Rate of Increase
Net Reproductive Rate R0 = SUMlxmx

(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 mx column. (It is easier to use qx the age specific mortality than mx
and even easier if you have px (px = 1-qx): px 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 = NT/N0
(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 Tc the mean cohort generation time.
Calculate Tc from a fecundity schedule (such as Table 18.1) by multiplying each lxmx by its approximate age: therefore x lxmx sum the whole column and divide by the sum of lxmx (which is R0 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].
Tc = _ (x lxmx /R0 ).

However, the really important value is r0 (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

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 r0, populations stay the same when r0 = 0, grow when r0 >0 and decline when r0 < 0.

[Check differences from R0]

Calculate growth
Nt = N0 ert
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 Nt = N0 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.

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.



(Why? because they are density-dependent)

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.

they affect growth, but can't regulate it
What are density independent causes of changes in the size of populations?

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.