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Modeling Populations:
an introduction
By Zelalem Nigussa
AIMS
the case of US population and Italy fish
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.
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
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
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.
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
 t0, t1, t2,
…, tN: equally-spaced times at which the
population is determined: Δt = 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
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.
The Malthus Model
Mathematical Equation:
(Pi + 1 - Pi) / Pi = r * Δt
r=b-d
or
Pi + 1 = Pi + r * Δt * Pi
ti+1 = ti + dt; i = 0, 1, ...
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 Δt = 1
year.
Example Calculation
P0 = 76.2; t0 = 1900; Δt = 1; r = 0.013
P1 = P0 + r* Δt *P0 = 76.2 + 0.013*1*76.2 = 77.3;
t1 = t0 + Δt = 1900 + 1 = 1901
P2 = P1 + r* Δt *P1 = 77.3 + 0.013*1*77.3 = 78.3;
t2 = t1 + Δt = 1901 + 1 = 1902
...
P2000 = 277.3 (284.5), t2000 = 2000
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 19002000
Malthus Plot
Pseudo Code
INPUT:
t0 – initial time
P0 – initial population
Δt – 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 + Δt
Set Pi = Pi + r* Δt * 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)] * Δt * Pi
 r0
is maximum possible population growth rate.
 K is called the population carrying capacity.
Logistics Model
Pi+1 = Pi + [r0 * (1 - Pi/K)] *Δt* 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.
Growth of Yeast Cells
Population of yeast cells grown under laboratory
conditions: P0 = 10, K = 665, r0 = .54, Δt = 0.02
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 intervals 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-totalcatch 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.
D’Amcona’ s Queries
D’Amcona was puzzled by the large increase 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 connection between the intensity of fishing and the
populations of food fish and predators?
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 for 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.
2.
3.
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 predatorprey interaction: predators consume prey
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 *Δt * Fi *(1 - Fi/K)
and in the absence of prey, the predator
population, S, is modeled by
Si+1 = Si –rS *Δt *Si
Stage 2: Predator-Prey Interaction
a is the prey kill rate due to encounters with
predators:
Fi+1 = Fi + rF*Δt*Fi*(1 - Fi/K) – a*Δt*Fi*Si
b is a parameter that converts prey-predator
encounters to predator birth rate:
Si+1 = Si - rS*Δt*Si + b*Δt*Fi*Si
Stage 3: Fishing
f is the effective fishing rate for both the
predator and prey populations:
Fi+1 = Fi + rF*Δt*Fi*(1 - Fi/K) - a*Δt*Fi*Si
- f*Δt*Fi
Si+1 = Si - rS*Δt*Si + b*Δt*Fi*Si
f*Δt*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