Robert M. May (1974): Biological
Populations with Nonoverlapping Generations: Stable Points, Stable Cycles and
Chaos
In the article Biological
Populations with Nonoverlapping Generations: Stable Points, Stable Cycles and
Chaos. May explores how dynamical behavior arises from seemingly simple
nonlinear difference equations that model population growth as intrinsic growth
rate r increases. When using equivalent logistic equation dN/dt
= rN(1-N/K) the only long term outcome is stability where N = K (population
= carrying capacity). In nature populations fail to reflect this stability.
However, when nonlinear difference equations are used, depending on the value
for r a spectrum of dynamic behavior
can be seen. For the two equations provided (the Ricker map and the logistic
map), stable equilibrium similar to that seen in the logistic equation can be
achieved for N = K, as long as N > 0 and 2 > r > 0. However as r increases
the a series of bifurcations occur resulting stable oscillations between 2, 4,
8, 16, 32… population points until the behavior becomes chaotic at a high
enough value of r. This chaotic behavior
is however not stochastic, but determinate. So long as the initial population
is known the population at any time can be determined using these equations however
the slightest differences in starting points can result in drastically different
outcomes. From this we can appreciate that the dynamic fluctuations seen in
nature may not just be the result of immeasurable interactions and influences but
a result of the system itself.
Lorenz is credited as
the first to “discover” these chaotic systems in weather modeling. However, his
papers presumably had little impact on the scientific community as a whole due
the journals they were published in. May can be credited though with bringing
these dynamic models into light revolutionizing how dynamic systems are looked
at in many fields of study.
Vito Volterra (1926): Fluctuations
in the Abundance of a Species considered Mathematically
In this article
Volterra explains fluctuations, or abundances in regards to predator prey
interactions. He states that one species given enough food would grow exponentially
if left to, while a second would perish without food if left alone. However,
when the second preys upon the first in a predator prey relationship the two
can co-exist in equilibrium. The number of prey decrease as the number of
predators increase. As the prey decrease the predators follow this trend
leading to an increase in prey. If plotted as population vs time the
populations of predators and prey would oscillate out of phase to each other
around a specific population size. From this Volterra makes three laws:
I.
The fluctuations seen are periodic
II.
The average number of predators and prey
tend to be a constant as long as nothing else is influencing the system.
III.
If a proportionate number of predators
and prey are destroyed, the number of prey tends to increase while the average number
of predators decreases. However if the prey are “protected” an increase in the
average number of both predator and prey will be seen.
Volterra also goes on
to explain two types of associations he has witnessed, conservative and dissipative
associations. In conservative
associations fluctuations in the numbers of predators and prey appear to remain
constant. While in dissipative
associations the fluctuations dampen and eventually stabilize.
It is interesting to note that while Volterra was
developing the differential equations to model these predator prey interactions
Nicholson and Baily introduced the difference equation modeling parasitoid-host
interactions. May argued the latter being applied more successfully.
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ReplyDeleteMay Paper
ReplyDeleteThis paper seems to take the discrete and continuous population modeling strategies and blend them, to note that even populations with well categorized parameters and all assumptions of particular models met, still show variability from predictions of growth and stable state. This chaos is actually a somewhat predictable pattern of oscillations around a stable value, and that with increasing r increasing oscillations occur. I find this concept pretty useful in explaining the many boom-bust cycles we see in nature, and just as a nice hedge for when predictions aren’t met in reality.
Voltera Paper
ReplyDeleteI liked how well this piece goes with May’s in noting and explaining that populations rarely follow predicted stable values. The simplicity of the system described (essentially a food chain) was really useful in visualizing what was being detailed. I also thought the detail on conservation/preservation was really useful in a practical sense and showed how a simple intervention in a natural system can have unintended consequences.
Voltera Paper
ReplyDeleteI liked how well this piece goes with May’s in noting and explaining that populations rarely follow predicted stable values. The simplicity of the system described (essentially a food chain) was really useful in visualizing what was being detailed. I also thought the detail on conservation/preservation was really useful in a practical sense and showed how a simple intervention in a natural system can have unintended consequences.
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ReplyDeleteBoth these papers are neat in the way that they take relatively simple equations representing theoretical population dynamics and explore their properties to come up with principles that could govern actual population dynamics found in nature. In this regard they fit in well with what we've read of Cole and Nicholson & Bailey. To me the fact that these papers use no data (a few minor example excepted) is both their beauty and their weakness. I guess that's how theoretical ecology works. A few additional thoughts and questions:
ReplyDelete- I found myself trying to distinguish between the approaches/assumptions taken by May, Volterra, Cole, and Nicholson & Bailey. The obvious ones are continuous (Volterra, Cole) v. discrete (May, Cole, Nicholson & Bailey) and single species (Cole, May) v. multi-species (Volterra, Nicholson & Bailey). There are certainly others, but I was having a hard time putting my finger on them. Maybe we can discuss in class.
- Given the theoretical nature of these papers, I'm wondering too what degree the hypotheses generated have been tested and confirmed/refuted by observation. Real & Levin state in the intro that little effort has gone into applying the equations of Lotka and Volterra to actual populations, yet they are conceptually useful. That made me wonder, what makes a theoretical study in ecology useful and/or more tractable for empirical verification?
- I've also been wondering, why is it that so much theoretical ecology is based around population biology? We've seem some devoted to community dynamics, but what about the theory behind other branches of ecology (e.g. ecosystem, landscape ecology)?
- A few point of confusion with Volterra: What is Fig. 2 about? I also didn't fully understand the description of conservative v. dissipative associations. I actually think this piece could've been clearer with a few equations!
May and Volterra: May takes advantage of his skills as physicist to highlight the importance of simple non linear equation in biology, which at that time were misunderstood. These equations have a large spectrum of behavior when growth increases (r) . The example that is used is population growth in man (continuous growth and generation overlap)and 13 years periodical cicadas (different interval and nonoverlap). He also uses other two equation for modeling competition between species.
ReplyDeleteMay: One of the best things about this paper is the presence of an Abstract that has been lacking in the other papers so far. Although I did get a little overwhelmed by the constant mathematical equations, I was able to understand what the author was trying to do because of this abstract. I also liked May's attempt to point out how many difference equations of population biology are not suitable in their usage.
ReplyDeleteVolterra: What I liked most about the Volterra paper was the attempt at an explanation of the role of the species in the food chain as not merely searching for food but its role as prey itself. He seems to consider all inter- and intra- species interactions as well.
ReplyDeleteMay seems to be the first person to use “dynamic” to describe populations. I think this is an appropriate term because prior to Pearl and May’s papers in 1974 most ecologists “characterized population growth as basically simple pattern… Without intervening forces, populations would smoothly approach this limit and then stay there” (p. 183). Like Eric described, populations do not follow a simple pattern and often populations dynamically fluctuate between boom and bust cycles. May’s paper paved the way for the study of theoretical ecology and population dynamics.
ReplyDeleteIt is interesting to me that no one has been able to apply Lotka and Volterra’s multispecies interaction models to actual populations. My question is: why hasn’t anyone been able to replicate or test these models in situ? What are the limiting factors of determining the parameters that would enable the application Lotka and Volterra’s models/equations to real life?
I liked that in the introduction it was noted that the concepts behind chaos theory can be directly traced to work in population dynamics done by ecologists. That means that Malcolm in Jurassic Park has ecology to thank for his rock star standing in the world of mathematics.
ReplyDeleteIn response to why the Lotka/Volterra models don't actually apply to real populations due to the shear scope of how many parameters are involved in multispecies interactions. The Volterra paper kind of lacked equations, but in the introduction there were three equations that helped to visualize what Volterra was saying. The most striking one though was the equation showing that any change in one species is a function of every single other species in the environment. I think if you don't take into account all of these other parameters than the models fail to accurately reflect reality.
Good to know I wasn't the only one reading the intro in Malcolm's voice...
DeleteMay: I appreciate the simplicity of starting with a discrete-time model for populations with non-overlapping generations, as happens with cicadas. What seems amazing is that equations 1 and 2 produce population behavior recognizable from observation (as Eric writes, "the many boom-bust cycles we see in nature"), even though they depend only on the initial population size, the growth rate, and the ratio between the population and the carrying capacity, and not on any environmental variables. What about an example such as Hutchinson's "many larvae on a single mass would die of starvation, while a few larvae would survive" (Hutchinson 1957, p. 425)? Isn't the extinction or survival of the larvae in this case dependent on the availability of food, as well as upon the initial population and the growth rate? Yet the chaotic behavior of the graphs in figure one when the growth rate (r) >= 2.6 seems as though it could be a good description of this type of phenomenon, if the time period were right.
ReplyDeleteIt seems initially paradoxical that a process could be chaotic, yet fully deterministic. Is this a result of not taking environmental factors into account?
May: This article seems to make sense, but it is incredible to me how much of a difference the starting population makes in all of these equations. As the population grows, it is a nice idea to be able to predict the oscillations it will go through over a given amount of time. This being said, there is one part where Mays makes a comment about falsely analyzing the data and thinking it follows a three-point cycle. This is a downfall of not looking at data over a long enough period - it seems with population modeling, a long-term research period is required. I'll be interested in class to talk more about the consensus of this paper and how other people analyzed it.
ReplyDeleteVolterra: This was a very helpful article to try and decipher between the types of population growth models that we have looked at so far. Of all of the theoretical models, this seems to make the most intuitive sense to me - as predator population increases, it makes sense that the number of prey would decrease. I found the third law he made (regarding destroying a proportionate amount of each predator and prey) to be so interesting. Starting from new (or close to it) encourages the prey population to grow and the predator population to decrease. It makes me wonder how closely this follows the proportion of predator:prey in actual systems - for example, what is it in the African savannah, and would the population follow the trend of the third law if destroyed? I am very curious how these laws are actually embodied in today's natural systems.
I also agree with Dunbar's comment regarding why all of these papers are modeling population biology. I suppose it is an easily quantifiable thing to model, but I still wonder why there are not a lot of other theoretical papers in different topics.
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ReplyDeleteI really enjoyed both of these papers, particularly Volterra's. The descriptions of predator/prey dynamics are beautiful, perhaps especially so considering the dearth of paleontological data at the time of writing. The interactions follow surprisingly well when you look at... say... the late Pleistocene die off. The disappearance of the megafauna has a sizable effect on the predator communities, which get less varied, and more prey-specific.
ReplyDeleteI thought that the May and Volterra papers were very interesting and gave rise to many questions in my mind. first: why does an increase in species diversity decrease community stability? I guess this question falls in line with the introductory text in asking what do we mean when we categorize something as "stable" or "complex?" what is the true definition of "stability" in a community, is there such a thing in reality? For Volterras work, How can predator/ prey relationships reach stability? It seems that there are so many factors and interactions (increased complexity) that we still do not fully understand, making me think that stability cannot truly be attained.
ReplyDeleteVolterra: Although May's discrete model depends only on initial population, rate of growth, and density, Volterra's continuous model (while incorporating population size and rate of growth) fluctuates because of the interaction between predator and prey. This interaction could be seen as an environmental force, outside of the population being considered--very different from May's way of looking at populations! I find it interesting that Volterra's model includes the probability of the meeting of predator and prey--a parameter which could be estimated based on Grinnellian surveys.
ReplyDeleteI'm also interested in Volterra's thought process. He uses the increase of predator fish and decrease of prey fish after a period of non-fishing during WWII to illustrate the converse of his Law III (removing a destructive force, and going from the cycle with center omega' to the cycle with center omega, instead of adding a destructive force, and going from the cycle with center omega to center omega'). Volterra calls this "the best actual verification so far found of the theory," but if you find one example of the converse of a theory, does it verify the theory?
Correction to Volterra comment: WWI, not WWII!
ReplyDelete