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Exponential Growth

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					   Overview of                              discrete        continuous
population growth:           density     Geometric        Exponential
                           independent

                            density      Discrete           Logistic
New Concepts:              dependent     Logistic

- Stability
- DI (non-regulating)
            vs.
  DD (regulating) growth

- equilibrium
                            Variability in growth

                            (1) Individual variation in births and deaths
                            (2) Environmental (extrinsic variability)
                            (3) Intrinsic variability
How do populations grow – a derivation of geometric growth


       Growth rate (r) = birth rate – death rate
                               (express as per individual)


           N1 = N0 + rN0




         N0 = initial population density (time = 0)
         N1 = population density 1 year later (time =1)
       How do populations grow?


Growth rate (r) = birth rate – death rate


   N1 = N0 + rN0 = N0 (1 + r)
       How do populations grow?


Growth rate (r) = birth rate – death rate


   N1 = N0 + rN0 = N0 (1 + r)

   N2 = N1 + rN1 = N1 (1 + r)
       How do populations grow?


Growth rate (r) = birth rate – death rate


   N1 = N0 + rN0 = N0 (1 + r)

   N2 = N1 + rN1 = N1 (1 + r)

   Can we rewrite N2 in terms of N0 ???
       How do populations grow?


Growth rate (r) = birth rate – death rate


   N1 = N0 + rN0 = N0 (1 + r)
                          substitute
   N2 = N1 + rN1 = N1 (1 + r)
              How do populations grow?


    Growth rate (r) = birth rate – death rate


         N1 = N0 + rN0 = N0 (1 + r)
                                 substitute
         N2 = N1 + rN1 = N1 (1 + r)

rewrite: N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2
        How do populations grow?


Growth rate (r) = birth rate – death rate


   N1 = N0 + rN0 = N0 (1 + r)
                           substitute
   N2 = N1 + rN1 = N1 (1 + r)

   N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2

                   or

         Nt = N0 (1 + r)t
                        }

                        = , finite rate of increase
     Discrete (geometric) growth



                                      5



 Nt = N0t
                 N
= finite rate                    4



  of increase                3

                         2
                     1



                                 time
    Continuous (exponential) growth




                     5




                          Nt = N0ert
N                4
                         r = intrinsic
            3            growth rate
        2
    1



                time
                   Continuous (exponential) growth
                        5

                                 population             per capita
                                 growth rate           growth rate

 N                  4            dN = rN;               1 dN = r
               3
                                 dt                     N dt
           2
       1
                                 dN    Read as change in N (density)
                   time          dt        over change in time.




                                          1 dN = r
1 dN                                      N dt
N dt
                                               Y   = b + mX

                   N             Per capita growth is constant and
                                         independent of N
Comparison              Discrete                       Continuous
                        Nt = N0t                      Nt = N0ert
              Where:  = er                              r = ln 
         Increasing:  > 1                               r>0
         Decreasing:  < 1                               r<0

                      Every time-step                   None
        Time lag:     (e.g., generation)       Compounded instantaneously

                       Populations w/        No breeding season - at any time
  Applications:   discrete breeding season   there are individuals in all stages
                                                      of reproduction
                      Most temperate
    Examples:       vertebrates and plants
                                              Humans, bacteria, protozoa


                     Often intractable;
  Mathematics:          simulations
                                               Mathematically convenient
Geometric (or close to it)
growth in wildebeest population
of the Serengeti following
Rinderpest inoculation
Exponential growth in the total human population
       The Take Home Message:
Simplest expression of population growth:
1 parameter, e.g., r = intrinsic growth rate

Population grows geometrically/exponentially,
but the Per capita growth rate is constant

First Law of Ecology: All populations possess
the capacity to grow exponentially


 Exponential/geometric growth is a model
          to which we build on
   Overview of                              discrete        continuous
population growth:           density        X
                                         Geometric            X
                                                          Exponential
                           independent

                            density      Discrete           Logistic
New Concepts:              dependent     Logistic

- Stability
- DI (non-regulating)
            vs.
  DD (regulating) growth

- equilibrium
                            Variability in growth

                            (1) Individual variation in births and deaths
                            (2) Environmental (extrinsic variability)
                            (3) Intrinsic variability
Variability in space   In time
Variability in space    In time




Source-sink structure
Variability in space     In time




Source-sink structure



 (arithmetic)

Source-sink structure
with the rescue effect
Variability in space             In time

                          (geometric)
                        G < A
                         G declines with
Source-sink structure       increasing variance


 (arithmetic)

Source-sink structure
with the rescue effect
Variability in space                     In time

                                 (geometric)
                               G < A
                                G declines with
Source-sink structure              increasing variance

                                 (arith & geom)
 (arithmetic)                  Increase the number of
                                subpopulations increases
Source-sink structure           the growth rate (to a point),
                                and slows the time to
with the rescue effect          extinction

 Temporal variability reduces population growth rates

 Cure – populations decoupled with respect to variability,
        but coupled with respect to sharing individuals

				
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