topic 3- exponential pop growth

Question 1

what is the exponential (cont) pop growth model

Answer 1

Nt=N0e^rt  
-dN/dt for this one is integrated 
- pop size at time 
dN/dt=rN
- cont. differential equation pop growth rate

Question 2

what are we interested with? assumptions?

Answer 2

Interested in continuous growth

Assume time interval is infinitely small (t approaches 0)

Question 3

discrete time model vs exponential

Answer 3

deltaN/deltat=(b-d)N from discrete model is replaced with dN/dt=(b-d)N
it is derivative - describes how a function changes as input changes

Question 4

what is pop growth now descirbed by? variables?

Answer 4

population growth is now described by the change in population size (dN) occurring
during an infinitely small time interval (dt)
b & d are now “instantaneous” rates
b-d=r
r = instantaneous rate Of increase
(aka: intrinsic rate of increase, finite rate of increase, Malthusian parameter

Question 5

solve differential equation?

Answer 5

dN/dt=rN - differential equation, output = instantaneous pop growth rate
solve by integration ->
Nt=N0e^rt

Question 6

geometric vs exponential model? look at the graphs

Answer 6

geometric
- pop size in intervals
-pop growth rate (slope) deltaN/deltaT is the same from (t) to (t+1)
exponential
-smooth line
pop cont. reproducing/increasing
-cont changing slope/ pop growth rate (dN/dt)

Question 7

how do geometric and exponential behave similarly?

Answer 7

follow same curve
λ=e^r
r= loge(λ)= In(λ)

Question 8

how is the pop growth rate approximated? how does it change?

Answer 8

• Population growth rate 
	Population growth rate (dN/dt) is 
	approximated by the slope of the 
	line tangent to the curve at any point 
	in time (t) 
	Population growth rate changes 
	continually over time 
	Populations increase at an 
	accelerating rate for the exponential 
model

Question 9

pop growth for different values of r or λ

Answer 9

r>0 or λ>1 = pop grows
r=0 or λ=1 = pop constant
r<0 or λ<1 = pop decline

Question 10

convert exponential model to linear model

Answer 10

Nt=N0e^rt becomes In(Nt)=In(N0) +rt
In(No) is y intercept
r is slope
if r is contant, the log of N will be a linear function of ti,e

Question 11

exponential doubling time?

Answer 11

In(2)/r=t
if r is 0 = undefined
if d>b, r is negative

Question 12

assumptions?

Answer 12

• Closed population
• Unlimited resources & constant
environment (b, d, r constant)
• All individuals identical (b & d) or average b
& d is constant throughout time (stable age-
class distribution)
• Continuous growth with no time lags