### Low-carbon growth in Brazil?

```Predator-Prey Models
Gilberto Câmara
Acknowledgments and thanks
 Many thanks to the following professors for making
slides available on the internet that were reused by
us
 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
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
increases
prey population
increases
predator
population
decreases
as less food
predator population
increases
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
with:
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
```