Download Computational Models

COMPUTATIONAL
MODELS
Calc I in One Slide….
• Consider the function:
• We denote the derivative:
• In this case:
dy
 2x
dx
y  x2
dy
dx
Unconstrained Population Growth
dp
 rP
dt
• r is the growth rate. P is the population at a given time.
• This is the differential equation because it includes a
derivative.
• Solving for this differential equation means finding an
equation for P given an initial population and a time.
Solving
dp
 rP
dt
• This one can be solved:
P  P0e rt
• Unfortunately, it is usually impossible.
• In those cases we approximate with a finite difference
equation: population(t )  population(t  t )  growth * t
• This looks like Python code!
Constrained Population Growth
• It’s unrealistic to suppose that a population can grow
exponentially forever.
• Assume the system has some carrying capacity, call it K.
(This is the max # of organisms that can be supported.)
Then the differential equation/rate of change of the
population looks like:
dP
P

 rP1  
dt
 K
Lotka-Volterra Equations: Multiple
Interacting Species
• Track the population of two species:
• V (prey)
dV
• P (predator)
dt
 kVV  kVPVP
dP
 k PVVP  k P P
dt
• Here, kv is the growth rate of V, kVP is the proportionality constant
for the reduction of V interacting with P, kPV is the constant for the
increase of P interacting with V, and kP is the death rate of P.
Analyzing Lotka-Volterra
• This is getting complicated. What can we do to
understand the system?
Analyzing Lotka-Volterra
• This is getting complicated. What can we do to
understand the system?
• Solve for the equilibrium points!
(These are points where the derivative is always zero.)
• We find:
V
kP
k PV
kV
P
kVP
Equilibrium Points
• A system may or may not have equilibrium points.
• Three different kinds:
• Unstable: system heads off to 0 or infinity if it is perturbed
• Stable: system returns to the equilibrium point if it is perturbed
• Marginally stable: system oscillates when perturbed
Activity: Look at Lotka-Volterra module
Differential Equations: They’re not just for
Population Modeling!
• An incredibly versatile tool:
• Epidemic modeling
• Modeling of physical systems
• Planets
• Pendulums
• Connonballs
• The list is endless.
Differential Equations: They’re not always
the Right Tool
• Some examples….
• Empirical models (based on data, used to make predictions)
• Simulations
• Randomness
• Cellular automaton
Cellular Automaton (CA)
• Discrete model
• Consists of grid (of any finite dimension) of cells, each in
one of a finite # of states.
• Each cell has a neighborhood consisting of a specific set
of cells relative to it.
• New generations are created based on a set of rules
determining states of cells. Rules applied to each cell
simultaneously.
• Conway’s Game of Life!
Lattice Models
• Similar to CA
• Except, cells are selected at random, and randomly
interact with neighbors.
• Example:
r
V O 

2V
p
V P

2P
d
P O 

2O
(Reproduction)
(Predation)
(Starvation)
Mean Field Approximation
• Differential Equation Approximation of this lattice model:
dV
 rVO  pVP
dt
dP
 pVP  dPO
dt
dO
  rVO  dPO
dt
• This would hold if:
• The environment were infinitely large.
• Every individual could interact with every other individual
regardless of location.
Equilibrium Points
• These give rise to one non-trivial steady state (equilibrium
pt):
V
d
d r p
P
r
d r p
• These are marginally stable!
Wouldn’t This Make More Sense?
r
V O 

2V
(Reproduction)
p
V P

2 P (Predation)
d
(Starvation)
P

2O
• Probably, but it has boring stable equilibria.