On the Stability of Four Dimensional Lotka-Volterra Prey-Predator System

The aim of this work is to study a modified version of the four-dimensional Lotka-Volterra model. In this model, all of the four species grow logistically. This model has at most sixteen possible equilibrium points. Five of them always exist without any restriction on the parameters of the model, while the existence of the other points is subject to the fulfillment of some necessary and sufficient conditions. Eight of the points of equilibrium are unstable and the rest are locally asymptotically stable under certain conditions, In addition, a basin of attraction found for each point that can be asymptotically locally stable. Conditions are provided to ensure that all solutions are bounded. Finally, numerical simulations are given to verify and support the obtained theoretical results


INTRODUCTION
The food chain can be described as a transfer of energy from one type of living organism to another, while the energy transferred is the food that ensures the continuity of life is ecological balance. Chemical, physical and biological systems are inherently nonlinear. A large class of ISSN: 0067-2904 models that represent predator-prey population dynamics can be described by ordinary differential system or partial differential system. One of the simplest model is the Lotka-Volterra equations, also called predator and prey equations which are two of the first-order nonlinear differential equations. Populations change over time according to the following equations [1]: where ( ) represents the prey species and ( ) is the density of predator species at time t. All parameters , , and are nonnegative constants. The parameter represents the natural growth rate of the prey in the absence of the predator. Parameter represents the effect of the predator on the prey population. Moreover, if is the only decreasing factor for the prey population, then prey will be eaten by predators. Parameter represents the effect of prey on the predator population. Moreover, if is the only increasing factor for the predator population, then the population growth is proportional to the food available. Parameter represents the natural death rate of the predator in the absence of prey. In [2], the predator-prey model with at least one predator and two prey has been investigated.
In [3], the following population dynamics of the Lotka-Volterra model consists of three species: two predators and one prey were studied. ̇= − − ,̇= − + ,̇= − + , where ( ) ≥ 0, represents prey, ( ) ≥ 0 and ( ) ≥ 0 represent predators, and a,b,c are positive parameters. The parameters , and are positive and are interpreted as follows: represents the natural growth rate of the prey in the absence of predators, represents the natural death rate of the predator y in the absence of prey, represents the natural death rate of the predator z in the absence of prey. Many authors modified and investigate the classic Lotka-Volterra model . For more details, see [4] and [5].
Global dynamics of 3-dimensional Lotka-Volterra models with two predators competing for a single prey species in a constant and uniform environment has been studied in [6]. It is assumed that the two predator species compete purely exploitatively with no interference between rivals, the growth rate of the prey species is logistic or linear in the absence of predation, respectively and the predator's functional response is linear [6]. In [7], the two preyone predator system are also discussed and studied. In [8] and [9], the authors investigated the following mathematical model : where represents the density of the prey in their two divers' habitats; represents the density of the predator. The two species of prey are supposed to grow logistically with a certain growth rate and carrying environmental capacity to ; represents the rate of predation by the predator 1 , on prey ; represents the rate of predation by the predator 2 on 1 ; represents the mortality rate of predators such that = 1,2 , and 1 , 2 and γ are the corresponding conversion rates. The two functions 1 2 1 ( 1 + 2 ) −1 and 1 1 1 ( 1 + More details on the modified Lotka-Volterra models and prey-predator models can be found in [10][11][12][13][14]. In [15], a modified for Lotka-Volterra model was proposed and studied, and it represents a food chain consisting of three species. They all grow logistically, which means that the absence or decimation of one species does not cause the death of the others: Lions are at the top of the food chain. They do not differentiate between hyenas and deer when they are hungry. Tigers are in the second place; they prey on every animal weaker than them in the wild. In the third-place are hyenas; they live on less powerful animals such as deer or zebra. In our work, a modification of the last mathematical model was introduced such that another predator was added to the model that studied in [14]. The present model has four firstorder nonlinear differential equations describing the dynamic behavior for four species, in which all the species grow logistically. The current model is considered more comprehensive than the model studied in [14]. Because of the increase in the number of parameters as well as the number of differential equations, and because of that this model contains sixteen possible equilibrium points, while the number of the possible equilibrium points in the previous model is eight. In the following section, the modification of the four-dimensional Lotka-Volterra model, such that the four species grow logistically is formulated, and the boundedness of the positive solutions of the model is studied. In Section 3, it is shown that the model has at most sixteen possible equilibrium points. Five of them always exist without any restriction on the parameters of the model, while the existence of the other points is subject to the fulfillment of some necessary and sufficient conditions, moreover the local stability of all possible equilibrium points is discussed, and it is shown that eight of the equilibrium points are unstable while the rest equilibrium points are locally asymptotically stable under certain conditions. In sections 4 a basin of attraction for all equilibrium points of the system (1), which are locally asymptotically stable, will be discussed by finding a suitable Lyapunov function for the mentioned points. In section 5 a numerical simulations are given to verify and support the obtained theoretical results. Finally, a brief conclusion was presented about the findings of this work.

THE MATHEMATICAL MODEL
In this section, we modify the Lotka-Volterra model to include four species. The mathematical model is given as follows: where x, y, z and w represent the densities of the species at time t . We assume that the four species grow logistically. The parameters , , and are positive constants which represent the growth's rates of the species x, y, z, and w, respectively. The nonnegative constants , , and are the change's rates of with respect to , , w, respectively. While the nonnegative constants and w respectively. The nonnegative , and f are the change's rates of y with respect to , , and respectively; , ℎ, and m are the change's rates of with respect to w,y, and x respectively; , , and are the change's rates of with respect to , , and , respectively.
Hence, all these functions are Lipschitizion functions on R +4 . Therefore, the solution of the model (1) exists and is unique. Some sufficient conditions are provided in the next theorem which ensures all solutions of system (1) are bounded.
It is easy to show that the right hand of the last inequality is less than So that ̇( ) + ( ) ≤ Γ. By using Grönwall's inequality, we obtain that ( ) ≤ Γ + (0) − Now, when approaches to infinity, it follows that Thus, the proof is completed.

THE EQUILIBRIUM POINTS WITH ITS LOCAL STABILITY:
The system (1) contains at most sixteen equilibrium points, some of them exist regardless of the parameter's values. While the other need the fulfillment of some necessary and sufficient conditions to exist. The existence conditions and the local stability analyses of them are given and shown in this section. The possible equilibria of the system(1) are: 1. The equilibrium points 0 (0,0,0,0), 1 ( , 0,0,0) , 2 (0, , 0,0) , 3 exists, if and only if | ̅ || ̅ | > 0, = 1,2,3.
exist if and only if | || | > 0, = 1,2,3,4. Now, we study the local stability of all possible equilibrium points of the system. This will be done by evaluating the Jacobian matrix of system (1) at each equilibrium point. Recall that an equilibrium point x * of the system (1) is said to be locally asymptotically stable if all eigenvalues of the Jacobian matrix evaluated at x * has negative real part. If one or more has a positive real part, then x * is an unstable point. The Jacobian matrix of the system (1)   Proof: 1. It is clear that the eigenvalues of the at the point 0 (0,0,0,0) are λ 1 = α, 2 = , 3 = , and 4 = . Therefore, the point 0 (0,0,0,0) is always unstable point .
. The Jacobian matrix of the system (1) at the equilibrium point 10 (0,0,̂1,̂2) is given Thus, the proof is completed.
THEOREM 3: Consider the system (1), then we have the following: . the equilibrium point 12 is locally asymptotically stable point if the following conditions hold ].
Proof: 1. The Jacobian matrix of the system (1) at the equilibrium point 12 (̃1,̃2, 0,̃3) Is given as follows , Therefore, the characteristic equation of the 12 is λ − Η] = 0, Where, H , = 1,2,3 are the diagonal minors of the following matrix According to the Routh-Hurwitz principle, the equilibrium point 12 (̃1,̃2, 0,̃3) is to be locally asymptotically stable if: Note that, the trace of the matrix 12 is (−̃1 −̃2 − ̃3 ), which is always negative. So that the previous conditions can be written as follows: trΗ < Η.
According to the Routh-Hurwitz principle, the equilibrium point 13 (̌1, 0,̌2,̌3)is to be locally asymptotically stable if: Note that, the trace of the matrix 13 is (−̃1 −̌2 − ̃3 ), which is always negative. So that the previous conditions can be written as . The Jacobian matrix of the system (1) at the equilibrium point 14 (0,̂1,̂2,̂3)is given by −̂3 ] Therefore, the characteristic equation of the 14 ( 1 − +̂1 +̂2 +̂3)(λ 3 − λ 2 trΗ + ∑ |Ĥ | 3 =1 λ − |Η|) = 0, where,Ĥ , = 1,2,3 are the diagonal minors of the following matrix According to the Routh-Hurwitz principle, the equilibrium point 14 (0,̂1,̂2,̂3) is to be locally asymptotically stable if the following the conditions hold : Note that, the trace of the matrix 14 is (−̃1 −̌2 − ̃3 ), which is always negative, it follows that the previous conditions can be written as . The Jacobian matrix of the system (1) at the point 15 Thus, the proof is completed. THEOREM 4: 1. If 4 is locally asymptotically stable, then the equilibrium points 7 , 9 and 10 can not be existed.

2.
If one of the equilibrium points 7 , 9 and 10 does not exist, then the 4 is not stable. Proof: 1. Let 4 be a locally asymptotically stable, then according to Theorem 2, the following condition must be satisfied: > { , , }. Therefore, we have − < 0, − < 0 and − < 0. Hence, the conditions of the existence of the equilibrium points 7 , 9 and 10 can not be satisfied so that these points are not existed. 2. Now, if one of the equilibrium points 7 , 9 , or 10 exists. Therefore, − > 0, − > 0 or − > 0, that is < , < < . So that 4 is locally not stable. Thus, the proof is completed.
To prove that ̇1 3 is negative 13  So that 13 is a Lyapunov function. Therefore 13 is a basin of attraction for the equilibrium points 13 (̌1, 0,̌2,̌3). Thus, the proof of Theorem 10 is done. So that 14 is a Lyapunov function. Therefore 14 is a basin of attraction for the equilibrium points 14 (0,̂1,̂2,̂3). Thus, the proof of Theorem11 is done. Proof: Consider the following real valued function It is clear that 15  So that 15 is a Lyapunov function. Therefore 15 is a basin of attraction for the equilibrium points 15 ( 1 , 2 , 3 , 4 ) . Thus, the proof of Theorem12 is done.

CONCLOSIONS
In this paper, a modified model of the Lotka-Volterra model was presented such that the proposed model is a complete food chain consisting of four species. The model has sixteen possible equilibrium points; five of them always exist, whatever the values of the model parameters. The number of unstable equilibrium points is eight, while the rest are locally asymptotically stable if they meet the conditions specified in this paper. For each of the equilibrium points that can be locally asymptotically stable, a basin of attraction was found using the Lyapunov function. In a numerical example, it is found that the number of equilibrium points for the system (1) was fourteen, all of which were unstable, except the coexistence point, which was locally asymptotically stable and, two points, do not exist in the mentioned examples due to not meeting some specific conditions. Changing two value parameters eliminates eleven equilibrium points.