Exponential Growth (PowerPoint download)

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					      A Introduction to
Modeling Population Dynamics




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        Why Model Populations?

• The size of the human population impacts
  our lives as both citizens of the world and
  members of our local communities.
  – Will the population of my town, region, or state
    grow or decrease and what will be the
    economic impact of those changes?
  – Will the world population grow to a point that
    is not sustainable?

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         Why Model Populations
• The size variations of non-human populations
  have relevance to us.
  – Will our once-abundant fish species exist in
    sufficient numbers to be harvested for food?
  – Will populations of endangered animals continue
    to be affected by human interventions?
• For more population research topics, check
     forest.bio.ic.ac.uk/cpb/cpb/programme.html

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          Presentation Overview
• Illustrate the development of some basic
  one- and two-species population models.
• Present four models of population growth:
  –   Malthusian (exponential)growth
  –   Logistics growth
  –   Logistics growth with harvesting
  –   Predator-Prey interaction

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          The Malthus Model of
           Population Growth
• Proposed in 1798 by the Englishman, Thomas R.
  Malthus.
• By observing human populations, he conjectured
  that
  Populations appeared to increase by a fixed
  proportion over a given period of time, and that,
  in the absence of constraints, this proportion is
  not affected by the size of the population.

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 Motivation for Malthus’ Model




Graphs of US population available to Malthus.

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          The Malthus Model

To develop a mathematical representation
of Malthus’ model, we use the following
hypothesis as our governing principle.

Populations appeared to increase by a fixed
proportion over a given period of time, and that,
in the absence of constraints, this proportion is
not affected by the size of the population.

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            The Malthus Model
Notation:
– t0, t1, t2, …, tN: discrete times at which the population is
  determined; all ti are equally spaced with time step
  h = ti+1 - ti for all i.
– P0, P1, P2, …, PN: populations at times t0, t1, t2, …, tN,
  respectively.
– b and d: birth and death rates, respectively (with units
  consistent with those used to measure time);
– c = b – d: effective growth rate.


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             The Malthus Model
Mathematical Equation:
     (Pi + 1 - Pi) / Pi = c*h
or
     Pi + 1 = Pi + c*h*Pi
for i = 0, 1, …, N.

The initial population, P0, is given at the initial
time, t0.

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                  An Example

Example:
Let t0 = 1900, P0 = 76.1 million (US
population in 1900) and c = 0.01 (a per
capita growth rate of 1.0% per year).
Determine the population at the end of 1, 2,
and 3 years, assuming the time step h = 1

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              Example Calculation
P0 = 76.1; t0 = 1900; h = 1; c = 0.01

P1 = P0 + c*h*P0 = 76.1 + 0.01*1*76.1 = 76.9;
t1 = t0 + h = 1900 + 1 = 1901

P2 = P1 + c*h*P1 = 76.9 + 0.01*1*76.9 = 77.7;
t2 = 1901 + 1 = 1902

P3 = P2 + c*h*P2 = 77.7 + 0.01*1*77.7 = 78.5;
t3 = 1902 + 1 = 1903
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                   Pseudo Code

INPUT:
 T0 – initial time
 P0 – population at t0
 H – length of time interval
 N – number of time steps
 C – population growth rate



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                    Pseudo Code
OUTPUT
 Ti – ith time value
 Pi – population at ti
 for i = 0, 1, …, N
ALGORITHM:
 Set T = T0
 Set P = P0
 Print T, P

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                 Pseudo Code
For i = 1, 2, …, N
    Set T = T + i*h
    Set P = P + C*H*P
    Print T, P
End For




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 Malthusian Curves




    Malthus model plots: c > 0.

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 Malthusian Curves




    Malthus model plots: c < 0.

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A Malthusian Population Prediction
• Malthus model prediction of the US population for
  the period 1750 – 1998, with initial data taken in
  1750:
         t0 = 1750, P0 = 1,000,000, c = 0.025
  Prediction is plotted against actual US population
  for period 1900-1998.

                         Malthus Plot

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                 Logistic Growth

• Results from Malthus model are not valid over long
  time periods for humans or for other biological
  forms.
• A new model was proposed in 1838 by Belgium
  mathematician Pierre-Francois Vehulst: growth rate
  depended on population size.
• Result was rediscovered in 1920 by Raymond Pearl
  and Lowell Reed and used to model US population,
  as well as populations of fruit flies and yeast cells.

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              Logistic Growth

Pearl-Reed growth rate model in which rate
depends on population:
       b = b0 - b1*P and d = d0 + d1*P
Assigning
      c = b0 - d0, r = b1 + d1, and K = c/r,
and substituting new values for b and d into the
Malthus equation gives a logistics equation:

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                  Logistic Growth

             Pi+1 = Pi + c*h *Pi*(1 - Pi/K)

• K is the population carrying capacity.
   – If any Pi = K, equation gives Pi+1 = Pi for all future
     times.
   – For Pi = K, birth and death rates are equal.
• K has biological meaning for populations with
  strong interaction among individuals that control
  their reproduction

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  Example: Growth of Yeast Cells




Population of yeast cells grown under laboratory conditions.

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Logistics Growth with Harvesting
Harvesting populations, removing members of a
population from their environment, is a real-world
phenomenon.
Assumptions:
– Harvesting intensity constant.
– Per unit time, each member of the population has
  chance of being harvested.
– In time period h, expected number of harvests is
  f*h*Pi. where f is harvesting intensity factor.


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Logistics Growth with Harvesting
The logistic model can easily by modified to
include the effect of harvesting:
     Pi+1 = Pi + c*h*Pi*(1 - Pi/K) - f*h*Pi
or
     Pi+1 = Pi + c*h*Pi*(1 - Pi/K),
where
     c = c - f, K = [(c – f)/c]*K
                       Harvesting Plots

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      A Predator-Prey Model: two
      competing fish populations
An early predator-prey model
• In the mid 1920’s the Italian biologist Umberto
  D’Ancona was studying the population variations
  of species of fish that interact with each other.
• He came across data on the percentage-of-total-
  catch of several species of fish that were brought
  to different Mediterrian ports in the years that
  spanned World War I

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 Two competing fish populations

Data for the port of Fiume, Italy for the years 1914-
1923: percentage-of-total-catch of predator fish
(sharks, skates, rays, etc), not desirable as food fish.
                                     Fiume, Italy

                     40
        selachians
         Percent




                     30
                     20                                              Fiume, Italy
                     10
                      0
                      1910    1915     1920      1925
                                  Years



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   Two Competing Fish Populations
• The level of fishing and its effect on the two fish
  populations was also of concern to the fishing
  industry, since it would affect the way fishing was
  done.
• As any good scientist would do, D’Amcona
  contacted Vito Volterra, a local mathematician, to
  formulate a model of the growth of predators and
  their prey and the effect of fishing on the overall
  population.
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 Strategy for model development

 The model development is divided into three
 stages:
1. In the absence of predators, prey population follows a
   logistics model and in the absence of prey, predators die
   out. Predator and prey do not interact with each other
   and no fishing is allowed.
2. The model is enhanced to allow for predator-prey
   interaction: predators consume prey
3. Fishing is included in the model

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     Overall Model Assumptions

Simplifications
• Only two groups of fish:
  – prey (food fish) and
  – predators.
• No competing effects among predators
• No change in fish populations due to
  immigration into or emigration out of the
  physical region occupied by the fish.
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             Model Development

Notation
• ti denote specific instances in time
• Fi denotes the prey population at time ti
• Si denotes the predator population at time ti
• cF denotes the growth rate of the prey in the absence of
  predators
• cS denotes the growth rate of the predators in the absence
  of prey
• K denotes the carrying capacity of prey


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         Stage 1: Basic Model

In the absence of predators, the fish population, F,
is modeled by
          Fi+1 = Fi + cF*h*Fi*(1-Fi/K)

and in the absence of prey, the predator population,
S, is modeled by
                Si+1 = Si -cS*h*Si



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Stage 2: Predator-Prey Interaction

a is the prey kill rate due to encounters with
predators.
   Fi+1 = Fi + cF*h*Fi*(1-Fi/K) - a*h*Fi*Si

b is a parameter that converts prey-predator
encounters to a predator birth rate.
        Si+1 = Si -cS*h*Si + b*h*Fi*Si

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               Stage 3: Fishing
f is the effective fishing rate for both the
predator and prey populations:

Fi+1 = Fi + cF*h*Fi*(1-Fi/K) - a*h*Fi*Si - f*h*Fi

     Si+1 = Si -cS*h*Si + b*h*Fi*Si - f*h*Si




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                     Pseudo Code
INPUT:
 T0 – initial time
 F0 – prey population at t0
 S0 – predator population at t0
 H – length of time interval
 N – number of time steps
 FR – effective removal rate for fishing
 CS - growth rate of the prey in the absence of predators
 CF - growth rate of the predators in the absence of prey


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                      Pseudo Code
 A – prey kill rate
 B – predator birth rate
 K – prey carrying capacity
OUTPUT
 Ti – ith time value
 Fi – prey population at ti
 Si – predator population at ti
 for i = 0, 1, …, N


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                      Pseudo Code
ALGORITHM:
Set T = t0
Set F = F0
Set S = S0
Print T, F, S
For i = 1, 2, …, N
   Set T = T + i*h
   Set FTEMP = F
   Set F = F*(1 + H*(CF*(1 – F/K) – A*S – FR))


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                     Pseudo Code
  Set S = S*(1 + H*(B*FTEMP – CS – FR))
  Print T, F, S
End For




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   Model Initial Conditions and
           Parameters
           Plots for the input values:
T0 = 0.0               S0 = 100.0                          F0 = 1000.0
H = 0.02               N = 6000.0                          FR = 0.005
CS = 0.5               CF = 0.3                            A = 0.002
B = 0.0005             K = 4000.0


                Predator-Prey Plots
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 D’Ancona’s Question Answered
       (Model Solution)
A decrease in fishing, f, during WWI
decreased the equilibrium prey population,
F, and increased the equilibrium predator
population, P.
   f                   Prey                    Predators
   0.1                 800                      150
   0.01                617                      205
   0.001               597                      211

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            General Questions
What are some of the weaknesses in our approach
to modeling populations that will likely cause the
mathematical models to differ from physical
reality?
– Mathematical modeling is, at some level, an idealistic
  approach to reality and it is often difficult, if not
  impossible, to take into account all of the elements in a
  model that play a role in the real phenomenon.



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            General Questions
– The key to developing a good mathematical model is to
  determine the essential parameters that govern the
  phenomenon and incorporate these into the model. A
  model with say, 100 variables is of little value. Again,
  a good knowledge of the phenomenon being studied is
  essential.
– In all of our models, we have, for the most part,
  assumed that constants and parameters are fixed;
  however, these quantities generally depend on time and
  hence to achieve more realism, time dependent
  parameters must be included in the model.

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                General Questions
• What role could population modeling play in such
  areas as the spread of an infectious disease, the
  control an insect population, the management of a
  fish hatchery?
   – Control of insects: Many insect populations suffer from
     large populations shifts, due to diseases caused by
     micro-parasites (e.g., viruses, bacteria, fungi, protozoa).
   – These pathogens (micro-organisms or viruses that can
     cause disease) have been used successfully in a variety
     of pest control measures, as bio-insecticides, and as
     longer-term bio-control agents.
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          General Questions
- With the correct model, we can ask under what
circumstances does the pathogen regulate the host. If
we have a choice of pathogens, we could ask which
would be the most suitable as a bio-control agent?




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posted:2/14/2012
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