We suppose the climate of our world is determined by the temperature
T(x,t) only, and the dynamics is described by the equation
where is the world ``area'', A is the albedo of planet surface at a
given point x, is the solar constant, so that the term is the
insolation, and the term (where is the Stephan-Boltzmann
constant) is the erradiation. The latters can be presented in the
Budyko's form [5] also: a+bT, but it is not principally.
The Budyko form together with the empirical constants a,b would give us a better agreement with the real radiation data of the earth.
For a review of the so-called energy-balance models see, e.g., [6].
Let us
rewrite eq.(1) in the form:
Further we suppose that the planet surface can be covered by the vegetation with
the density N(x,t), its dynamics is described by a logistic equation:
where is the Malthusian parameter [7] of the growth function.
Now we formulate two hypotheses, which are the description of feedbacks
between the climate and vegetation of our world.
Figure 1: Albedo A as a function of vegetation density N
according to hypothesis 1. is the albedo of
the naked surface, of surface fully covered by vegetation.
Then the function will be the following (see Fig. 2). Obviously that .
Figure 2: The function in (2).
We assume for any N>0.
Figure 3: The Malthusian growth function according to hypothesis 2.
Obviously that if , and if , if .
On the ecological point of view, the describes the ecological niche, defined as the interval of T with , for vegetation in the space of climatic factors.
And finally, the equations (2) and (3) together with the functions A(N) and and the corresponding initial and boundary conditions make up the biosphere for our world.