Constrained Growth

CS 170:Computing for the Sciences

and Mathematics

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

Rate of change of population

Rate of change of B proportional to P

dB

dt rP

dP

dt rP

dD

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

If population is nearing the carrying capacity, what should the behavior look like?

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?

Continuous logistic equations

dD

dt(rP)

P

M

dP

dt(rP) (rP)

P

M r 1

P

M

P

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

If initial population < M, S-shaped graph

If initial population > M

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

HOMEWORK!

READ Module 3.3 in the textbook

Homework 3 Vensim Tutorial #2 Due NEXT Monday

Thursday class in P115 (Lab) Chance to work on HW #3 and ask questions