p-Adic Model for Population Growth

Abstract

We use p-adic numbers to describe a model with limited growth of population (see, for example, [1] and the first section of this paper for p-adic numbers). Dynamics of growth is described by the well-known logistic differential equation:

$$ N'(t) = kN(t)\left( {1 - N(t)} \right), $$

where N(t) is a population number at the moment t, k = k ₊ - k _- is a coefficient of growth. The solution of the equation (1) is the map:

$$ {N_k}(t) = \frac{1}{{1 - {e^{{ - kt}}}}}. $$

(2)

Using p-adic analysis (integration with respect to a Haar measure with p-adic values), we get a representation of the solution (2) as a mixture of maps: nij(t) = e ^jkt , j = 1, 2,… These maps are the solutions of the equation:

$$ {n'_{\alpha }}(t) = \alpha {n_{\alpha }}(t) $$

(3)

with α = jk. They describe the dynamics of population for a model with unlimited growth.