On the dynamical behavior of three species food web model
Introduction
The multi-species continuous time models consisting of interactions among three or more species may have a very complex dynamics, including cycle, quasi-periodic and even chaotic. The research studies of the last three decades have focus on the dynamical behavior in three species food chain model [1], [2], [3], [4]. However, less attention has been given to the study of food webs and their dynamical behavior. An investigation, by Gilpin 1979 [5], demonstrated that a food web model consisting of one predator and two competing preys with linear functional response exhibits chaotic behavior. He is shown the existence of chaos by looking at numerical solutions for particular set of parameter values. More complete numerical studies of the same model, including Poincare sections, were performed by Schaffer [6]. Later on, Klebanoff and Hastings [7] studied the dynamical behavior of Gilpin model through the bifurcation theory. The persistence and extinction conditions of one predator and two preys system with ratio-dependent functional response are discussed [8]. Recently, a food web model consisting of a predator and two preys with modified Holling type-II functional response has been extensively studied [9]. In this paper, the dynamical behavior and persistence of a food web model consisting of two preys and one predator system with Beddington–DeAngelis functional response is investigated analytically as well as numerically.
Section snippets
The mathematical model
The three level food web models, consisting of two competing preys (X1 and X2) and a predator (Y), can be represented mathematically by the following system of differential equations:With .
In model (1), Ri, Ki, Ei, hij (i, j = 1, 2, i ≠ j) and D are the model parameters which assuming only positive values. The prey Xi grows with
Existence and dissipativeness
Obviously, the interaction functions Gi (i = 1, 2, 3) of system (8) are continuous and have continuous partial derivatives on the state space . Therefore the solution of system (8) with non-negative initial condition exists and is unique, as the solution of system (8) initiating in the non-negative octant is bounded. Further more, the system is said to be dissipative if all population initiating in are uniformly limited by their environment [11]. Accordingly, the
Kolmogorov analyses and stability
For a biological meaningful system, system (8) has to be a Kolmogorov system. Since the Kolmogorov theorem is applicable in (2D) dynamical system only [13]. Therefore, the two subsystems (9), (10) should be qualified as the Kolmogorov system. An application of Kolmogorov theorem for both the subsystems shows that, the subsystem (9) is a Kolmogorov system under the conditionsWhile subsystem (10) is a Kolmogorov system under the conditions
The dynamical analysis and persistence
In this section, the existence of the equilibrium points of system (8) and the local stability analysis of each one are investigated. The persistence conditions of all the species are established.
At most there are seven possible non-negative equilibrium points for system (8), the existence conditions of them are given as the following:
- 1.
The equilibrium points E0 = (0, 0, 0), E1 = (1, 0, 0) and E2 = (0, 1, 0) are always exist. However, there are no equilibrium points on the y axes.
- 2.
The non-negative point
Numerical simulations
In this section, the global dynamical behavior of the system (8) is investigated numerically. Extensive numerical simulations are carried out for various values of parameters and for different sets of initial conditions. The objective is limited to examine different ranges of parametric space (biologically feasible) to explore the possibilities of dynamical behavior for the food web.
There are number of diagnostic tools to detect the qualitative behavior of the dynamical systems. We shall use in
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