Class Notes:
02.06.03 today: jam bars, population growth questions to be asking every time you encounter a model: 1. what is the phenomenon we?re claiming to represent?? 2. what are the actors we?ve decided to focus on, ignoring all ?distractions? 3. what are the assumptions that make it easier to 4. representation a. actors b. dynamic 5. spatial or temporal scales 6. precision, realism, generality: which are being sacrificed in order to maximize the other one/two
population growth phenomenon of growth happens at the level (of organization) of the population the variations that affect growth occur one level down, at the level of individuals.
the python code for a pop. growth model ?import Tkinter? brings in a bunch of commands that somebody already wrote and put in a library. this one has to do with graphics N = # of individuals, starts = to 1 parameter R = 1.0125 = avg. # of kids each individual has time steps running from 1 to 500 for every time step, the program calculates the #of individuals in the next generation heart of the model ?Nnext = N * R? this is called a difference equation. it is like a little piece of a differential equation, taking it one step at a time. changing this to ?Nnext = N + N * R? removes death from the model. but we do want to include death. you can interpret R as including death. it simply says that in the next generation, there will be one and a quarter percent more individuals that in the current generation. maybe not all of them are kids. the model doesn?t say. you can predict the population size. the size after n generations is 1.0125 to the nth power. [insert code here]
populations can grow in many patterns. the human population is growing exponentially. in this model, note again that the processes affecting the progress of the model occur at the level of the individual. the actors are individuals. this program models a pop. growing with polynomial growth.
these equations confused us because they didn?t seem to take into account all the factors necessary to be thinking about. more generally, population @ nxt. gen. = pop. now +(pop. now * birth rate) - (pop. now * death rate), where the birth rate is the expected number of kids that EACH individual will have. you can also write the code as: Nnext = N + N*R where R = 0.0125 with the same result
so, now we want to add a parameter to account for the carrying capacity of the environment. although the human population has been growing exponentially, most pop.s grow to a certain point, where they are using all the resources their environment can provide. to model this, we want R to approach zero as the population approaches the carrying capacity. note the assumption here, this decrease in R is modeled as a straight line. easy math, right? that doesn?t mean we actually think the population will decrease that way. this model isn?t necessarily accurate as to how R approaches zero. by the way, this is called the logistic growth equation. R is called the ?population response?.
for exponential growth, change the equation. we want Nsub tplusone/ Nsubt to equal some R. this R = R from before * (1 + Nsub t* ((Rfrom before minus 1)/k)) where k is the carrying capacity. so as N approaches K, R approaches 1. this creates an ?s-curve? graph. it behaves like an exponential curve for small N, but approaches an asymptote later on, while the exp. curve doesn?t. again, the main simplifying assumption is using a straight line for the decrease of R. there is not empirical basis for choosing this relationship.
there are no spatial scales. in this model they are a ?distraction?. one oversimplification is taking k as a constant (natural disasters). is it o.k. to use an average R? maybe we need to take age @ reproduction into account. R is probably a function of lots of things. we?re assuming that, if this is sexual reproduction, that all individuals can find a mate.
if the model fits the data, doesn?t it take into account all the factors above? maybe it does, through a simplification, account for these factors. a long-term question: what does it mean if a model fits reality? does it necissarily mean the model is true? what about if a model doesn?t fit reality? does it mean the model is useless? note: this model doesn?t fit the data.
another model (Malthus). food supply (ie. carrying capacity) incr. linearly, while the food supply incr. exponentially.
changing above code to include a linear increase in carrying capacity ... somebody should do this.
let?s talk about time scale. what is the time step -- how much time? a year? a week (for bacteria)? if we had a certain species in mind, we?d want to have an accurate time step. in the above model, there is an implicit relationship between R and the assumed length of the time step. changing R is equiv. to changing t.
we looked at graphs for different values of R R = 24.0.0125 => s-curve incr. quickly to a horiz. line R = 7x above R => pop. rapidly oscillates rapidly upon reaching something above k where R should = 1. ... so there is different long-term behavior there can be a crash (an extinction), a damped oscillation (overshoot, undershoot, stabilization). stable oscillations occur when 2<R<2.57. above 2.57, the oscillations are non-repeating (chaotic).
question: how accurate is this model?? trapping records show oscillations. this is a record of how many pelts people came back with and was assumed to represent population variation. you can measure R for lab. populations. if you got all the conditions the same, couldn?t you get a perfect test?
a problem: in empirical data, we can?t tell the difference between chaos, and random oscillations (caused by variations in k, for example). we don?t know if actual oscillations in nature are really chaotic. does the fact that our assumptions (constant r, k) don?t hold in the lab or in nature mean that our model is useless?
(Or, put Popperianly, do lab observations falsify the model? If so, is the model unscientific? - Ted)
debate over usefulness of this model. maybe this model isn?t useful unless it can make precise predictions. this is a very simple model, yet it produces seemingly chaotic oscillations. so the model shows us that complicated behavior doesn?t require complicated explanations. that?s useful, isn?t it? so, this model sacrifices precision in order to gain generality and realism. it scores high on generality, because it?s so simple. note: the next step (if this were a class in human population modeling) is to add age structure, then social structure. realism? ok, but the straight line thing isn?t realistic. delta N as a function of N usually isn?t a straight line.
ex. in annual plants, which die every year and leave seeds, have a very natural time scale.
note: this model is very sensitive wrt. the temporal scale. we, as the new generation of modelers, should be trained to think of time as spatial scales.
