Low-carbon growth in Brazil?

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Transcript Low-carbon growth in Brazil?

Predator-Prey Models
Pedro Ribeiro de Andrade
Gilberto Câmara
Acknowledgments and thanks
 Many thanks to the following professors for making
slides available on the internet that were reused by
 Abdessamad Tridane (ASU)
 Gleen Ledder (Univ of Nebraska)
 Roger Day (Illinois State University)
“nature red in tooth and claw”
One species uses another as a food resource: lynx
and hare.
The Hudson’s Bay Company
hare and lynx populations (Canada)
Note regular periodicity, and lag by lynx population peaks just after hare peaks
Predator-prey systems
The principal cause of death among the prey
is being eaten by a predator.
The birth and survival rates of the predators depend
on their available food supply—namely, the prey.
Predator-prey systems
Two species encounter each other at a rate that is
proportional to both populations
Predator-prey cycles
normal prey population
prey population
prey population
as less food
predator population
as more food
prey population decreases
because of more predators
Generic Model
• f(x) prey growth term
• g(y) predator mortality term
• h(x,y) predation term
• e - prey into predator biomass conversion coefficient
Lotka-Volterra Model
r - prey growth rate : Malthus law
m - predator mortality rate : natural mortality
a and b predation coefficients : b=ea
e prey into predator biomass conversion coefficient
Predator-prey population fluctuations in
Lotka-Volterra model
Predator-prey systems
Suppose that populations of rabbits and wolves
are described by the Lotka-Volterra equations
k = 0.08, a = 0.001, r = 0.02, b = 0.00002
The time t is measured in months.
There are 40 wolfes and 1000 rabbits
Phase plane
Variation of one species in relation to the other
Phase trajectories: solution curve
A phase trajectory is a path traced out by solutions (R, W)
as time goes by.
Equilibrium point
The point (1000, 80) is inside all the solution curves. It
corresponds to the equilibrium solution R = 1000, W = 80.
Hare-lynx data
Hare-lynx data