next up previous
Next: Do dissipative structures exist? Up: A minimal model of Previous: The model description

Preliminary analysis: uniform biosphere

First we will consider solutions, which do not depend on the spatial coordinate x. These solutions correspond to the so-called ``uniform biosphere'', i.e. the biosphere with characteristics which are identical for any point of the planet. The evolution in time of the uniform biosphere derived from eqs. 2 and 3 is determined by the following nonlinear system:
 eqnarray94
Further we can see that the most important properties of the general problem can be reduced from analysis of the system (4).

Let us consider the equilibrium points tex2html_wrap_inline879, tex2html_wrap_inline881.of eq. 4. Then they must satisfy the equations:
 eqnarray107
It is better to consider them graphically on the plane tex2html_wrap_inline883. Depending on the value tex2html_wrap_inline803 6 cases can be differentiated in respect to the number and position of the intersections between curve (I):tex2html_wrap_inline887 and (II):tex2html_wrap_inline889 (see Fig. 4a-f). These intersections fulfil eq. (5) and are equilibria of (4).

  figure122
Figure 4: Graphical representation of (5) at different tex2html_wrap_inline803.I:tex2html_wrap_inline887, II:T=tex2html_wrap_inline895. The different stationary points tex2html_wrap_inline897 are denoted by letters a-c.

Up to three equilibria denoted as tex2html_wrap_inline899, tex2html_wrap_inline901, and tex2html_wrap_inline903 can be found. The points a in Fig. 4a-f are semi-trivial equilibria, where
equation142
If tex2html_wrap_inline907 then we can speak about our planet as the ``cold desert'', if tex2html_wrap_inline909 then about the ``hot desert''.

Let us calculate the eigenvalues tex2html_wrap_inline911 of the Jacobi matrix
equation154
for (4):
equation172
The value of the tex2html_wrap_inline913 characterizes the behaviour of the system in the vicinity of the equilibria tex2html_wrap_inline915:

  1. tex2html_wrap_inline917: tex2html_wrap_inline915 is an attractor and is called a stable node.
  2. tex2html_wrap_inline921: tex2html_wrap_inline915 is called a focus. The local phase portrait is a spiral that winds into the node.
  3. tex2html_wrap_inline925: tex2html_wrap_inline915 is a repeller and is called a saddle point with one stable direction.
  4. tex2html_wrap_inline929: tex2html_wrap_inline915 is a repeller and is called a unstable node.
If one of the tex2html_wrap_inline913 is equal to zero then an analysis of next order is necessary. A comprehensive introduction to the analysis of ordinary differential equations and dynamical systems can be found in [8, 9, 10].

For the ``semi-trivial'' points with tex2html_wrap_inline935 we have
equation211
and if either tex2html_wrap_inline907 or tex2html_wrap_inline909 the equilibrium tex2html_wrap_inline941 is a stable node (tex2html_wrap_inline943), if tex2html_wrap_inline945 then the equilibrium is a saddle point (tex2html_wrap_inline925).

For the ``non-trivial'' equilibrium with tex2html_wrap_inline949 we have the following:

According to the Poincaré-Bendixson theorem the system ends up either in a stable node or a limit cycle (a one-dimensional attractor) because we have only two phase variables T and N. If we apply to the system (4) the Dulac criterion as a proof of the absence of limit cycles (see, e.g., [11]) in the form
eqnarray247
then we can see that there is even not a limit cycle inside the positive quadrant N>0, T>0. Therefore we conclude:

The structure of phase plane for the system (4) is sufficiently simple.

Let us come back to the Figs. 4a-f: Now phase portraits of system (4) are plotted for parameters equivalent to the corresponding Figs. 4 in respect to the number and positions of the equilibria. Their stability is analyzed in the following.

  figure266
Figure 5: Phase portrait of system (4) corresponding to Figs. 4.

In Fig. 5a the point a is a stable node, the final state of this planet (for any initial conditions) is the ``cold desert'' (tex2html_wrap_inline969, i.e. no vegetation can occur). It exists always if
equation278
If tex2html_wrap_inline971 as before, but tex2html_wrap_inline973, we have the following picture (Fig. 5b): In this case the point a is a stable node, the point b is a saddle and the point c is a stable node, since tex2html_wrap_inline981.

It is interesting that there are two stable final states (point a and c), and the singular trajectory of saddle point b divides the quadrant N>0,T>0 into two domains of attractivity. The initial condition of temperature and amount of vegetation determines whether vegetation can exist or not. Let tex2html_wrap_inline993 (Fig. 5c), and, in addition, the inequality (10) is valid (note if tex2html_wrap_inline995 this inequality is not valid), then the point c is a stable focus. As before, the point a is a stable node, the point b is a saddle. If tex2html_wrap_inline1003 the phase picture (Fig. 5d,e) changes: The point a becomes a saddle point, the point c is, as before, either a stable node or a stable focus. In Fig. 5f the point a is a stable node that corresponds to the ``hot desert'' (tex2html_wrap_inline1011).


next up previous
Next: Do dissipative structures exist? Up: A minimal model of Previous: The model description

Werner von Bloh (Data & Computation)
Thu Jul 13 15:02:47 MEST 2000