## Info

hold. As is seen in Fig. 8.3 these inequalities cut out a very narrow domain of the positive orthant, where only the thermodynamic evolution of the system is possible.

If we look at Fig. 8.3a, then we see that the upper boundary D = N2 (the curve 0a1b) contains both the unstable node (0,0) and the saddle point (1,1), and the curve D = N2 is the unique (so-called homoclinical) trajectory, going out from the origin of the coordinates (it corresponds to thermodynamic equilibrium) and coming into the saddle point.

The latter corresponds to the equilibrium containing only single species (either the first or the second). In this equilibrium the system can be ambiguous in time but always leaves it, moving along the separatrix (1,2). The latter comes into the stable equilibrium D*(2)= N*(2) = 2/(1 + y) = 4/3. It is interesting that the transition along the curve 0a 1 does not depend on y, i.e. it is true for any system with y < 1. The next transition along the separatrix depends naturally on y. Other possible transitions from thermodynamic equilibrium strictly to dynamic stable equilibrium 2 can be realised along the trajectories, lying lower than D = N2. It is interesting that in this case the system begins its evolution as a community of two populations, while in the previous case it started as a single population. Also note that along these trajectories the dissipation is less and the total size is greater than on D = N2. Clearly, the system passes from thermodynamic to stable dynamic equilibrium (0! 1) with minimal current values of D and maximal values of N along the lower boundary D = ((1 + y)/2)N2 (0c2d).

So, there are two extreme ways leading from zero point to stable equilibrium: the first is along 0a12, and the second is along 0c2. The latter seems more "optimal" from both the thermodynamic and ecological points of view, since current values of dissipation are minimal and the current total size is maximal. What is an "optimality" of the first way? Our hypothesis is: if one excludes the ambiguous residence time in the saddle point, then the transition time will be minimal for the first way and maximal for the second one (certainly, we consider the transition between vicinities of these points). The system pays for its reliability by the slowness of its evolution.

Fig. 8.3. The phase portrait of Eq. (5.2) in the space {D, N} for two values of y: y — 0:5 (a) and y — 1:5 (b). D is the function of dissipation, and N the total size of community.

Look at the upper part of the graph where D > D* (2). If we move along the line N = N* (2) towards the stable point, then the value of D decreases, illustrating by the same token the Prigogine principle (minimum of entropy production). If we now consider the movement along the line D = D* (2) towards the stable point, then the value