An organism living in a fluctuating environment with a periodicity somewhat longer than its generation time (an ILC situation) could adapt to this environment by passing information about the current state of the environmental conditions to its offspring. We shall first consider the simplest situation, in which the environment does not induce the change in state, but acts only as the selective agent. This is classical Darwinian evolution, the difference between our model and the conventional one being that for cellular inheritance, the information carrier in our model is usually an EIS rather than DNA, and for behavioral inheritance, it is the nervous system. The model for phenotypic transmission is described in Figure 1.
Note that there is no modifier locus which affects the transition rate in the genome; the transition rate is local and intrinsic to the locus. The assumptions about the regularity and symmetry of the fluctuating environment simplify the model but can be relaxed without changing its basic results (results not shown). The model is suitable for a chromatin marking EIS, where information is transmitted as epigenetic marks carried by chromosomal DNA. Variations in chromatin marks are formally most similar to variations in DNA sequences. The model is equally suitable for describing transitions between behavioral phenotypes transmitted by social learning.
With these assumptions the population sizes of 
carriers (
)
and of 
carriers (
) change every generation according to
- the transition rate, 
and 
. In environment 
, in the next generation
their new values will be:
where 
are the population sizes, is the transition
rate and 
are the fitnesses for the ``good'' or the ``bad''
phenotype.
In environment 
:
thus we have two linear transformations 
where 
is 
, and
The transformation for a whole cycle will be
where 2n is the number of generations
in one cycle. The population growth rate for a phenotype with
transition rate will be the maximal eigenvalue of the
transformation.
This will be the growth rate for 2n generations. The growth rate per
generation will be the 2n-th root of this
(see also Leigh 1970, and Ishii et al. 1989).
Figure 2
shows the growth rate per generation plotted against n, the number
of generations, and , the transition rate. It shows, for
example, that for n=27, 
and 
, there will be a
growth rate of 1.03 for types with 
, and 0.99 for
types with 
. In this case the selection for the optimal
transition rate is quite high (4%). Appendix 2 shows that this result
is general. For big enough n, selection for the optimal transition
rate is of the order of 
(s is the selection coefficient).
As can be seen in Figure 2, for each n there is a
transition rate 
which is selectively the most favorable.
Figure 3 shows an example: a graph of n vs. (the best transition rate), when and are
again 1.1 and 0.9 . One can see that for small n, the best
transition rate is very high compared to classical mutation rates
(that are in the range of 
to 
). In cases of
asymmetric environmental fluctuations between and (n is
different in each environment) the best transition rates will not be
equal in both directions. The basic result, however, is unchanged: for each environment will still be approximately equal to
(see Appendix 1).
So far we have described the simplest situation, in which the transition from one state to another is insensitive to environmental induction. The optimum transition rate depends on the length of the fluctuation cycle and on the selection coefficients. Clearly, another option to cope with a fluctuating environment is to adapt to it phenotypically in every generation without transmission to the offspring. However, if the phenotypic change is not instantaneous, there will still be some time in which the organism is not adapted, so there will still be a selection pressure towards an optimal transition rate.
Another option, which is adopted by organism that can learn, in the
cell lineages of multicellular organisms, and also in some unicellular
organisms, is to have induced transitions. The environment will then
enhance transitions from the ``bad'' to the ``good'' phenotype. We
shall denote the probability for such an induction as 
. It is
clear that the closer is to 1 the better. An organism will
always benefit from making the transition as fast as possible.
To describe this model we used the following matrices ( in this model
it is assumed there are no random transitions ) :
with the transformation for the whole cycle again being 
. The growth rate under inducing conditions has been explored by
Jablonka et al. (in press).
Figure 4 is a graph of the change in the growth rate
plotted against n and (the induced transition rate). To
describe induced variation, an induction coefficient is included in
the matrices describing the basic model. For each mutation rate
there is a corresponding induced transition rate , which causes
the same population growth for a given cycle-length.
Figure 5 shows, for n=20 and fitnesses 0.9 and 1.1,
the values of and , that result in the same rate of
population growth.