Population Ecology
I. Attributes
II.Distribution
III. Population Growth – changes in size through time
A. Calculating Growth Rates
1. Measuring Rate of Geometric Growth – Discrete Time (breeding seasons)
- compare population size at same point in each ‘generation’ (after breeding season).
- one generation to the next: N(t+1) = N(t)λ
- extrapolating from initial generation to any generation in the future: N(t) = N(0)λt
2. Measuring Exponential Growth – Continuous Growth
- can measure population size at any two times; it is increasing continuously: N (t) = N(0)ert
3. Equivalency
- they can both describe the same data and the same pattern of growth; the difference in WHEN the
growth is occurring: all at once (discrete) or continuously.
- λ = er, or ln (λ) = r
4. Attributes of Growth Rate
- per capita estimate, averaged over the population
- function of per capita birth and death rates, averaged over population r = (b – d)
- more complex models include migration: r = (b + i) – (d + e)
5. Changes in Population Size
- exponential models are simpler to use, so these are generally preferred
- N(t) = N(0)ert ….
- We can measure the rate of instantaneous change by taking the derivative with respect to time
- dN/dt = rN so, the change in pop size depends on the intrinsic per capita rate of growth and the size of
the population; larger pops will add more individuals per unit time than smaller pops with the same growth rate..
it is like an interest rate, with the population size being the principal upon which interest (offspring) is generated.
B. The Effects of Age Structure
- Organisms may not be equal in their probability of reproducing ( birth rate, b, or fecundity) or of dying (d). So,
to more closely approximate how a population will grow, the age specific birth and death rates must be taken
into account. This is done in a “life table”
1. Life Table
- Types
- static: measure parameters over different age classes for one time period
- dynamic (“cohort”) – follow a group through their life.
- Components
- survivorship is proportion surviving to next birthday. (lx) – three prototypic survivorship curves
- mortality rate is proportion dying in that age class. (qx)
- fecundity is number of offspring produced (on average), per capita, for that age class (bx)
- compute age-class specific contribution of offspring to next generation.
- Lm = average number alive during the age class interval = (nx + nx+1)/2
- Ro = net rate of reproduction = Σ(lxbx)
- Generation Time (T) = Σ(xlxbx)/ Σ(lxbx)
- Results
- As a population grows, it will create a ‘stable age class distribution’ in which the proportional
representation of each age class equilibrates, and each age class grows at the same rate. When this happens,
there is no need to compute age-specific growth rates… they are the same! So, we should be able to compute r
for the whole population. This is called the “intrinsic rate of increase” and is estimated as rm = ln(Ro)/T when the
population has reached a stable age class distribution.
- Doubling time = t2 = ln(2)/r = 0.69/r
2. Age Class Distributions
- after time, a stable age class distribution (ACD) is reached.
- when comparing populations of similar organisms with the same survivorship curves, the shape of the
ACD can determine how fast the population may expand.
- However, the shape of the ACD is NOT an index of whether it is stable or not – a bottom-heavy ACD can
be stable if there is high juvenile mortality, for instance.
C. Growth Potential
- A population’s growth potential is a function of its survivorship and
fecundity schedules, and is calculated as Net Reproductive Rate (Ro) and
intrinsic rate of increase (r).
- Ro = Σlxbx and represents the mean number of daughters produced
over the lifetime of each female.
“Replacement”, resulting in no growth of the population occurs when Ro = 1. Anything >1 means the population
has the potential to increase exponentially.
- T = generation time = T = Σ(xlxbx)/ Σ(lxbx)
- r = ln(Ro)/T
- so, a population will grow more rapidly if R increases or if T decreases (shorter generation times)
- Malthus - All populations have the capacity to expand exponentially. Even those with a slow growth
rate because of delayed reproduction and low fecundity, like large mammals, can increase to huge populations
rapidly if unrestricted.
D. Life History Redux
1. Review
- Selection is, by definition, “differential reproductive success”.
- So, organisms (or subpopulations) that reproduce the most will dominate the population over time.
- Selection will favor a particular survivorship and fecundity schedule in a given environment
2. Maximizing Reproduction
- Increase fecundity (birth rate), increase rate of population growth (obviously).
- Shorter generation time – increase rate of population growth.
- increasing survivorship only pays off, reproductively, if:
- survival to a particular age is a required prerequisite for reproduction.
- by surviving and growing, an organism increases the total energy budget so much that, when it
does reproduce, it can produce many many more offspring at ‘catch up’ to the production of early reproducers.
D. Limits on Population Growth: Density Dependence
1. Malthus
P1: All populations have the capacity to increase exponentially
P2: But eventually, as the population increases, a necessary resource will become limiting
C1: At this point, there will be a struggle for existence and the population will stop growing
2. Effects of density on the intrinsic rate of increase r = (b-d)
- as the population increase is size, competition may cause an increase
in mortality rate (d), a decrease in birth rate (b), or both.
- This occurs in animals (song sparrow) and plants
- competitive effects in plants also produce the “-3/2 self-thinning
pattern”, in which the slope of the line describing the relationship between plant size
and plant density has a slope of -3/2.
- Result
– produce an equilibrial density where b= d. This is called the “carrying capacity” and it is a
function of a particular population in a specific environment; it is not a characteristic of one (population) or the
other. In a different environment with more resources, for instance, the population may equilibrate at a higher
density.
- The Model: Logistic Growth Curve
dN/dt = rN((K-N)/K)
- when the population is very small, (K-N)/K goes to one and the growth rate approximates the
exponential maximum of rN.
- When the population is large and almost as large as K. then (K-N)/K becomes a very small
fraction, and the growth rate of the population slows.
- When the population is at the carrying capacity, N=K, (K-N)K goes to zero, and the population
does not grow… it has reached equilibrium where b = d.
- When the population is larger than the carrying capacity, then (K-N)/K becomes negative and
the population declines in size.
The first bullet point here seems unrealistic. Can we believe that the smaller a population gets,
the faster its rate of growth? Wouldn’t it be more difficult to find a sexual partner? Might there not be a
minimum size that a population can reach and still maintain itself in the community without necessarily going
extinct? Ye = Minimum Population Size, and it is modeled by Dn/dt = rN[(K-n)/K] [(N-m)/N].
E. Temporal Dynamics
- Populations are not in equilibrium; their numbers fluctuate for a variety
of reasons both extrinsic (good and bad years in the environment) and intrinsic
(relationships between r and K). These fluctuations in population size often affect
members of particular age classes differently, so fluctuations usually cause
changes in the age class distribution of a population, further complicating
projections of future growth.
1. Discrete Growth
- roughly speaking, the change in population size will be a function of its intrinsic rate of growth (r) and
the size of the current population relative to the carrying capacity.
- this can be represented fairly easily in terms of logs: : log(Nt+1) - log(Nt) = r [log(K) - log(Nt)]
- if r < 1, then it will grow a fraction of the difference between N and K each generation, creating a
smooth, asymptotic approach to K.
- if r = 1, then it will grow by the exact difference between N and K and reach K in one generation.
- if 1 < r < 2, then N will converge on K in a damped oscillation. Overshooting and undershooting by less
and less.
- if r = 2, then the population will oscillate above and below K at the same level.
- if 2 < r < 2.5, then N will bounce above and below K within a bounded area, in a stable limit cycle.
- if r > 2.5 or so, then these oscillations will be unbounded and chaotic.
2. Continuous Growth
- Populations that are growing continuously should be able to match
reproduction to resources more closely. However, these responses are not
instantaneous – there are “time lags” due to the biological processes of metabolism
and development. In addition, if age classes may not be affected equally by density
dependent effects in the same way. So, if the increase in mortality that occurs as a
population increases in size falls disproportionately on the young, then the
population will have fewer reproductive adults in the future and may not match its growth rate based on
population size, alone. In this case, the growth rate would be appropriate for a population size in the past –
there would be a time lag. Lags can also occur because individuals accumulate resource at one time, and
reproduce later. So, consider some individuals that are harvesting resources when the population is small. They
harvest lots of resource and will eventually produce lots of offspring – commensurate with the low population
density and lack of density dependent limitations. During the interim, in a continuously breeding population,
other organisms will have reproduced, making the population larger. Now when those target individuals get
around to having their bunch of offspring, their high rate of reproduction is no longer appropriate for the larger
population, and the population overshoots K.
- What is critical is the product of the Reproductive Potential and time lag:
- if these are both low, then the potential to overshoot is small.
- as either of both of these increase, then the magnitude of ‘overshoot’ increases. When the product Rt>
1.6, there are huge swings in population size because:
- with long lags, the population is not matching reproductive potential to current environmental
Limitations
- and with a high R, they produce lots of offspring and overshoot by a lot!
F. Spatial Dynamics and Metapopulations
The persistence of a population composed of small isolated subpopulations depends on the balance between
extinction rates of these small populations and the colonization rate needed to ‘resuce’ these populations or
recolonize open habitats.
IF:
p = proportion of suitable habitats occupied
e = extinction rate
ep = prop of habitats being vacated
Rate of colonization depends on fraction of patches that are empty (1-p), and number of occupied patches
dispersing colonists.
c = migration rate, and so rate of colonization = cp(1-p)
Change in patch occupancy: dp/dt = cp(1-p) - ep
Equilibrium patch occupancy Peq = 1 – e/c. So, what is critical is the relationship between e and c.
If e> c, then population will go extinct.
Extinction rates increase in small populations.
Colonnization rates decline as populations become more isolated.
So, species consisting of small, isolated populations are more likely to go extinct than species consisting of large,
connected populations.
Why is this Important?
We are a population, and we are dependent on other populations. All our densities vary through time. These
simple models show us that some very complicated behaviors and dynamics (different types of oscillations, for
instance) can be produced by some very simple changes (changes in reproductive potential of a population).
None of these values/parameters (r, K, lx, mx) are STATIC through time and space for a species. A population’s
reproductive potential and carry capacity will change from one environment to the next, or may change as it
adapts to a new environment or is confronted with a new predator, pathogen, or pest. Populations that are
stable can lose that stability. Populations that are growing one minute can crash the next. And it doesn’t take
much – just some very small changes in the principal attributes of a population. The “balance of nature” is a
façade – populations are almost always in flux – in a very dynamic equilibrium that can be very sensitive to even
small changes. And we are dependent on that dynamic system.
Study Questions:
1) Describe how net reproductive rate and generation time are calculated, and describe the effects they have
growth potential (intrinsic rate of increase) of a population.
2) In terms of life history adaptations and maximizing reproductive success, when will delaying reproduction be
favored? Why is early reproduction such a boon to long-term genetic domination in a population?
3) How can changes in the birth rate or death rate with increasing population size result in the population
stabilizing at an equilibrium, rather than continuing to increase exponentially?
4) Outline Malthus’s argument for the importantce of density dependence.
5) Write the logistic equation and explain how the new term (K-N)/K works to damp growth at high N.
6) What is a minimum viable population, why (biologically) might it occur, and how might we model it’s effects
on population growth?
7) With discrete growth, whow and why does a change in r affect patterns of oscillation?
8) With continous growth, how can oscillations occur, and why does an increase in the length of the time lag
increase the amplitude of the oscillation?
Study Questions:
1) Why do source-sink models predict that certain populations, exploiting certain habitat patches, would more
likely be donors (sources) than other patches? (This makes it more complex than simple metapopulation
models).
2) Landscape models consider the effects that a heterogenous matrix might have on 1) patch quality and 2)
migration rates between patches. Explain the two effects.
3) Brown found a weak but significant relationship between a species’ density at the center of its range and the
size of the range. How did he explain this pattern?
4) What is the ‘energy equivalence rule’ and what affect does it have on the relationship between the densities
of large and small animals per unit area?
5) In the U.S. large ranges of abundant species are oriented east-west while smaller ranges of rare species are
often oriented N-S. How does this related to a species being a generalist or specialist?
6)When would you use a continuous or a discrete model to describe the growth of a particular population?
7) If all age classes have the same survivorship and fecundity schedules, what is the simplest equation for
continuous growth that you could use to predict the change in population size over time?
8) Why does the growth rate of a population oscillate and then stabilize over time?
9) How does a cohort life table differ from a static life table?
10) Consider the cohort life table on the following page:
X
0
1
2
3
Nx
150
100
50
0
lx
dx
qx
lm
Fill in the table and calculate the life expectancies for eache age class (ex).
11) Now calculate the generation time and doubling time
X
Nx
lx
mx
0
150
0
1
100
2
2
50
1
3
0
0
lxmx
xlxmx
ex