ABSTRACT
Let us consider an information diffusion process without a central information source. We assume:
∂ ∂
αn t
kn N n r n n N
= − − −
( ) 1 (8.1)
Here N is population size, n is the number of knowers, (N – n)is the number of learners, k is the growth rate, r is the removal rate. The last term is the removal rate or forgetting rate. Its form differs from the conventional constant loss rate (Bartholomew 1976). We may consider r as the measure of learning ability or degree of difficulty in learning a new technology. Here we introduce a new factor α, the degree of sensitivity. If α > 0, it is a measure of strangeness aversion. When few people accept the new information, the removal rate is large. When most people accept it, the forgetting rate decreases. This is the characteristic of conservatism. On the contrary, if α < 0, this term is a measure of adventure loving. The absolute value of α is less or equal to unity. Different α represent different behavior or cultures, such as, social or solitary animals, conservative or progressive cultures. We easily find the equilibrium solution to equation (8.1):
n N r kN
r kN
* /= −
−
1 1 α
We see that
If we have a fluctuating environment, we may consider the following stochastic equation:
∂ ∂
α s xx t
kx N x rx x N
kx t= − − −
+( ) ( )1 (8.2)
Here x is a random variable, x(t) is white noise with zero mean and unit variance, s is the variance of white noise. It is easy to find the extrema of the stationary probability density of the
Fokker-Planck equation obtained from (8.2) (Horsthemke and Lefever 1984). They are:
x N r kN
k N
r kNm
= − −
−
1 2
1 2α α
when s s< c
xm = 0 when s s< c
where
sc k N r
k 2 2= −
We may compare two species: one is conservative in learning (α1 > 0) and another is progressive (α2 < 0). For conservative species, their steady size n1 is larger. So, in order to maintain the same population size, they need smaller resource. And for progressive species n2, they need larger subsistence space. For stability against fluctuating environment, the conservative one is more stable than the progressive. It is especially true if there is some survival threshold population (Zurek and Schieve 1982). But when new information comes, the conservative species has less potential to absorb new technology than progressive, if the learning ability is limited for every individual.
