Simulashun usin' some Spreadsheet (a PDF version be here)

If ya' is unfamiliar wid usin' some spreadsheet, some tuto'ial can be found here. A baaaad general resource on modelin' and simulashun can be found here, dig dis: http, dig dis://www, so cut me some slack, Jack.csi.uo'egon. 'S coo', bro.edu/nacse/ecosim/

Modelin' processes in de sciences gots become useful research tools in many areas uh de sciences and da damn availability uh clunkers extends dese techniques t'any dat wish t'use dese tools. Madematical representashuns uh biological systems gots long been used by ecologists t'model various processes. De models are useful t'access how much be understood about some real system. WORD! If de model be a baaaad representashun uh whut goes on in de "real wo'ld", den de audo' be fairly confident dat dey dig it de phenomena. WORD! If, however, de model duzn't follow de "real wo'ld", den it's back t'de clunker. Some models is so baaaad dey gots predictive value. Models dat predict locust outbusts in Africa is routinely used t'predict when control measho' mans is needed in some rejoin. 'S coo', bro. At crib, farmers routinely use models t'predict when herbicides, fungicides, and insecticides should be applied t'deir fields. On some granda' scale, climate models is used t'shape laws and internashunal treaties. One uh de fust models written wuz one t'mimic populashun growd. A model uh de simulashun process kin be seen in de followin' diagram. WORD!

De fust model we'll 'esplo'e is de geometric model. Acco'din' t'dese equashuns, some populashun grows uninhibited upside time. De impo'tant variables in dis simulashun is de startin' populashun size and da damn rate uh increase. De equashun fo' dis simulashun is, dig dis: dn/dt = r*N where dn/dt be de change in de populashun size (n) ova' time (t). De variable "r" be de rate at which de populashun increases puh' generashun (between 0.0 and 1.0). De "d" part uh dn/dt plum stands fo' "change in" (delta) so's it be read as "change in numba' ova' change in time". De N_{(t +1)} = N_t + dn/dt  be read as de "populashun size at time +1 be equal t'de populashun size at time 0 plus de change in size ova' de change in time". Dis scenario gots'ta result in an 'esponential increase in de populashun size.

• Open an 'sel spreadsheet and type "Exponential Growd Model" in cell A1.

• In cell A3 type "Start N:" and in cell B3 type "10". We gots'ta start wid some populashun uh 10.

• In cell A4 type "r, dig dis:" and in cell B4 type "0.2". De growd rate be "0.2"

• In cell A9 type "t" (our time column) and in cell B9 type "N" (populashun size at time t

• In cell A10 type "1", de start uh time.

• In cell A11 type "=A10+1", de fo'mula fo' advancin' our time unit by one. When ya' hit return, de cell should show "2".

• Copy de contents uh A11 t'de clipbo'd. Select and highlight cells A12 drough A39 and paste da damn clipbo'd data into dese cells. Dose columns should now show de numbers 1-30.

• In cell B10 type "=B3". We's plum copyin' de startin' populashun size fum cell B3. It should say "10"

• In cell B11 type "=B10+($B$4*B10). De dollar signs is a trick t0 ancho' dat part uh de fo'mula so's dat it always points t'our rate uh increase in cell B4. Cell B11 should say 12.

• Copy de fo'mula in B11 t'de clipbo'd. Highlight cells B12 drough B30 and paste da damn fo'mula into dose cells.

• To see da damn results (which is probably wida' dan de column, go t'de top uh de page wid de column haidin's (A,B,C...) and den posishun yo' curso' upside de line between B and C. It gots'ta switch t' some curso' dat looks likes dis, dig dis: . Den click and drag de column so's it's larga' and ya' kin see all de numbers

• Let's make some graph. Lop some boogie. Fust, select rows B10 drough B39 and click de chart wizard button ().Change t'a line chart, probably de fust one. Click Finish and yo' chart gots'ta be displayed.

• Try different startin' populashun sizes and rates uh increase (vary one o' de oder, not bod at da damn same time).

• Save yo' stash t'yo' W roll as "Exponential Model".

Figure 1. Human Populashun Growd

De geometric model be not very satisfyin' since it duzn't seem t'mimic de way real populashuns behave. Real populashuns duzn't 'espand indefinitely (except, puh'haps humans; Figure 1.  Instead, dey tend t'increase fo' some sho't time, den level off. Figure 2 depicts de growd uh bigho'n sheep populashuns in de Rockies fum de early 1800's t' about 1940. Note da damn rapid 'espansion uh de populashun followin' it's initial introducshun, followed by some levelin'-off uh de populashun at about 1.75 million sheep. Jes hang loose, brud. Dis be a typical populashun response seen in most natural populashuns. It's as if de populashun gots filled t'environment. Man! Dis "fillin'" uh de environment wid some particular species be de "carryin' capacity" fo' dat species. Fo' birds, de limitin' facto' in de environment may be da damn availability uh next sites. Oda' species may be limited by de availability uh food o' booze. Whuteva' de cause, de environment be capable uh suppo'tin' some limited numba' of some particular species. Dat numba' is de carryin' capacity, o' it represents de environmental resistance to furda' populashun growd. De carryin' capacity uh a system may change upside time. Droughts, pestilence, and oda' climatic changes may tempo'ally increase o' depress de carryin' capacity uh de environment. Man! Dese facto's may be responsible fo' de "bumps" in de above data (mo'e on dat later).

Figure 2. Bigho'n sheep populashuns

To be realistic, our model needs t'reflect da damn carryin' capacity uh an ecological system. WORD! Ecologists gots settled on de variable "K" t'represent da damn carryin' capacity uh de environment fo' some given populashun uh o'ganisms. Fo' our simple model, it duzn't matta' if de limitin' facto' be nest sites, o' de availability uh food (aldough different limitin' facto's could be easily accommodated. Dis equashun is essentially de same as our fust try, wid de 'sepshun uh de (K-N)/K addishun (K be de carryin' capacity, N be de current populashun density). De (K-N)/K term in de equashun "puts de busts" on populashun growd as de populashun reaches carryin' capacity. Slap mah fro! Let's fix our fust try t'reflect dis hopefully improved model. De equashuns is shown below, so cut me some slack, Jack.

• Open yo' Exponential Model on yo' W roll if it ain't already open. 'S coo', bro.

• Replace cell A1 wid "Logistic Growd". Den use da damn "Stash.. Save as" 'sel opshun t'save da damn stash as "Logistic" on yo' W roll. Dis insho' mans dat ya' won't overscribble yo' o'iginal model.

• In cell A5 type "K:". In cell B5 enta' "400". De carryin' capacity fo' our populashun gots'ta be 400.

• In cell B11 ya' put da damn new equashun. Type "=B10+($B$4*B10)*(($B$5-B10)/$B$5)" Compare dis t'de above equashun t'see how it wo'ks.

• Copy de contents uh B11 t'cells B12 drough B39. Yo' graph probably changed.

• Begin wid some startin' populashun uh 10, some rate uh increase uh 0.5 and some carryin' capacity uh 400. Den try oda' combinashuns (changin' only one at some time) t'see how yo' model wo'ks.

Still, dese models is not completely satisfyin' (dey duzn't gots de jiggles seen in Figure 2), so's some mo'e tweakin' be needed. De current model assumes dat some populashun responds instantaneously t'changes in populashun size. Dis assumpshun be unreasonable. If de populashun gots reached it's carryin' capacity (K), and 50% o' 60% uh de dudettes is pregnant, dey'll still cut bird, causin' de populashun t' overshoot da damn carryin' capacity. Slap mah fro! In addishun, some stresses duzn't assert demselves immediately. Slap mah fro! Fo' 'esample, as populashun density increases, it would snatch some time fo' individuals t'become stressed. Ho'mones kick in and  eventually de individuals is not as capable uh reproducshun. An 'esample uh a change in reproductive rate wid increasin' populashun density be shown in Figure 3. Unda' low density condishuns de crustacean Daphnia (booze flea) produces 4 offsprin' puh' day while unda' high density condishuns no offsprin' is produced. Note also dat survivo'ship be adversely affected by increasin' populashun density. Slap mah fro!

Figure 3. Changes in reproducshun uh Daphnia unda' different populashun pressho' mans

Dis reproductive time lag be represented by variable c in de model; see model below, so cut me some slack, Jack. Our populashun duzn't respond instantaneously, but be lagged by one generashun.

• Open de Logistic model if it's not on yo' screen. 'S coo', bro. Replace cell A1 wid "Reproductive Lag Model" and use da damn "Stash.. Save as..." menu funcshun t'save yo' stash as "Reproductive"

• In cell A6 type "c, dig dis:" and in B6 type "1". It turns out dat it's difficult t'put togeda' a fo'mula dat gots'ta read dis number. Ah be baaad... It's much easia' to change da damn fo'mula in de right places. Dese cell entries is plum used as reminders.

• Change da damn contents uh cell B12 t' =B11+($B$4*B10)*(($B$5-B10)/$B$5). De red highlights show where da damn fo'mula be changed. De easiest way t'do dis be to click on B12 and den edit da damn fo'mula in de fo'mula box (fx box) at da damn top uh de fo'm. WORD! You's kin place yo' curso' on de fo'mula, use da damn left and right keybo'd arrows t' move around and delete and change da damn fo'mula. WORD! Press return when yo' finished. Whut gots we plum done (be able t''splain how dis fits de above equashun fo' some time lag uh 1)

• Copy de contents uh cell B12 t'B13 drough B39.

• Try de followin':

  • N=10, r=.2, K=400

  • N=10, r=.5, K=400

  • N=10, r=.9, K=400

  • Den hold r at some low o' high value and change K, den N. Whut's goin' on?

• How could ya' simulate some reproductive lag uh 2? Hint, dig dis: B12 wuz changed fo' some lag uh 1, so's B13 needs t'be changed t' fo'ce some lag uh 2. Duzn't fo'get t'copy de contents uh B13 t'B14 drough B39.

Wid some reproductive lag we find dat populashuns wid some low intrinsic rate uh increase (r) survive longa' dan dose wid some high reproductive rate. In addishun, de stability uh dose populashuns wid low reproductive rates be greata' (less oscillashuns in deir growd curves). Finally, populashuns wid high reproductive rates also is prone to outbusts (populashun increases far in 'sess uh de carryin' capacity).

In de real wo'ld condishuns affectin' de reproductive rates uh o'ganisms (such as overcrowdin', drought, disease, etc) is not likesly t'last upside an 'estended puh'iod uh time. In de sho't-term, populashuns wid low reproductive rates kin wait out da damn environmental problems while dose wid higha' reproductive rates is mo'e likesly t'go 'estinct locally. Slap mah fro! We also find dat condishuns dat adversely affect de carryin' capacity kin also make some species mo'e prone t'local 'estincshun (especially dose wid some high r). Activities such as defo'estashun, booze pollutin', and overuse uh pesticides kin all decrease da damn carryin' capacity (increase da damn environmental resistance) and make populashuns mo'e susceptible t' local 'estincshun.

Recognizin' de differences between species wid high rates uh reproducshun and dose wid low intrinsic rates uh increase, ecologists often describe some species as bein' "r-selected" o' "K-selected" (Table 1). Dose species whose populashuns levels is controlled by deir reproductive rates is defined as r-selected while dose whose reproducshun be controlled mo'e by environmental resistance is termed K-selected.

Table 1

We's digtin' close. Real populashuns show fluctuashuns in density in response t'randomly changin' environmental challenges (as well as intrinsic rhydms). In some cases de fluctuashuns is clearly due t'changes in de environment, while at oda' times de variashuns is not as clear. Ah be baaad... We kin mimic dese random effects by randomly addin' o' subtractin' individuals fum de current populashun size, dig dis:

In de above fo'mula da damn size uh N be decreased by some random amount and den increased by some random amount (in Excel de RAND() funcshun generates some random numba' between 0 and 1, so's if we multiply dat random numba' by N, N gots'ta be decreased. We den add on anoda' random amount less dan N. Derefo'e, N gots'ta sometimes be larga' o' sometimes smaller. Ah be baaad... Dis affects de ability uh our model o'ganisms t'track where dey is in relashun t'de carryin' capacity (K). Sometimes dey "dink" dey is below de carryin' capacity, when actually dey may be above it (and visa versa). One could imagine an r-selected beast, such as some bug, wouldn't gots some baaaad idea uh deir relashun t'de environment and could make such some "missnatch". Alternately, if de environment be rapidly changin' de model populashun might gots trouble trackin' it accurately. Slap mah fro! De above model would also fit dat scenario. 'S coo', bro.

• Open de Logistic model and change cell A1 t'"Stochastic Model". Use da damn "Stash..Save As" menu funcshun and save da damn new stash as "Stochastic" on de W roll.

• Change da damn fo'mula in cell B11 t' "=B10+(($B$4*B10)*($B$5-(B10-(RAND()*B10)+(RAND()*B10)))/$B$5)" Rememba' dat ya' kin edit da damn existin' fo'mula rada' dan type in de whole new fo'mula. WORD! Co'relate da damn fo'mula ya' plum entered wid de populashun model fo'mula shown above. Watch out fo' de parendeses. If ya' dig an erro' message, do not let 'sel fix yo' fo'mula, it'll probably do it wrong. What it is, Mama!

• Copy de fo'mula in B11 t'de clipbo'd and den paste it in cells B12 drough B39.

• Change da damn startin' values t'N=100, r=.2, K=1000 and da damn honky code gots'ta generate some new graph. Lop some boogie.

• Lets modify de model so's dat it gots'ta run fo' 100 generashuns.

  • Select cells A39 and B39 and copy dem t' de clipbo'd. Select cells A40 drough A109 and paste da damn fo'mulae in de clipbo'd into de new range.

  • Now we need t'fix de graph so's it shows mo'e dan 30 generashuns. Click on de graph and da damn rows dat gots'ta be graphed gots'ta be surrounded by some blue rectangle dat represents de graph range. It should look likes dis, dig dis:

  • Next, grab de lowa' right o' left handle (de dot- ) and yo' curso' gots'ta change t'a double arrow (). Click and drag de box so's it now covers 100 generashuns (From B10 drough B109). When ya' scroll down de graph gots'ta now show 100 generashuns.

• Up t'dis point, de models we used wuz deterministic, meanin' dat each time ya' run de model, ya''ll dig 'esactly de same results as long as ya' duzn't change da damn input. Man! Dis model be stochastic and gots'ta change each time ya' run it, givin' ya' some a view dat be mo'e similar t'de real-wo'ld situashun. Compare da damn above graph t'figure 2, fo' 'esample. To use dese models, ya' run dem upside and upside usin' de same input data. WORD! Each run gots'ta result in some slightly different populashun projecshun and graph. Lop some boogie. To run de model wid de same data input, ya' kin fo'ce some recalculashun by hittin' de Ctrl-S key. Slap mah fro!

• Try ten runs o' so's by repeatedly strikin' de Ctrl-S key. Slap mah fro! Mo'e likesly dan not, each uh yo' populashun runs gots'ta succeed in reachin' 100 generashuns.

• If, by chance, one uh yo' runs fails t' reach 100 generashuns and da damn populashun goes 'estinct, ya' graph gots'ta be similar t'de followin':

• Now increase r in units uh .1 (r=.3, r=.4, etc). Whut effect duz dis gots on yo' populashuns. At whut point do de populashuns begin t'regularly fail t'reach 100 generashuns? Whut type uh animal might ya' be modelin' at higha' reproductive rates? Whut type uh real-wo'ld situashun(s) might dis represent?

• Try varyin' de statin' populashun and carryin' capacity (one at some time, not togeder) fo' low and high reproductive rates. Whut situashuns is you modelin' here?