The Complex Dynamics in a Food Chain Involving Different Functional Responses

A food chain model in which the top predator growing logistically has been proposed and studied. Two types of Holling’s functional responses type IV and type II have been used in the first trophic level and second trophic level respectively, in addition to Leslie-Gower in the third level. The properties of the solution are discussed. Since the boundary dynamics are affecting the dynamical behavior of the whole dynamical system, the linearization technique is used to study the stability of the subsystem of the proposed model. The persistence conditions of the obtained subsystem of the food chain are established. Finally, the model is simulated numerically to understand the global dynamics of the food chain under study.


Introduction
In ecological modeling, many researchers and scientists supported this field of study with brand new ideas. May [1] with Hasting and Powell [2] put a new base to propose and describe the new generation of complex ecological models. The complex behaviors of various ISSN: 0067-2904 ecological models involving different factors namely, predation, switching, and competition, is the most challenging task in such studies and thus it receives good attention from many scientists [3][4][5][6][7][8][9][10]. The functional response defined by Lotka and Volterra [11] for a predatorprey model is linear and unbounded while studying the complexity in model ecosystems needs reasonable functional responses that should be nonlinear and bounded [12]. In 1959 Holling [13] used a type II functional response. Collings [14] proposed a new function and called it Holling type IV response. This response function describes a situation in which the predator's per capita rate of predation decreases at sufficiently high prey densities. Moreover, both Holling type IV [5][6][7] and Leslie-Gower [15,16] functional responses are relatively less studied in population ecology. In their experiments about the kinetics of phenol oxidation Sokol and Howell [17] suggested a simplified Holling type IV function and found that it is simpler and better than the original function of Holling type IV. In this paper, the three species food chain model proposed by Alaoui [16] is modified so that it contains three different types of functional responses.

The Mathematical Model
Consider a three-species food chain model consisting of the prey that denoted to their density at time t by x(t), the intermediate predator that denoted to their density at time t by y(t), and the top predator, which denoted to their density at time t by z(t). It is assumed that the intermediate predator preys at the lower level according to Holling type IV response, and the top predator preys upon the intermediate predator at the second level according to Holling type II and growing according to modified Leslie-Gower response. The dynamics of the above food chain model can be represented by the following ( ) with ( ) ( ) ( ) . Obviously, system (1) is continuous and have continuous partial derivatives on the positive octant *( ) ( ) ( ) ( ) +, and hence the solution of the system (1) exists and is unique. Here the positive constants ; , , , can be described as: a is the growth rate of the prey x , b represents the intraspecific competition of prey x , k w 's are the maximum values attainable by each per capita rate, measures the extent to which the environment provides protection to the prey and respectively, represent the death rate of the intermediate predator.
While is the value of at which the per capita removal rate of becomes , the constant represents the growth rate of by sexual reproduction, however represents the residual loss in population due to serve scarcity of its favorite food . Moreover, it is easy to verify that model (1) is uniformly bounded.

Analysis of the subsystem
To study the dynamical behavior of the model (1), it is important to study their subsystem in the plane. Many characteristics of the model (1) (such as persistence) depend on the dynamical behavior of their subsystem in the plane, see [17,19]. Now in the absence of the top predator , the system (1) reduces to the following subsystem.
(2) Now, it is easy to verify that the next condition shows that the system (2) is a persistence Kolmogorov system (3) In addition to the above it is observed that, the Kolomogrov model (2) has the following nonnegative equilibrium points. The equilibrium points ( ) and ( ) always exist. However, the positive equilibrium point, say ( ̅ ̅) in the ( ) of the plane can be determined by solving the first two equations of the model (2), such that (4) which gives (5a) and then it is obtain that Clearly, ̅ provided that ̅ . Therefore, there are many cases about the existence of the positive equilibrium point, these are given below.

Case 1. If
, then there is no positive roots for Eq.(4). This implies that the specialist predator goes extinct too, and the system (2) have just two equilibrium points ( ) and ( ).

Case 2. If
, then there is one positive root for Eq.(4), that is given by ̅ , and then substitute ̅ in the first equation of the system (2) gives the positive equilibrium point, say This exists provided that .
In addition to that, the system (2) has a unique positive equilibrium if ̅ ̅ , so in addition to ( ) and ( ), the model has three equilibrium points.
In this paper, it is assumed that the system (2) has at most three equilibrium points and the positive equilibrium point ( ̅ ̅) Further, the stability analysis of the Kolmogorov model (2) is carried out and according to the following Jacobian's matrices of , and ,respectively the following results are obtained: On the other hand, the positive planar equilibrium point ( ̅ ̅) is a locally asymptotically stable in the ( ) of the plane if the following condition holds while is unstable saddle if the opposite of condition (9) holds. Moreover, the global stability of ( ̅ ̅) is discussed in the following theorem. Obviously, the function ( ) be in the ( ) of the plane. Also we have that Clearly if condition (9)  Therefore Recall that, since ( ̅ ̅) is globally asymptotically stable in the ( ) of the plane. Therefore, there are no periodic orbits in this boundary plane. So to prove that is persistence function, in the sense of Gard [20], and hence model (2) is uniform persists, it is enough to show that there are no omega limit sets on the boundary planes of or equivalently the following conditions should be satisfied: ( ) , ( ) * + Note that by choosing sufficiently large value and keeping fixed at small positive value then ( ) holds. Also, due to the Kolmogorov condition (3) the inequality ( ) is satisfied for any positive value of . Therefore represents persistence function and hence system (2) is uniformly persists. Now, before go further, we have to mention that the function will be an extinction function, in the sense of Gard [18], if condition (3) violated, then the solution of model (2) approaches to an omega limit point on the boundary planes of the and hence model (2) does not persist.

Numerical exploration
The food chain model (1) is simulated numerically to study the global dynamics of it by using six order Runge-Kutta method. For the following set of fixed parameters values (10) Figure 1-Bifurcation diagram of model (1) for data set (10) that shows the successive maxima of as a function of ( ) and .
Bifurcation diagram and the typical 3D attractors of model (1) are plotted with their time series. Our target here is to investigate the behavior of model (1) depending upon the parameters and with keeping other parameters of (10) fixed. Now, the first case is by fixing and varying the value of in the range . It is observed for the value the dynamics of model (1) with data (10) is chaotic as shown in Fig.1 and Fig.2. Decreasing the value of change the behavior of model (1) to period-doubling as shown in Fig.3, and for the value the model will be stable as shown in Fig.4.
The second case by fixing and varying the value of in the range . It is observed for data (10) that model (1) behavior is chaotic as it is blotted in Fig.2 and periodic with approach to extinction as it is shown in Fig.5. Moreover, the stable case appears in model (1) for data (13) when and with approach to extinction as it is shown in Fig.6.      (1) stable point for data (10) with with approach to extinction b) Time series of Fig.6a Conclusions A food chain model with different functional responses including Holling type IV and type II in additional to Leslie-Gower is proposed and studied. In order to explain the dynamical behavior of the proposed food chain model (1) is local as well as global stability analyses are carried out for the subsystem. Persistence of the subsystem is discussed. Global stability for the food chain model (1) is analyzed numerically. According to our study above we obtained that the parameters and are controlling parameters and they are responsible about the chaotic, periodic and asymptotic stable of Leslie-Gower food chain model (1) with simplified Holling type IV functional response.