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Life Tables
Concept 9.1
Life tables show how survival and reproductive rates vary with ___, ___, or ___ ____ ____.
age, size, life cycle stage
Nx=
# of individuals alive at age x
Sx=
=proportion of individuals of age x that survive to age x+1;
Sx = Nx+1/Nx
(survivorship)
lx=
=proportion of individuals that survive from birth (age 0) to age x;
lx = Nx/No
(survival rate)
Fx=
=average # of offspring born to a female while she is of age x.
(fecundity)
Age Structure
Concept 9.2
Life table data can be used to project the future ___ ____, ___, and ____ ____ of a population.
age structure, size, growth rate
9.2
Members of a population whose ages fall within a specified range are said to be part of the same ___ ____.
age class
9.2
Age structure influences how ___ populations grow
rapidly(at least initially)
9.2
Age structure and population size can be predicted from __ ___ ___
life table data
ratio that provides a measure of the yearly population growth rate
Nt+1
_____
λ =
Nt
9.2
Populations grow at ___ ___ when age-specific birth and death rates ___ change over time
fixed rates
don't
Exponential Growth
Concept 9.3
Populations can grow exponentially when conditions are ___, but exponential growth cannot continue ____.
favorable
indefinitely
9.3
Populations grow ____ when reproduction occurs at regular time intervals
geometrically
9.3
Geometric Growth Equation (9.1)
Nt+1 = λNt

Nt is population size after t generations or, equivalently, after t discrete time periods (e.g., t years if there is one generation per year),
λ is any # greater than zero.
9.3
λ is referred to as the ____ ____ ____ ____;
λ is also known as the (__ ___) ____ ___ _ ______
geometric population growth rate;
(per capita) finite rate of increase.
9.3
when the population "growth" rate λ is between 0 and 1, the population _________, but rather _____ in size over time.
does not grow,
decreases
9.3
2nd equation for geometric growth(9.2)
Nt = λ^tNo

No initial population size
9.3
rearrange Equation 9.1 to get an estimate for λ
λ=Nt+1/Nt
9.3
If you want to predict the population size in the next time period, and you know λ and the current population size, you can use Equation
9.1
9.3
If you know the population size in both the current and previous time periods, you can rearrange equation 9.* to get an estimate for __
9.1
λ
9.3
use Equation __ to predict the size of the population after any number of discrete time periods.
9.2
Nt=λ^tNo
9.3
Populations grow exponentially when reproduction occurs _____
continuously
9.3
When individuals within a species reproduce at varying points in time, generations can overlap, and the species is said to reproduce ____
continuously
9.3
When a population of a species with continuous reproduction changes in size by a constant proportion at each instant in time, we refer to the growth that results as _____ ___
exponential growth
9.3
2 Equations for Exponential Growth
9.3 & 9.4
dN/dt = rN (9.3)

and

N(t) = N(0)e^(rt) (9.4)

N(t) is pop. size at each moment in time, t.
9.3
In equation 9.3, dN/dt represents the ____ ___ _____ in population size at each instant in time;
rate of change
9.3
we see from the equation that dN/dt equals a ____ ____ multiplied by the ____ population size, __.
constant rate (r)
current
N
9.3
r is called the ___ ___ ___ __ or the ______________
exponential population growth rate
or the
(per capita) intrinsic rate of increase.
9.3
Geometric and exponential growth curves overlap because equations 9.2&9.4 are similar in form, except that __ in equation 9.2 is replaced by e^r in equation 9.4.
λ
9.3
Thus, if we want to compare the results of discrete&continuous time growth models, we can calculate λ from r, or vice versa:
λ=
r=
λ=e^r
r=ln(λ)
ln(λ) is the natural logarithm of λ,
or LOGe(λ)
e.g., if λ=2, an equivalent value for r would be r=ln(2)
9.3
Look at equations 9.1&9.3. In equation 9.1, which value of λ will ensure that the population doe snot change in size from one time period to the next? Similarly, in equation 9.3, which value of r causes the population to remain fixed in size?
The answers are λ=1(because then Nt+1=Nt) & r=0(because then the rate at which the population size changes is 0).
9.3
When λ<1 (or r<0), the population will decline to _____, whereas when λ>1 (or r>0), the population will ____ _____ (or ______) to form a _______ curve.
extinction,
increase exponentially(or geometrically)
J-shaped
9.3
_____ times and ___ reproductive rates provide useful measures of population growth
Doubling and net
9.3
by solving equation 9.4 for the time it takes a population to increase from its initial size, N(0), to 2N(0), doubling times can be estimated as
(9.5)
tｄ=ln(2)/r
r is the exponential growth rate
9.3
We can use life table data (broken down by age class) to calculate the
Net Reproductive Rate (Ro), the mean # of offspring produced by an individual during its lifetime. Ro is calculated as ________(9.6)
Xlast
R₀ = ∑ lx Fx
Xfirst

where x is age, Xfirst is the age of 1st reproduction, Xlast is the age of last reproduction, and lx&Fx are survivorship&fecundity.
Note that to estimate Ro, we multiply by lx because the likelihood of surviving to each reproductive age is just as important as the # of offspring produced at that age (Fx).
9.3
Whenever Ro is greater than _, measured from one generation to the next, λ will be greater than 1 (and r>0).
1
9.3
Populations can grow rapidly because they increase by _____
multiplication
9.3
Eqs 9.1&9.3 show that populations increase by multiplication, not addition: at each point in time, the population changes in size according to the multiplier _ or _.
λ or r.

As a result, populations have the potential to add large #s of individuals rapidly whenever λ>1 or r>0.
Effects of Density
Concept 9.4
Population size can be determined by ______ and ______ factors.
density-dependent
density-independent
9.4
Density-independent factors can ____ population size
determine
9.4
Factors such as temperature & precipitation, as well as catastrophic events such as floods or hurricanes, are often referred to as _______ factors,
density-independent factors, meaning that their effects on birth&death rates are independent of the # of individuals in the population.
9.4
In terms of population growth rates, per capita growth rates (λ or r) are _____ when they are not a function of population density.
density-independent
9.4
Density-dependent factors ______ population size
regulate
9.4
As densities ____, it is common for birth rates to ____, death rates to ____, & dispersal from the population(emigration) to _____
increase,
decrease,
increase,
increase---
all of which tend to decrease population size.
9.4
When densities ____, birth rates tend to ___ and death & emigration rates ____.
decrease,
increase
decrease.
9.4
When one or more density-dependent factors cause population to increase when #s are low & decrease when #s are high, ____ _____ is said to occur.
population regulation
9.4
Density-independent factors can have large effects on population size, but they do not ____ population size because they do not consistently increase population size when it is small & decrease population size when it is large.
regulate

Thus, by definition, only density-dependent factors can regulate population size.
9.4
When birth, death or dispersal rates show strong density dependence, population growth rates (λ or r) may ___ as densities ____.
decline
increase.
9.4
If densities become high enough to cause λ to equal _ (or r to equal _), the population ____ ____ entirely;
if λ becomes less than _(or r<_), the population _____.
λ=1 or r=0
stops growing

λ<1 or r<0
declines.
Logistic Growth
Concept 9.5
The logistic equation incorporates limits to growth & shows how a population may ____ at a maximum size, the ____ _____.
stabilize
carrying capacity.
9.5
In some cases, the growth of a population can be represented by an S-shaped curve. Such a population exhibits ____ ____
logistic growth
in which its abundance increases rapidly at 1st, then stabilizes at a population size known as the carrying capacity(the maximum population size that can be supported indefinitely by the environment).
9.5
The logistic equation models _________ population growth
density-dependent
9.5
To modify the exponential growth equation to make it more ____, we replace the assumption that r is _____ with the assumption that r declines in a straight line as density (N) _____. When we do this, we obtain the _____ _____:
realistic,
constant
increases
logistic equation: (9.7)
9.5
Logistic Equation (9.7)
dN/dt = rN [1 - (N/K)]

N is population density
r is the (per capita) population growth rate under ideal conditions, & K is the density at which the population stops increasing in size.
9.5
K can be interpreted as the ____ __ of the population, while the term (1 - N/K) can be viewed as representing the effect of factors that ____ the population growth rate from the constant rate (r) seen in exponential growth.
carrying capacity
reduce
9.5
Exponential growth: dN/dt = __

Logistic growth: dN/dt = _______
Exp dN/dt = rN

Logstc dN/dt = rN (1 - N/K)