Chaos in a Hybrid Food Chain Model

In this paper, chaotic and periodic dynamics in a hybrid food chain system with Holling type IV and Lotka-Volterra responses are discussed. The system is observed to be dissipative. The global stability of the equilibrium points is analyzed using Routh-Hurwitz criterion and Lyapunov direct method. Chaos phenomena is characterized by attractors and bifurcation diagram. The effect of the controlling parameter of the model is investigated theoretically and numerically.


Introduction
Nonlinear phenomena play important rules in design systems of engineering and structures. The works of Lotka [1] and Volterra [2] are the most major works in mathematical models and dynamical systems. Researchers in ecology and engineering reported the essence of chaos [3][4][5][6][7][8][9][10][11][12]. In the analysis of the mechanical model, the geometrical non-linearity, represented by the pendulum, may lead to chaotic dynamics [4,13]. Continuous three species food chain models from dissipative dynamics are accounted for to have complicated patterns, such chaos limit cycle, and stable cycle [14][15][16][17]. Two
Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2362-2368 2363 types of hybrid functional responses of a food chain model were investigated [18] and the numerical works proved the existence of chaotic dynamics. Upadhyay et al. [19] proposed and studied a tritrophic food chain model with Sokol-Howell functional response, incorporating the multiple gestation time delays. They explored the existence of various dynamical structures, such as Hopf-bifurcation, periodic solutions, and chaotic dynamics. Fayeldi et al. [20] applied Euler method to discretize a SIR epidemic model with non-monotone incidence rate. They showed analytically and numerically that the discrete model is dynamically consistent with its continuous model. Liu and Huang [21] investigated the dynamical behavior of a predator-prey system with Holling type IV functional response, in which both species are subject to harvesting. They illustrated that using the harvesting effort as a control parameter can change the behaviors of the system, which may be useful for the biological management.
The primary motivation behind this work is to examine the dynamics of a three species model. A food chain model with Lotka-Volterra and Holling type IV functional responses is proposed and studied both theoretically and numerically. The food chain model is modified to the food chain model studied in [9], so that it involves Lotka-Volterra type of functional response in the second and third levels, instead of Holling type-II functional response that was used in the previous work. The idea of such modification comes from the fact of availability of food at the second level in the environment. The second reason for such modification is that we want to reduce the intensity of nonlinearity in the system and then study the effects of such reduction on the existence of chaos.

The Mathematical Model
While is a positive root of the following equation ( ) According to Descartes's rule of signs, Eq. (6) has either one positive root (given by and two complex conjugate roots or three positive roots. Straightforward computation using Mathematica software shows that Eq. (6) has only one positive root with two complex conjugate roots. Now, the local dynamical behavior of the food chain system (1) near the above equilibrium points is investigated. It is observed that the eigenvalues of the variational matrix of system (1) near the trivial equilibrium point ( ) are determined as (7) while the eigenvalues near the axial equilibrium point ( ) are given by (8) Hence, the trivial equilibrium point is a saddle point, while the axial equilibrium point is locally asymptotically stable provided that the following condition holds (9) Moreover, the eigenvalues of system (1) near the planar equilibrium point can be written as: (13b) Therefore, according to the Routh-Hurwitz criterion [22], the locally asymptotically stable takes place. However, the region of the global stability of the positive equilibrium point is investigated in the following theorem. Theorem 1. Assume that of system (1) is locally asymptotically stable and the following condition holds . Then, the basin of attraction in the of satisfies the above conditions. Proof. . Consider the positive definite function ∫ ∫ ∫ Taking the derivative of with respect to t along the solutions of system (1), we get after some algebraic manipulation that * + * + Clearly, under the given conditions that is negative semi definite and then by using Lyapunov method, the positive equilibrium point is stable. Moreover, it is easy to verify that, by using LaSalle's invariance principle , is the only invariant set for which the solution is approached asymptotically and hence it becomes an attracting point. Therefore, is an asymptotically stable point for any initial point that belongs to the region that is satisfying the above conditions.

Numerical exploration
The global dynamics of system (1) is studied numerically using six order Runge-Kutta method. Many cases of system (1) are obtained for the following set of fixed parameters values (15) Bifurcation diagram and the typical 3D attractors of system (1) are plotted with their projections and time series in the boundary planes. The fundamental target is to investigate the possibility of chaotic behavior in system (1) by relying upon the controlling parameter and keeping different parameters of (15) fixed.  Now, in the first case when with the data set (15), it is observed that system (1) is chaotic and approaches to periodic attractor, as shown in Figure- (15), where we observe the change in the dynamics of system (1) from limit cycles attractor to stable point, as blotted in Figure-    Conclusions A food chain system with modified Holling type IV and Lotka-Volterra functional responses is proposed and discussed. The global stability of the system is investigated using Routh-Hurwitz criterion and the second method of Lyapunov. Bifurcation diagram and attractors are blotted to study the behavior of the hybrid continuous time model. The parameter controlling the dynamics of this food chain model. Moreover, replacing Holling type-II functional response in [9] by Lotka-Volterra response in system (1) above reduces the effect of nonlinearity and makes the dynamics less chaotic. In fact, the system approaches clearly a periodic behavior.