A system of semilinear parabolic equations with a manifold of steady states is considered and the conditions of stabilizability of this manifold are obtained in the paper. Consider the system of differential equations:
=
= ( , , ),
+ ( , , ),
+ ( , , ),
where , ∈ ; , ∈ ; , ∈ . Assume that ( , 0,0)≡ 0, ( , 0,0)≡ 0, ( , 0,0)≡ 0. Then the system (
Following [1, 2], let say that the manifold is stable with respect to variable = ( , ), if for any point ∈ and any neighborhood of zero W in phase space × we can specify such a neighborhood of zero 0 ⊂ × , that for any point 0 = ( 0, 0)∈ the corresponding solution = ( , 0, 0), = ( , 0, 0)( (0, 0, 0)= 0, (0, 0, 0)= 0) satisfies the ratio = ( , 0, 0)∈ when ≥ 0.
Let say that is asymptotically stable with respect to variable = ( , ), if it is stable with respect to variable and, moreover, lim →∞ ( , 0, 0)= 0 for all 0 ∈ .
Let say that is stabilized, if it is asymptotically stable with respect to variable and when → ∞ { ( , 0, 0), ( , 0, 0)} converge to some point of diversity , if 0 ∈ 0.
M.A. Ayzerman and F.R. Gantmakher established that the state of equilibrium of nonholonomic system is stable, if all roots of the characteristic equation, except for the zero roots, the number of which equals the number of equations of nonholonomic connections, have negative real parts [3, 4]. Each perturbed motion, which is close enough to unperturbed motion, is converge to one of the possible established motions, belong to a given manifold, when → ∞. [5]
Let consider the model of interaction of two populations of microorganisms in one-dimensional case. This system is based on the equations of Fisher-Kolmogorov-Petrovsky-Piskunov. Let ( , )and ( , )be concentrations of the two sub-types of a virus at a point and a time . Consider the problem on the interval ∈ [0; 1]. The system has the form: ( , ) { ( , ) = 1 2 (2, )+ 1 ( , )(1 − 1 ( , ))(1 − ( , )− ( , )); = 2 2 (2, )+ 2 ( , )(1 − 2 ( , ))(1 − ( , )− ( , )), where a1,a2 - the replacement rates for populations u and v accordingly, D1,D2 – the coefficients of diffusion, q1, q2 − the coefficients of the interaction between individuals of different populations.
The condition of impermeability at the ends of the considered interval are considered as the boundary conditions in this problem. They look like: ( , )|
=0
( , )|
=0
= ( , )| =1
= ( , )| =1
= 0;
= 0.
(
Let consider the Jacobian matrix for the system (10). The signs of the eigenvalues of this matrix will give the conditions of stability of the stationary solutions.
Let find the conditions of stability for the model (
Consider the approximation of functions 1( , ), 2( , )near any equilibrium states ( , ). Multivariable calculus
may be used to obtain the following approximations:
1( , )≈ 1( , )+
2( , )≈ 2( , )+
1
2
+
+
consideration equations (
{ +1− = 2
1+1− −+11 = 0; 1 +1− −+11 = 0.
The boundary conditions will take the form: 0 = { Define the initial conditions as follows:
0,9(−5( − 1)2 + 1), 0 > 0, 0 = { 0, 0 ≤ 0; 0, 0 ≤ 0.
0,9(−5 2 + 1), 0 > 0,
Their graphs are presented in figure1. these functions on variables , . Then, the Jacobian matrix Α takes the form:
2 ( 2( Α = ( 1(1 − 1 )(1 − 2 − ) 1 ( 1( + 2 − 1)− 1) + 2 − 1)− 1) 2(1 − 2 )(1 − − 2 ) ). equilibrium ( 3, 3)= (0.5, 0.5).
Substitute ( 3, 3)= (0.5, 0.5)in (12): Α = ( −0.5 1(1 − 0.5 1) −0.5 2(1 − 0.5 2) −0.5 1(1 − 0.5 1)). solution is stable. And if 1 1 + 2 2 > 2( 1 + 2), the solution is not stable.
To solve the problem (
To solve the problem (14)-(16) the program was realised in Matlab, which calculates the values of the grid functions on the time interval 0 ≤ t ≤ 600.
Consider the case when the coefficients of the first and the second equations are equal, i. e. 1 = 2 = 1, 1 = 2 = 0.001, q1 = q2 . Separating the variables and solving the task on eigenvalues, find the value of parameters q1 = q2 = 2, in the transition through which the bifurcation happens in the system. To illustrate this phenomenon, consider the three different cases: 1. q1 = q2 < 2 2. q1 = q2 ≈ 2 3. q1 = q2 > 2
In the first case, the trajectories of system converge to the equilibrium (0,5;0,5), belonging to the manifold of equilibrium states of the system. By Ayzerman-Gantmacher`s theorem, the state of equilibrium of system is stable. Thus, manifold is stabilized. In the second case, there is a soft loss of stability of the system when passing through the critical value, and in the third case, it is possible to observe a complete loss of stability. 5.1. Case, when q1 = q2 < 2.
For the first case , when q1 = q2 = 1.5, the dynamics of function u(x,t) is presented in figure 2. The dynamics of function v(x,t) is presented in figure 3. In figure 4 a solution in a finite time t=600 is presented. 5.2. Case, when q1 = q2 ≈ 2.
For the second case, when q1 = q2 = 2.05, the dynamics of function u(x,t) is presented in figure 5. The dynamics of function v(x,t) is presented in figure 6. In figure 7 a solution in a finite time t=600 is presented. 5.3. Case, when q1=q2>2.
For case 3, when 1 = 2 = 2.5, the dynamics of function u(x,t) is presented in figure 8. The dynamics of function v(x,t) is presented in figure 9. In figure 10 a solution in a finite time t=600 is presented.
Hence, it is shown that for q1 = q2 < 2 the manifold of equilibrium states of the system is stabilized, and when passing through the value of the coefficients of the interaction q1 = q2 = 2 loss of stability occurs in the system.
The paper was supported by the Russian Foundation for Basic Research and the government of the Samara region in the framework of a research project № 16-41-630529.