Upload
barbara-snider
View
22
Download
0
Embed Size (px)
DESCRIPTION
CS 170: Computing for the Sciences and Mathematics. Constrained Growth. Administrivia. Last time Unconstrained Growth Today Unconstrained Growth HW3 assigned Thursday’s class will be in P115. Constrained Growth. Population growth usually has constraints Limits include: Food available - PowerPoint PPT Presentation
Citation preview
Administrivia
Last time Unconstrained Growth
Today Unconstrained Growth HW3 assigned
Thursday’s class will be in P115
Constrained Growth
Population growth usually has constraints
Limits include: Food available Shelter/”Room” Disease
These all can be encapsulated in the concept of “Carrying Capacity” (M) The population an environment is capable of supporting
Unconstrained Growth
Rate of change of population is directly proportional to number of individuals in the population (P)
where r is the growth rate.
rPdt
dP
Rate of change of population
D = number of deathsB = number of birthsrate of change of P =
(rate of change of B) – (rate of change of D)
dP
dtdB
dtdD
dt
Death
If population is much less than carrying capacity, what should the behavior look like? No limiting pressure!
Behavior
If population is much less than carrying capacity, almost unconstrained model
Rate of change of D (dD/dt)0
dD
dt 0
dP
dt
dB
dt
Death, part 2
If population is less than but close to carrying capacity, growth is dampened, almost 0
Rate of change of D larger, almost rate of change B
dD
dt
dB
dt rP
Behavior, part 2
For dD/dt = f(rP), multiply rP by something so that
dD/dt 0 for P much less than M In this situation, f 0
dD/dt dB/dt = rP for P less than but close to M In this situation, f 1
What is a possible factor f? One possibility is P/M
dP
dt rP
dD
dt
If population is greater than M…
What is the sign of growth? Negative
How does the rate of change of D compare to the rate of change of B? Greater
Does this situation fit the model?
Discrete logistic equations
D D t D(t t) rP t t P t t M
t
Pbirths deaths
P rP t t t rP t t P t t M
t
Pk(1P t t M
)P t t , where k rt
Equilibrium
Equilibrium solution to differential equation Solution where derivative is always 0
M is an equilibrium point for this model Population remains steady at that value Derivative = 0
Population size tends to M, regardless of non-zero value of population For small displacement from M, P M
Stability
Solution q is stable if there is interval (a, b) containing q, such that if initial population P(0) is in that interval then P(t) is finite for all t > 0 P q
P = M is stable equilibrium
There is an unstable equilibrium point as well… P = 0 is unstable equilibrium Violates P q