the system (3.12) reads (3.13) (3.14) Taylor expanding (3.14) at Assume on the center manifold where ρ 8 = X 2 k + δ 2 k . It is easy to derive m 20 = m 11 = m 02 = 0. So, the system (3.15) restricted to the center manifold takes Noticing that (3.16) and (3.11) are the same, the results of judging the occurrence and stability of the flip bifurcation also are the same, so the other steps are omitted here. The proof is complete.
Finally one consider Case IV: m = 2, n ∈ N + , μ 1 = 1, μ 2 > 0, α > 0. At this time, the two eigenvalues of the linearized matrix of the system (1.3) at the fixed point E 1 satisfy |λ 1,2 | = 1, so, no bifurcations occur at this case. E 2 = (0, 1) From Theorem 2.3, one sees that the dimensional numbers for the stable manifold and the unstable manifold of the fixed point E 2 change when the corresponding parameter nα or μ 2 goes through the critical value 2 or 1. Then a bifurcation will take place at each case. Thereout, there are the following four cases to be consider:

For Fixed Point
Through calculation, we find that the Taylor expansions of the system (1.3) at the fixed point E 2 for m = 1 and m ≥ 2 are different, so we study its bifurcation problems according to the following two different situations: m = 1 and m ≥ 2.
For convenience of statement, now define First consider Case I where the parameters (m, n, μ 1 , μ 2 , α) ∈ 3 . 3 can be further divided into 1 3 ∪ 2 3 , where When the parameters (m, n, μ 1 , μ 2 , α) ∈ 2 3 , by analysis, one finds that the conditions for the birth of a bifurcation do not hold although |λ 1 | = 1 and λ 2 = 1 at this time, which means that no bifurcations occur. Therefore, we only formulate this case of the parameters (m, n, μ 1 , μ 2 , α) ∈ 1 3 when considering the bifurcation problem of the system (1.3) at the fixed point E 2 for the parameters (m, n, μ 1 , μ 2 , α) ∈ 3 . One has the following result.
Give a small perturbation μ * 2 of the parameter μ 2 around the critical value μ 0 2 to get the perturbation model of the system (3.17) as follows: Expanding (3.19) as a Taylor series at (u k , v k , μ * k ) = (0, 0, 0) to the third order produces the following model: The transformation
Because the center manifold equation of the system (3.29) and that of the system (3.25) are the same, the results for judging the occurrence of period doubling bifurcation are also the same, omitted here. The proof is complete.

For Fixed Point
In addition, the resonance 1:2 bifurcation problem of the system (1.3) at the fixed point E 3 , namely, in the case of λ 1 = λ 2 = −1, is relatively complicate and left as the future work.
Noticing when m > 2 1−μ 2 or m < 2(1−μ 1 μ 2 ) 1−μ 2 with α = α 0 , Theorem 2.4 with Lemma 1.2 (i.2) indicates λ 1 = −1 and λ 2 = −1. At this time we show that the system (1.3) at the fixed point E 3 may undergo a period-doubling bifurcation for the parameters (m, n, μ 1 , μ 2 , α) in the space In order to show the process clearly, we carry out the following steps. Take the changes of variables u k = x k − x 0 and v k = y k − y 0 to transform the fixed point E 3 = (x 0 , y 0 ) to the origin O(0, 0) and the system (1.3) into Choose the parameter α as bifurcation parameter. Giving a small perturbation α * of the parameter α around the critical value α 0 , the perturbation of the system (3.30) reads (3.32) Expanding the system (3.32) as a Taylor series at (u k , v k , α * k ) = (0, 0, 0) up to the third order deduces the following map where Suppose on this center manifold So, the system (3.34) restricted to the center manifold is given by Next we calculate the following quantities to judge the occurrence of a period-doubling bifurcation. One can derive f 2 Therefore, the period-doubling orbit bifurcated from E 3 lies on the right (left) side of α 0 for β 3 > (<)0.
Summarizing the above analysis, we have the following consequences.

Numerical Simulation
In this section, we present the bifurcation diagrams, phase portraits, and maximum Lyapunov exponents for the system (1.3) to illustrate the above theoretical analysis and show some new interesting complex dynamical behaviors by using numerical simulations.

For the Bifurcation of Fixed Point E 2 = (0, 1)
The occurrence of bifurcation for the system (1.3) at the fixed point E 2 = (0, 1) can be discussed from the following cases:  Fig. 3a and b display the dynamical behavior of the system (1.3) when α changes from 0.3 to 1. We see that when α = α 0 = 0.5 the flip bifurcation appears for the system (1.3). With the increase of α beginning from 0.5, the system (1.3) undergoes a series of perioddoubling bifurcations. In general, the positive maximal Lyapunov exponents can be considered as an evidence for chaos occurring. Figure 3b displays the birth of chaotic phenomenon in the system.
For Case (4), Fig. 4a shows the bifurcation diagram in the (α, x)-plane of the system (1.3), from which we see that the fixed point E 2 is stable for α < 0.4, loses its stability for α > 0.4 and a period-doubling bifurcation takes place for α = 0.4. Moreover, Fig. 4b also tells us that there is a chaotic set to emerge with the increase of α.  The bifurcation diagram is plotted in Fig. 5a, from which we can see that the fixed point E 3 is stable for 0.1 < α < 0.4, and loses its stability on the right of the critical value α = α 0 = 0.4. Figure 5b depicts the corresponding maximal Lyapunov exponents, from which one can easily see that the maximal Lyapunov exponents are positive when the parameter α ∈ (0.68, 0.8), indicating the birth of a chaos.

Discussion and Conclusion
We reconsider in this paper the dynamical behaviors of a known two-species discrete predatorprey model with ratio-dependent functional response under its equivalent and simpler form. According to given parametric conditions, the new discrete system, i.e., the system (1.3) in this paper, has four nonnegative fixed points E 0 = (0, 0), E 1 = (1, 0), E 2 = (0, 1) and Firstly, we analyze the stability of four fixed points. which not only supplements the stability of the equilibria E 0 , E 1 and E 2 , but also gives more complete results for the stability of the fixed point E 3 than the ones in the previous paper. Then we present some sufficient conditions for the emergences of its transcritical bifurcation and period-doubling bifurcation, which have not been considered in the previous paper.
Considering that the Taylor expansions of the system (1.3) at the fixed points E 1 and E 2 are different for the cases n = 1 and n ≥ 2 and the cases m = 1 and m ≥ 2, we respectively divide them into two different cases to discuss its dynamical behaviors, and obtain complete results.
Finally, numerical simulations illustrate the system (1.3) to have new dynamics -chaos may occur, which is waiting for further theoretical studies. Our careful analysis to the parameters in the system (1.3) displays that this new system may produce a flip-fold bifurcation and 1:2 resonance bifurcation, which are worthy further investigations.