Grub Chains, Webs, an' th' Cybernetic Ecosystem

A grub chain dexcribes how inergy an' nutrients move through an ecosystem (Reckon 1). In most terrestrial grub chains inergy is cappured by plants thet also inco'po'ate ino'ganic materials (nitrojun, phospho'us,su'fur, etc.) as chemical nutrients. Th' git-up-and-git an' nutrients pass fum one level t't'other o' are returned t'th' environment by decomposers. Th' prodoocers is foun' at th' base of th' grub chain (also known as th' 1^st tropic (feedin') level). Git-up-and-git tempo'arily held in th' plants is then transferred t'th' 2^nd trophic level, inhabited by herbivo'es an' omnivo'es. Th' herbivo'e level is also referred t'as th' primary cornsoomrs. Git-up-and-git then flows t'th' 3rd an' sometimes higher trophic levels whar it is inco'po'ated into th' secondary an' tertiary consoomrs (not shown). Carnivo'es an' top carnivo'es occupy th' upper levels. Th' second law of thermodynamics prohibits th' recyclin' of inergy in this hyar system, dawgone it. In addishun, it makes it impostible t'transfer 100% of th' git-up-and-git fum one level t't'other. On account o' of this, less inergy is available th' higher yo' move up th' grub chain which puts limits on th' number of tropic levels fo' a particular ecosystem, dawgone it.

Thar is three types of pyramids (center of figger 1) thet dexcribe th' relashunships among th' various trophic levels:

• Pyramid of Numbers: This hyar is a pyramid based on th' number of o'ganisms at etch trophic level, ah reckon. Howevah, it does not show th' mass of grub cornsoomd at etch level, ah reckon.
• Pyramid of Biomass:This hyar is a pyramid based on th' total mass of o'ganisms at etch trophic level, ah reckon. This hyar model is also limited on account o' it does not reveal th' git-up-and-git corntributed by etch o'ganism in th' ecosystem, dawgone it.
• Pyramid of Git-up-and-git:This hyar is a pyramid which shows how much grub inergy is available at etch trophic level in an ecosystem, dawgone it. This pyramid is the dawgoned-est complex of th' three on account o' calculashuns of th' number, masses, an' inergy corntent fo' etch type of o'ganism muss be cornsidered, cuss it all t' tarnation.

A second grub chain is shown in figger 2. Fo' both these examples, we haf emphasized th' grazin' grub chain. Mo'e impo'tant, howevah, fum an inergy point of view, is th' decomposer grub chain (Reckon 3), whar most of th' git-up-and-git transfer takes place (takin' care of all grazin' git-up-and-git as fine as it's own). While th' decomposer grub chain kin't recycle inergy, nutrients is recycled at this hyar point. Mo'e on th' decomposin' grub chain later.

A dexcripshun of most interackshuns among o'ganisms in th' environment is rarely as simple as shown in these two figgers (although figger 2 purdy much covahs it. Fo' figger 1, other birds'd sartinly be feedin' on th' insecks an' spiders an' these'd be potential prey fo' th' hawk. Shet mah mouth! Th' puma (mountain lion, cougar,wild cat)'d also feed on a wider variety of prey. When all these grub chains are added up, we haf a grub web (figger 4). Even this hyar representashun is an on over simplificashun on account o' menny o'ganisms will occupy mo'e than one trophic level (Table 1).

One reason ecologists is interested in grub webs is th' apparent relashunship between stability of an ecosystem an' its complexity. Junerally, systems wif fewer interconneckshuns among species is less stable than them wif mo'e conneckshuns (Reckon 5). Thar is two ways of lookin' at stability:

• Stability of th' Fust Type: On over time thar is a increasin' cornstancy of numbers. Th' system will eventually retch a steady state unner cornstant corndishuns (th' next state is predickable fum wifin th' system
• Stability of th' Second Type: The system has greater resistance t'changes thet is external (Reckon 7). This hyar is usually accompanied by higher inergy flow.

Modelin' grub webs is a-gonna require a diffrunt approach fum thet we've used fo' our two-species interackshuns, in part on account o' we need to model a greater number of species, but also on account o' we need a way t'apply all of the potential species interackshuns (Table 2).

Cybernetic Systems
Cybernetics is corncerned wif corntrol an' communicashun in systems fo'med by livin' o' non-livin' indivijools an' their artifacks (No'bert Wiener). A system is a set of diffrunt elements, etch in diffrunt states, etch state influenced by t'others. Cybernetics dexcribes th' interackshuns. Th' various elements (species, eleckronic systems, etc.) is linked by feedback loops. Negative feedback loops are stabilizin' fo' th' system (blood glucose levels, a thermostat), while positive feedback loops is destabilizin' (epidemics, high fevahs, o'gasm- no thet warn't a miss-spellin').

Negative feedback among th' components leads to stability, not only fo' th' entire system, but also fo' selecked components. Thus, the system as a whole shows persissence. Cybernetic systems kin influence their own futures on account o' th' present state holds info'mashun (number of species, interackshuns among the species, populashun sizes, etc.). Thus, info'mashun in th' current state kin be used to predick th' next junerashun.

T'other quality of cybernetic systems is thet they're se'f-o'ganizin'. Th' negative feedback loops among th' various species o' components will eventually "settle down" t'an equilibrium state. Th' likelihood of se'f-o'ganizashun is greater when thar is a larger number of intities in th' system on account o' th' interackshuns (feedback) among th' entities (species) is weaker.

Info'mashun about th' states of th' entities in the system increases wif time an' results in an increase in complexity. Unlike simple machines, which is a final produck, cybernetic systems change wif time an' show persissence. Wif all th' sto'ed info'mashun, these systems haf a resistance t'external events an' show stability (of both th' fust an' second type).

Reckon 8 is an example of a complex cybernetic system; a po'shun of th' internet routes surroun'in' Urbana, Illinois. This hyar system retains an' transfers info'mashun (even eff'n ha'f of it is po'n), an' etch node sarves as a feedback system (through sof'ware). These chareeckeristics, along wif th' size qualify th' internet as a cybernetic system, dawgone it. It's impo'tant t'note thet this hyar system is se'f-assemblin'. As users is added, logged on, logged off, an' send o' receive info'mashun, no one is in charge! Th' only intervenshun is t'make sho'nuff ev'ryone has their own unique address. Reckon 9 shows internet cornneckshuns as separeete nodes, indicatin' how adjacent local systems interconneck an' communicut wif one t'other. Wif a li'l imaginashun, it's not difficult t'see th' similarities between our grub web problem an' this hyar diagram, dawgone it.

Th' Cybernetic Ecosystem Model
We kin start th' model wif some familiar concepps:

• Etch species (i) will haf a populashun size o' density:: N_i
• Fo' etch species (i) thar will be a rate of change in th' density: dN_i /dt
• Thar is selecked invironmental facko's thet   kin affeck th' outcome of th' above: E
• These facko's will also change on over time: dE/dt

Th' above interackshuns will be made propo'shunal to th' sum of th' producks of th' interackshuns t'prodooce a set of diffruntial equashuns to model th' states of th' species at etch point in time. A three-species model is shown in Table 3.

This hyar model satisfies th' needs of our cybernetic system, dawgone it. Species kin interack wif th' environment, other species o' wifin a species. As an example, runnin' down th' column fo' species 1 (N₁):

• Species 1 cuzs a change in th' environment (dE /dt). Th' effeck of species 1 on th' environment is shown in th' matrix as EN₁
• Species 1 kin interack wif itse'f: dN₁ /dt = -f N₁²
• Species 1 also interacks wif species two (dN₂ /dt = +i N₂N₁) an' three (dN₃ /dt = +l N₃N₁).
• Th' model kin be easily expan'ed (jest add mo'e rows an' columns).

Th' model also allers fo' diffrunt interackshuns among th' species. Th' coefficients (a,b,c....) is th' equashuns dexcribin' the stren'th of th' interackshuns between th' two species. Th' sign of th' coefficients shows how it will affeck th' change in species numbers fo' enny of th' i^th species (dN_i /dt). These also satisfy our requirement t'be able to model all potential species interackshuns as laid out in table 2:

• Species 2 is a predato' on species 1 (in pink in table 3) on account o' species 1 has a positive effeck on 2 (+i   N₂N₁)an' species 2 has a negative effeck on 1(-g N₁N₂).
• Species 2 an' 3 is competin' (in yeller) on account o' both their coefficients is negative (-k an' -m)
• Some interackshuns is non-exissent (species 3 has no effeck on th' environment).
• Enny of th' potential interackshuns in table two c'd be built into th' model, ah reckon.
• Dependin' on th' amplitude of th' coefficients (a,b,c...), th' interackshuns between two species may be strong o' weak. Shet mah mouth! In juneral, species thet interack weakly wif other species does so wif a large number of species. Them thet interack strongly does so wif fewer number of species.

Fo' th' system shown in table 3:

• Th' rates of change (Column 1) depend on th' values of th' equashuns in th' assosheeated rows an' determine th' values in th' next junerashun (th' model is predickive).
• Th' process is iterative wif a steady-state goal (lackin' external changes).
• Th' steady state of th' system depends on the coefficients of th' interackshuns.
• Transient states depend on E an' N_i.
• Transient states may change th' coefficients of the interackshuns (not built into th' model we'll be usin'; th' idea is thet changes in the invironment o' numbers of a completely diffrunt species c'd change a competitive situashun, fo' example).

Expan'in' Niche Theo'y
Th' concepps we've developed hyar kin be used t'expan' our explanashuns of th' ecological niche. Thar is sevahal ways t'reckon about th' niche thet we've already developed:

• Odum niche: Th' oldess of th' niche corncepps. Essentially, th' habitat of th' o'ganism, dawgone it.
• Ecological niche: Th' physical space occupied by th' o'ganism an' th' funckshunal role of th' species in thet space. Thar are three components: spatial (habitat), trophic (whar on th' grub chain), an' other (posishun on an invironmental gradient- whar th' species fits in).
• Huchensonian niche: This hyar is a mathematical corncepp. To dexcribe th' Huchensonian niche, yo' measure th' tolerances fo' th' species acrost all postible invironmental dimenshuns (temperature, hoomidity, wind, various nutrients, etc.) an' then, wif these n measurements, cornstruck an n-dimenshunal hyperspace thet dexcribes all of th' postible invironmental tolerances on o'thoganal axes (at right angles t'one t'other; it's OK on account o' it's a mathematical corncepp). Th' resultin' hyperdimenshunal data cloud is th' dexcripshun of th' species' tolerances an' is the Huchensonian niche.
• Cybernetic niche: Dexcribes the species through their coefficients of interackshun in th' system matrix. Fo' example, species thet is competin' wif one t'other will haf negative coefficients of interackshun in th' matrix. Conceppually, this hyar can be see as two pareellel negative feedback systems along th' resource corntinuum (Reckon 10). In th' figger, both species B an' C are cornnecked through negative feedback loops (solid arrows) t'a limitin' resource A. Assumin' equal competitive abilities, eff'n th' species on overeat th' resource, it will become unavailable an' th' species populashuns will drop. Eff'n howevah,  species B is better equipped t'take th' resource, species C will drop in populashun size, resultin' in makin' th' resource mo'e available t'B which then fo'ces C even lower; eventually indin' in the local extinckshun of species C. Thus, a run-away positive feedback loop is implied between th' two species (dotted lines in th' figger). Similar loops kin be cornstrucked fo' enny potential species interackshun.

Usin' th' Model
Two vahshuns of th' stan'-alone web simulato' is available fo' downloadin' an' installin'. Th' large fo'mat simulato' (Reckon 11) requires a screen resolushun of at least 1024 X 768 an' allers yo' t'view all display an' corntrol panels at th' same time. Th' trimenjus grub web simulashun is har. A smaller footprint vahshun is also available as a download (Reckon 12) which is identical t'the web-based plug-in, as enny fool kin plainly see. Yo' kin git th' small screen vahshun har.

Controllin' th' Simulashun.

• Reckon 12, area A, allers yo' t'view various po'shuns of th' model, ah reckon. Eff'n th' data input radio button is selecked, yer in data input mode. Data kin be direckly intered into th' matrix (D) o' kin be junerated, cuss it all t' tarnation. Eff'n yo' seleck th' Graphs & Data radio button then th' display will shif' t'thet shown in th' small inset figger (B). Th' data grid keeps track of etch of yer runs an' kin be copied t'the clipboard fo' pluggin' into a Wo'd docoomnt o' fo' further analysis in Excel, ah reckon. A graph of th' populashun size fo' etch of th' species is also shown, as enny fool kin plainly see. Species is colo'-coded acco'din' t'th' colo's shown in area G (N₁, N₂, N₃...). By seleckin' th' Trimenjus Graph radio button yo' kin git a larger graph (as in inset C).
• Data kin be intered by han' in area D o' kin be junerated usin' th' controls in area E. Th' number of species kin be ennywhar between 3 an' 10. Th' maximum number of junerashuns th' simulashun will helter-skelter can also be corntrolled hyar. Eff'n th' Auto Junerate check box is selecked, a noo data set will be junerated follerin' etch simulashun. Eff'n th' Reload Last check box is set, th' simulato' will reload th' origeenal data set. Eff'n neifer is set then pressin' th' show me button will corntinue t'other 25 junerashuns fum whar th' last simulashun stopped, cuss it all t' tarnation. Yo' kin also set the relative levels of species interackshun (species interat weakly, medium, o' strongly) as fine as th' startin' populashun size.
• Area F allers yo' t'add external fo'cin' facko's to th' simulashun.
• Area G shows some simple statistics fo' th' run includin' yer startin' numbers, th' end values, percent success (percent of species thet made it t'th' end of th' simulashun, a measure of divahsity (H), an' equitability (I)

Reckon 13 shows a junerated data set in the interackshun matrix fo' 10 species, wif low interackshun, high populashun density, an' no external fo'cin' facko's. Note thet th' populashun sizes is ran'omly set, so thet some of th' populashuns may start off wif low numbers. Th' populashun density jest sets the highess postible level, ah reckon. Yeller blocks in th' matrix represent null interashuns (no effeck of th' column species on row species growth. Species 1, fo' example does not affeck the growth of species 4 in th' figger. Pink cells is negative interackshuns. White cells are positive interackshuns. Reckon 14 depicks th' results of a typical simulashun  Fo' the data grid, S is th' number of species thet yo' started wif, jun is the number of junerashuns th' simulashun ran, int is th' speceis interackshun level (1=low, 3=high), P den is th' startin' populashun density (1=low, 3=high), ext is th' external fo'cers (0=none, 3=high), H is th' divahsity index (Shannon) at th' end of th' run, an' I is th' equitability.

Use th' simulato' t'answer th' follerin' quesshuns. Remember, th' results of etch simulashun is independent of all other simulashuns. Assumin' yer autyjuneratin' noo data, ev'ry simulashun will be diffrunt (even eff'n all th' radio buttons an' check marks is th' same). Fo' this hyar reason yo' kin't simply helter-skelter a simulashun once wif a low interackshun, an' then once wif a high interackshun t'determine th' effeck of thet variable. Yo' sh'd does etch helter-skelter at least five times t'see th' trends (thass whut th' grid is fo'). Eff'n yer usin' th' web-bhased simulato', it kin be foun' at th' bottom of th' top frame.

1. Determine th' effeck of number of species by tryin' five runs wif 3 species, then 5 wif 10 species. (Eff'n yer havin' trouble changin' the Num Species text box t'10, leave th' 3 in th' box an' add a 0 af'er it. It'll change to th' maximum allered).
2. Whut in tarnation effeck does changin' th' species interackshun haf on small webs (3) vs. large webs (10)?
3. Whut in tarnation effeck does changin' th' populashun density have on small vs. large simulashuns?
4. How is species interackshun an' populashun density interack fo' small vs. large webs? (interackshun low + density low; interackshun high + density low, etc.)
5. Fo' th' wo'st an' bess case4 simulashuns determined fum exercises 1 - 4, determine th' resistance t'environmental facko's (use all four).
6. Discuss how this hyar simulashun relates t'real-wo'ld ecosystems. Whut in tarnation situashuns is like th' tundra, a grasslan', a tropical rainfo'est? Which simulashuns might be related t'effecks thet hoomins haf on th' environment? How does yer simulashuns related t'r- vs. K-selecked o'ganisms?
7. Reckon 15 shows how yo' kin manually change the interackshuns among th' species. In this hyar example ah have changes th' interackshun between species 1 an' 2 so thet they is strongly competin'. Make sho'nuff t'set th' Reload Last check box so thet yer allus startin' wif th' same initial pareemeters. Thet way yer changes kin be direckly compared fum one helter-skelter to th' next. IMPORTANT: Th' next three quesshuns sh'd be done all at th' same sittin', otherwise th' inital values will change an' yo'll hafta start this hyar seckshun on over! Fry mah hide!! Fry mah hide!! Fry mah hide!! Fry mah hide!
   Change th' interackshun between two species so thet they will compete as shown in figger 15. Determine th' effeck of this hyar strong competishun on th' two species an' th' ress of th' ecosystem, dawgone it. Change th' pareemeters so thet one species wins, an' so they both co-exist. Remember t'discuss th' effecks on the ress of th' species. Figger of a real-wo'ld situashun thet'd fit this hyar scenario an' explain it.
8. Now set up a strong predato'-prey interackshun an' splore th' results as in quesshun 7.
9. Finally, simulate a mutualism an' explain whut yo' haf foun'..