Population Ecology - Size and Stage structure
Basics of size and stage structured populations
Life cycle graphs
Projection Matrix approach
Sensitivity analysis
Time varying models
Density dependent models
Papers to Read:
Crowder, L. B., D. T. Crouse, S. S. Heppell and T. H. Martin. 1994.
Predicting the Impact of Turtle Excluder Devices on Loggerhead Sea
Turtle Populations Ecological Applications 4:437-445.
Olmsted, I. and Elena R. Alvarez-Buylla. 1995. Sustainable Harvesting
of Tropical Trees: Demography and Matrix Models of Two Palm Species in
Mexico. Ecological Applications 5:484-500.
When does size/stage matter?
Characterizing population dynamics can be done from an
individual-perspective (i-state) in which we track individuals with
different characteristics (age, stage, size, etc.) or can be done at
population-perspective (p state) in which we characterize a population
as having a density function (size or age distribution for example).
Basic population models we've seen characterize a population by a
single p-state variable (population abundance). Demography classically
used age as an i-stae variable, and characterized the population with
the p-state variable of the age distribution. For many populations,
this characterization is inadequate - when knwing the age of an
individual doesn't provide a good basis for predicting its survival or
reproduction. In this case, knowing the population distribution of age
doesn't allow projection of population dynamics. Typical alternative
i-state variables are size, stage (e.g. developmental stage such as
instar), physiological state (e.g. lipid content), or location, or some
combination of these.
Whe is age inadequate? This depends upon how much relevant information
age provides about survival and reproduction of individuals. Caswell
gives 3 main cases for which age alone is not sufficient:
(i) Size-dependent vital rates and growth very plastic. Here
individuals of the same age can vary significantly in size, so age
tells us little about an idividual's size and reproduction. This is
most readily apparent in species with large ranges of adult body size
due to indeterminant growth. Examples include:
Species with a threshold size to reproduce (many perennial plants,
crabs, fish)
Arthropods developing through discrete stages (instars) that must reach
a certain stage before becoming reproductive.
Once reproduction begins, it may be allometrically related to body size
(herbaceous plants, trees, kelp, molluscs, fish, amphibians, turtles)
Can be complex interaction between age, size, etc on reproduction -
size dependence of sexual changes in fish (sequential hermaphroditism)
Size dependent mortality - modular organisms such as coral in which
mortality decreases significantly with size (corals, herbaceous
plants).
Note that size-based vital rates are not sufficient by themselves to
rule out the use of age to characterize population demographics. If
size is very well estimated by age, then use of age-structure would be
adequate. It is growth plasticity which leads to individual vbariation
in size as a function of age that limits the use of age alone.
(ii) Multiple modes of reproduction: many organisms have both sexual
and vegetation reproduction, and if similar-aged offspring derived from
each of these differ greatly in their survival and fertility, then age
is an inadequate descriptor of demography.
(iii) Population subdivision and multistate demography: Here if a
population is subdivided and the vital rates in the different
subpopulations are stongly linked to local environmental factors, then
an appropriate state variable would include age and the local
environmental conditions. - multistate demography means that i-states
are multidimensional.
Life Cycle graphs:
This is a graphical representation of the life stages and flow of
individuals between them. Procedure is to:
(a) Define a projection interval (this affects the structure of the
graph)
(b) Define a node and number them for each life stage
(c) Put a directed line or arc from one node say n1to another say n2 if
there is some flow of individuals from n1 to n2 during a projection
interval - that is if an individual in state I at time t can contribute
individauls (by change in state itself or by reprocduction) to state j,
draw a directed arc from I to j.
(d) Assign each arc with a coefficient a i,j giving the per individual
contribution og individuals from state I to state j in a time period.
These coefficients can therefore be transition probabilities (e.g. the
probability that an individual in state I grows into state j in one
time period) or they can be re[productive outputs from state I
individuals creating state j individuals.
The projection matrix is then made up of these coefficients, and just
as in the age-structured case, allows projection of structure changes
in the population from one timeto another. Unless there is some
inherent periodicities in the life cycle of the organisms (e.g. bamboos
which only reproduce very occasionally), projecting a population
forward in time will lead to a stable stage distribution (similar to
stable age distribution) after a long time. There are a number of
relatively simple conditions to check for the projection matrix or the
life cycle graph to ensure this occurs (irreducibility meaning no
subgroup of stages that is self contained such as a postreproductive
class, and primitivity meaning that it is not cyclic). The rate of
approach to the stable stage distribution depends upon the ratio of the
dominant eigenvalue of the projection matrix to the magnitude of the
first sub-dominant eigenvalue (the so called damping factor). For the
vast majority of estimated life cycle graphs, the population would be
projected to grow exponentially eventuially with a stage distribution
given by the dominant eigenvector and growth rate the dominant
eigenvalue.
Sensitivity analysis:
This refers to investigating how the results of analysis of the
projection matrix (such as the asymptotic grwoth rate and stage
structure) would vary if the matrix were changed. This may be of
interest to do for:
(1) measuring how imporatant a particular vital rate is to overall
population growth - how much does a change in mortality or reproduction
of onme stage affect overall population growth?
(2) evaluating the effect of errors in estimating the vital rates on
the population grwoth rate and stable stage distribution
(3) quantifying the effects of environmental perturbations which can
modify the mortalities and fertilities in the projection matrix - life
table response experiments
(4) evaluating alternative management strategies which lead to
increases in mortality in some stages (say due to harvesting) but not
in others.
(5) predicting the intensity of natural selection. This views the
growth rate as a measure of fitness and investigates the effects of
didderent genotypes with different stage dependent vital rates on
population fitness.
A sensitivity analysis typically involves determining the derivative of
the dominant eigenvalue as a function of the elements of the projection
matrix. There is a simple analytical form for these, determined just
by knowing the left and right eigenvectors associated with the dominant
eigenvalue (v = vector of reproductive values and w = vector of stable
stage distribution) by
dl/da ij = v I w j / (v,w)
where this is a partial derivative and the denominator on the right is
the dot or inner product and l is the dominant eigenvalue. To compare
these though, they must be put on similar scales by normalizing them,
giving the elasticities:
e ij = (a ij / l ) dl/da ij = d log(l) / d log(a ij)
which gives the proportional change in the population growth rate with
a proprotional change in the vital parameter a ij.
Time-varying models:
Here we view the projection matrices as time varying due to external
environmental factors affecting the individual demographic parameters.
These variations can be:
(a) deterministic - periodic environments, in which case the analysis
isn't too complex as long as the focus is on the population at the
period associated with the environmental periodicity.
(b) deterministic - aperiodic - here the sequence of environments does
not repeat and therefore the demographic parameters do not either.
For both (a) and (b) there are a number of ergodic results under most
circumstances - i.e. leading to long-term approach of the population
structure to a stable stage distribution.
(c) stochastic environments with either time-homogeneous distribution
(e.g distribution of random environmental stattes does not vary with
time) or non-time homogeneous distribution. The latter would occur for
example in the global warming scenario.
(d) stochastic environments with or without some temporal correlation
structure, so that successive environments are like independent coin
tosses or not.
Again there are some ergodicity results here, which hold in some
average sense - see Tuljapurkar's book.
Density dependent models: Here the projection matrix depends either
upon overall population density or upon the whole distribution of
densities across stages. So linear analysis no longer applies and there
can be a very complex set of behaviors of the projection model.
Topics from 1990's search on size or stage structure and demography in
4 ecology journals:
Aquatic species 11
corals species 3
Terrestrial plants 13
Fish 2
Terrestrial mammals 2
Marine mammals 1
Reptiles and Herps 3
Modeling 3
Total 38