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Modeling Populations: an introduction. AiS Challenge Summer Teacher Institute 2004 Richard Allen. Population Dynamics. Studies how populations change over time
Summer Teacher Institute
Models should reflect biological reality, yet be simple enough that insight maybegained into the population being studied.
Illustrate the development of some basic one- and two-species population models.
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.
P0 P1 P2 … PN
t0 t1 t2 … tN
(0.01/year)*(1 year) * (1,000,000 persons) = 10,000 persons.
(Pi + 1 - Pi) / Pi = r * Δt
r = b - d
Pi + 1 = Pi + r * Δt * Pi
ti+1 = ti + dt; i = 0, 1, ...
The initial population, P0, is given at the initial time, t0.
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.
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
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
t0 – initial time
P0 – initial population
Δt – length of time interval
N – number of time steps
r – population growth rate
ti – ith time value
Pi – population at ti for i = 0, 1, …, N
Set ti = t0
Set Pi = P0
Print ti, Pi
for i = 1, 2, …, N
Set ti = ti + Δt
Set Pi = Pi + r* Δt * Pi
Print ti, Pi
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
Pi+1 = Pi + [r0* (1 - Pi/K)] *Δt* Pi
Population of yeast cells grown under laboratory conditions: P0 = 10, K = 665, r0 = .54, Δt = 0.02
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.
Harvesting populations, removing members from their environment, is a real-world phenomenon.
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
Pi+1 = Pi + rh * (1 – Pi / Kh) *Δt * Pi
rh= r0 - f,Kh= [(r0 – f) / r0] * K
An early predator-prey model
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.
The model development is divided into three stages:
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
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
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
Plots for the input values:
t0 = 0.0 S0 = 100.0 F0 = 1000.0
dt = 0.02 N = 6000.0 f = 0.005
rS = 0.3rF = 0.5 a = 0.002
b = 0.0005 K = 4000.0 S0 = 100.0
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