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In this paper, the problem of chaos, stability and estimation of unknown parameters of the stochastic lattice gas for prey-predator model with pair-approximation is studied. The result shows that this dynamical system exhibits an oscillatory behavior of the population densities of prey and predator. Using Liapunov stability technique, the estimators of the unknown probabilities are derived, and also the updating rules for stability around its steady states are derived. Furthermore the feedback control law has been as non-linear functions of the population densities. Numerical simulation study is presented graphically.

A lattice model usually represents the motion of a network of particles, where the motion is produced by forces acting between the neighboring particles. Lattice models also are used to simulate the structure of polymers and can exhibit its dynamic behaviors.

The interaction between particles of the systems, which is the subject of this study, is continuous-time Masrkov process on certain spaces of configurations of particles. These systems began as a branch of probability theory in the 1960’s. Most of the inputs came from the work of Spitzer in United States [

Many of models of interacting particles system have shown a chaos behavior. The problems of estimating and controlling stochastic systems are far from solved, and a considerable amount of research is under way. Estimation of the internal states of a stochastic dynamical system is a topic with important applications in different fields such as physics, biology and medicine [

A spatial stochastic model to discuss strategies to control the epidemic was introduced by Schinazi [

A stochastic lattice gas model is proposed to describe the dynamics of two animals populations, one being a prey and the other a predator [

This paper has the following structure. In section 2, the stochastic probability model and its proposed rules will be discussed, and also the pair approximation mathematical model and its analytical solution will be presented. In section 3, stability analysis of the system will be studied and presented graphically. Estimation of the unknown parameters and the updating rules are derived in section 4. In section 5, numerical solutions are derived and presented graphically. Finally, conclusions are provided in Section 6.

In this section, we will describe the stochastic rules of the proposed stochastic lattice gas model for prey-predator.

The lattice gas models describing special chemical reaction. Let us consider a lattice of N sites, every site can be either empty (O) or occupied by a prey (1) or occupied by a predator (2). At any time step a site is randomly chosen. For that site, we suppose that

This Markov process contains three probability parameters

O | 1 | 2 | |
---|---|---|---|

O | 0 | ||

1 | 0 | ||

2 | 0 |

Let us consider the system state as

In this section, the two-site approximation will be presented and the final mathematical model will be written.

In the hierarchical system Equation (6) in [

where sites i and k are the nearest neighbors of site j. We also seek for spatially homogeneous and isotropic solutions of Equation (6) in [

where

and S is the total number of nearest neighbors of the site. This is a nonlinear system of differential equations. The analytical solution for this system is given by solving the system of equation:

By Maple program and [

where,

(for more details, see [

Study of the stability and the chaos of the system will be discussed in this section, also some of the equilibrium point will be presented.

For the case of

where

The Jacobian matrix in Equation (8) of the system in Equation (4) evaluated at the vacuum-absorbing state

and its eigenvalues just are the elements of the main diameter, which are

Using the linear stability analysis, since

Similarly, we get the Jacobian matrix in Equation (8) of the system in Equation (4) evaluated at the prey-absorbing state

Such behavior for the system in prey, predator and vacuum densities in Figures 3(a)-(c) respectively, represents an oscillatory behavior. For limit-cycle that appear in Figures 3(d)-(f) where all the neighboring trajectories tend to a limit-cycle at time tends to infinity, causing the so-called a stable limit-cycle, which indicates that the system stochastic lattice gas of prey-predator system according to the pair-approximation has an oscillatory behavior.

In this section we will derive the dynamic estimators of the unknown probabilities

At the beginning, let us assume the modified model with unknown probabilities in Equation (4) to become as follows

where

where

To solve the problem of this stabilization, we will use the Liapunov stability technique. Constructive upon this, let us consider the following positive definite form of Liapunov function

then

By substituting

and the following update rules of the unknown probabilities

the total time derivative of the Liapunov function in takes the form:

where

By substituting Equation (15) in the modified controlled system in Equation (11), in addition the update rules in Equation (16) we get the final system as follows

where

This section presents some numerical solutions of the controlled nonlinear system of the stochastic lattice gas of prey-predator model in Equation (18) and the estimators of the system unknown probabilities to show how the control for this system is possible. Also, numerical examples for controlled stochastic lattice gas of prey-predator model were carried out for various probabilities values and different initial densities. For illustration purpose, we display the numerical solutions of the system graphically. Furthermore, the percentage error in the estimate for real values of the parameters will be calculated. The percentage error

The following figures display two examples of numerical solutions of the non-linear system in Equation (18). The first solution is shown below.

Clearly, the densities

The second numerical example of solution is given below.

The densities

2 | 5 | 3 | 2 | 1 | 1 | 2 | 2 | 0.4 | 0.4 | 0.2 |

In the following, numerical calculation for the percentage error in each estimate for different real values of the system unknown parameters.

In this paper, we have introduced the mathematical model of stochastic lattice gas of prey-predator model with pair-approximation. The stability for this model has been discussed and it is found that this system has a chaos behavior. The estimators of the unknown parameters and the updating rules are derived according the conditions of the asymptotic stability. Some numerical solutions to show the stable system are presented graphically.

Alwan, S.M. (2016) Chaos Behavior and Estimation of the Unknown Parameters of Stochastic Lattice Gas for Prey-Predator Model with Pair-Approxi- mation. Applied Mathematics, 7, 1765-1779. http://dx.doi.org/10.4236/am.2016.715148