The Existence of at most Twenty Seven Nonnegative Equilibrium Points in a Class of 3-D Competitive Cubic Systems

Th is paper presents the stability analysis of equilibrium po ints of a model involving competition between three species subject to a strong Allee effect which occurs at low population density . By using the software of MAPLE 10, we prove that, under certain conditions, the model has at most twenty seven nonnegative equilibrium points and, via Lyapunov function, we derive criteria for the asymptotical stability of the unique positive equilibrium point.


Introduction
The Allee model of gro wth has been widely and successfully used as a simp le, yet adequate descriptor of the dynamics of small populations or critical depensation model [1], and many theoretical studies (e.g. [2], [3]) have been achieved. The Allee effect refers to reduced fitness or decline in population growth at low population densities. In population models, the Allee effect is often modeled as a threshold, below which there is population ext inction.
In the present paper, we consider the Allee effect within the context of the symmetric model of three co mpeting species. We wish to point out that an model of two competing species with Allee effect was proposed and studied in [2][3][4], and some papers dealing with experiments, simu lations, or combinations of these competitive systems among others are described in [5][6]. In Section 2, we introduce the symmetric model of three competing species subject to the Allee effects. The main analytical results on stability analysis of the equilibriu m points, are presented in Section 3. Section 4 is devoted to a discussion, in the context of nu merical simulat ion, of the analytical results obtained in this paper. Concluding remarks on the paper are made at the end.
In our model we consider that the intrinsic growth rates are quadratic and we prove that, under certain conditions, system (2.1) has at most twenty seven nonnegative equilibria. By using the software of MAPLE 10, a numerical examp le is provided to illustrate the behavior of the system (2.1) for a bio logically reasonable range of parameters with only one asymptotically stable equilibriu m point and seven unstable equilibriu m points in + 3 . We believe that this is the first time that the three-species competition system (2.1) has been formulated and analyzed in the literature.

Boundedness of the Solutions
Consider the system (2.1). Obviously the functions , , and are continuous and Lipschitzian with respect to all independent variables on + 3 = { , , / ≥ 0, ≥ 0, ≥ 0}. Therefore, a solution of the system (2.1) with nonnegative initial conditions exists and is unique. The basic existence and uniqueness theorem for differential equations ensures that which g ive the points of inflection of the graph of versus t. The solutions are increasing and concave down when * 2 < < 1 ; increasing and concave up when < < * 2 ; decreasing and concave down when * 1 < < ; decreasing and concave up when 0 = < < * 1 or > 1. We conclude that = 0 and + 2 = 1 are sinks; and + 1 = is a source. Then if the initial population size is below , the population = ( ) will d ie out.
Similarly to and , respectively.

Existence and Stability of Equilibrium Points
Co mputations of the boundary equilibria and the analysis of the existence of positive equilibriu m po ints and their stability for system (2.1), provide the info rmation needed to determine the coexistence or ext inction of species. To do so, we compute the Jacobian matrix of (2.1). The signs of the real parts of the eigenvalues of evaluated at a given equilibriu m po int = ( , , ) determine its stability. Here where , , and are as in (2.1), < 1, < 1, < 1, < 1 4 and < 1 4 , and all the parameters are pos itive.
When system (2.1) is restricted to 2 , we obtain the following subsystem: Using the rule of signs of Descartes it follo ws: (a ) There are four sign changes in ( ), so there are 4, 2 or 0 positive roots; (b) There are no sign changes at (− ), so there are no negative roots. Hence, at most four positive equilibriu m points are possible in the − plane [15].
Under certain conditions on the parameters we have the following geometric interpretation (see Fig.3

Existence, Stability and Linearization of Positi ve Equili brium Points
Let = * , * , * denote an interior equilibriu m point of + 3 , if it exists. It fo llo ws fro m d irect substitution and algebraic manipulation: Proposition 3.3. System (2.1) has at most eight equilibriu m po ints in the interior of + 3 . Their equilibriu m values * , * and * are given by 12 ,    It is always informative to draw the set of positive equilibriu m points of the system (2.1) in + 3 . Here the set is defined by the intersection of the surfaces: Under certain conditions on the parameters of the system (2.1), we obtain (see Fig 3.2): To determine the stability of a positive equilib riu m point of (2.1), we will use the direct method of Lyapunov:

Direct Method of Lyapunov
Next let us consider the local stability of a positive equilibriu m point = ( * , * , * ) є ⊂ + 3 , where is a neighborhood of to be determined. Based on the "direct method" of Ly apunov, we construct a continuous function , , ln ln ln ln ln ln where (i=1,2,3) are positive constant numbers which are yet unspecified, satisfying the following properties: , , > 0 for ∖ { } , that is, the equilibriu m point is an isolated min imu m of . In fact, where the partial derivatives are calculated at .
If we prove that = ( * , * , * ) is an isolated maximu m of ′ , , = , , , then (c) follows easily, that is: (c1) We note that is a crit ical point of the function , , , that is

Numerical example
By using the software of MAPLE 10, a nu merical example has been provided to illustrate the behavior of the system (2.1) for a bio logically reasonable range of parameters. Choosing the following set of values for the parameters in (2. is a locally asymptotically stable equilibriu m point. Here, we observe that the equilibriu m points +++ ( ≠ 5) are unstable.
In the absence of a co mpetitor, we have: (a) ++0

Concluding Remarks
In this paper, a mathemat ical model o f co mpetition between three populations with lower threshold sizes has been proposed and investigated. The main focus was to analyze the question of existence and stability of nonnegative equilibria. Our results show that there exist at most twenty-seven equilibriu m points for the system under consideration and, by using the software of MAPLE 10, a numerical examp le has been provided to illustrate the behavior of the system (2.1) for a bio logically reasonable range of parameters with only one positive equilibriu m asymptotically stable and 7 positive unstable equilibriu m points.