AiS Challenge

1
Modeling Populations an introduction

• AiS Challenge
• Summer Teacher Institute
• 2004
• Richard Allen

2
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 interaction among themselves.
• Models should reflect biological reality,
  yet be simple enough that insight may be
  gained into the population being studied.

3
Overview

• Illustrate the development of some basic one-
  and two-species population models.
• Malthusian (exponential) growth human
  populations
• Logistics growth human populations and yeast
  cell growth
• Logistics growth with harvesting.
• Predator-Prey interaction two fish populations

4
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 popu-lations to double
  was essentially constant (about 25 years at the
  time), regardless of the initial population size.

5
US Population 1650-1800

• Data for U.S. population probably available to
  Malthus.
• The nearly-linear character of the right graph
  indicates good agreement after 1700 with the
  "uninhibited growth" model he produced.

6
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.

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Discrete-in-time Model

• t0, t1, t2, , tN equally-spaced times at which
  the population is determined ?t ti1 - 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
• ----------------------------------
  -----gt t
• t0 t1 t2
  tN

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Note on units.

• The units on birth rate, b, and death rate, d,
  are (1/time) and must be consistent with units on
  dt.
• For example, suppose the time interval, dt 1
  yr, and the growth rate, r, was 1 per year.
• Then, for a population of P 1,000,000 persons,
  the expected number of additions to the
  population in one year would be
• (0.01/year)(1 year) (1,000,000 persons)
  10,000 persons.

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The Malthus Model

• Mathematical Equation
• (Pi 1 - Pi) / Pi r ?t
• r b - d
• or
• Pi 1 Pi r ?t Pi
• ti1 ti dt i 0, 1, ...
• The initial population, P0, is given at the
  initial time, t0.

10
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 ?t 1 year.

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Example Calculation

• P0 76.2 t0 1900 ?t 1 r 0.013
• P1 P0 r ?t P0 76.2 0.013176.2 77.3
  
• t1 t0 ?t 1900 1 1901
• P2 P1 r ?t P1 77.3 0.013177.3 78.3
• t2 t1 ?t 1901 1 1902
• ...
• P2000 277.3 (284.5), t2000 2000

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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

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Pseudo Code

• INPUT
• t0 initial time
• P0 initial population
• ?t length of time interval
• N number of time steps
• r 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 ti t0
• Set Pi P0
• Print ti, Pi

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Pseudo Code

• for i 1, 2, , N
• Set ti ti ?t
• Set Pi Pi r ?t Pi
• Print ti, Pi
• end for

16
Logistics Model

• In 1838, Belgian mathematician Pierre Verhulst
  modified Malthus model to allow growth rate to
  depend on population
• r r0 (1 P/K)
• Pi1 Pi r0 (1 - Pi/K) ?t Pi
• r0 is maximum possible population growth rate.
• K is called the population carrying capacity.

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Logistics Model

• Pi1 Pi r0 (1 - Pi/K) ?t Pi
• ro controls not only population growth rate, but
  population decline rate (P gt 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.

18
Growth of Yeast Cells

• Population of yeast cells grown under laboratory
  conditions P0 10, K 665, r0 .54, ?t 0.02

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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

20
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 fdtP where f is a harvesting intensity
  factor.

21
Logistics Growth with Harvesting

• The logistic model can easily by modified to
  include the effect of harvesting
• Pi1 Pi r0 (1 Pi / K) ?t Pi - f
  ?t Pi
• or
• Pi1 Pi rh (1 Pi / Kh) ?t Pi
• where
• rh r0 - f, Kh (r0 f) / r0 K
• Harvesting

22
A Predator-Prey Model two competing fish
populations

• An early predator-prey model
• In the mid 1920s the Italian biologist Umberto
  DAncona 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-cat
  ch of several species of fish that were brought
  to different Mediterrian ports in the years that
  spanned World War I

23
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.

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DAmcona s Queries

• DAmcona was puzzled by the large in-crease of
  predators during the war.
• He reasoned that this increase was due to the
  decrease in fishing during this period.
• Was this the case? What was the connec-tion
  between the intensity of fishing and the
  populations of food fish and predators?

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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, DAmcona
  con-tacted Vito Volterra, a local mathematician,
  to formulate a model for the growth of predators
  and their prey and the effect of fishing on the
  overall fish population.

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Strategy for Model Development

• The model development is divided into three
  stages
• 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.
  
• The model is enhanced to allow for predator-prey
  interaction predators consume prey
• 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 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

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Stage 1 Basic Model

• In the absence of predators, the fish
  population, F, is modeled by
• Fi1 Fi rF ?t Fi (1 - Fi/K)
• and in the absence of prey, the predator
  population, S, is modeled by
• Si1 Si rS ?t Si

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Stage 2 Predator-Prey Interaction

• a is the prey kill rate due to encounters with
  predators
• Fi1 Fi rF?tFi(1 - Fi/K) a?tFiSi
• b is a parameter that converts prey-predator
  encounters to predator birth rate
• Si1 Si - rS?tSi b?tFiSi

31
Stage 3 Fishing

• f is the effective fishing rate for both the
  predator and prey populations
• Fi1 Fi rF?tFi(1 - Fi/K) - a?tFiSi
  - f?tFi
• Si1 Si - rS?tSi b?tFiSi
  - f?tSi

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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

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DAnconas 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