Exponential Growth

what are the components of a closed system

no immigration, no emigration, births increase pop size, deaths decrease pop size

population growth models are used to estimate…

population size many generations in the future

life tables used to determine

population growth from one generation to the next generation

to determine a change in population size (N) from one point in time to sometime later:

N(later)=N(now) + Births - Deaths

what are the underlying assumptions of the exponential model

• all individuals have the same average birth (b) and death (d) rates

• b and d are constant through time

• births and deaths occur continuously throughout year

• no migration (I or E) - the pop is closed

• resources are unlimited

what is the name

exponential population growth

what does the geometric model assume

discrete breeding (population only grows during breeding season) but similar equation to exponential model

variable for population size

N

variable for change in time

∆t

variable for birth

b

variable for death

d

variable for number of inds in the pop at t=0

N_o

per capita growth rate

r=b-d

exponential growth rate

r is an instantaneous per capita rate of increase

exponential growth equation

dN/dt = rN

for the per capita growth rate, rate of change in the pop size changes _____

continuously

the rate of change in pop size over a _____ time interval is equal to the ____ multiplied by ______

short, per capita rate of change, the number of inds in population N at t

r=+ _____ r=- _____ r=0______

increase, decrease, constant

do we typically observe exponential growth in nature

no because underlying assumption are often violated

what is violated in the assumption that all individuals have the same average b and d rates

different age classes generally have different b and d (life tables)

what is violated in the assumption that r(b and d) are constant through time

it’s rare because

• age structure of a pop changes through time(ie the number of inds in each age class change through time)

• environmental conditions change through time and continuously influence b, d, and r

(r the highest growth rate that the pop has potential to achieve if a stable age distribution and constant b and d)

what is violated in the assumption that there is no migration (I and E) - the population is closed

populations are rarely closed (except in the laboratory)

what is violated in the assumption that resources are unlimited

resources usually become limited

when population size is low:

• no competition (resources are unlimited)

• population grows at the intrinsic rate of increase, r (exponential growth) = highest growth potential

when population size is high

• intraspecific competition increases (resources become limited)

• competition decreases r (b ↓ and d ↑) as N increases (=density dependence)

• N levels off and r=0

why has the human pop size not leveled off

• medical advances (less death more birth)

• technological advances (agriculture, aquaculture, unlimited food supply)

we can determine whether a population grows from one generation to the next using age specific survival and fecundity data from

life tables

to determine population size many generations in the future, we must use mathematical models such as

exponential (& geometric) models

the exponential (& geometric) models require the knowledge of ____, _____ and ______

average b, average d, pop size (N)

most species violate the assumptions in the exponential model for many reasons: ______, _______, ________

• b, d ( r ) vary through time

• populations are not closed

• resources are usually limited

exponential growth rates, or intrinsic rates of increase ( r ), will only be accurate over the long term if ____ and ______

• age structure is stable

• b and d do not vary