On the dynamical behavior of three species food web model

doi:10.1016/j.chaos.2006.04.064

Chaos, Solitons & Fractals

Volume 34, Issue 5, December 2007, Pages 1636-1648

https://doi.org/10.1016/j.chaos.2006.04.064 Get rights and content

Abstract

In this paper, a mathematical model consisting of two preys one predator with Beddington–DeAngelis functional response is proposed and analyzed. The local stability analysis of the system is carried out. The necessary and sufficient conditions for the persistence of three species food web model are obtained. For the biologically reasonable range of parameter values, the global dynamics of the system has been investigated numerically. Number of bifurcation diagrams has been obtained; Lyapunov exponents have been computed for different attractor sets. It is observed that the model has different types of attractors including chaos.

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: $\begin{matrix} \frac{d X_{1}}{d T} = R_{1} X_{1} (1 - \frac{X_{1}}{K_{1}} - \frac{h_{12} X_{2}}{K_{1}}) - F_{1} (X_{1}, X_{2}, Y) Y, \\ \frac{d X_{2}}{d T} = R_{2} X_{2} (1 - \frac{X_{2}}{K_{2}} - \frac{h_{21} X_{1}}{K_{2}}) - F_{2} (X_{1}, X_{2}, Y) Y, \\ \frac{d Y}{d T} = E_{1} F_{1} (X_{1}, X_{2}, Y) Y + E_{2} F_{2} (X_{1}, X_{2}, Y) Y - DY, \\ X_{i} (0) ⩾ 0, Y (0) ⩾ 0, i = 1, 2 . \end{matrix}$ With $F_{i} (X_{1}, X_{2}, Y) = \frac{A_{i} X_{i}}{BY + X_{1} + {mX}_{2} + C}, i = 1, 2$ .

In model (1), R_i, K_i, E_i, h_ij (i, j = 1, 2, i ≠ j) and D are the model parameters which assuming only positive values. The prey X_i grows with

Existence and dissipativeness

Obviously, the interaction functions G_i (i = 1, 2, 3) of system (8) are continuous and have continuous partial derivatives on the state space $R_{+}^{3} = {(x_{1}, x_{2}, y) : x_{1} ⩾ 0, x_{2} ⩾ 0, y ⩾ 0}$ . 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 $R_{+}^{3}$ 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 conditions $w_{2} < 1 and 0 < \frac{w_{4} w_{10}}{w_{8} - w_{10}} < 1 .$ While subsystem (10) is a Kolmogorov system under the conditions $w_{2} w_{5} < w_{7} and 0 < \frac{w_{4} w_{10}}{w_{9} - w_{3} w_{10}} < 1 .$

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 $E_{3} = (\overset{⌢}{x}$

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

References (16)

C. Letellier et al.
Analysis of the dynamics of a realistic ecological model
Chaos, Solitons & Fractals
(2002)
S. Gakkhar et al.
Order and chaos in predator to prey ratio-dependent food chain
Chaos, Solitons & Fractals
(2003)
S. Gakkhar et al.
Existence of chaos in two-prey, one-predator system
Chaos, Solitons & Fractals
(2003)
S. Gakkhar et al.
Order and chaos in a food web consisting of a predator and two independent preys
Commun Nonlinear Sci Numer Simul
(2005)
H.I. Freedman et al.
Interactions leading to persistence in predator–prey system with group defense
Bull Math Biol
(1988)
T.W. Hwang
Global Analysis of the predator–prey system with Beddington–DeAngekis functional response
J Math Anal Appl
(2003)
T.W. Hwang
Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngekis functional response
J Math Anal Appl
(2004)
H.I. Freedman et al.
persistence in models of three interacting predator prey populations
Math Biosci
(1984)

There are more references available in the full text version of this article.

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On the dynamical behavior of three species food web model

Abstract

Introduction

Section snippets

The mathematical model

Existence and dissipativeness

Kolmogorov analyses and stability

The dynamical analysis and persistence

Numerical simulations

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Commun Nonlinear Sci Numer Simul

Bull Math Biol

J Math Anal Appl

J Math Anal Appl

Math Biosci