Abstract

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in . 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.

1. 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? The 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 and the predator population . Then, 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: for positive constant . Next, Volterra reasoned the number of contacts per unit time between selachians (predators) and food fish (prey) is for positive constant . Therefore, , and so Volterra reasoned that predators have a characteristic pace of lessening relative to their present number and that they likewise increment at a rate relative to their present number and their nourishment supply , and hence, one has [1–5]

Now include the effects of fishing in (1). It is observed that fishing decreases the population of food fish at a rate and decreases the population of selachians at a rate , and the parameter 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 discrete-time systems, and in recent years, many authors gave great contribution towards discrete models [6–15]. The 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, 16–21]. 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

This 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. The 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. The conclusion of the paper is given in Section 8.

2. 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 is the fixed point of (4):

It is noted that , satisfying (5) obviously, and so is a trivial fixed point of (4). For a positive fixed point, system (5) reduces into the following form:

From (6), one gets and . Therefore, (4) has the unique positive fixed point: if .

Hereafter, we will study the topological classifications at and of the model (4) by method of linearization. So, the Jacobian matrix evaluated at becomes as follows:

Now, the auxiliary equation of at is as follows:where

Now, in the rest of the section, we will give topological classification at equilibria: and as follows:

2.1. Topological Classifications at

It is noted that, at , (7) takes the following form:whose characteristic roots are and . So based on stability theory, one can conclude the topological classifications at as the following result.

Proposition 1. For , the following classifications hold:(I)If then is a sink.(II)If then is a source.(III)If then is a saddle.(IV)If or then is nonhyperbolic.

2.2. Topological Classifications at

Now, we will give topological classifications at for the considered system. After some manipulations, at (7) takes the following form:

Furthermore, at , (8) becomeswhere

Finally, from (17) one getswhere

Hereafter, at equilibrium: , we will summarize the topological classifications by allowing sign of discriminant quantity, i.e., , as a following proposition.

Proposition 2. If , then for , the following classifications hold:(I) is never stable focus.(II)If then is unstable focus.(III)If then is nonhyperbolic.

Proposition 3. If , then for , the following classifications hold:(I)If with then is a stable node.(II)If along with (24) holds, then is an unstable node.(III)If then is nonhyperbolic.

3. Existence of Possible Bifurcations at Fixed Points: and

In view of obtained results in Section 2 regarding topological classifications at equilibria and , we will study the existence of possible bifurcations in this section, as follows:(I)Recall that, at , has two characteristic roots in which , but . This implies that (4) may undergo the flip bifurcation if locate in the set: or Again under the nonhyperbolic condition, which is depicted in (15), one can obtain that , but . This implies that model (4) may undergo the flip bifurcation if locate in the set:(II)Under the hypothesis of Proposition 2 and obtained nonhyperbolic condition, which is depicted in (22), one gets . Hence, model (4) may undergo a Neimark–Sacker bifurcation if parameters locate in the set:(III)Under the hypothesis of Proposition 3 and obtained nonhyperbolic condition, which is depicted in (26), one gets , but . So model (4) may undergo the flip bifurcation if locate in the set:

Note: for case I, it is easy to see that model does not undergo flip bifurcation if , and hence, is degenerated with a higher codimension.

In the subsequent section, we will present comprehensive N-S and flip bifurcations analysis when parameters, respectively, .

4. Comprehensive Bifurcation Analysis at

4.1. N-S Bifurcation at

In the subsequent section, we will discuss N-S bifurcation of model (4) at if , and hence, the result can be stated as the following theorem.

Theorem 1. If , then at , model (4) does not undergo N-S bifurcation.

Proof. Since and hence if bifurcation parameter varies in a small neighborhood of , that is, , where , then (4) becomes as follows:with if . The auxiliary equation of at becomeswhereFinally, from (32), one getsFrom (34), we obtainHereafter by using the following transformationtransform in origin. In view of (36), from (31), one getsNow, we will study normal form of (37) if . After Taylor series expansion about , one getsIn order to transform liner part of (38) into canonical form, we use the following transformation that can be easily constructed by computation:In view of (39) and (38),whereFrom (41), one getsIn order to undergo the said bifurcation, the following quantity should be nonzero [22–29]:whereUtilizing (42) in (44), one getsUsing (45) in (43), one gets . Finally, the model considered undergoing N-S bifurcation requires that if which corresponds to . However, which contradicts to the fact that , and hence, eigenvalues of fixed point lay in the interaction of the unit circle with the coordinate axes when . Therefore, model (4) does not undergo N-S bifurcation.