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In this paper, we consider Lotka-Volterra predator-prey model between one and three species. Two cases are distinguished. The first is Lotka-Volterra model of one prey-three predators and the second is Lotka-Volterra model of one predator-three preys. The existence conditions of nonnega-tive equilibrium points are established. The local stability analysis of the system is carried out.

The Lotka-Volterra model provides a nice mathematical device to study and understand complex systems of mutually interacting species or agent [

A Lotka-Volterra system of n-dimensions is expressed by the ordinary differential equations [

where

where

The existence and stability of a nonnegative equilibrium point of system (1.1) or subsystems of (1.1) has been investigated by many authors [

In this paper, we shall concentrate on Lotka-Volterra systems of the fourth dimension. A Lotka-Volterra two preys-two predators system is studied by Takeuchi and Adachi [

This work is organized as follows: In Section 2, we describe our model. In Section 3, the existence conditions of nonnegative equilibrium points are established. The local stability analysis of the system is carried out in Section 4.

In Section 5, we present an example to clarify each case.

In this section we consider Lotka-Volterra predator-prey model between one and three species and assume that there is no interspicific competition between the three species x_{2}, x_{3} and x_{4}. This is represented by the following system of differential equations:

where

where

is a 4 × 4 community matrix.

The system (2.1) is a prey-predator system if the following assumption is satisfied.

Two cases of system (2.1) can be distinguished:

The first case describes a one prey-three predators system where x_{1} represents the prey and x_{2}, x_{3}, x_{4} represent the predators. In this case we assume that the following conditions are satisfied in addition to (H1):

The second case describes a one predator-three preys system where x_{1} represents the predator and x_{2}, x_{3}, x_{4} represent the preys. In this case we assume that the following conditions are satisfied in addition to (H1).

(H2)'

(H3)'

In this section, the existence of the equilibrium points of system (2.1) in each case is investigated. At most there are nine possible non-negative equilibrium points for system (2.1) in the first case, the existence conditions of them are given as the following:

1) The equilibrium points _{1} is the equilibrium point in the absence of predation and

2) The positive equilibrium point

(H4):

where

3) The positive equilibrium point

(H5):

where

4) The positive equilibrium point

where

where

A is the interaction matrix defined in (2.2).

For the second case of system (2.1), at most there are fifteen possible nonnegative equilibrium points. The existence conditions of them are given as the following:

1) The equilibrium points

are always exist where E_{2}, E_{3}, E_{4} are the equilibrium points in the absence of predation and

2) The positive equilibrium point

(H4)':

where

3) In the absence of predator and one prey species, both the other two prey species grow. Thus, the equilibrium point

4) The positive equilibrium point

(H5)'

where

5) In the absence of predator, all three prey species grow. Thus, the positive equilibrium point

6) The positive equilibrium point

(H6)':

where

We will use the symbols

In this section, the local stability analysis of equilibrium points is investigated. Assuming that all previous equilibrium points existing.

The Jacobian matrix J of system (2.1) is given by:

Computing the variation matrixes corresponding to each equilibrium point and then using Routh-Hurwitz criteria [

1) Substituting by E_{0} in the variation matrix (4.1), we get the eigenvalues

So for the first case, E_{0} is a saddle point with locally stable manifold in the x_{2}x_{3}x_{4} space and with unstable manifold in the x_{1} direction. Near E_{0} the prey’s population x_{1} grows while the predators’ populations x_{2}, x_{3} and x_{4} decline.

For the second case, E_{0} is a saddle point with locally stable manifold in the x_{1} direction and with unstable manifold in the x_{2}x_{3}x_{4} space. Near E_{0} the predator population x_{1} decline while the preys’ populations x_{2}, x_{3} and x_{4} grow.

2) Substituting by E_{1} in the variation matrix (4.1), we get the eigenvalues

(By using (H3) and (H4)).

So E_{1} is a saddle point with locally stable manifold in the x_{1} direction and with unstable manifold in the x_{2}x_{3}x_{4} space. Near E_{1} the prey species x_{1} remains close to

Similarly, E_{2} has three positive eigenvalues

(By using (H3)' and (H4)')

So E_{2} is a saddle point with locally stable manifold in the x_{2} direction and with unstable manifold in the x_{1}x_{3}x_{4} space. Near E_{2} the prey species x_{2} remains close to

E_{3} and E_{4} have the same stability behavior of E_{2}.

We now state the local stability behavior of other equilibrium points in the form of Theorems. The proofs of these theorems follow directly from the Routh-Hurwitz criteria [

Theorem 4.1

1) E_{12} is locally asymptotically stable in the x_{1}x_{2} plane.

2) If E_{1230} and E_{1204} exist, then E_{1200} is a saddle point with locally stable manifold in the x_{1}x_{2} plane and with unstable manifold in the x_{3}x_{4} plane.

Proof

Consider the following subsystem from (2.1)

Evaluating the variation matrix of system (4.2) at E_{12}, we have

The characteristic polynomial is

Since

Then, _{12} is locally asymptotically stable in the x_{1}x_{2} plane.

Computing the variation matrix (4.1) at E_{1200}, we have

The characteristic equation of matrix V_{12} is

Comparing with (4.3) we get that

If E_{1230} and E_{1204} exist, then

Therefore, E_{1200} is a saddle point with locally stable manifold in the x_{1}x_{2} plane and with unstable manifold in the x_{3}x_{4} plane.

1) Behavior of solutions near the equilibrium points E_{13} and E_{14} are the same behavior of solutions near the equilibrium point E_{12}.

2) Behavior of solutions near E_{1030} and E_{1004} are the same behavior of solutions near E_{1200}.

Theorem 4.1

a) E_{23} is locally asymptotically stable in the x_{2}x_{3} plane.

b) If E_{1230} exists, then E_{0230} is a saddle point with locally stable manifold in the x_{2}x_{3} plane and with unstable manifold in the x_{1}x_{4} plane.

Proof

Consider the following subsystem from (2.1)

Evaluating the variation matrix of system (4.4) at E_{23}, we have

which have the eigenvalues

Therefore, E_{23} is locally asymptotically stable in the x_{2}x_{3} plane.

Substituting by E_{0230} in the variation matrix (4.1), we get the eigenvalues

If E_{1230} exists, then

(By using (H3)' and (H5)').

Hence E_{0230} is a saddle point with locally stable manifold in the x_{2}x_{3} plane and with unstable manifold in the x_{1}x_{4} plane.

Theorem 4.2

a) E_{123} is locally asymptotically stable in the x_{1}x_{2}x_{3} space.

b) If _{1230} is a saddle point with locally stable manifold in the x_{1}x_{2}x_{3} space and with unstable manifold in the x_{4} direction.

Proof

Consider the following subsystem from (2.1)

Evaluating the variation matrix of system (4.5) at E_{123}, we have

which has the characteristic polynomial

where

From Routh-Hurwitz criterion, E_{123} is locally asymptotically stable if and only if

It is clear that all the coefficients c_{1}, c_{2} and c_{3} are positive and

Therefore E_{123} is locally asymptotically stable in the x_{1}x_{2}x_{3} space.

Substituting by E_{1230} in the variation matrix (4.1), we get the characteristic equation

Comparing with (4.6), we obtain that

Therefore, E_{1230} is a saddle point with locally stable manifold in the x_{1}x_{2}x_{3} space and with unstable manifold in the x_{4} direction.

Remark 4.1

1) Behavior of solutions near E_{124} and E_{134} are the same behavior of solutions near the equilibrium point E_{123}.

2) Behavior of solutions near E_{1204} and E_{1034} are the same behavior of solutions near E_{1230}.

Theorem 4.3

a) E_{234} is locally asymptotically stable in the x_{2}x_{3}x_{4} space.

b) If _{0234} is a saddle point with locally stable manifold in the x_{2}x_{3}x_{4} space and with unstable manifold in the x_{4} direction.

Proof

Proof of this theorem follows directly as proof of Theorem 4.2

Now, we study asymptotic stability of the positive equilibrium

Substituting by

where

From Routh-Hurwitz criterion [

It is clear that all the coefficients c_{1}, c_{2}, c_{3} and c_{3} are positive and if

Then

Theorem 4.4

Proof.

We define the Liapunov function

where

In the region

It is clear that

Then calculating the time derivative of V along the positive solutions of system (2.1), we have

Then, we can choose

Hence, we obtain

Therefore, it follows from well-known Liapunov-LaSalle theorem that the positive equilibrium

The reader can be check local asymptotic stability of the system 2.1 for:

Example 5.1

Example 5.2

The authors would like to thank all staff members who help me in this article.