**POPULATION REGULATION**

**Rate of Increase **

Net Reproductive Rate
R_{0} = SUM*l*_{x}m_{x}

(I still haven't
got a method of inserting Greek letters. See book for proper presentation: where
I wrote SUM you should use the Greek letter sigma.)

Smith constructs a population projection table by
mechanically multiplying for each year the number of new individuals, putting
them in 0, and then moving each age group up 0 ->1, 1->2, 2->3 etc.
while reducing them by the % indicated in the m_{x} column. (It is
easier to use q_{x} the age specific mortality than
m_{x} **and even easier if you have
p**_{x} (p_{x} = 1-q_{x}): p_{x} is age-specific
proportion surviving.

But notice that they are all ways of presenting the
same information).

From this information you get the number of each age in
the next year. Do it again using year 2 numbers to generate the third year, etc.
(Table 18.2)

**The reason for this mechanical exercise is to let you
see how population growth in a multi-age population works. **

Lambda (
**) = finite multiplication rate, the rate of growth in one year (or one time
period). From a population projection Table (Table 18.2), divide the total
number of year x + 1 by the total for year x.** **Or, to generalize: T =
N**_{T}/N_{0}

(T is generation time)

Note that you can go
the other way: if you know or can estimate , you can calculate N for some
generation in the future.

**Generation time - the time from the
birth of the female to the birth of her middle child.**

**with
nonoverlapping generations a unit of time can = a generation**

**but with
overlapping generation, ***T* is *T*_{c} the mean cohort
generation time.

**Calculate ***T*_{c} from a fecundity
schedule (such as Table 18.1) by multiplying each
*l*_{x}m_{x} by its approximate age: therefore *x
lxmx* sum the whole column and divide by the sum of
*l*_{x}m_{x} (which is *R*_{0} of course).
*[Spend a few minutes thinking through what they're trying to do. Its actually
pretty simple tho it gets confused in the jargon].*

*T*_{c} = _ **(***x l*_{x}m_{x}
/R_{0} ).

**However, the really important value is
***r*_{0} (note its a lower case r!!) the intrinsic rate of natural
increase.

**Smith discusses how to estimate it (p. 392) **

we'll come
back to that

**POPULATION GROWTH**

**Exponential Growth**

**All populations are capable of rapid increases.**

*dN/dt =
(b-d)N* is the instantaneous rate of growth for a population: the change in
number over a period of time is the difference between births and deaths times
the number (N) in the population.

*r *the intrinsic rate of
natural increase = (b-d)

**Note population increase is proportional to
population size and the "growth rate", the difference between births and
deaths.**

**This is an instantaneous rate of growth as you reduce the
intervals --> 0 (as in calculus). **

Note: for *r*_{0},
populations stay the same when *r*_{0} = 0, grow when
*r*_{0} >0 and decline when* r*_{0} < 0.

*[Check differences from R*_{0}]

**Calculate
growth**

*N*_{t} = N_{0} e^{rt}

where* N *is the number in the population at times* 0 *and
t*, e *is natural log* *2.718..,* r* is the instantaneous rate of
increase. Complex calculation because populations with overlapping generations
growth with "compound interest".

**[This will be nearly the same as
***N*_{t} = N_{0} *t* provided R0 is
approximately 1. Overlapping generations calculating growth with calculus reduce
the intervals to almost 0, nonoverlapping generations average the same
information, so the numbers will not be quite identical. *Nonoverlapping
generations are easier to visualize, tho very few populations have
nonoverlapping generations. Population projection tables are very tedius:
computers or formulas generally get this information quickly. Thus, these are
used here to teach the the relationships more than they are used in real
situations*.]

**If you plug in*** r0* >0 or an R0 > 1
and determine the population size (N) year after year, you get an exponential
growth curve. also called J-shaped curve. Fig. 18.1, 18.2, 18.3

this
kind of curve seen where resources are not limiting

**While there
is some difference between the rate of increase of elephants and bacteria, both
could cover the earth in their own species based on the physiological natality
rate.**

**But they don't! **

Central to Population Biology and
Ecology generally is the difference between different populations in the way
they behave over time.

**Some are quite constant around an average
value, for others the average is meaningless--some years they are very rare,
other years there's an "outbreak" and they are very abundant.**

**LOGISTIC GROWTH**

**An exponentially growing population
eventually experiences limits: food or space runs out, diseases proliferate,
predators home in on them...**

**The basic equation for this is the
logistic equation**

**dN/dt = rN (K - N)/K**

**dN/dt is the
instantaneous rate of change, r is the intrinsic rate of natural increase, N is
the number in the population and K is carrying capacity.**

**This is just
the expontial growth equation with a negative feedback feature: as N approaches
K, rN is multiplied by a smaller and smaller number. At K, K-N/K = 0 so dN/dt=0
ie the population is not growing. Above K, the population declines until it
stops changing at K.**

**A logistic curve gives and "S shaped
curve"**

**Slow growth at the beginning because N is low, **

slow growth
at the end as N approaches K.

**This is a simple model and assumes:
a relatively stable age distribution, immigration or emigration insignificant,
no time lags. Linear relation between population size and growth rate (ie
constant ***r). *Predetermined level of K.

**So you can see its
an oversimplification, but its 1) useful (some populations act this way) and 2)
a good starting point for analysing a population.**

**Time lags -
frequently real populations overshoot K, drop back below K, rise above next time
etc. because the controls on the population take some time to actually work.**

**(If the 1998 human population all agreed to 2.1 children (lifetime)
today, the population would still double before coming to a stable number
because of all the people under 15 who haven't yet started their families.)**

**To make the logistic equasion more realistic, add reaction time lag (W) a
lag between environmental change (eg reduced food) and population growth**

**dN/dt = rN (K-Nt-w/K)**

**or reproductive time lag (g) a lag
between environmental change and change in length of gestation (or whatever
changes)**

**dN/dt = rNt-g (K-N/K)**

**Fig. 18.6 a) classical
logistic growth**

**e) strong overshoot--as when herbivores eat all the
plants and the population starves. I don't think this one is the plot of an
particular equation.**

**Fig. 18. a, d, c, b Chaos (the mathematical
field; read ***Jurassic Park* for Michael Cricton's popularization)

Chaos
theory is about the behavior of deterministic equations that are unpredictable
in that from where you are now, you can't predict the next value. (They are
predictable in staying within a range of values and in other ways.)

**The logistic equation is such an equation. At real-world r values, it
gives an S shaped curve. As r increases, the values start to oscillate but
dampen to K (Fig. 18.6d) at higher r's there are stable limit cycles (Fig.
18.6c) and at very high r's chaos.**

**(To see this for yourself, using a
computer or calculator plug in values in this rearrangement of the equation
**

The ecological question is: do populations ever really behave
chaotically?

**The probable answer is yes, but having a long enough
valid time series to prove it has proven virtually impossible. Note that since
the logistic equation is very simplistic, adding variables simultaneously makes
it more realistic and more susceptible to chaos dynamics.**

**[You are
not responsible for chaos in this course. The explanation is because Fig. 18.8
is unclearly labled.]**

**Looking at the logistic growth curve: its
confusing because for many species you never see the j part, for others they
don't seem to be at K but rather returning to K or crashing**

*r-selected species and K-selected species* (covered by Smith on
pp. 448-9)

McArthur and Wilson looked at the logistic equation and concluded
that it explained the life histories of species.

Early succession - enter a
rich environment, grow fast (high *r*). But as the environment becomes
filled and competitive, they don't do well, lose out to other species that are
good competitors (*K*-selected). For r-selected species, a little addition
to *r* will make a big difference. For species at *K, r* not
important, but a change in efficiency to raise *K*, will give them an edge.

annual plants vs dominants; short-lived species like mice vs longer-lived
species like weasles, tho it works best for comparing relatives. And you might
want to say "relatively r-selected" and "relatively K-selected" because its a
comparison, not an absolute.

**POPULATION REGULATION**

**WHAT LIMITS POPULATIONS--WHAT STOPS THEIR GROWTH?**

*LIST
THEM!*

**WHICH OF THESE WILL CAUSE A POPULATION TO BE SHAPED
LIKE THE **

LOGISTIC EQUATION?

**(***Why? because they are
density-dependent)*

**DENSITY DEPENDENT POPULATION
REGULATION**

**Consider "regulation". This term gets used to mean
"whatever stops growth" a rather odd kind of regulation.**

**The fact is:
every population stops growing exponentially.**

**NOTE: Because the
environment is variable, K is also generally variable. The best estimate of K is
the population size over time.**

**Note that: Population regulation
requires that there be density dependent birthrates, deathrates, or both. (Fig.
18.7 is a common way ecologists express this sort of thing.)**

**Density-dependence = the change is related to the size of the population
(usually linearly) (and they want it to be positive density dependence at low
numbers and negative density-dependence at high numbers).**

**Intraspecific competition - one of the important ways density
dependent regulation works**

*Note: in ecology it isn't competition
if there's enough of the resource to go around or if the resource is not
essential.*

**There are several patterns of competition:**

**scramble competition - each organism gets a part of the resource, but not
enough**

**exploitative competition - use up the resource before the
others get to it **

contest competition - some get plenty, others get none

**interference competition - Smith says this is a name for contest
competition but not everyone uses it that way.**

**Plant communities
are in pretty constant general competition, intraspecific and interspecific, for
light, water and space. The result is that a particular area will turn out about
the same biomass whether its a few large or many small individuals -- law of
constant final yield. The slope of -3/2 on a log/log plot as competition reduces
the number of individuals is a good description of a lot of plant communities
but the causes are controversial.**

**DENSITY INDEPENDENT
INFLUENCES**

**they affect growth, but can't ***regulate *it

**What are density independent causes of changes in the size of
populations?**

**POPULATION FLUCTUATIONS AND CYCLES**

**The
goal is usually to predict or control populations**

**so we end up
considering multiyear patterns**

**Resilience - the rate at which a
population returns to equilibrium after being disturbed away from
equilibrium**

**note that r selected species should be more resilient;
small animals are more resilient,.**

**(tho it might be useful to look at
resilience in proportion to generation time, not just years).**

**Cycles - when we get to predator prey relations, we'll see that in
simple two species systems, predator and prey would cycle (or herbivore and
plant, etc). Real situations have time lags that make that difficult. There is
an ongoing argument of whether the cycles seen (Fig. 18.17, 18.18, 18.20, 18,21,
18.22) should be attributed to predator prey or other density-dependent biotic
interactions, or to simple inefficient tracking of a changing physical
environment (dry, semi dry, wet, years etc)., or some other cause. About every 5
years someone else analyses the snowshoe hare/lynx relationships to give us a
new explanation. **

**
**