Systems Paleoecology – Allee Effects I

Tags

Allee effect, extinction, paleoecology, stochastic extinction

WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series

1. Welcome Back Video
2. Introduction
3. Malthusian Populations
4. Logistic Populations
5. Logistic Populations II
6. Deviations from Equilibrium
7. r, R, and Bifurcations
8. Quasiperiodicity and Chaos
9. Chaotic Stability

10. Environmental Variation: Expectations and Averages
11. Nonlinearity and Inequality
12. States, Transitions and Extinction
13. Regime Shifts I
14. Regime Shifts II

In the previous post, we discussed the dramatic decline of the Atlantic cod (Gadus morhua) off Newfoundland over the past 60 years. I left us with the question of why, given the very limited catch sizes since the 1990’s, there was little evidence of population recovery (at least up until 2005). An Allee effect is a likely explanation for the failure of the population to recover during that extended period of reduced fishing pressure.

Beginning around 1994, the population may have become limited by an Allee phenomenon, or more appropriately mechanism, where a population’s size is limited far below the presumed carrying capacity, or observed maximum population size, because of reduced population size itself. Analogous to carrying capacity, where an upper limit is set on population growth rate by the effects of a relatively large population size, an Allee effect is an upper limit set by relatively small population size. Intuitive examples are easy to find, e.g. (1) species that require sufficient numbers for successful defense against predators will be increasingly limited by predation at low population size; (2) species for which habitat engineering by a sufficient number of individuals is necessary for offspring success; (3) species that depend on a minimum number of participants for the formation of successful mating assemblages. G. morhua, in which individual fecundity increases with age and body size (to a limit) (Fudge and Rose, 2008), is known to form, or have formed, large pelagic assemblages during spawning. Allee effects, therefore, describe situations where individual fitness depends on the presence of conspecifics, and is positively correlated with population size.

One vulnerability of populations subject to Allee effects is that small population size becomes an inescapable trap, with the likelihood of extinction increasing as population size declines. The reasons for this are twofold. First, if growth rates decline to zero or even become negative below an Allee threshold, then the state of zero population size becomes a stable state and extinction is assured. If you recall, our earlier models of population growth considered X= 0 (extinction) to be an unstable steady state; unstable because the addition of reproducing individuals to the population would result in divergence away from the zero state —population growth. Second, even if growth rate never becomes negative below the Allee threshold, a sufficiently large or sustained decline of population size increases the probability of extinction due to random events, a phenomenon termed stochastic extinction. Stochastic extinction, the probability of which could increase with deteriorating environmental conditions, is of interest to anyone studying extinction, including paleontologists, and will be discussed in a later section. Here, however, we will first explore several simple models of Allee effects.

Models of Allee effects

In the logistic model (Eq. 1 here), mortality rate increases as population size, X, approaches carrying capacity K, and population growth rate subsequently declines. The logistic model has two alternative steady states, X=K and X= 0, the latter of which is unstable as discussed above. The extinct state is a stable attractor, however, in the presence of an Allee effect. There are several simple models that demonstrate the effect, but to appreciate them, and the Allee effect itself, let us first examine the relationship between population size and growth rate under the logistic model. If we plot growth rate (dX/dt) against population size in the logistic model (Fig. 1), we see that the rate increases steadily at small population size, reaches a maximum when population size is half of the carrying capacity —X(t) =K/2— and declines steadily thereafter, reaching zero at carrying capacity. This value can be arrived at analytically because what we are visualizing is the rate of change of growth rate itself, technically the second derivative of the logistic growth equation. If we expand the logistic growth rate equation


and take the derivative, we derive the acceleration (or deceleration) of the rate of change of population size as a function of population size itself.

Setting d²X/dt equal to zero —the point at which growth rate is neither accelerating nor decelerating— we get the maximum that is illustrated in Fig. 1.


The important thing to note here is that growth rate is always positive when 0<X(t)<K, that is, when population size lies between zero and the carrying capacity.

Fig. 1: The relationship between population growth rate and population size under a logistic model. In this example carrying capacity K=100.

There are several ways in which an Allee effect can be modelled in a logistically growing population. For example, if the Allee threshold is represented as a specific population size A, then the effect can be incorporated into the logistic formula as

(Lewis and Kareiva, 1993; Boukal and Berec, 2002). The first term on the RHS of the equation is the logistic function, where growth declines to zero as X approaches K. The second term introduces the threshold, A, with growth rate declining if X < A, and increasing when X > A. Here, the effect is treated as the difference between population size and the threshold, taken as a fraction of carrying capacity, or maximum population size. Note that if A=0 —there is no Allee effect— the model reduces to the logistic growth model. A more nuanced model, where A must be greater than zero —an Allee effect always exists— treats the Allee threshold as equivalent yet opposite to K, representing a lower bound on growth rate (Courchamp et al., 1999).

If A=1 —in which a population comprising a single individual is compromised under all circumstances— then the strength of the Allee effect depends on the size of the population. In both models, growth rate becomes negative below the threshold A, effectively dooming the population to extinction (Fig. 2). This condition is often termed a “strong” Allee effect.

Negative growth rates, a feature that is common to many models of the Allee effect, can be somewhat problematic from a conceptual viewpoint because of their determinism. We’ll pick this point up in the next post, and also discuss why paleontologists might care about both Allee effects, and model determinism.

Fig. 2: Two models of strong Allee effects illustrates as plots of population growth rate vs. population size. K=100. Red shows the first model where growth rate is relative to the Allee threshold A as a function of K. Blue shows the second model where growth rate is relative to the threshold A itself.

Vocabulary
Allee effect — A positive correlation between individual fitness, or population growth rate, and population size. This means that fitness and/or growth rates decrease with declining population size.
Second derivative — The derivative of a function’s derivative (the first derivative), thus the acceleration (deceleration) of a rate. E.g. the first derivative of a body in motion, described by position and time, is velocity or speed. The second derivative is acceleration, or the rate at which the speed is changing.
Stochastic extinction — A relationship between the probability of a population’s extinction, and population size and/or environmental variability. In general, the risk of extinction increases due to random fluctuations of either factor.
Strong Allee effect — Population growth rate becomes negative below some threshold of population size.

References
Boukal, D. S. and Berec, L. (2002). Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. Journal of Theoretical Biology, 218(3):375–394.
Courchamp, F., Clutton-Brock, T., and Grenfell, B. (1999). Inverse density dependence and the Allee effect. Trends in Ecology & Evolution, 14(10):405–410
Fudge, S. B. and Rose, G. A. (2008). Changes in fecundity in a stressed population: Northern cod (Gadus morhua) off Newfoundland. Resiliency of gadid stocks to fishing and climate change. Alaska Sea Grant, University of Alaska Fairbanks.
Lewis, M. and Kareiva, P. (1993). Allee dynamics and the spread of invading organisms.Theoretical Population Biology, 43(2):141–158

Ecosystems, Epidemics, and Economies

Yesterday I gave a talk for the “Breakfast Club” series at the Academy (California Academy of Sciences). The club is a twice weekly series of online talks started by the Academy in response to the widespread shelter-in-place and shutdown orders. It’s intended to bring a bit of our science and other activities to those interested who, like so many of us, find ourselves mostly limited these days to online interaction.

My talk focused on some new work that we are doing in the lab, related to the COVID-19 pandemic, but inspired by and built partly on our paleoecological and modelling work. I hope that you find it interesting! Oh, and while you’re there, check out the other talks in the series (link above)!

Systems Paleoecology – Logistic Populations II

Tags

ecology, logistic growth, mathematical model, paleoecology, population growth

WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series:

1. Welcome Back Video
2. Introduction
3. Malthusian Populations

4. Logistic Populations

The logistic equation, covered in the previous post, is a differential equation, where time is divided up into infinitesimal bits to model the growth and size of X (“infinitesimal” knots your stomach? I cannot recommend Steven Strogatz’s “Infinite Powers” enough!). We can also simulate the logistic model in discrete time to get a better feeling for it, where X(t + 1) is population size in the next “time step” or generation. This approach is instructive because anyone can play with the calculations using a calculator or spreadsheet! Here is an example of a discrete version of logistic growth, the Ricker difference equation (Ricker, 1954).

EQ. 1: (FUTURE POPULATION SIZE) = (CURRENT POPULATION SIZE) x (EXPONENTIAL REPRODUCTION LIMITED BY CARRYING CAPACITY AS IN THE LOGISTIC MODEL)

r has been replaced by R, the main difference between the logistic equation and Eq. 1 being that, because we no longer measure time as continuous but instead step discretely from one generation to the next, we measure the intrinsic rate of increase as the “net population replacement rate”. The population again, after an initial interval of near-exponential growth, settles down to a fixed value at K (Fig. 1). Both the continuous and discrete logistic patterns of population growth are, by all definitions, stable populations in equilibrium. They are stable because, at least within the scope of the models, once a population attains its carrying capacity there is no more variation of population size. This brings us to our first definition of stability.

Stability: An absence of change.

Discrete time logistic growth, showing population sizes per discrete generation. X(0) = 1 and K = 100. R = 1.0.

In the following sections we will cover a real-world example of logistic growth, and then go through the derivation of the logistic function itself.

An Example of Logistic Growth

The state of Washington in the United States employed a program of harbor seal (Phoca vitulina) culling during the first half of the twentieth century. The seals were considered to be direct competitors to commercial and sport fishermen. The state sponsored monetary bounties for the killing of seals until 1960, by which time seal populations must have been reduced significantly below historical levels. Additional relief arrived for the seals in 1972 with passage of the United States Marine Mammal Protection Act. Monitoring of seal populations along the coast, estuaries and inlets of Washington, primarily by the Washington Department of Fish and Wildlife, and the National Marine Mammal Laboratory provided a time series of seal population size, spanning the beginning of recovery in the 1970’s to the end of the century (Jeffries et al., 2003). Population sizes from one region of the coastal stock, the “Coastal Estuaries”, show a logistic pattern of growth (Fig. 2). The function fitted to the data (using a nonlinear least squares regression) is y = 7511.541/[1 + exp[−0.265(x − 1980.63)]] (r-squared = 0.98; p < 0.0001; note that “r-squared” is the coefficient of correlation, not our intrinsic rate of increase). Given an initial population size of X(0) = 1,694 in year 1975, the function yields estimates of r = 0.265 and K = 7,511. This excellent example of logistic growth in the wild, or recovery in this case, was unfortunately brought to us courtesy of the ill-informed belief that the success of human commercial pursuits necessitate, or even benefit from, the destruction of wild species.

Logistic recovery of a harbor seal (Phoca vitulina) population in Washington state, U.S.A., after the cessation of culling and passing of the Marine Mammal Protection Act. Orange circles are observed population sizes, the blue line is the fitted logistic curve, and the red horizontal line is the estimate carrying capacity.

Deriving the logistic equation
Equation 1 in the previous post is the logistic growth rate of the population, but it is not the logistic function itself. That function is obtained by integrating the growth rate dX/dt, and the process is instructive because, as illustrated in later sections, our ability to do so with more complicated dynamic equations is quite limited.

The logistic growth rate is first re-written to eliminate the X/K ratio (makes it easier to proceed)


and then re-arranged to separate variables,

The logistic function is derived by integrating both sides, but doing so with the left hand side (LHS) requires simplification using partial fractions (some of you might remember those from high school math; or not).




The solutions to the final equation are A=1 and B-A=0, yielding B=1. Therefore

Now if we wish to integrate our logistic differential equation,
,
we can substitute our partial fractions solution and proceed as follows.


And if you recall our integration of the Malthusian Equation, the solution is


Let , a constant. Then



which is the equation for logistic population growth! Whew.

References
Jeffries, S., Huber, H., Calambokidis, J., and Laake, J. (2003). Trends and status of harbor seals in Washington State: 1978-1999. The Journal of Wildlife Management, 67:207–218.
Ricker, W. E. (1954). Stock and recruitment. Journal of the Fisheries Board of Canada, 11:559–623.

Forest fires

Tags

Forest fires

Four patches of trees in a forest. Time is shown on the x-axis, and relative population size on the y-axis. The local patch populations are often prevented from reaching carrying capacity, or staying there, but frequent forest fires.

Myself and several colleagues at the Academy have been working with scientists at the United States Forest Service, exploring how resilience may be maintained in forests of the Sierra Nevada in the coming century. The major concern is fire and its management. The natural regime of the Sierra Nevada is one of frequent fire, promoted by highly seasonal precipitation (intensely wet winters, very dry the rest of the year), thus with many fires during a year. The high frequency of fires would maintain a sparser density of trees, and fuel accumulation would be regulated by regular burning. The first human residents of the Sierra Nevada, Native American peoples, utilized natural resources in the Sierra Nevada, and developed a system of management using frequent fires, thus maintaining to a great extent the pre-anthropogenic regime. In later historic times, however, beginning in the late 19th century, fire management was implemented in various forms and levels in the interest of the timber industry, as well as the mining and expanding high elevation communities. Significant deforestation was replaced in many areas by increasing tree densities driven by fire suppression, conservation and re-planting efforts. Unfortunately, the result has been unnaturally high tree densities in many areas of the Sierra Nevada, and the dangerous accumulation of fuels. This is a common problem throughout forested areas of the United States, and one of the dangerous outcomes are fires of sizes and intensities well beyond what would be natural for the forest systems. California is at high risk because of the extent of its montane forested regions, its seasonal precipitation, drying and drought due to climate change (global warming), and extensive human use of the forests. The USFS and other agencies, as well as residents, communities and industries, are therefore engaged in developing strategies to maintain resilience of the forests, and sustainable use of forest resources. Academy scientists have become involved both as advisors about communication and education efforts, as well as working on the ecology of the system.

Forest growth under fire suppression. Note the lower frequency of burns, but the larger loss of trees.

One of the first steps that we took was to begin work on a simple conceptual model of tree growth, fuel and fire dynamics based on ordinary differential equations. Most recently I’ve been translating the model into a tractable system of difference equations for simulation on a landscape. The model is still too preliminary to get into much detail here (and not yet peer reviewed), but here are some cool visualizations. The plot at the top of the post shows a landscape of four forest patches. Each patch consists of reproducing trees. Patches may export a small fraction of seedlings to neighbouring patches, including out into nothingness (tree bare patches), and will eventually asymptote density-dependently to a stable equilibrium (based on a standardized carrying capacity of 1). Trees also produce fuel, which accumulates unless a fire is ignited, at which point a large fraction of the fuel is lost. At each time step, trees reproduce and produce fuel, and there is a probability of fire. The top plot shows the forest (and individual patches), where the probability of ignition is fairly high, mimicking pre-historic conditions. The “jaggedness” of the population trajectories indicate fires and recovery. This second plot is the same forest, but this time with a much lower incidence of fire, corresponding to a modern regime of fire suppression. Notice the lower frequency of burns, but when they do occur they are larger and losses of trees are greater. These would correspond to our unfortunate current megafires.

Finally, the latest bit of work has been to scale the model up, and here are the results of two 10×10 forest grids run for 500 years (with the first 100 years discarded as transient burn in) (see the YouTube links below). “Greener” indicates higher tree density, and yellow indicate lower densities. The main thing to get from these is that the landscape is overall greener when the incidence of fire is lower, that when fires occur in that regime they are indicated by the immediate appearance of bright yellow patches of tree loss, and that there is a more moderate density of trees in the high fire landscape, and correspondingly greater frequency of smaller fires. That’s pretty much it for now, but working on this has been a lot of fun so far, and who doesn’t like nice visualizations?! (If you don’t like them, feel free to comment…)

Low suppression, high fire

High suppression, low fire

 

New paper: Comparing paleo-ecosystems

Tags

dynamics, ecology, evolution, modeling

Modeled ecological dynamics in South Africa 1 million years after the end Permian mass extinction, showing the highly uncertain response of the community to varying losses of primary production.

We have a new paper on paleo-food web dynamics in the Journal of Vertebrate Paleontology! The paper is one in a collection of 13 (and 27 authors), all focused on the “Vertebrate and Climatic Evolution in the Triassic Rift Basins of Tanzania and Zambia”. The collection covers work done in the Luangwa and Ruhuhu Basins of Zambia and Tanzania, surveying the vertebrates who lived there during the Middle Triassic, approximately 245 million years ago (mya). This is a very interesting period in the Earth’s history, being only a few million years after the devastating end Permian mass extinction (251 mya). They are also very interesting places, capturing some of our earliest evidence of the rise of the reptilian groups which would go on to dominate the terrestrial environment for the next 179 million years. The evidence includes Teleocrater, one of the earliest members of the evolutionary group that includes dinosaurs and modern birds.

Our paper, “Comparative Ecological Dynamics Of Permian-Triassic Communities From The Karoo, Luangwa And Ruhuhu Basins Of Southern Africa” is exactly that, a comparison of the ecological communities of southern Africa before, during and after the mass extinction. Most of our knowledge of how the terrestrial world was affected by, and recovered from the mass extinction comes for extensive work on the excellent fossil record in the Karoo Basin of South Africa, but that leaves us wondering how applicable that knowledge is to the rest of the world. We therefore set out to discover how similar or varied the ecosystems were over this large region, comparing both the functional structures (what were the ecological roles and ecosystem functions) and modeling ecological dynamics across the relevant times and spaces of southern Africa. We discovered that during the late Permian, before the extinction, the three regions (South Africa, Tanzania, Zambia) were very similar. In the years leading up to the extinction, however, communities in South Africa were changing, becoming more robust to disturbances, but the change seemed slower to happen further to the north. The record becomes silent during the mass extinction, and for millions of years afterward, but when it does pick up again in the Middle Triassic of Tanzania, the communities in South Africa and Tanzania are quite distinct in their composition. The ecosystem in South Africa was dominated by amphibians and ancient relatives of ours, whereas to the north we see the earliest evidence of the coming Age of Reptiles. Yet, and this is where modeling can become so cool, the two systems seemed to function quite similarly. We believe that this a result of how the regions recovered from the mass extinction. Evolutionarily, they took divergent paths, but the organization of new ecosystems under the conditions which prevailed after the mass extinction lead to two different sets of evolutionary players, in two different geographic regions, playing the same ecological game. As we say in the paper, “This implies that ecological recovery of the communities in both areas proceeded in a similar way, despite the different identities of the taxa involved, corroborating our hypothesis that there are taxon-independent norms of community assembly.”

And finally, this work would not have been possible without the generous support of the United States National Science Foundation’s Earth Life Transitions program.

Abstract spaces for unknown ecologies

Tags

food webs, metanetwork, paleo-food web, Permian-Triassic extinction, Scientific models

A guild-level food web for the Karoo Basin ecosystem.

At the heart of our paper lies a model framework which we devised for analyzing fossil food webs. I stated in the previous post that our main question was “How would those food webs (important parts of the paleoecosystems) have responded to everyday types of disturbances, on the short-term, as the planet was busily falling apart?” We could approach this question in several different ways if we were working with modern food webs. We could conduct manipulative experiments with simple mock-ups of the food web, using for example some of what we believe to be the key species to represent the community. Or, we could conduct large scale manipulations, such as removing a species entirely; but that is very difficult to do, or to obtain permission! Or, we could measure variables such as species population sizes, how species interact with each other, and so forth, to then conduct numerical analyses and simulations. None of those approaches are available to us when dealing with ancient, extinct ecosystems. Therefore, what we did instead was to use the most accurate information that we have for each paleoecosystem, which consisted of categorizing species into “guilds”. Here, a guild is a group of species who shared the same habitat, and  potentially shared the same predators and prey. The “potentially” is based on our best interpretations of the ecologies of those extinct species, because without actually being there to witness their interactions (back to that Tardis again), we cannot be sure. The result is usually something like the box figure above. Even with species lumped, you can see how complex and busy the system would have been! And from this guild-level model, we can then construct many many different food webs, tweaking the specific links between species. An example is shown in the second figure.

Now, the number of different food webs that you could generate based on even a modest number of species, say 20, and a few guilds, is astronomically large (in fact beyond astronomical). The important thing, however, is that all of them would be consistent with the guild scheme. Let me give an example. Say we did a guild scheme for the modern African savannah. We would be justified to some extent to place lions and hyaenas in the same guild. We might not know exactly which antelope species (for example) each predator species was preying on, but we would never draw a food web where antelope were preying on lions, hyaenas or each other! So, what we have done for our food webs is constructed a mathematical space that contains all the food webs which could possibly have existed in our paleoecosystem. In other words, we have taken the full set of food webs that could be constructed for a certain number of species, and constrained ourselves to consider only those that are consistent with our accurate knowledge of the guild structure.

Detail of one possible food web just prior to the mass extinction.

This still doesn’t solve the problem of how those food webs would have responded to various types of disturbances in the distant past. And in fact, we really cannot solve that problem, so we did what we think is the next best thing. We asked if there was anything special about those food webs, compared to any others that were not consistent with our guild structure. In other words, what if the ecosystem had evolved a bit differently, and comprised species a bit different from what we actually observe in the fossil record? We considered a number of such alternative models, differing from the real ecosystem in ways such as moving species around in the guilds, or moving guilds and the interactions between them, or removing guilds altogether. And each time we did that, and generated a food web from the new guild scheme, we examined the stability of the food web. Exactly what we mean by stability, and how we measured it, will be the subject of the next post.

A new paper on the Permian-Triassic mass extinction

Tags

biodiversity, extinction, food webs, modeling, paleo-food web, paleontology, Permian-Triassic extinction, Scientific models

Dicynodon, an ancient relative of mammals, at the end of the Permian. (Marlene Hill Donnelly).

Yesterday, Ken Angielczyk and I published our most recent paper on the Permian-Triassic mass extinction (PTME) in the journal Science. In a nutshell, we examined a series of paleocommunities spanning the extinction, from the Late Permian to the Middle Triassic, and modelled the stability of their food webs. We compared the models to hypothetical alternatives, where we varied parameters such as how species are divided among guilds, or ecological “jobs”, and the numbers of interactions that species have. One of our very interesting discoveries is that the real food webs were always the most stable, or amongst the most stable of the models, even during the height of the extinction! That’s remarkable, given the devastating loss of species at the end of the Permian. Our other discovery is that the ability to remain highly stable during the extinction stemmed from the more rapid extinction of small, terrestrial vertebrate species. That’s not something we would predict given our experience with modern and ongoing extinctions, where larger vertebrate species are considered to be at greater risk. And finally, our last interesting observation is that the early recovery, the immediate aftermath during the Early Triassic, was an exception to the above. That community was not particularly stable, which seems to have been the result of the rapid evolutionary diversification of the extinction survivors, and the arrival of immigrants from neighbouring regions.

Some aspects of the paper are quite technical, and take advantage of fantastic new paleontological data and recent developments in theoretical ecology. Therefore, over the next few posts I’ll go through what we did, and how we did it, using a more “plain language” approach. In the meanwhile, the paper was covered by a number of news outlets, and here’s my favourite!

“5 things we learned from the mass extinction study that’s “the first of its kind”“, The Irish Examiner.

Simulating a Tragedy of the Commons I

Tags

Code, simulations, tragedy of the commons

I introduced a recent paper, in the previous post, on the Tragedy of the Commons. The paper is published in Sustainability, an open access paper and is therefore available to everyone. One bit that I did not include in the paper, however, is the code that I used for the simulations therein. On the suggestion of a friend, I’ll publish them here in a series of posts. (Thanks Mauro)

The simulations are very simple, being based on difference equations, and were all coded in Octave. The code is not pretty, but I cleaned it up and added some comments. If you have questions, just drop me a comment on the blog. This first simulation is of a basic tragedy involving two users, and corresponds to Figures 1B-C in the paper. A resource is simulated to a steady state based on a Ricker model, for 100 steps, and then users begin to utilize it. They increase their utilization per time step at a steady rate (acceleration of utilization is constant), and the simulation is run until the resource is exhausted. The basic output of resource, one user’s benefit, and total utilization, are plotted in the figure here. The equations simulated are outlined in the excerpt from pg. 754 of the paper:
“We now introduce a term to represent consumption or utilization by our human TOC agents.

where U(t) is the total fraction of R utilized by human users at time t, and ranges from 0 to 1. The term in square brackets on the right hand side of the equation is the modified growth rate of R. The resource is stable when this term is positive or zero, that is, the rate of resource renewal exceeds or is equal to utilization. Since U is the total standardized utilization rate of all users, it may be expanded to

where there are N users and is the standardized utilization rate of the user at time t. The benefit gained from utilization is modelled as

where is the benefit to user i, f is a factor that converts resources utilized into another commodity, for example converting harvested food to energy or currency, and B is the total benefit to all users.”

The code follows, and can be executed by running it in an interactive Octave session. Please be aware that WordPress has somewhat limited options for formatting code. I’ve used Matlab formatting here, but wrapped lines might not be obvious. Single lines are all terminated with a semicolon, “;”, so just look out for those.

#RESOURCE DATA
#initial resource level
R0 = 1;
R = zeros(1,2);
R(1, 1) = 1;
R(1, 2) = 1;
#carrying capacity
K = 50;
#intrinsic growth rate r
r = 1.5;

#USER DATA
#There are 2 users
#initial acceleration of utilization
du0 = 1.05;
#individual user parameters, users 1 and 2
#initialize arrays
u1 = zeros(1,2);
u2 = zeros(1,2);
b1 = zeros(1,2);
b2 = zeros(1,2);
u1(1,1) = 1;
#initial utilization rate
u1(1,2) = .001;
#conversion factor of resource to benefit
f1 = 1;
b1(1,1) = 1;
b1(1,2) = 0;
u2(1,1) = 1;
#initial utilization rate
u2(1,2) = .001;
#conversion factor of resource to benefit
f2 = 1;
b2(1,1) = 1;
b2(1,2) = 0;
U = zeros(1,2);
U(1,1) = 1;
#total utilization
U(1,2) = b1(1,2) + b2(1,2);
du = zeros(1,2);
du(1,1) = 0;
du(1,2) = du0;

#BEGIN SIMULATION

#RUN RESOURCE TO STEADY STATE, NO UTILIZATION
for count2 = 2:100
  R(count2, 1) = count2;
  R(count2, 2) = R(count2-1, 2) * (exp(r*(1-(R(count2-1, 2)/K))));
  u1(count2, 1) = count2;
  u1(count2, 2) = u1(1,2);
  u2(count2, 1) = count2;
  u2(count2, 2) = u2(1,2);
  du(count2,1) = count2;
  du(count2,2) = du0;
endfor

#INITIATE UTILIZATION
for count1 = 101:265
  R(count1, 1) = count1;
  #modified Ricker model with utilization
  R(count1, 2) = R(count1-1, 2) * (exp(r*(1-(R(count1-1, 2)/K))) - (u1(count1-1,2)+u2(count1-1,2)));
  u1(count1, 1) = count1;
  #increase utilization
  u1(count1, 2) = u1(count1-1, 2) * du(count1-1,2);
  #benefits (equal for both users in this case)
  b1(count1, 1) = count1;
  b1(count1, 2) = f1 * u1(count1-1, 2) * R(count1-1, 2);
  b2(count1, 1) = count1;
  b2(count1, 2) = f2 * u2(count1-1, 2) * R(count1-1, 2);
  u2(count1, 1) = count1;
  u2(count1, 2) = u2(count1-1, 2) * du(count1-1,2);
  #total benefits
  U(count1, 1) = count1;
  U(count1, 2) = b1(count1, 2) + b2(count1, 2);
  du(count1,1) = count1;
  du(count1,2) = du(count1-1,2);
endfor

Ecology and the Tragedy of the Commons

Tags

real world networks, Robustness, Scientific models, tragedy of the commons

Well, it’s been quite some time since the last post, but I’ve been busy! This post is just a short notice of a new paper, just published today. The paper is part of a special issue on the Tragedy of the Commons in the journal Sustainability. My paper takes a comparative look at the Tragedy in ecological communities and human societies, and the potential of human mutualisms for avoiding tragedies. The situation is not a very hopeful one, however, given our ever-growing human population. Hardin did note this in his original essay. Finally, this paper was inspired by an earlier paper by myself and Ken Angielczyk.

Here’s a link to the paper, as well as the abstract.

Roopnarine, P. Ecology and the Tragedy of the Commons. Sustainability 2013, 5, 749-773.

Abstract

This paper develops mathematical models of the tragedy of the commons analogous to ecological models of resource consumption. Tragedies differ fundamentally from predator–prey relationships in nature because human consumers of a resource are rarely controlled solely by that resource. Tragedies do occur, however, at the level of the ecosystem, where multiple species interactions are involved. Human resource systems are converging rapidly toward ecosystem-type systems as the number of exploited resources increase, raising the probability of system-wide tragedies in the human world. Nevertheless, common interests exclusive of exploited commons provide feasible options for avoiding tragedy in a converged world.

One more time: Why model?

Tags

modeling, Scientific models, simulations

A wonderful statement by Jay Melosh in a recent interview with Physics Today. It really underscores the approach of much of the research presented in this blog; just replace “physics” with “biology” or “ecology”. And while Melosh is a master of planetary geology and hence the history of the Solar System, we’re dealing here with evolution and the histories of species and ecosystems. Okay, here’s the quote:

“Computer modeling, of course, plays a big role in evaluating the consequences of different hypotheses, which we then compare to observations. While the physics of an individual process may be simple, Nature is messy and computers are one of our principal tools for combining simple processes into the complex fabrics necessary to mimic observations and thus either validate or refute different hypotheses about what we see.“