Life History Models by hcj

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									Life History Models [Lecture 1]

A. Two Fundamental facts of population dynamics (behavior):

___1. Geometric increase is possible




____where N is the density of individual plants e.g. plants/m2
      Nt is the population density at the current time and Nt+1 is density at the next time
        r is the reproductive rate i.e. the number of new individuals coming from each
       individual in the population

2. Populations cannot continue unrestricted growth because of resource limitations.




_    Thus we introduce a carrying capacity term = K
                                                                                                2


B. Demography = the numerical changes in a population through various stages of
development.

Nt+1 = Nt + B - D + I - E Difference Equation

        Nt+1 = Population density after a specified time (t), usually one generation
        Nt = Population density at the current time
        B = number of births during time t
        D = number of deaths during time t
        I = number of individuals that immigrate (come in) into the population during time t
        E = number of individuals that emigrate (leave) out of the population during time t

Population growth rate can be expressed in 2 ways:

                               if negative then population is going extinct
        1. dN/dt = Nt - Nt-1   if positive population is growing
                               0 = population is at equilibrium
                               if 0 < < 1.0 then population is going extinct
   2.      = Nt/Nt-1           if > 1.0 then population is growing
                               if = 0 then population is at equilibrium




C. Life cycle = a series of processes that govern the basic behavior of organisms. For plants
this includes the following processes:

Germination                                       Reproduction
                                                         Fertilization
Seedling establishment                                   Embryonic growth
                                                  Seed dispersal
Development into adulthood
                                                  Senescence and death

                                                  Seed survival to next generation




The processes that link stages in the life cycle can be thought of as vital rates.
                                                                                       3


By focusing on the vital rates of the life cycle, demography addresses both temporal
dynamics and the age (or size) structure of plant populations.

Temporal dynamics = changes in population density (e.g. plants/m2) over time

Spatial dynamics = changes in population density over space

Population structure = numbers of individuals at different life history stages




Diagram from John Harper, 1977
                                                                              4


Diagrammatic Plant Population Model
Boxes = state variables

Arrows = demographic processes (vital rates)




                                               Mathematical Model

                                               SDL = SB t * emrg

                                               FP = SDL * s

                                               SP = FP * spp

                                               SBt+1=[(SP*m)+(SBt-SDL)]*sbs
                                                                                                  5


The model described above predicts exponential growth or decline, thus it is not realistic but
has some useful properties regardless. The useful properties to a weed scientist is the ability
to calculate a population growth rate (λ) for a given species in a given environment. The λs
can then be used to compare among species or populations of a species in different
environments and thus can be useful for prioritizing management. The λ used for these
comparisons is the one calculated at the stable age distribution. That is when the proportion
of individuals in each life cycle stage stabilizes over time (where λ becomes constant).

Demogrphic Parameter Inputs
      Default values
Parm  Mean      Std Dev
ss       0.3          0.1
emrg     0.35         0.1
gf       0.22        0.05
spp      100          10
sr       0.52         0.1

                                                     Population
                                                     Growth Rates
    t        SB        SDL        FR        SP        dN/dt      λ
   0             50        18          3       300
   1             56        20          4       434         6      1.128
   2             79        28          6       606        22     1.3962
   3            110        38          8       847        31     1.3962
   4            154        54         12      1182        44     1.3962
   5            214        75         17      1650        61     1.3962
   6            299       105         23      2304        85     1.3962
   7            418       146         32      3217       119     1.3962
   8            583       204         45      4492       166     1.3962
   9            814       285         63      6271       231     1.3962
   10          1137       398         88      8756       323     1.3962
   11          1588       556        122     12225       451     1.3962
   12          2217       776        171     17068       629     1.3962
   13          3095      1083        238     23831       878     1.3962
   14          4321      1512        333     33273      1226     1.3962
   15          6033      2112        465     46455      1712     1.3962
   16          8424      2948        649     64861      2390     1.3962
   17         11761      4116        906     90559      3337     1.3962
   18         16421      5747       1264    126439      4660     1.3962
   19         22926      8024       1765    176534      6506     1.3962
   20         32010     11203       2465    246476      9083     1.3962

In order to get an estimate of the variation in λ one can use values of the demographic
parameters observed in each replicate plot. If one has made more than one year of
observations then the demographic parameters for each year in a plot can be selected at
random to predict stochastic population dynamics and λ. For example, 6 years of data were
used to predict λs for Senecio jacobae in 3 different forest environments (Burn, Burn and
Logged and Meadow) to help prioritize management in the invaded region.
                                                                                                                                                         6




                                                                           6


                                                                           5
                                         Population Growth Rate (lambda)




                                                                           4


                                                                           3


                                                                           2


                                                                           1                                                                         1


                                                                           0

                                                                                         Burn                   Burn and Logged             Meadow

Distribution of λ values from replicate plots in 3 environments where Senecio jacobae was
found.

The next level of model complexity adds density dependence to create a more realistic
simulation of population dynamics over time. The reproductive rate (seed produced per
plant) will decrease as intraspecific density increases. Thus, spp get replaced by
                     b N 
spp  sppmax 1                in the model.
                1 b  N / a 
                              

                                             350
   Number of seed p rodu ced per plant




                                             300

                                             250

                                             200

                                             150

                                             100

                                                          50

                                                                       0
                                                                                                                            8

                                                                                                                                  0

                                                                                                                                        2
                                                                               12

                                                                                    24

                                                                                         36

                                                                                                48

                                                                                                     60

                                                                                                          72

                                                                                                                84

                                                                                                                     96
                                                                       0




                                                                                                                          10

                                                                                                                                12

                                                                                                                                      13




                                                                                              Plant density, N (plants/m 2)
                                                                                                     7


With the inclusion of density dependent feedback into the model the population will grow to
some equilibrium density and then stabilize over time.

                                   1000

                                   900

                                   800
   Plant density, N (plants/m 2)




                                   700

                                   600
                                                                                               SB
                                   500                                                         SDL
                                                                                               FR
                                   400

                                   300

                                   200

                                   100

                                     0
                                          0   1   2   3     4     5    6      7   8   9   10
                                                          Generation (year)


Population dynamics with density dependence.

So far models that predict plant population density over time have been described. The next
level of complexity is to consider how population density changes over space and time. One
can create a grid map and make specific assumptions about how plant propagules (seeds and
vegetative structures like rhizomes and lateral roots) may move between the grid cells,
otherwise maintaining a population dynamics model, like the ones above, in each cell of the
grid map. This model type is a cellular automata.
                                                                                              8


In the simplest form of this type of model one may assume that the propagules will disperse
completely at random to any of the other cells at each time step (Levins 1969).

How one might use these spatial-temporal models will be discussed in Seminar.



References:

Caswell, H. 2001. Matrix Population Models: Construction, Analysis, and Interpretation.
Sinaur Press.

Menges, E.S. 2000. Population viability analyses in plants: challenges and opportunities.
Trends in Ecology & Evolution 15:51-56.


Firbank L.G. and A.R. Watkinson. 1986. Modelling the population dynamics of an arable
weed and its effects upon crop yield. J. of Applied Ecology 23:147-159.

								
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