Food Chains, Webs, and the Cybernetic Ecosystem (PDF VERSION IS HERE)

A food chain describes how energy and nutrients move through an ecosystem (Figure 1). In most terrestrial food chains energy is captured by plants that also incorporate inorganic materials (nitrogen, phosphorus,sulfur, etc.) as chemical nutrients. The energy and nutrients pass from one level to another or are returned to the environment by decomposers. The producers are found at the base of the food chain (also known as the 1^st tropic (feeding) level). Energy temporarily held in the plants is then transferred to the 2^nd trophic level, inhabited by herbivores and omnivores. The herbivore level is also referred to as the primary consumers. Energy then flows to the 3rd and sometimes higher trophic levels where it is incorporated into the secondary and tertiary consumers (not shown). Carnivores and top carnivores occupy the upper levels. The second law of thermodynamics prohibits the recycling of energy in this system. In addition, it makes it impossible to transfer 100% of the energy from one level to another. Because of this, less energy is available the higher you move up the food chain which puts limits on the number of tropic levels for a particular ecosystem.

There are three types of pyramids (center of figure 1) that describe the relationships among the various trophic levels:

• Pyramid of Numbers: This is a pyramid based on the number of organisms at each trophic level. However, it does not show the mass of food consumed at each level.
• Pyramid of Biomass: This is a pyramid based on the total mass of organisms at each trophic level. This model is also limited since it does not reveal the energy contributed by each organism in the ecosystem.
• Pyramid of Energy: This is a pyramid which shows how much food energy is available at each trophic level in an ecosystem. This pyramid is the most complex of the three since calculations of the number, masses, and energy content for each type of organism must be considered.

A second food chain is shown in figure 2. For both these examples, we have emphasized the grazing food chain. More important, however, from an energy point of view, is the decomposer food chain (Figure 3), where most of the energy transfer takes place (taking care of all grazing energy as well as it's own). While the decomposer food chain can't recycle energy, nutrients are recycled at this point. More on the decomposing food chain later.

A description of most interactions among organisms in the environment is rarely as simple as shown in these two figures (although figure 2 pretty much covers it. For figure 1, other birds would certainly be feeding on the insects and spiders and these would be potential prey for the hawk. The puma (mountain lion, cougar,wild cat) would also feed on a wider variety of prey. When all these food chains are added up, we have a food web (figure 4). Even this representation is an over simplification since many organisms will occupy more than one trophic level (Table 1).

One reason ecologists are interested in food webs is the apparent relationship between stability of an ecosystem and its complexity. An introduction to the stability/complexity controversy is here. Generally, systems with fewer interconnections among species are less stable than those with more connections (Figure 5). There is two ways of looking at stability:

• Stability of the First Type: Over time there is a increasing constancy of numbers. The system will eventually reach a steady state under constant conditions (the next state is predictable from within the system)
• Stability of the Second Type: The system has greater resistance to changes that are external (Figure 7). This is usually accompanied by higher energy flow.

Modeling food webs is going to require a different approach from that we've used for our two-species interactions, in part because we need to model a greater number of species, but also because we need a way to apply all of the potential species interactions (Table 2).

Cybernetic Systems
Cybernetics is concerned with control and communication in systems formed by living or non-living individuals and their artifacts (Norbert Wiener). A system is a set of different elements, each in different states, each state influenced by the others. Cybernetics describes the interactions. The various elements (species, electronic systems, etc.) are linked by feedback loops. Negative feedback loops are stabilizing for the system (blood glucose levels, a thermostat), while positive feedback loops are destabilizing (epidemics, high fevers, orgasm- no that wasn't a miss-spelling).

Negative feedback among the components leads to stability, not only for the entire system, but also for selected components. Thus, the system as a whole shows persistence. Cybernetic systems can influence their own futures since the present state holds information (number of species, interactions among the species, population sizes, etc.). Thus, information in the current state can be used to predict the next generation.

Another quality of cybernetic systems is that they're self-organizing. The negative feedback loops among the various species or components will eventually "settle down" to an equilibrium state. The likelihood of self-organization is greater when there are a larger number of entities in the system since the interactions (feedback) among the entities (species) are weaker.

Information about the states of the entities in the system increases with time and results in an increase in complexity. Unlike simple machines, which are a final product, cybernetic systems change with time and show persistence. With all the stored information, these systems have a resistance to external events and show stability (of both the first and second type).

Figure 8 is an example of a complex cybernetic system; a portion of the internet routes surrounding Urbana, Illinois. This system retains and transfers information (even if half of it is porn), and each node serves as a feedback system (through software). These characteristics, along with the size qualify the internet as a cybernetic system. It's important to note that this system is self-assembling. As users are added, logged on, logged off, and send or receive information, no one is in charge! The only intervention is to make sure everyone has their own unique address. Figure 9 shows internet connections as separate nodes, indicating how adjacent local systems interconnect and communicate with one another. With a little imagination, it's not difficult to see the similarities between our food web problem and this diagram. See the power of cybernetic systems here.

The Cybernetic Ecosystem Model
We can start the model with some familiar concepts:

• Each species (i) will have a population size or density:: N_i
• For each species (i) there will be a rate of change in the density: dN_i /dt
• There are selected environmental factors that   can affect the outcome of the above: E
• These factors will also change over time: dE/dt

The above interactions will be made proportional to the sum of the products of the interactions to produce a set of differential equations to model the states of the species at each point in time. A three-species model is shown in Table 3.

This model satisfies the needs of our cybernetic system. Species can interact with the environment, other species or within a species. As an example, running down the column for species 1 (N₁):

• Species 1 causes a change in the environment (dE /dt). The effect of species 1 on the environment is shown in the matrix as EN₁
• Species 1 can interact with itself: dN₁ /dt = -f N₁²
• Species 1 also interacts with species two (dN₂ /dt = +i N₂N₁) and three (dN₃ /dt = +l N₃N₁).
• The model can be easily expanded (just add more rows and columns).

The model also allows for different interactions among the species. The coefficients (a,b,c....) are the equations describing the strength of the interactions between the two species. The sign of the coefficients shows how it will affect the change in species numbers for any of the i^th species (dN_i /dt). These also satisfy our requirement to be able to model all potential species interactions as laid out in table 2:

• Species 2 is a predator on species 1 (in pink in table 3) since species 1 has a positive effect on 2 (+i   N₂N₁)and species 2 has a negative effect on 1(-g N₁N₂).
• Species 2 and 3 are competing (in yellow) since both their coefficients are negative (-k and -m)
• Some interactions are non-existent (species 3 has no effect on the environment).
• Any of the potential interactions in table two could be built into the model.
• Depending on the amplitude of the coefficients (a,b,c...), the interactions between two species may be strong or weak. In general, species that interact weakly with other species do so with a large number of species. Those that interact strongly do so with fewer number of species.

For the system shown in table 3:

• The rates of change (Column 1) depend on the values of the equations in the associated rows and determine the values in the next generation (the model is predictive).
• The process is iterative with a steady-state goal (lacking external changes).
• The steady state of the system depends on the coefficients of the interactions.
• Transient states depend on E and N_i.
• Transient states may change the coefficients of the interactions (not built into the model we'll be using; the idea is that changes in the environment or numbers of a completely different species could change a competitive situation, for example).

Expanding Niche Theory
The concepts we've developed here can be used to expand our explanations of the ecological niche. There are several ways to think about the niche that we've already developed:

• Odum niche: The oldest of the niche concepts. Essentially, the habitat of the organism.
• Ecological niche: The physical space occupied by the organism and the functional role of the species in that space. There are three components: spatial (habitat), trophic (where on the food chain), and other (position on an environmental gradient- where the species fits in).
• Huchensonian niche: This is a mathematical concept. To describe the Huchensonian niche, you measure the tolerances for the species across all possible environmental dimensions (temperature, humidity, wind, various nutrients, etc.) and then, with these n measurements, construct an n-dimensional hyperspace that describes all of the possible environmental tolerances on orthogonal axes (at right angles to one another; it's OK since it's a mathematical concept). The resulting hyperdimensional data cloud is the description of the species' tolerances and is the Huchensonian niche.
• Cybernetic niche: Describes the species through their coefficients of interaction in the system matrix. For example, species that are competing with one another will have negative coefficients of interaction in the matrix. Conceptually, this can be seen as two parallel negative feedback systems along the resource continuum (Figure 10). In the figure, both species B and C are connected through negative feedback loops (solid arrows) to a limiting resource A. Assuming equal competitive abilities, if the species overeat the resource, it will become unavailable and the species populations will drop. If however,  species B is better equipped to take the resource, species C will drop in population size, resulting in making the resource more available to B which then forces C even lower; eventually ending in the local extinction of species C. Thus, a run-away positive feedback loop is implied between the two species (dotted lines in the figure). Similar loops can be constructed for any potential species interaction.

Using the Model
Two versions of the stand-alone web simulator are available for downloading and installing. The large format simulator (Figure 11) requires a screen resolution of at least 1024 X 768 and allows you to view all display and control panels at the same time. The big food web simulation is here. A smaller footprint version is also available as a download (Figure 12) which is identical to the web-based plug-in. You can get the small screen version here.

Controlling the Simulation.

• Figure 12, area A, allows you to view various portions of the model. If the data input radio button is selected, you are in data input mode. Data can be directly entered into the matrix (D) or can be generated. If you select the Graphs & Data radio button then the display will shift to that shown in the small inset figure (B). The data grid keeps track of each of your runs and can be copied to the clipboard for plugging into a Word document or for further analysis in Excel. A graph of the population size for each of the species is also shown. Species are color-coded according to the colors shown in area G (N₁, N₂, N₃...). By selecting the Big Graph radio button you can get a larger graph (as in inset C).
• Data can be entered by hand in area D or can be generated using the controls in area E. The number of species can be anywhere between 3 and 10. The maximum number of generations the simulation will run can also be controlled here. If the Auto Generate check box is selected, a new data set will be generated following each simulation. If the Reload Last check box is set, the simulator will reload the original data set. If neither is set then pressing the show me button will continue another 25 generations from where the last simulation stopped. You can also set the relative levels of species interaction (species interact weakly, medium, or strongly) as well as the starting population size.
• Area F allows you to add external forcing factors to the simulation.
• Area G shows some simple statistics for the run including your starting numbers, the end values, percent success (percent of species that made it to the end of the simulation, a measure of diversity (H), and equitability (I)

Figure 13 shows a generated data set in the interaction matrix for 10 species, with low interaction, high population density, and no external forcing factors. Note that the population sizes are randomly set, so that some of the populations may start off with low numbers. The population density just sets the highest possible level. Yellow blocks in the matrix represent null interactions (no effect of the column species on row species growth. Species 1, for example does not affect the growth of species 4 in the figure. Pink cells are negative interactions. White cells are positive interactions. Figure 14 depicts the results of a typical simulation  For the data grid, S is the number of species that you started with, gen is the number of generations the simulation ran, int is the species interaction level (1=low, 3=high), P den is the starting population density (1=low, 3=high), ext is the external forcers (0=none, 3=high), H is the diversity index (Shannon) at the end of the run, and I is the equitability.

Use the simulator to answer the following questions. Remember, the results of each simulation are independent of all other simulations. Assuming you're auto-generating new data, every simulation will be different (even if all the radio buttons and check marks are the same). For this reason you can't simply run a simulation once with a low interaction, and then once with a high interaction to determine the effect of that variable. You should do each run at least five times to see the trends (that's what the grid is for). If you are using the web-based simulator, it can be found at the bottom of the top frame.

1. Determine the effect of number of species by trying five runs with 3 species, then 5 with 10 species. (If you're having trouble changing the Num Species text box to 10, leave the 3 in the box and add a 0 after it. It'll change to the maximum allowed).
2. What effect does changing the species interaction have on small webs (3) vs. large webs (10)?
3. What effect does changing the population density have on small vs. large simulations?
4. How are species interaction and population density interact for small vs. large webs? (interaction low + density low; interaction high + density low, etc.)
5. For the worst and best case4 simulations determined from exercises 1 - 4, determine the resistance to environmental factors (use all four).
6. Discuss how this simulation relates to real-world ecosystems. What situations are like the tundra, a grassland, a tropical rainforest? Which simulations might be related to effects that humans have on the environment? How do your simulations related to r- vs. K-selected organisms?
7. Figure 15 shows how you can manually change the interactions among the species. In this example I have changes the interaction between species 1 and 2 so that they are strongly competing. Make sure to set the Reload Last check box so that you are always starting with the same initial parameters. That way your changes can be directly compared from one run to the next. IMPORTANT: The next three questions should be done all at the same sitting, otherwise the inital values will change and you'll have to start this section over!!!!
   Change the interaction between two species so that they will compete as shown in figure 15. Determine the effect of this strong competition on the two species and the rest of the ecosystem. Change the parameters so that one species wins, and so they both co-exist. Remember to discuss the effects on the rest of the species. Think of a real-world situation that would fit this scenario and explain it.
8. Now set up a strong predator-prey interaction and explore the results as in question 7.
9. Finally, simulate a mutualism and explain what you have found..