Discrete-Time Predator-Prey Model with Bifurcations and Chaos

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-preymodel have been explored inR+. It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.


Introduction
In the mid-1920s, an Italian Biologist U. D'Ancona studied the population variations in different species of fish that were interacting with one another. In his investigation, he found some percentage data of total catch of different fish species during the World War I which was brought into various Mediterranean ports. In precise, the data determine the percentage (%) of total catch of selachians, which are not adorable as food fish, and during the years 1914-1923, the percentage data for the port of Fiume, Italy, is given in Table 1.
D'Ancona was surprised by a very large increase in the percentage of selachians during the World War I. He reasoned the increase in the percentage of selachians was because of decline in fishing during this period. But how does the intensity of fishing affect the fish populations? e answer was obviously the struggle for existence between competing species, which was of great concern to D'Ancona and also to the fishing industry. Naturally, selachians depend on food fish for their survival as selachians are predators and food fish are prey. D'Ancona thought that this accounted for the large increase in number of selachians during the war period. Since the level of fishing was less, there were more food fish available to selachians, which therefore increased rapidly. D'Ancona just shows the increase in number of selachians when fishing's level is reduced. D'Ancona did not explain why a declining level of fishing is more helpful to predators than to their prey. Later, paying all possible biological clarifications to this phenomenon, D'Ancona twisted to his coworker, the well-known mathematician Vito Volterra. With hope, Volterra would come up with a mathematical model of the growth of the selachians and food fish, their prey, and his model would provide the answer to D'Ancona's question. Volterra started his analysis on this problem by separating all the fish into the prey population x(t) and the predator population y(t). en, he reasoned that the food fish do not compete very fast among themselves for their food supply since this is very plentiful, and the density of fish population is not very much. Hence, in the absence of predators, their prey would grow according to the Malthusian law of population growth: (dx/dt) � rx for positive constant r. Next, Volterra reasoned the number of contacts per unit time between selachians (predators) and food fish (prey) is bxy for positive constant b. erefore, (dx/dt) � rx − bxy, and so Volterra reasoned that predators have a characteristic pace of lessening − dy relative to their present number and that they likewise increment at a rate cxy relative to their present number y and their nourishment supply x, and hence, one has [1][2][3][4][5] Now include the effects of fishing in (1). It is observed that fishing decreases the population of food fish at a rate εx(t) and decreases the population of selachians at a rate εy(t), and the parameter ϵ reflects the intensity of fishing. So, by the effect of fishing, the continuous-time model which is depicted in (1) becomes as follows [1]: It is important here to note that generally many biological models are directed by continuous as well as discretetime systems, and in recent years, many authors gave great contribution towards discrete models [6][7][8][9][10][11][12][13][14][15]. e reasons are that, for nonoverlapping generation, discrete models are much convincing than continuous models, and moreover, these models provide more effective computational results for numerical simulations [6,[16][17][18][19][20][21]. Due to scientific computation, in the present study, our aim is to explore the qualitative behavior of the discrete-time model corresponding to (2). It is predicted that the discrete model is dynamically consistent with the continuous-time model. It is also mentioned in [15] that if population has a nonoverlapping generation, as stated above, it is essential to write a discrete system that corresponds to model (2). So, applying the forward Euler scheme, (2) becomes as follows: After some manipulations, from (3), one gets is paper is organized as follows: Section 2 is about the topological classifications at fixed points of the model (4), whereas existence of possible bifurcations at respective fixed points is given in Section 3. e comprehensive bifurcation analysis at a positive fixed point is investigated in Section 4. In Section 5, some simulations are performed to demonstrate obtained theoretical results, and this is also about the study of fractal demission that characterized the strange attractors. In Section 6, we investigated the chaos control by the feedback control method, whereas existence of periodic points is studied in Section 7. e conclusion of the paper is given in Section 8.

Topological Classifications at Fixed Points
We will study the topological classifications at fixed points of the model (4) in this section. In order to determine fixed points, one need to solve the following system, where (x, y) is the fixed point of (4): It is noted that (x, y) � (0, 0), satisfying (5) obviously, and so E trival � (0, 0) is a trivial fixed point of (4). For a positive fixed point, system (5) reduces into the following form: From (6), one gets x � ((d + ε)/c) and y � ((r − ε)/b). erefore, (4) has the unique positive fixed point: Hereafter, we will study the topological classifications at E trival and E positive of the model (4) by method of linearization. So, the Jacobian matrix Ω E (x,y) evaluated at E (x,y) becomes as follows: Now, the auxiliary equation of Ω E (x,y) at E (x,y) is as follows: where Now, in the rest of the section, we will give topological classification at equilibria: E trival and E positive as follows:

Topological Classifications at E trival .
It is noted that, at E trival , (7) takes the following form: whose characteristic roots are ζ 1 � 1 + h(r − ε) and . So based on stability theory, one can conclude the topological classifications at E trival as the following result.

Proposition 1.
For E trival , the following classifications hold: then E trival is a sink.
then E trival is a source.
then E trival is a saddle.
then E trival is nonhyperbolic.

Topological
Classifications at E positive . Now, we will give topological classifications at E positive for the considered system. After some manipulations, at E positive , (7) takes the following form: Furthermore, at E positive , (8) becomes where T E positive � 2, Finally, from (17) one gets where Hereafter, at equilibrium: E positive , we will summarize the topological classifications by allowing sign of discriminant quantity, i.e., Disc � 4h 2 (ε − r)(d + ε) < 0 (respectively ≥ 0), as a following proposition.
then E positive is nonhyperbolic.
then for E positive , the following classifications hold: with Mathematical Problems in Engineering then E positive is a stable node.
along with (24) holds, then E positive is an unstable node.
then E positive is nonhyperbolic.

Existence of Possible Bifurcations at Fixed Points: E trival and E positive
In view of obtained results in Section 2 regarding topological classifications at equilibria E trival and E positive , we will study the existence of possible bifurcations in this section, as follows: (I) Recall that, at E trival , Ω E trival has two characteristic roots in which ζ 1 (14) � − 1, but or Again under the nonhyperbolic condition, which is depicted in (15), one can obtain that (II) Under the hypothesis of Proposition 2 and obtained nonhyperbolic condition, which is depicted in (22), (III) Under the hypothesis of Proposition 3 and obtained nonhyperbolic condition, which is depicted in (26), . (30) Note: for case I, it is easy to see that model does not undergo flip bifurcation if (b, c, d, h, r, ε) ∈ FB|E trival , and hence, E trival is degenerated with a higher codimension.
In the subsequent section, we will present comprehensive N-S and flip bifurcations analysis when parameters, respectively,

Comprehensive Bifurcation
Analysis at E positive

N-S Bifurcation at E positive .
In the subsequent section, we will discuss N-S bifurcation of model (4) at E positive if (b, c, d, h, r, ε) ∈ HB| E positive , and hence, the result can be stated as the following theorem.

Flip Bifurcation at E positive .
In the subsequent section, we will discuss flip bifurcation of model (4) at E positive if (b, c, d, h, r, ε) ∈ FB| E positive , and hence, the result can be stated as the following theorem.

Theorem 2. If (b, c, d, h, r, ε) ∈ FB| E positive , then at E positive , model (4) undergoes the flip bifurcation if p varies in a small neighborhood of the origin.
Proof. It is noted that if r in a small nbhd of r * , i.e, r � r * + p, where p ≪ 1, then model (4) becomes Now, one can transform E positive (p) in origin by using the following transformations: From (47), (46) takes the following form: By Taylor series about (Φ t , Ψ t ) � (0, 0), one gets Now by using following transformation 6 Mathematical Problems in Engineering linear part of (49) transforms into canonical form. In view of (50), (49) becomes where (52) Now, for (51), center manifold M c E trival at E trival is explored in a small nbhd of p.
erefore, M c E trival can be expressed as follows: After some manipulations, one gets (54) Finally, the map (51) restricting to M c E trival is with In order for the map (51) to undergo flip bifurcation, the following discriminatory quantities are required to be nonzero [28,29]: On computation, one gets ∧

Numerical Simulations
Here, some simulations will be presented in order to verify the obtained results in Sections 2 and 4. For instance, choose ε � 1.9, h � 1, then from (24), one gets d > 0.2052631578947369. Furthermore, if d � 0.5 > 0.2052631578947369, then from (23), one gets r > 0.23333333333333317. But for the existence of unique positive fixed point, it is required that r > ε � 1.9. So, for these numerical values and the condition on parameter r where unique positive fixed point is stable focus is r > max 0.23333333333333317, 1.9 { }. Hence, if one choose parametric value r � 2.1 which satisfies r > max 0.23333333333333317, 1.9 { } and b � 0.4, c � 0.5, then it is clear from Figure 1(a) that E positive � (4.8, 0.5) is stable focus. Similarly, Figure 1(b) also shows that E positive of (4) is stable focus. Now, from (25), one can say that E positive of (4) is unstable focus if r < 0.23333333333333317, and hence, flip bifurcation takes place if r < 0.23333333333333317. For instance, if r � 0.09 < 0.23333333333333317, then � 0.026666666666666672 > 0 which implies that stable period-2 points bifurcating from E positive . Hence, this simulation agrees with theoretical results in Section 4. e 2 D bifurcation diagrams with corresponding maximum Lyapunov exponents for said parametric values are plotted in Figure 2. Finally, the trajectories associated with Figures 2(a) and 2(b) are also plotted in Figures 3(a)-3(f ) that indicates (4) exhibits complex dynamics having orbits of period-4, period-6, period-8, period-12, period-17, and period-18.

Chaos Control
By a state feedback control method, we will stabilize chaotic orbits at an unstable fixed point motivated from existing literature [32,33]. By adding control force C t to model (4), then with g 1 and g 2 being feedback gains. Now, Ω C | E positive at E positive for the controlled system (61) is whose auxiliary equation is with If ζ 1,2 are roots of (63), then Here, it is noted that solution of the following equations gives the lines of marginal stability, which further grantee the fact that |ζ 1,2 | < 1. From (66) and (69), one gets From (65), (66), and (67), one gets From (65) us, L 1 , L 2 , and L 3 in (g 1 , g 2 )-plane determines the triangular region that gives |ζ 1,2 | < 1 (see Figure 5(a)), whereas Figures 5(b) and 5(c) tell about E positive that the chaotic trajectories are stabilized. For the qualitative behavior of continuous dynamical systems, we refer the interested readers to [34][35][36] and references cited therein.

Periodic Points of Model (4)
Existence of periodic points at E trival and E positive is investigated in this section.

Theorem 3. E trival is a periodic point having prime period-1.
Proof. From (4), we have where From (73) along with (74), one gets the required results: Theorem 4. E positive is a periodic point having prime period-1.
Proof. From (77), one gets the required statement:

Conclusion
In this paper, we have investigated the topological classifications at fixed points, bifurcation analysis, and chaos in a model, which is depicted in (4). It is examined that ∀d, ε, h, c, b, r model has a trivial fixed point: E trival and the unique positive fixed point: E positive if r > ε. By existing linear theory of stability, we have studied the topological classifications at E trival and E positive , and conclusion is reported in Table 2. Furthermore, we have examined the existence of possible bifurcations at E trival and E positive and proved that model (4) does not undergo flip bifurcation if (b, c, d, h, r, ε) ∈ FB| E trival , and hence, E trival is degenerated  + ε)); unstable node if r < (εh 2 (d + ε) − 4/h 2 (d + ε)) Nonhyperbolic if r � (εh 2 (d + ε) − 4/h 2 (d + ε)) with a higher codimension. We have also examined that if (b, c, d, h, r, ε) ∈ HB| E positive , then at E positive , model (4) does not undergo N-S bifurcation. Moreover, at fixed point E positive , we have proved that model (4) undergoes the flip bifurcation if (b, c, d, h, r, ε) ∈ FB| E positive . Some numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have also computed maximum Lyapunov exponents numerically. By the feedback control method, we have stabilized chaos existing in the considered model. Finally, existence of periodic points at E trival and E positive of model (4) is explored.

Data Availability
All the data utilized in this article have been included, and the sources adopted are cited accordingly.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.