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Modeling Populations: an introduction. AiS Challenge Summer Teacher Institute 2004 Richard Allen. Population Dynamics. Studies how populations change over time
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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