organics and mathematics: great together

mad librettist · May 4, 2012

I wanted to start a little thread dedicated to maths!

this may seem off topic, but bear with me.

why math? because with mathematics we can express a pretty complex idea quite simply, and we can also make predictions as welll as reconstruct a model of what has already happened.

Let's start with exponential growth, which is what we see in most pest populations with no controls. For anyone who has seen mites "appear out of nowhere in great numbers", here is your explanation:

from http://www.saburchill.com/IBbiology/chapters02/049.html

Modelling population growth, the maths
Population growth follows the numbers of individuals in a population through time. So the models try to trace what will happen little by little as time passes by.
A small change in time is given by

which is usually reduced to dt.
In population models time may be measured in regular units such as years or even days or it may be measured in units such as generations.
A small change in numbers is given by

which is usually reduced to dN

A change in numbers as time passes by is given by:

mad librettist · May 4, 2012

cont...

Exponential growth
We know the population growth of species in a newly colonised habitat will start exponentially.
This is an example of positive feedback. The more individuals there are in a population the faster they will breed. The growth curve looks like this (often called the J-shaped curve):

The curve gets steeper and steeper with time.
Even Darwin was aware of this. He calculated that 1 pair of elephants could produce 19 million elephants in 700 years.
Modelling the curve
The exponential growth curve can be modelled by the equation:

r is the rate of increase (the rate of reproduction aka biotic potential) of a species.
For example if a population increases by 4% per year we would write:

Of course the higher N becomes the bigger the increase in the population becomes. This leads to exponential growth.
Real examples of exponential growth
The periodic explosion of pest species populations, such as locusts, are modelled by ecologists to predict their growth.
Pest species show exponential growth because humans provide them with a perfect environment for population growth.
Alien species, which often become pest species, also show this pattern. When a new species is introduced accidentally or deliberately into a new environment it has no natural predators or diseases to keep it under control.
European starling (Sturnus vulgaris), a bird, was introduced into the United States. Between 1890 and 1891, 160 of these birds were released in Central Park New York. By 1942 they had spread as far as California. An estimate population of between 140 and 200 million starlings now exist in North America, making it one of the commonest species of bird on Earth.
r-species
These species often have a biology which specialises in maximum reproductive potential when the opportunity arrives. They show periodic population explosions. Pests and pathogens (disease causing organisms) are often r-species.

mad librettist · May 4, 2012

using these formulas, you can actually estimate how long it will take a mite population, for instance, to reach numbers great enough to cause major damage.

as you can see from the shape of the curve describing exponential growth, at the beginning there is very little change in population size, but then all of a sudden the graph curves upwards very quickly.

When most people estimate how a population will grow, they do not account for exponential growth, and are always surprised by how pest numbers can explode seemingly out of nowhere. In fact what has happened is simple and predictable exponential growth.

A wise grower, on the other hand, understands exponential growth and extrapolates what he or she sees more or less according to the formula above in order to predict how the pest population will grow over time.

Jungle.Microbes · May 4, 2012

i like where this is going, I always learn much better when there is a real life application put into it

mad librettist · May 4, 2012

now for a little closer look at the early part of infestation, and why things are slow at the beginning:

The Allee Effect and the living dead

In our example of starlings above, it should be added that there were several attempts to introduce starlings to the US during the 19th century. In Pennsylvania 1850, in Ohio 1872 and in Oregon in 1889. These all failed. The starlings in Central Park New York were introduced in two waves April 1890 and March 1891. It took them10 years to establish themselves in New York city and it’s suburbs but only 40 more years to cover the rest of the country (7.8 million km²).

This slow start is called the Allee Effect and it seems to be related to population density. For animals showing sexual reproduction they need to find a partner. This is not easy when the population density is low. This gives the growth curve a slower start than exponential growth would predict.

This effect of under population can explain why when species threatened with extinction are pushed below a certain level, they cannot recover and eventually die out. These species on the verge of extinction are called the living dead by conservationists.

if we think of the example of the starlings and apply it to a borg scenario, we can see easily why it's easy to beat mites back when their numbers are low, but much harder to do the same once their numbers are high.

in organic pest control, we are trying create the "living dead" scenario for the pest species. when we do, the pest seems to "give up". in fact the pest has not given up, and math shows us exactly what has happened.

mad librettist · May 4, 2012

carrying capacity - the secret to biological controls

here is where the rubber meets the road. carrying capacity is a a great way to model what happens when we use IPM to put pressure on a mite population without eliminating it entirely:

The Carrying Capacity
One of Darwin’s important observations was that a population never continues to grow exponentially for ever. There is a resistance from the environment as the food supply nesting sites decrease (i.e. competition increases) and the numbers of predators and pathogens increase. This resistance results from negative feedback.
This leads to the classic s-shaped or sigmoid population curve (below):

This too can be modelled but it needs a component in it that will slow down the population growth as it reaches a certain point, the carrying capacity of the environment (K).
The equation is called the logistic equation:

Whilst N<K then

will be positive and the population will increase in size. When N=K then

will be zero and the population growth will stop. Should N>K then

will become negative and the population will decrease.

whenever we alter the environment, introduce predators, introduce competition, introduce pathogens, or reduce food sources, we are manipulating the carrying capacity of the system.

mad librettist · May 4, 2012

now let's get some realistic scenarios going. we could model exponential growth in any number of pests, how long it will take for biological controls to work, when to stop spraying, the best time to for preventive spraying, and all kinds of other neat stuff.

can anyone in here volunteer to graph? I don't know how to get it done on a computer.

jaykush · May 5, 2012

ugh i hate math.....

mad librettist · May 5, 2012

math is beautiful!

look at the curve representing exponential growth. it's beautiful!

math is just one enlightened way to look at nature

LouDog420 · May 5, 2012

Good info mad... I don't care to rip out my diff eq book, so I'm staying away from the math. Don't even know where to start in estimating reproduction rates for mites and such... I'm sure it's out there if you look hard enough though, but your explanation is a good description of the concepts

Pest populations are often over exaggerated in regards to your carrying capacity curve. Due to their quick reproduction rates, if left alone, they will greatly overshoot that carrying capacity. Eventually the ebb and flow of overshooting and dying off from lack of resources will level off at the carrying capacity of the room, but I would definitely say anyone seeing pests in their room, can safely assume they are still reproducing exponentially... I like to think a half way decent grower will catch it sooner than the time it takes to hit the carrying capacity!!

What to take away from it all?? Pests will grow exponentially in your grow room, so lock that shit down as soon as you see the little buggers !!

heady blunts · May 5, 2012

nerd alert!

i could probably make you a graph but you have to do all the numbers.

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organics and mathematics: great together

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