Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R2+ are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, themaximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors.


Introduction
In population dynamics, continuous-time dynamical systems, such as a Lotka-Volterra system, have been used to understand the interaction between ecological species [1][2][3][4][5][6][7][8].In recent years, discrete-time population systems also come to the fore due to the following reasons [9][10][11][12][13][14][15][16]: Firstly, discrete-time systems are more suitable than continuoustime systems to describe populations with nonoverlapping generations.Secondly, they can produce more complex and rich dynamical behaviors than continuous-time systems.Finally, we can get more accurate numerical simulation results from discrete-time systems compared with continuous-time systems.
In order to investigate the dynamical relationship between two species, predator and prey, one of important components we should pay attention to is the predator's rate of feeding on the prey, which is called functional response.In this context, Holling [17] suggested three different kinds of functional response for different kinds of species to model the phenomena of predation, which are called Holling's type I, II, and III.These functional responses are not only monotonically increasing, but also uniformly bounded functions in the first quadratic.The authors [18,19] introduced another Holling-type functional response called Holling-type IV functional response or Monod-Haldane functional response.Ivlev [20] also suggested a well-known functional response, called Ivlev functional response, in the following form; where ℎ,  are positive constants and represent the maximum rate of predation and the decrease in motivation to hunt, respectively.
Taking into account this functional response, we will consider the following predator-prey system with Ivlev functional response: where () and () are functions of time representing the population densities of the prey and the predator, respectively.

Mathematical Problems in Engineering
All constants are positive.The parameter  means the growth rate of the prey and  is the carrying capacity.The constant  denotes the death rate of the predator, and  is the rate of conversion of a consumed prey to a predator.Now, we will apply the forward Euler scheme to system (2) to obtain the following discrete-time predator-prey system with Ivlev functional response: where  > 0 is the time step size.Until now, many researchers have concerned themselves with discrete-time predator-prey systems with Holling-type functional response ( [9, 12-16, 29, 30]).Most of these works have dealt with the existence and stability of fixed points and bifurcation phenomena such as flip bifurcation, saddlenode bifurcation, or Hopf bifurcation.On the contrary, in this paper, we will take into account a discrete predator-prey system with Ivlev functional response which has not been studied well until now.
The purpose of this paper is to analyze dynamical behaviors of system (3) and to observe a flip bifurcation and Hopf bifurcation by using the center manifold theorem and bifurcation theory.Moreover, numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamics of system (3) which are difficult to explain theoretically.This paper is organized as follows.In Section 2, we investigate the existence and local stability of the fixed points of system (3).In Section 3, we show that there exist some values of parameters such that system (3) undergoes a flip bifurcation and Hopf bifurcation.In Section 4, we display the numerical simulation to provide numerical evidence for theoretical results and to exhibit numerically various dynamical behaviors including periodic doubling and chaotic phenomena.In addition, we calculate the maximum Lyapunov exponents to confirm the dynamical complexity of system (3).In Section 5, we summarize the results of this paper and compare system (3) to the discrete systems with Holling-type functional response with respect to dynamical behaviors.

The Existence and Stability of the Fixed Points
In this section, from the point of view of biology, we will focus on the dynamical behaviors of system (3) in the closed first quadrant R 2 + .In order to determine the fixed points of system (3) we have to solve the nonlinear system given by Simple calculation yields that there are at most three fixed points: (i)  0 (0, 0) is the origin; (ii)  1 (, 0) is the axial fixed point in the absence of the predator; (iii)  2 ( * ,  * ) is the interior fixed point, where exists if and only if the following condition is satisfied: We will study the stability of these fixed points by computing the variation matrix corresponding to each fixed point.The Jacobian matrix of system (3) at the state variable (, ) is given by ) .
The characteristic equation of the Jacobian matrix (, ) can be obtained as where (, ) is the trace and (, ) is the determinant of the Jacobian matrix (, ) which are defined as In order to determine the stability of the fixed points of system (3), we give the following lemma which can be easily proved by the relations between roots and coefficients of the quadratic equation (8) [9,30].Let  1 and  2 be the two roots of (8), which are called eigenvalues of the fixed point (, ).We recall some definitions of topological types for a fixed point (, ) [12,14,32]

Lemma 1. Let 𝐹(𝜆) = 𝜆
Theorem 2. For the fixed point  0 (0, 0), one has the following topological types: Proof.It is easy to get that the Jacobian matrix  at  0 is given by Hence the eigenvalues of the matrix are 1 +  and 1 − .Therefore, it is easy to show that (1)-(3) hold.
Proof.The Jacobian matrix  at  1 can be obtained as Then we have the eigenvalues of the characteristic equation ( 8) as follows: Therefore elementary calculation yields that the theorem holds.
It follows from Theorem 3 that for the fixed point  1 (, 0), if (, , , , , ) ∈  1 , where then one of the eigenvalues of the fixed point  1 (, 0) is −1 and the other is neither −1 nor 1. Thus system (3) can undergo a flip bifurcation due to the stability of the fixed point  1 (, 0) when parameters vary in the small neighborhood of  1 .In fact, if the parameters of system (3) are in  1 , then a center manifold of system (3) at  1 (, 0) is  = 0, and if system (3) restricted to this center manifold is the same as the logistic system, / = (1 − /).Thus in this case the predator becomes extinction; on the other hand, the prey undergoes the period-doubling bifurcation to chaos in the sense of Li-Yorke for suitable bifurcation parameter .(1)  2 is a sink if one of the following conditions is satisfied: (2)  2 is a source if one of the following conditions is satisfied: ( 4 Mathematical Problems in Engineering (4)  2 is nonhyperbolic if one of the following conditions is satisfied: Here ).Thus Δ 2 < 0 by the hypotheses.From elementary calculation, we obtain that Then we can easily see that It follows form Lemma 1 that one of roots of () = 0 is −1 and the other is neither −1 nor 1 if and only if Δ 2 < −2√Δ 1 and  = (−Δ 2 ± √Δ 2 2 − 4Δ 1 )/Δ 1 , and the roots of () = 0 are a pair of conjugate complex numbers with modulus 1 if and only if −2√Δ 1 < Δ 2 and  = −Δ 2 /Δ 1 .Thus, it is easy to see that the theorem holds.Now consider the following sets; From Theorem 4 (4), we see that if (, , , , , ) ∈  2 (or  3 ) then one of the eigenvalues of the Jacobian matrix ( 2 ) at the fixed point  2 ( * ,  * ) is −1 and the other is neither −1 nor 1, and if (, , , , , ) ∈   , then the eigenvalues of the matrix ( 2 ) are a pair of conjugate complex numbers with modulus 1.
In the following section, we will investigate a flip bifurcation and Hopf bifurcation of the fixed point  2 ( * ,  * ) when parameters vary in the small neighborhood of  2 (or  3 ) and   , respectively.

Bifurcation Analysis
In this section, we will discuss two kinds of bifurcation phenomena of system (3), flip bifurcation and Hopf bifurcation, according to the parameter .System (3) has no saddle-node bifurcation since the system has at most one positive fixed point  2 ( * ,  * ) as mentioned in Section 2.
First we investigate a flip (or period-doubling) bifurcation of system (3) at the fixed point  2 ( * ,  * ) when parameters vary in a small neighborhood of  2 .Similar arguments can be applied to the case  3 .
In order to transform the fixed point  2 ( * ,  * ) of system (17) into the origin, let  =  −  * and V =  −  * , and then, expanding  and  as a Taylor series at (, V,  * ) = (0, 0, 0) to the second order, system (17) where We construct an invertible matrix and using the transformation for system (18), then system (18) can be written as where It follows from the center manifold theorem [32] that there exists a center manifold   (0, 0) of system (18) at the fixed point (0, 0) in a small neighborhood of  * = 0, which can be approximately represented as follows: where Thus we can obtain the map, which is restricted to the center manifold   (0, 0): where In order for the map (26) to undergo flip bifurcation, we require that two discriminatory quantities  1 and  2 are not zero, where From the above analysis and the theorems in [32] we have the following theorem.Theorem 5.If  1 ̸ = 0 and  2 ̸ = 0, then system (3) undergoes a flip bifurcation at the fixed point  2 ( * ,  * ) when  varies in a small neighborhood of  * .Moreover, if  2 > 0 (resp. 2 < 0), then the period-2 points that bifurcate from  2 ( * ,  * ) are stable (resp.unstable).
For case (ii), it follows from Theorem 4 that system (4) has only one positive fixed point  2 ( * ,  * ) = (0.5108, 0.1041), which is sink when  <  2 = −Δ 2 /Δ 1 = 2.6444, since  In order to investigate various dynamical phenomena for this case, the bifurcation diagrams of system (3) are displayed in Figures 5(a) and 5(c).It is observed from Figure 5(a) that the fixed point (0.5108, 0.1041) of system (3) is sink for  <  2 , and an attracting invariant close curve appears, as shown in Figure 7, when the parameter  exceeds  2 .After    the invariant circle occurs, its diameter is bigger and bigger according the value of .In addition, period-doubling and period-halving phenomena are also observed in Figures 5(b 8, orbits of period 25 and 50 appear.Also, periodic-halving phenomenon is detected in Figure 9.The maximum Lyapunov exponents corresponding to Figure 5(a) are calculated and plotted in Figure 6 to display the dynamical complexity of system (3).

Conclusion and Discussion
In summary, we considered the discrete-time predator-prey system with the step size  obtained from a continuous predator-prey system with Ivlev functional response.Firstly, we found all fixed points of the system and investigated their stability via the eigenvalues of Jacobian matrix.Based on this analysis, we provided the parameter sets for which the system undergoes a flip bifurcation and Hopf bifurcation according to the value  by using the center manifold theorem and bifurcation theory.In addition, we demonstrated numerical simulations including phase portraits and bifurcation diagrams of the system in order to validate our mathematical results and show that the system has complex dynamical behaviors such as a cascade of periodic doubling, an attracting invariant circle, or chaotic attractors.Also, we computed the Lyapunov exponents to confirm the dynamical complexity of the system.
In fact, it follows from Theorem 5 that system (3) can have a flip bifurcation, which results in periodic doubling and chaotic phenomena as shown in Figures 1 and 4.Here the chaotic phenomenon indicates biologically that the biological system (3) is not stable and, particularly, if the prey is in chaotic, then the predator will eventually go extinct or go to a stable equilibrium.Also Theorem 6 guarantees the existence of a closed curve as shown in Figure 7, which means biologically that the predator coexists with the prey with at least two-frequency oscillation.
There is an interesting thing that the parameter ℎ, the maximum rate of predation, has no effect on qualitative dynamical behaviors including the stability of the fixed points and the existence of bifurcation phenomena.For instant, if we take ℎ = 0.1 and ℎ = 0.2 instead of ℎ = 0.3 and ℎ = 0.6 in cases (i) and (ii), respectively, we can obtain qualitatively similar bifurcation diagrams in Figure 10 to Figures 1 and 5.
As mentioned in the introduction, there are many works about discrete-time predator-prey systems with Holling-type functional responses.Now we will compare the dynamics of these systems with that of system (3).If a predator-prey  system has stable positive fixed points, then all species of the system could coexist, which is a biologically significant fact from the point of view of biodiversity.In this context, we will focus on positive fixed points.We summarize the dynamical behavior of systems around positive fixed points in Table 1 by letting LAS, SNB, FB, and HB stand for locally asymptotically stable, saddle-node bifurcation, flip bifurcation, and Hopf bifurcation, respectively.
Comparing the first row to the second row of Table 1, we figure out that the discrete-time predator-prey system exhibits much richer dynamics than the continuous-time system.From the second row to the fifth row, we assert that the discrete-time predator-prey systems have similar dynamical behaviors around their positive fixed points even if their functional responses are different from each other.Moreover, we conclude that the discrete-time systems have various dynamical behaviors regardless of the functional responses.
It can be inferred from the results obtained in this paper that the Ivlev functional response may be a cause of chaoslike oscillation of populations and can make the behavior of population systems more complex.Nevertheless, identifying complicated, possibly chaotic dynamics in population data has remained a major challenge in ecological studies [34,35].However, this work could be very helpful for the biologists who work with discrete-time prey-predator systems.
) and 5(d) which are the local amplification of Figures 5(a) and 5(c), respectively, for  ∈ [4.04, 4.09].In fact, if the value of  exceeds 4.048, as shown in Figure

Table 1 :
This table lists the stability and bifurcations of systems according to functional response.