Population Growth

1
Population Growth Regulation

• Chpt. 11, pp 181-193 6th edition
• Chpt. 18, pp 389-410 5th edition

2
Population Dynamics
Remember, Ro net reproductive rate
? Ro ? lx my x0 If
Ro 1 then births (b) deaths (d) If
Ro gt 1 then b gt d and N is increasing If
Ro lt 1 then b lt d and N is decreasing.
3
Population Dynamics
With the survivorship schedule (lx), fecundity
schedule (mx or fx) and an additional
parameter.. (px) survival proportion in each
age class We can begin to chart population
growth by constructing a population projection
table. Note these come from the Life Table.
4
Population Growth
Biologists and ecologists typically describe the
manner in which populations grow using some
simple mathematical models. The first type you
should all know..
I. Exponential Growth - is a continual increase
in population size that may occur when a
non-native species invades a suitable, but
unoccupied habitat. (Also, re-introductions,
stocking of reservoirs, etc.)
5
Exponential Growth

• Exponential growth is described by the
  differential equation
• dN/dt (b - d) N (1)
• or
• dN/dt r N (2)
• Integrating equation 1, we get the more useful
  equation..
• Nt No ert (3)
• Remember, e base log s 2.7183 r rate of
  popl. Increase t unit of time

6
II. Geometric Growth

• This equation (3) ..
• Nt No ert (3)
• is very similar to the equation for Geometric
  growth (4)
• Nt No ?t (4)
• In the geometric growth equation, ? replaces er
  
• If N r and ? gt 1
• If N r and ? 0
• If N r and ? lt 0

7
A Comparison of Geometric and Exponential Growth

• For exponential, r 0.186 and for geometric ?
  0.120.
• No 10 for both models.
• Exponential Geometric models yield very similar
  results.
• Examples zebra mussels invading Lake Erie
  Nutria in Atlantic Gulf swamps.

8
III. Logistic Growth

• We know that no population can grow exponentially
  forever, eventually resources such as space and
  food become limiting.
• At increasing density
• births can decline
• mortality can increase
• or emigration can occur
• These changes cause population growth to slow ( r
  ).
• THEORETICALLY r 0 and the population may
  reach some sort of equilibrium.

9
III. Logistic Growth
When r 0, the population is considered to be
at its carrying capacity. The exp-growth model
is modified for the logistic equation dN r
N K - N dt K where, K carrying capacity (I
guess early modelers didnt know how to
spell). SO, (K-N)/K or 1 - N/K is the
unutilized opportunity for population growth.
10
III. Logistic Growth
dN r N K - N dt K When N is low
relative to K most resources are unutilized and
growth is near r, acts like an exponential
model. As N K K - N 0 K If
N gt K dN/dt is negative and N declines
K
11
Logistic Growth Example

• In an imaginary woodlot we have a squirrel
  population with carrying capacity (K) of 50.
• Starting population size is 10, r 0.186.
• Year N Year N
• 0 10 50 49.98
• 1 11 70 49.99
• 5 19
• 8 26
• 10 30
• 15 40
• 25 48
• 40 49.2

K
Maximal growth rate of N is achieved at K/2
12
Assumptions of the Logistic Growth Model
1. Age distribution is at least initially
stable. 2. No immigration or emigration. 3.
Increasing density immediately supresses rate of
population growth (w/o time lags in reproductive
responses). 4. Assume a predetermined value for
K.
13
An Example of the Logistic Growth Model
For a population of nutria escaped from a fur
ranch and invading a VA salt marsh, initial
population size was 12 with an r 0.304 and
assuming K 10,000 dN r N K - N
dt K So, dN 0.304 (12) 10000 - 12
dt 10000 solve for year 1. 0.304
(12) 0.9988 3.64 Thus, N1 123.64
15.64
14

• The logistic equation suggests that populations
  act as systems regulated by and - feedback.
• Growth is stimulated by feedback (e.g.
  exponential growth curve).
• Growth is slowed by - feedback of competition and
  dwindling resources.
• In the Logistic Equation as N K the
  population theoretically responds instantaneously
  -- This is rarely the case in reality.
• Often times population adjustments lag, the
  availability of resources may allow populations
  to overshoot K.

15
Factoring Realism into the Logistic Growth Model
factoring in a reaction time lag (W) into the
model dN r N K - N becomes. dt K
dN r N K - Nt - W dt K
16
Factoring Realism into the Logistic Growth Model
factoring in a reaction time lag (W) AND a
reproductive time lag (g) into the model dN
r N K - Nt - W dt
K becomes.. dN r N t - g K - Nt -
W dt K here g a
lag between the environmental change and a change
in the length of the gestation or its equivalent
(fecundity). Time lags result in fluctuations
in populations
17
Time lags result in fluctuations in populations
18
Chaos
(Chaotic fluctuations) sudden extinction. No
population regulation.
19
Stable Limit Cycle
Population fluctuations about some equilibrium
level (with period and amplitude)
20
Dampened Oscillations
Oscillations dampen over time after initially
overshooting K.
21
Crash
Population overshoots K, then crashes.
Population can either go extinct, or recover.
22
Can you think of any populations that behave like
these models?
1. Logistic Growth 2. Chaos 3. Stable Limit
Cycle 4. Dampened Oscillations 5. Crash
23
It is important to remember that since the
environment is variable, that (K) carrying
capacity is variable, too. Carrying Capacity -
while some resources others are . -
carrying capacity is influenced by the most
limiting resource (LR). - IF LR then K . -
Our best estimate of K comes from averaging the
population size over time, or N . -
Populations tend towards K due to
density-dependent regulation.
24
How do density-dependent feedbacks operate?
Density-dependence leads to increases or
decreases in mortality or reproduction. But,
mortality and births do not both have to be
density-dependent.
B no den-dep. M den-dep.
B M den-dep.
B den-dep. M no den-dep.
B
B
M
M
B
M
K
K
K
Population density
25
Density-Dependent Effects

• Influence a population in proportion to its
  size.
• At N den-dep. Has very little effect.
• As N den-dep. Effects become stronger.

26
Intraspecific Competition
Refers to competition with others of the same
species. Examples competition for food,
nesting sites, or mates, become in short supply
realtive to the demand for them..
demand
supply
Density-dep. competition
No Density-dep.
27
Some other competition terms...
Scramble - when food is limited and the meelee
for food is so great that none get enough food to
survive. Exploitative competition - competition
where each individual is affected by the amount
of the shared resource remaining. Interference
competition - some individuals receive all the
resources needed to survive while most perish or
produce no offspring.
28
Growth Fecundity
When N and resources become insufficient
(particularly food), something has to
give.usually it is growth rate, age at maturity,
or fecundity.

• Rearing density affected the time to
  metamorphosis in tadpoles.
• Since S is linked to size/growth, the longer it
  takes to get big, the less the chance of survival!

S
size
29
More Density-Dependent Feedbacks

• Increased density leads to reduced Growth in
  anchovy larvae and delayed metamorphosis (into
  fish).
• This delay increases Mortality because the
  smaller you are, the more things that can eat you
  and the less able you are to avoid predators.

M
growth
anchovy larvae metamorphose at 12 mm TL.
30
More Density-Dependent Feedbacks

• Harp seals get mature when they achieve 87 of
  mature body weight.
• Delays in reaching this weight (120 kg) will
  result in delays in age-at-maturity, leading to
  reduced population birth rate (r).

R
growth
Mean age at Whelping
31
More Density-Dependent Feedbacks

• Fertility can also be reduced as populations get
  large.

fertility
density
32
More Density-Dependent Feedbacks

• Density-dependent declines in population growth
  occur in bobwhite quail, as well..

• In quail and bison, the response is not linear,
  but curvilinear.

33
More Density-Dependent Feedbacks

• In quail and bison, the response is not linear,
  but curvilinear.

34
More Density-Dependent Feedbacks

• In elk in Yellowstone, the decline in
  reproduction is expressed as a linear decline in
  the proportion of the population of females with
  calves as population size increases.

35
Plant Biomass

• Consider the thinned and unthinned
  plantations in the WVU Forest (from the Forest
  Stand Dynamics Lab).
• Thinned Plantation Unthinned Plantation
• fewer, but larger trees more, but smaller
  trees
• Overall, biomass is similar between these two
  stands of different density, but similar age.
• Regardless of the initial density, all stands
  (theoretically) tend towards a state where the
  canopy will close thinning will occur as some
  trees cannot compete for light.

36
Plant Biomass

• The progressive decline in density and a shift
  from many small, to fewer large individuals is
  called self thinning.
• As density declines, individual weight increases
  and Yield increases in a stand.
• Law of Constant Final Yield
• Regardless of initial density, populations of
  plants converge on a common density that will
  decline over time..thus, a common final yield
  (biomass) from all starting densities!

Individual size
density
37
Law of Self Thinning

• Further, plots of the biomass per individual
  (y-axis) vs. density of survivors (m2, x-axis)
  tends towards a constant slope.
• Plant weight vs. log of Survivor density shows
  similar patterns universally.
• Thus, the -3/2 Power Law of Self Thinning!

100
Slope -3/2
10-1
Dry Wt. Per plant (g)
10-2
103
104
105
Density of Survivors / m2
38
Density-Independent Influences

• During the 1950s - 60s there was much debate
  among terrestrial ecologists as to whether
  density-dependent or density-independent
  processes control populations.
• Insect ecologists believed weather and stochastic
  events controlled populations (den-indep).
• Evolutionary ecologists believed in den-dep.
• Most ecologists now agree that both factors
  (den-dep den-indep) interact to shape
  populations.

39
Population Fluctuations Cycles

• Population ecologists have often noticed cycles
  in populations
• E.g. Ruffed grouse lemmings
• Inherent in these cycles is that the population
  will rebound from some low level to return to
  some equilibrium (?) level.
• Resilience is the rate at which a population
  returns to equilibrium.

40
Lynx and Hare in NW Territories of Canada

• Population fluctuations that are more regular or
  common than chance are said to be cyclic or have
  cycles or oscillations.

41
Key Factor Analysis

• To determine which density-dependent influences
  are at work in a population, KFA is often
  conducted.
• A key factor is a biological or environmental
  condition associated with mortality that causes
  major fluctuations in population size.
• This comes from the Life table - the k-value.
• k log 10 lx - log 10 lx-1
• K x0?? k sum of all ks over all ages
  K or killing power.

42
Key Factor Analysis

• Most of the mortality occurs in
  larvae-to-prepupae (1.22) and pupae-to-adult
  (0.922) stages.
• Thus, this is a key life stage to focus on for
  control of gypse moths.
• Would now look further at causes of this high
  mortality in these stages.

43
Extinction

• When d gt b r lt 1 and N
• .unless the population does something to reverse
  this N 0 (extinction)
• Creatures with narrow ranges and very narrow
  habitat are more prone to extinction.
• At low N, it may be difficult to locate a mature
  mate so females may go unfertilized.
• Today, we see an accelerated mass extinction of
  species via human activities.

44
Two Types of Extinction
Deterministic extinction - some force or change
from which there is no escape. (E.g. dinosaur
theories). Stoachastic extinction - due to
normal random changes in a population or the
environment. This usually only thins a global
population.
45
Review of Models to Describe Population Growth.

• 1. Exponential Growth Nt No ert
• 2. Geometric Growth Nt No ?t
• 3. Logistic Growth dN r N K - N
• dt K

46
Why do we care about population growth?
Management Conservation
N
time
to
If we are studying this population starting at
t0, we say it crashed and we try to manage it---
when in fact the population is in a stable limit
cycle.