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

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

Up to three equilibria denoted as , , and can be found. The points a in Fig. 4a-f are semi-trivial equilibria, where

If then we can speak about our planet as the ``cold desert'', if then about the ``hot desert''.

Let us calculate the eigenvalues of the Jacobi matrix

for (4):

The value of the characterizes the behaviour of the system in the vicinity of the equilibria :

1. : is an attractor and is called a stable node.
2. : is called a focus. The local phase portrait is a spiral that winds into the node.
3. : is a repeller and is called a saddle point with one stable direction.
4. : is a repeller and is called a unstable node.

If one of the 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 we have

and if either or the equilibrium is a stable node (), if then the equilibrium is a saddle point ().

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

• if then the point is either a stable node or focus,
• if then this point is a saddle point.
• if
   
  then the point is a focus, in the opposite case we have either node or saddle.

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

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.

  
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'' (, i.e. no vegetation can occur). It exists always if

If as before, but , 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 .

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 (Fig. 5c), and, in addition, the inequality (10) is valid (note if 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 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'' ().