A study of a three-dimensional competitive Lotka–Volterra system

In this paper we will consider a community of three mutually competing species modeled by the Lotka–Volterra system: { ẋi = xi ( bi − 3 ∑ i=1 ai j x j ) , i = 1, 2, 3 where xi(t) is the population size of the i-th species at time t, ẋi denote dxi dt and ai j, bi are all strictly positive real numbers. This system of ordinary differential equations represent a class of Kolmogorov systems. This kind of systems is widely used in the mathematical models for the dynamics of population, like predator-prey models or different models for the spread of diseases. A qualitative analysis of this Lotka-Volterra system based on dynamical systems theory will be performed, by studying the local behavior in equilibrium points and obtaining local dynamics properties. 2010 Mathematics Subject Classification: 37C25, 37D20, 37G35, 92D25.


Introduction
Although the systems of differential equations that represent the competition between species, the so called Lotka-Volterra systems, were introduced by Lotka ( [8]) and Volterra ([10], [11]) almost 90 years ago, their qualitative study, from the point of view of dynamical systems, is still of great interest for mathematical research and more.
In the present paper an algebraic study of the dynamics for a competitive Lotka-Volterra system for two and three competing species will be performed. For the system with two competing species, four equilibrium points with positive coordinates and four distinct phase portraits are obtained, exactly as in [1] and [2]. For the three dimensional competitive system, the algebraic study discovers very important results about the dynamics of the competition between three species like in [3], [12] and [13]. Under some conditions this 3-dimensional system of ordinary differential equations has eight equilibrium points with positive coordinates and represent a class of Kolmogorov systems.
The competitive LotkaVolterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to include trophic interactions. The form is similar to the LotkaVolterra equations for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species. In the equations for predation, the base population model is exponential. For the competition equations, the logistic equation is the basis. The logistic population model, when used by ecologists often takes the following form: where x is the size of the population at a given time t, r is inherent per-capita growth rate, and K is the carrying capacity.
Given two populations, x 1 and x 2 , with logistic dynamics, the LotkaVolterra formulation adds an additional term to account for the species' interactions. Thus the competitive LotkaVolterra equations are: where α 12 represents the effect species 2 has on the population of species 1 and α 21 represents the effect species 1 has on the population of species 2. Because this is the competitive version of the model, all interactions must be harmful (competition) and therefore all values of α are positive. Also, note that each species can have its own growth rate and carrying capacity.
In [1] and [2], I.M. Bomze presented a classification of phase portraits of the twodimensional Lotka-Volterra system using the dynamics of 3-type replicator. In [12] and [13], M.L. Zeeman used a three-dimensional geometric analysis to obtain a partial classification of the dynamics of three-dimensional competitive Lotka-Volterra systems and obtain 33 type of stable equivalent classes. For 25 of these classes it was obtained that the limit sets are equilibrium points and for the rest of 8 classes it is possible to have Hopf bifurcations and, consequently, periodic orbits.
This model can be generalized to any number of species competing against each other. One can think of the populations x i and growth rates r i as vectors and the interaction α i j as a matrix. Then the competitive system for n species is if the carrying capacity is pulled into the interaction matrix. If we take b i = r i and a i j = r i α i j , then the competitive Lotka-Volterra n-dimensional system is In general, an autonomous system of ordinary differential equations of the form where x i = x i (t), F i smooth functions, is called Kolmogorov system. If F i are polynomials in x j j = 1, . . . , n, then this system is a Lotka-Volterra type system which is widely used in modeling interacting species of predator-prey type arising, for example, in biology, ecology, epidemiology or even in economy. The class of Kolmogorov systems is widely used in the mathematical models for the dynamics of population. So, many predator-prey models or SIR models for spread of disease are particular classes of Kolmogorov systems.
The functions F i are called fitness functions and for many models F i are considered to be affine functions, i.e. of the form . . , n. Let us remark that the two classes have something in common. Further, taking into account this definition, a Lotka-Volterra n-dimensional system (5) is competitive if all a i j are positive.
In [10,11] Volterra studies dissipative Lotka-Volterra systems as generalizations of the classical predator-prey model. A Lotka-Volterra system with interaction matrix a i j is called dissipative, respectively conservative, if there are constants d i > 0, i = 1, . . . , n, such that the quadratic form q(x) = n i, j=1 a i j d j x i x j is negative semi-definite, respectively zero.
Note that the meaning of the term dissipative is not strict because dissipative Lotka-Volterra systems include conservative Lotka-Volterra systems. Moreover, we remark that conservative Lotka-Volterra models are in some sense Hamiltonian systems, a fact that was well known and explored by Volterra in [10,11].
The Lotka-Volterra competition model describes the outcome of competition between two or more species over ecological time. Because one species can competitively exclude another species in ecological time, the competitively-inferior species may increase the range food types that it eats in order to survive.
Generally, the definition of a competitive Lotka-Volterra system (5) assumes that all values in the interaction matrix a i j are striclty positive ( [5]). If it is also assumed that the population of any species will increase in the absence of competition unless the population is already at the carrying capacity b i a ii , then it results that b i > 0 for all i. This assumption is considered because we consider a competitive version of the model and so all interactions must be harmful, with a strong competition between the two species. Because a i j represents the effect of species j on the population of species i and a ji represents the effect of species i on the population of species j (these values do not have to be equal), the mutual competition between the species dictates that a i j > 0 for i j. In addition, each species is assumed to be self-regulating (a ii > 0), and in the absence of other species, to have a positive density independent growth rate constant b i > 0, up to the carrying capacity b i a ii .

Local analysis of the 2D Lotka-Volterra ODE system
Let us consider the two dimensional LotkaVolterra system for which we suppose that all coefficients a i j and b i are strictly positive. Next, we restrict our attention to the first quadrant R 2 + and we denote the open first quadrant by intR 2 + .
Moreover, we assume that the determinant of the matrix (a i j ) of the species interaction's coefficients is non zero, i.e. d 1212 = a 11 a 22 − a 12 a 21 0.
Under these assumptions about the parameters a i j and b i (i, j = 1, 2, 3), the system of ordinary differential equations (7) has at most four equilibrium points with positive components as follows. Indeed, from In this section we will use the following notations:

if at least one of the components x i is strictly negative, then the equilibrium E is called virtual.
From practical reasons the virtual equilibrium points are not studied here.
The Jacobi matrix at ( , with eigenvalues λ 1 = −b 1 , For Proof. See the signs of the eigenvalues of each equilibrium point. ITM Web of Conferences 34, 03010 (2020) Third ICAMNM 2020 The characteristic polynomial is Then we obtain that if the interior equilibrium point E is virtual, then one of the axial equilibrium points is a saddle, while the other is an attractor. So, we recover the well known result that for the two species competitive in a Lotka-Volterra model with no equilibrium point in the open first quadrant, one of the species will go to extinction, while the other population stabilises at its own carrying capacity ( [12], [13]).
The types of these four equilibrium points are summarized in the table 1.
type r a or s a or s a or s Table 1: The eigenvalues and types of the equilibrium points of the system (7); the abbreviations a, r and s stand for attractor, repeller and saddle, respectively.
The configuration of the lines d 1 : a 11 x 1 + a 12 x 2 = b 1 and d 2 : a 21 x 1 + a 22 x 2 = b 2 determines the dynamic behaviour of the system in the first quadrant R 2 + . Example 2.1 Let us consider the following two dimensional competitive Lotka-Volterra system: The Jacobi matrix at (x 1 , x 2 ) is and we have four equilibrium points O(0, 0), E 1 2 3 , 0 , E 2 0, 3 4 and E 1 6 , 1 2 with corresponding eigenvalues λ 1 = 2 3 , 6 ITM Web of Conferences 34, 03010 (2020) Third ICAMNM 2020 The Jacobi matrix at (x 1 , x 2 ) is The Jacobi matrix at (

for E. Therefore the origin O is a repeller, E 1 is an attractor and E 2 is a saddle point, while the interior equilibrium point E is virtual attractor. In this case the first species wins and the second species disappears.
Example 2.5 Let us consider the following two dimensional competitive Lotka-Volterra system: The Jacobi matrix at (x 1 ,

Local analysis of the 3D Lotka-Volterra ODE system
In this section we will study a community of three mutually competing species modeled by the following Lotka-Volterra system for which we suppose that all coefficients a i j and b i are strictly positive.
Next, we restrict our attention to the closed positive octant R 3 + and we denote the open positive octant by intR 3 + . Moreover, we assume that all diagonal second order minors of the matrix (a i j ) and the determinant of the matrix of the species interactions's coefficients are non zero, i.e. d 1212 = a 11 a 22 − a 12 a 21 0, d 1313 = a 11 a 33 − a 13 a 31 0, d 2323 = a 22 a 33 − a 23 a 32 0, det a i j 0. Under this assumptions about the parameters a i j and b i (i, j = 1, 2, 3), the system of ordinary differential equations (13) has at most eight equilibrium points with positive components ITM Web of Conferences 34, 03010 (2020) Third ICAMNM 2020 we obtain the following eight equilibrium points O(0, 0, 0), a 12 a 21 , b 2 a 11 −b 1 a 21  a 11 a 22 −a 12 a 21 , 0 , E 13 ( b 1 a 33 −b 3 a 13  a 11 a 33 −a 13 a 31 , 0, b 3 a 11 −b 1 a 31  a 11 a 33 −a 13 a 31 ), E 23 (0, b 2 a 33 −b 3 a 23  a 22 a 33 −a 23 a 32 , b 3 a 22 −b 2 a 32  a 22 a 33 −a 23 a 32 ) and 12 a 13  a 21 a 22 a 23  a 31 a 32 a 33   = a 11 a 22 a 33 − a 11 a 23 a 32 − a 21 a 12 a 33 + a 21 a 13 a 32 + a 31 a 12 a 23 − a 31 a 13  Further, we will use the following notations for the minors of the extended matrix a i j . . .b i : Then the equilibrium points in the coordinate planes can be written as follows: From practical reasons the virtual equilibrium points are not studied here. The Jacobi's matrix at the point (x 1 , x 2 , x 3 ) has the form: For O(0, 0, 0) the Jacobian matrix is For E 1 ( b 1 a 11 , 0, 0), we have the Jacobian For E 2 (0, b 2 a 22 , 0) we have the Jacobian For E 3 (0, 0, b 3 a 33 ) we have the Jacobian   Table 2: The eigenvalues and types of first four equilibrium points of the system (13); the abbreviations a, r and s stand for attractor, repeller and saddle, respectively.
with eigenvalues: The characteristic polynomial is P 12 (X) = X 3 + a 2 X 2 + a 1 X + a 0 , where Taking into account these considerations, we can conclude that in example 3.1 we have a coexistence of the species 2 and 3, in example 3.2 only the species 1 resist, species 2 and 3 disappear, and in example 3.3 we have a stable coexistance of all three species.

Conclusions
In this paper we have made a complete study of the dynamics of the competitive Lotka-Volterra system with two species, together with illustrative examples. For the 3-dimensional competitive system the approach is more complicated and we obtain only partial results, for particular systems. It would be very interesting to obtain a complete study for the competitive Lotka-Volterra systems with three species or more.