.  life-cycle graph

:  at census date we have only seeds and rosettes; flowering plants have already gone
 
   Each arrow indicates what can happen with a given individuals after one year; each arrow represents the same
   length of time. Notice that if a rosette turns into a flowering plant, it produces seeds in the same year. That's
   why we don't have to represent flowering plants explicitly in the life-cycle graph.

:  same graph, but now with transition probabilities and fecundities
 
   s1 = survival probability in the seed bank
   s2 = survival probability of seedlings
   s3 = survival probability of seeds before and during dispersal
   s4 = survival probability of vegetative rosettes
   g   = germination probability
   f   = flowering probability
   r   = average per capita number of seeds produced per flowering plant
 

:  transition matrix equivalent to the life-cycle graph above

 
 

:  choice of sensus date can affects number of stages and hence the size of the transition matrix
:  e.g., same biennial life-cycle, but census date in mid-summer



:  at moment of census there are seeds, rosettes and flowering plants
:  the corresponding life-cycle graph is
 
:  this life-cycle graph and the previous one represent the same population and will give the same long-term
   population growth rate and elasticities (see below)
: one possible reason why one would like to choose a census date such as to maximize (rather than minimize) the
  number of state variables (i.e., number of seeds, number of rosettes, and number of flowering plants, in this
  example) is to increase the resolving power of the of the model: If we census in fall, so that we have only seeds
  and rosettes, we can only determine the importance of seeds and rosettes for the population growth rate; If we
  sensus in mid-summer, when there are seeds, rosettes and flowering plants, we can determine the relative
  impact of the flowering plants as well.
 

:  transition matrix
 
 

:  the transition matrix can be used to calculate the number of individuals in the different stages from one year to
   the next (see box 12.1, p. 375-376).
:  if we itterate this for many years, then the frequency distribution over the different stages converges to a fixed
   distribution: the stable stage-distribution:

 
:  Moreover, the annual growth rate (i.e. the ratio of the total population size in two succeeding years) of the
   population converges to a fixed value too. In the teasel-example the long-term annual population growth rate
   converges to a value of 2.322. As this is a value greater than one, we're looking at a population that is groing
   with time. Sooner or later, when the population densities increase, mortality rates and fecundity are likely to
   change, so that the (fixed) transition matrix will no longer be valid.
:  Usually, transition matrices obtained for natural populations give long-term population growth rates close to
   one, as it should be in a persistent population (see lecture 25). Deviations from one are the result of

.: observation over a period that is too short to take full account of the total range of environmental
   variability and resulting variation in mortality, plant growth and fecundity,
.: observation on a spatial scale on which local extinction and (re)colonization frequently takes place,
   instead of the larger spatial scale of a stable meta-population.
 

:  The long-term population growth rate calculated from a transition matrix depends (obviously) on the values of
   the matrix entrees. Some of these entrees have a larger contribution to the growth rate than others, however.
:  The 'elasticity' corresponding to a given element of the transition matrix is defined as the relative change in
   population growth rate resulting from a relative change in the value of that given matrix element. The elasticities
   indicate the relative importance of the various population fluxes (transition probabilities and fecundities) for the
   total population growth rate.
:  Elasticities for the teasel example:

 

:  E.g., the elasticity for the survival of large rosettes (x5 to x5) is 0.0227. This means that if we reduced the
   survival of large rosettes by a factor two, the relative change in population growth rate would approximately be
   (0.5)(0.0227) = 0.0114. For the population growth rate we found 2.322, so that the absolute change in growth
   rate would be (0.0114)(2.322) = 0.0265. The new growth rate of the population thus would be approximately
   2.322-0.0265 = 2.2955.
:  Notice that the sum of all elasticities together is equal to one.
:  Sums over columns in the matrix of elasticities give the total elasticity of the outputs (=out-going arrows) from
   the different stages. Sums over rows give the total elasticity of the inputs (= in-going arrows) in the different
   stages. E.g., the sum of elasticities of the all the inputs into the stage of large rosettes (x5) is 0.3156, and of the
   outputs is 0.316. Within rounding-off errors, the total elasticity of a stage's inputs is always equal to the total
   elasticity of the outputs of the same stage (check it out in the elsticity matrix!).
:  The sum of the elaticities of the inputs (or outputs) of a given stage is a measure of the significance of that
   particular stage for the total population growth rate. It can be seen from the elasticity matrix that the population
   flux through one-year old seeds, two-year old seeds and small rosettes together is less than 7%
   (0.659+0.0003+0.0027). This means that if we removed each year all seeds in the seed bank and all small
   rosettes, than the population growth would be reduced with less than 7%. In teasel, apparently, the seed bank
   and small rosettes play a very minor role in the population dynamics. The population dynamics is dominated by
   the growth, and survival of medium sized and large rosettes and the fecundity of the flowering plants.
:  The conclusion that the seed bank (x1 + x2) plays only a very minor role in the population dynamics of teasel
   may not have been obvious from the stable stage distribution (see figure above) where the seeds are
   numerically the dominant stages of the plant's life-cycle.
 

:  comparison of elasticities corresponding to growth (G), survival (L) and fecundity (F) has lead to the
   GLF-classification of species
:  the sum of the elasticities associated with G, L and F must add up to one, that's why we can plot the different
   species in a triangle (fig. 12.12, p 378):