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					Modeling Populations:
  an introduction
    AiS Challenge
Summer Teacher Institute
           2003
       Richard Allen
Population Dynamics
Studies how populations change over time
Involves knowledge about birth and death
rates, food supplies, social behaviors,
genetics, interaction of species with their
environments and among themselves.
  Models should reflect biological reality,
  yet be simple enough that insight may be
  gained into the population being studied.
Overview

Illustrate the development of some basic
 one- and two-species population models.
    Malthusian (exponential) growth – human
     populations
    Logistics growth – human populations
    Logistics growth with harvesting.
    Predator-Prey interaction – two fish
     populations
The Malthus Model

In 1798, the English political economist,
Thomas Malthus, proposed a model for
human populations.
His model was based on the observation
that the time required for human
populations to double was essentially
constant (about 25 years at the time),
regardless of the initial population size.
US Population: 1650-1800




U.S. population data available to Malthus.
Governing Principle

To develop a mathematical model, we
formulate Malthus’ observation as the
“governing principle” for our model:
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.
Discrete-in-time Model

            …, tN: equally-spaced times at which the
 t0, t1, t2,
  population is determined: dt = ti+1 - ti
 P0, P1, P2, …, PN: corresponding populations at
  times t0, t1, t2, …, tN
   b and d: birth and death rates; r = b – d, is the
    effective growth rate.
          P0        P1        P2        …      PN
           |---------|---------|----------------|-----> t
          t0        t1         t2      …        tN
The Malthus Model

Mathematical Equation:
  (Pi + 1 - Pi) / Pi = r * dt
     r=b-d
or
     Pi + 1 = Pi + r * dt * Pi
     ti+1 = ti + dt
The initial population, P0, is given at the
initial time, t0.
An Example

Example:
Let t0 = 1900, P0 = 76.2 million (US
population in 1900) and r = 0.013 (1.3%
per-capita growth rate per year).
Determine the population at the end of 1, 2,
and 3 years, assuming the time step dt = 1
year.
  Example Calculation
P0 = 76.2; t0 = 1900; dt = 1; r = 0.013
P1 = P0 + r* dt*P0 = 76.2 + 0.013*1*76.2 = 77.3;
t1 = t0 + dt = 1900 + 1 = 1901
P2 = P1 + r* dt*P1 = 77.3 + 0.013*1*77.3 = 78.3;
t2 = t1 + dt = 1901 + 1 = 1902
P3 = P2 + r* dt*P2 = 78.3 + 0.013*1*78.3 = 79.3;
t3 = t2 + dt = 1902 + 1 = 1903

Pi = ?, ti = ?, i = 4, 5, …
US Population Prediction: Malthus
Malthus model prediction of the US
population for the period 1900 - 2050,
with initial data taken in 1900:
 t0 = 1900; P0 = 76,200,000; r = 0.013
Actual US population given at 10-year
intervals is also plotted for the period 1900-
2000
                Malthus Plot
  Pseudo Code

INPUT:
 t0 – initial time
  P0 – initial population
  dt – length of time interval
  N – number of time steps
  r – population growth rate
  Pseudo Code

OUTPUT
 ti – ith time value
 Pi – population at ti for i = 0, 1, …, N
ALGORITHM:
 Set ti = t0
 Set Pi = P0
 Print ti, Pi
Pseudo Code

for i = 1, 2, …, N
    Set ti = ti + dt
    Set Pi = Pi + r*dt*Pi
    Print ti, Pi
 end for
 Logistics Model

In 1838, Belgian mathematician Pierre
Verhulst modified Malthus’ model to allow
growth rate to depend on population:
       r = [r0 * (1 – P/K)]
       Pi+1 = Pi + [r0 * (1 - Pi/K)] * dt * Pi
 r0is maximum possible population growth rate.
 K is called the population carrying capacity.
Logistics Model

       Pi+1 = Pi + [r0 * (1 - Pi/K)] * dt * Pi
ro controls not only population growth rate, but
population decline rate (P > K); if reproduction is
slow and mortality is fast, the logistic model will
not work.
K has biological meaning for populations with
strong interaction among individuals that control
their reproduction: birds have territoriality, plants
compete for space and light.
 US Population Prediction: Logistic

Logistic model prediction of the US
population for the period 1900 – 2050, with
initial data taken in 1900:
  t0 = 1900; P0 = 76.2M; r0 = 0.017, K = 661.9
Actual US population given at 10-year inter-
vals is also plotted for the period 1900-2000.
                 Logistic plot
Logistics Growth with Harvesting

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

The logistic model can easily by modified to
include the effect of harvesting:
 Pi+1 = Pi + r0 * (1 – Pi / K) * Δt * Pi - f * Δt * Pi
or
        Pi+1 = Pi + rh * (1 – Pi / Kh) *Δt * Pi
where
        rh = r0 - f, Kh = [(r0 – f) / r0] * K

                   Harvesting
  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 results of fishing on
 population variations of various 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
 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.
                              Fium e, Italy

               40
  selachians
   Percent




               30
               20                              Fium e, Italy
               10
                0
                1910   1915     1920    1925
                         Years
 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
fish population.
 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; no fishing allowed.
2.   The model is enhanced to allow for predator-
     prey interaction: predators consume prey
3.   Fishing is included in the model
  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.
  Model Variables

Notation
 ti - specific instances in time
 Fi - the prey population at time ti
 Si - the predator population at time ti
 rF - the growth rate of the prey in the absence of
 predators
 rS - the growth rate of the predators in the absence
 of prey
 K - the carrying capacity of prey
Stage 1: Basic Model

In the absence of predators, the fish
population, F, is modeled by
      Fi+1 = Fi + rF * dt * Fi * (1 - Fi/K)
and in the absence of prey, the predator
population, S, is modeled by
             Si+1 = Si –rS * dt * Si
Stage 2: Predator-Prey Interaction

a is the prey kill rate due to encounters with
predators:
  Fi+1 = Fi + rF*dt *Fi*(1 - Fi/K) – a*dt*Fi*Si

b is a parameter that converts prey-predator
encounters to predator birth rate:
        Si+1 = Si - rS*dt*Si + b*dt*Fi*Si
Stage 3: Fishing

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

 Fi+1 = Fi + rF*dt*Fi*(1 - Fi/K) - a*dt*Fi*Si -
                     f*Δt*Fi

  Si+1 = Si - rS* dt *Si + b*dt*Fi*Si - f*dt*Si
Model Initial Conditions and
Parameters

            Plots for the input values:
t0 = 0.0          S0 = 100.0         F0 = 1000.0
dt = 0.02         N = 6000.0         f = 0.005
rS = 0.3          rF = 0.5           a = 0.002
b = 0.0005        K = 4000.0         S0 = 100.0


               Predator-Prey Plots
  D’Ancona’s Question Answered
  (Model Solution)

A decrease in fishing, f, during WWI decreased the
percentage of equilibrium prey population, F, and
increased the percentage of equilibrium predator
population, P.
     f            Prey             Predators
    0.1        800 (82.1%)        175 (17.9%)+
    0.01       620 (74.9%)        208 (25.1%)
    0.001      602 (74.0%)        212 (26.0%)
    0.0001 600 (73.8%)            213 (26.2%)
   + (%) - percentage-of-total catch
Reference URLs
Shodor site: Predator-Prey models
www.shodor.org/scsi/handouts/twosp.html
More discussion about the Fiume fish catch
http://www.math.duke.edu/education/webfe
ats/Word2HTML/Predator.html
Google: Search for “population models”,
predator-prey models”, etc.

				
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