Development of a Vegetation Soil Consumer Model with Harvesting

Transcription

1 Developmen of a egeaion Soil onsumer Model wih Harvesing Plan B Projec in he Masers Program for ompuaional and Applied Mahemaics Universiy of Minnesoa Duluh Submied by Tom Sjoberg Advisor Dr. Harlan Sech Deparmen of Mahemaics and Saisics

2 Table of onens 1 Inroducion 1 2 Model Derivaion 4 3 Model Analysis. 8 4 Applicaion of he Model: The Effecs of Harvesing egeaion Harvesing onsumer Harvesing Soil Ferilizaion 20 5 Implicaions of he Model and Topics for Furher Sudy 21 6 Appendix: Mahemaica odes References

3 Absrac: Through he derivaion, parameerizaion, and simulaion of a differenial equaion model, we will analyze he relaional impacs in he vegeaion soil consumer ecosysem. There are numerous parameers ha affec his ecosysem. However, we will focus on he impac of harvesing he vegeaion, or harvesing he consumer, or he inroducion of ferilizaion o he soil o deermine he impac ha hese aciviies have on he sabiliy of he ecosysem. We will find ha here are parameer values ha allow he sysem o be a a favorable sable equilibrium, and here parameer values ha will creae an unsable ecosysem, or cause he sysem o crash o an unfavorable sae.

4 1. Inroducion: egeaion densiy and soil erosion are facors ha significanly affec waershed ecosysems. There are numerous consideraions in he riad relaionship beween consumer vegeaion soil ecosysems. Soil condiions influence vegeaion growh, which (hrough lier decay) increases soil deph. onsumer biomass levels are affeced by he vegeaion on which hey feed, and hey regulae vegeaion biomass levels. Moreover, harvesing and he processes represened by muliple parameers affec hese ecosysems. In recen years, mahemaical modeling has been used o demonsrae he dynamical processes in he consumer vegeaion soil ecosysem. Many effors have been made o model vegeaion dynamics under ecological sresses. The developmen of differenial equaions o express he dynamical process of he ecosysem is a common approach in he modeling mehod. In one of he aemps o model his sysem, Johan van de Koppel and Max Riekerk 2 (2000) sudied he mahemaical implicaions of vegeaion regulaed herbivore populaion dynamics. They concluded ha herbivore populaions impac he sanding crop of vegeaion and irreversible vegeaion change may occur hrough overgrazing. Their use of phase plane analysis was helpful o heir undersanding of he dynamics of a plan herbivore sysem. The phase plane gave a graphical inerpreaion of isoclines (nullclines) beween he vegeaion and consumer wih equilibria a he poins of inersecion. Their model suppored sable and unsable equilibria dependen on he condiions in he ecosysem. Thus, hey anicipaed differen resuls dependen on he soil condiions. A vegeaion erosion model was creaed by: Z.H. Wang, G.H. Huang, G.Q. Wang, and J. Gao 8 (2004). The model was applied o he Xiaojiang, Heishui, and Shengou Waersheds, ribuaries o he Yangze River in Souhern hina. Their sudy, using a differenial equaion model, allowed he researchers o quanify vegeaion erosion dynamics in order o improve he susainabiliy of he 1

5 waersheds. However, he model hey used demonsraed some limiaions due o inaccuraely modeling erosion processes. Their erosion process was exponenial, wihou limiaion. Livesock grass soil sysems were considered in he modeling of Javier Ibanez, Jaime Marinez, and Susanne Schnabel 1 (2007). They sudied deserificaion due o overgrazing in he dry lands of he Medierranean Region. In heir work hey used he logisic growh model of populaions o represen he primary producion of grass. The logisic growh model is an imporan consrucion ha will be used in his paper as well. I will be an imporan par of he differenial equaion represening he growh rae of vegeaion. As livesock grazed on he drylands of he Medierranean, consumpion of he vegeaion was observed and recognized as having an impac on he vegeaion, soil, and hydrology of he area. Parameric values were esablished for hree exensive livesock farming sysems in Spain. However, he noaion for heir model was somewha awkward and i seemed he noaion could be simplified. Anoher sudy of herbivore vegeaion dynamics in a semi arid grazing sysem is described in he work of Meza, Bhaya, Kaszkurewicz, and osa 4 (2006). They proposed a mahemaical sysem o explain he alernaive equilibrium poins in a dynamical sysem sensiive o soil degradaion. An onoff policy was developed o manage livesock grazing around sable equilibrium poins. The objecive was o manage a susainable herbivore vegeaion ecosysem and no allow he ecosysem o receive irreversible damage. They used simulaion models o graphically depic he no grazing regions and he grazing regions when livesock were allowed o forage he vegeaion. The goal of his projec is o design an alernae, process based, differenial equaion model ha realisically describes paricular vegeaion soil consumer ecosysems. We will use phase planes o inerpre boundaries o he sysem in secion 3, and numerical simulaions o demonsrae endencies when parameer values are alered. We would like o clarify previous models represening he 2

6 consumer vegeaion soil ecosysem and will creae a special purpose simulaor used for performing numerical experimens. Simulaions of he model are used in secion 4 o inerpre he resuls when consumer and vegeaion harvesing are inroduced. Also, we are ineresed in adding nuriens o he soil and deermining he impacs on equilibria and ecosysem sabiliy. In secion 5, we discuss implicaions and sugges a few opics for furher sudy regarding his model. The Mahemaica 7 (2009) sofware program has a number of differen ools for simulaing sysems of differenial equaions and graphing he soluions of wo and hree dimensional differenial equaions. In he Appendix are found he Mahemaica codes used o perform he simulaions. References are included in he final secion where here is a disribuion of work five years or older, and references o work wrien wihin he las five years. 3

7 2. Model Derivaion: There is a selec group of equaions and funcions ha allow us o mahemaically describe an ecosysem and deermine he ecological direcion he environmenal sysem is headed. The moivaion of our model is he classic producer/consumer model of Rosenzweig and MacArhur 6 (1963): d ( 1 ) k growh Pmax Pmax H foraging γpmax Pmax H growh m{ moraliy In heir work, hey assume logisic growh of he vegeaion, and a Holling Type II model of consumer harvesing of he vegeaion. The Holling Type II funcion (1959) was firs used in ecology by. S. Holling o deermine he processing rae of he consumer. We consider a sysem consising of vegeaion, (), a consumer, (), and soil, S() as measured by heir nuriional levels. The vegeaion growh rae is represened by a logisic growh model which is modulaed by soil nurien condiions of he soil vegeaion ecosysem. The consumer growh rae is direcly influenced by he amoun and qualiy of vegeaion. The consumer foraging rae is represened by a sauraing funcion of vegeaion quaniy. Soil grows by he decay of vegeaion lier and soil nurien addiion. We assume soil growh due o weahering is nil. 4

8 The following diagram describes he relaionships beween,, and S and parameers (described laer) ha affec hem. moraliy onsumer harvesing predaion egeaion harvesing lier decay nurien upake Soil erosion ferilizaion Figure 2.1: A diagram showing he flow of nuriens hrough he sysem. 5

9 The change over ime of he vegeaion, consumer, and soil componens are expressed in he foregoing differenial equaions: ds ( 1 ) b + S 4k growh Pmax Pmax H foraging { L h { 1 deah harvesing γpmax Pmax H growh h 2 m{ { moraliy harvesing { F α al { ( ) S addiion decomposiion β erosion The rae of growh of vegeaion over ime is a produc of a sauraing funcion of he amoun of soil presen, muliplied by he logisic growh of he vegeaion, minus consumer harvesing, minus vegeaion moraliy, minus managed vegeaion harvesing. egeaion growh is dependen on soil qualiy. The radiional logisic growh model, 1, is muliplied by a sauraing funcion of soil levels,. As S +, k is he maximum carrying capaciy of he vegeaion assuming high soil nurien. The growh rae of he vegeaion is reduced by he rae of vegeaion harvesed by he consumer. The half sauraion consan, HP max, affecs he rae of vegeaion processed. Non foraging moraliy loss of vegeaion is included, as is a possible means of including a vegeaion harvesing rae, h 1. The growh rae per capia for he consumer is modeled as he foraging rae, imes a coefficien, 0 1, he conversion efficiency of vegeaion ino consumer biomass. There is 6

10 moraliy of he consumers, m, subraced from he rae of growh, along wih a possibly managed consumer harvesing rae, h 2. Soil qualiy (nurien) has a growh rae dependen upon he decay of vegeaion, and on he addiion of nuriens o he soil, F. There is a loss of soil nurien due o erosion ha is assumed o be a decreasing funcion of vegeaion cover. For simpliciy, he rae of erosion of soil is assumed o ake he form S (, α he asympoic loss rae of soil coefficien as >0. The Wang, e.al 8 paper couples vegeaion and erosion rae as dynamic processes in heir model of waershed sysems. In our model he dynamic process is beween vegeaion, consumer, and soil, where we consider erosion as an inrinsic rae loss of soil reduced by he amoun of vegeaion. The Wang, e.al 8 paper reas he erosion rae as he mos imporan eniy ha affecs vegeaion growh. In conras, our model considers erosion as a rae loss of soil nurien, reducing he availabiliy of soil nurien o he vegeaion. Our use of he logisic growh funcion o represen vegeaion growh is similar o he Ibanez, e.al 1 sudy. They have he primary producion of grass represened by a logisic funcion where (in heir noaion) PP= GR x G [1 ( )]. GR is he inrinsic growh rae of grass and GK is he carrying capaciy of grass. Boh are assumed proporionae o a funcion of he Max( S msg,0) form: 1 Exp[ ]. A reducion in soil nurien availabiliy negaively affecs sm1 vegeaion growh rae and he carrying capaciy of he vegeaion. egeaion growh is dependen on soil nurien availabiliy as is he carrying capaciy of grass. The Ibanez, e.al 1 paper uses an awkward compuer code noaion and equaion seing. Noaion and equaions in our paper are mean o be more easily undersood. However, heir parameric values were useful o our work in seing inervals for he inrinsic maximum growh rae of grass and he maximum carrying capaciy of grass. 7

11 Koppel and Riekerk 2 sudied he effecs of herbivore regulaion on semi arid grazing sysems as a means o avoid irreversible vegeaion change. In heir model, soil qualiy is a funcion of waer availabiliy. Their equaions are similar in form o a 2 dimensional simplificaion of our model. Their use of phase plane represenaions clarified where equilibria occurred as densiies of herbivore and vegeaion increased. Their sysem was wo dimensional (as we aforemenioned) when we se soil qualiyo be a funcion of vegeaion densiy. oexisence equilibria have a spiral roaion ino he equilibrium poin if i is sable wih complex eigenvalues and a spiraling ou from he poin if i is unsable wih complex eigenvalues. The phase plane represenaions used by Koppel and Riekerk 2 were hand drawn, bu we have chosen o use a sofware program, Mahemaica, o illusrae he phase planes. Mesa, e.al 4 coninued he sudy of Koppel and Riekerk using an on off policy for he herbivore vegeaion dynamics in a semi arid grazing sysem. I has been recognized for a considerably long ime ha alernaive vegeaion saes occur in semi arid grazing sysems. Simulaions of he alernae vegeaion and herbivore densiies resuled in heir beer undersanding of he sudden and irreversible jumps beween vegeaion saes. 3. Model Analysis If we assume ha he rae of soil gain/loss is small relaive o ha of he vegeaive and consumer componens, we can se S = 0 in he hird equaion. We can le S = S, where 1, a funcion of. This reduced form shares he same essenial form as he Koppel, e.al 2 and Mesa, e.al 4 models. If we subsiue S ino he firs equaion, he model becomes wo dimensional: 8

12 _ d S (1 ) _ k b + S growh P max Pmax H foraging L h { 1 { deah harvesing γpmax Pmax H growh h 2 m{ { moraliy harvesing Parameer defaul values used in our simulaions of he differenial equaions for he wo dimensional sysem are based on Ibanez 1 (2007) and Meza 4 (2006). asympoic soil feriliy coefficien, d=1.0 half sauraion soil consan, b=0 carrying capaciy, k=0.95 maximum rae of processing, P max= =1.0 half sauraion deermining consan for maximum rae of processing, H= lier growh, L=0 harvesing vegeaion, h 1 = conversion of vegeaion ino consumer biomass, γ=0.88 moraliy, naural deah of consumer, m=4 harvesing consumer, h 2 = soil nurien addiion, F= conversion of lier ino soil nurien, a=4 asympoic loss of soil coefficien, α=1 soil erosion sensiiviy consan, β=100 9

13 We will use our differenial equaions o develop a phase plane graphical analysis. Like he Koppel and Riekerk sudy, phase plane graphical analysis is a ool used o visually locae saes of equilibrium in he consumer vegeaion coordinae sysem. The nullclines, or zero ne growh isoclines, of he consumer and vegeaion are lines where he saes of equilibria for boh eniies will lie a he nullclines inersecion. γpmax Seing, m h2 = 0, we can find he nullcline for he consumer. HP + max γpmax Specifically from, = m + h2, HP + max we obain he nullcline for he consumer: HP = γ P max max ( m + h 2 ( m + h 2 ) ) _ d S Pmax Seing, ( 1 ) L h1 = 0, we can find he nullcline for he vegeaion. _ k HPmax + b + S Pmax d S Tha is, = ( 1 ) L h _ 1 HPmax + k b + S _ from which we obain he nullcline for he vegeaion: HP P _ d S (1 ) L h ] b + S k max = [ 1 max + wih 1. 10

14 We will use hese nullclines o pariion he phase plane. We can analyze he phase planes o deermine sysem equilibrium poins under variaions of he parameric values. onsider he nullclines, he sysem always suppors a no life equilibrium (,) =(0,0) and ds vegeaion only equilibrium (,) = (*,0) when * solves (1 ) L h1 = 0, b+ S k wih 1 S = (1 + β)( F + αl). The following figures illusrae ha he model can suppor α wo vegeaion only equilibrium saes. The defaul values are used. 0 H0.70, L h1= Figure 3.1a: Allee effec; h 1 =5; ( 0, 0 )=(0.70,); H=; =0, he unimodal; =0, he verical line The simulaions illusrae ha he susainabiliy of he sysem depends on he sysems iniial sae. The vegeaion only model possesses an Allee effec, Sephens, e.al 7. The Allee effec refers o a phenomena in biology where a populaion canno susain iself below a cerain criical level. If is less han he lower valued *, hen ends oward zero as ends oward posiive infiniy, Figure 3.1a. 11

15 Oherwise, ends oward he larger valued *, which is he carrying capaciy of he consumerfree sysem, Figure 3.1b. 0 H0.70, 4L h1= Figure 3.1b: Allee effec; h 1 =5; ( 0, 0 )=(0.70,4); H= ; =0, he unimodal; =0, he verical line The lesser consumer populaion of =4 versus = allows he sysem o end o he larger * when here are no oher changes in he sysem. When he wo nullclines inersec in he region > 0 here is also a hird ecologically relevan coexisence equilibrium a he poin of inersecion. The following figures depic possible phase planes and soluion rajecories for our model. Also, he figures show how model behavior is influenced by he predaion half sauraion parameer, H. 12

16 Figure 3.2: H = ; ( 0, 0 )=(0.70,); =0, he unimodal; =0, he verical line The defaul parameric values are used, and (0)=0.70 and (0)=. We find an inward spiraling sable equilibrium poin when he half sauraion deermining consan is H = Figure 3.3: H=9; ( 0, 0 )=(0.70,); =0, he unimodal; =0, he verical line The same parameric values are used excep we reduced he half sauraion deermining consan, H, by.01 o.09. A periodic orbi formed by a Hopf Bifurcaion is found by our iniial condiions for he vegeaion and consumer Figure 3.4:H=8; ( 0, 0 )=(0.70,); =0, he unimodal; =0, he verical line We changed H o 8 moving he consumer nullcline o he lef. The vegeaion and consumer boh are seen o crash o = 0, = 0, where vegeaion and consumer have boh died off. Addiional simulaions (no shown) indicae he sysem does no suppor periodic orbis for hese parameer values. 13

17 To summarize, in he preceding Figure 3.2, we have all parameric values se a defaul values creaing an inward spiraling coexisence equilibrium for he vegeaion and consumer nurien populaions. The verical line is he consumer nullcline where he rae of change over ime of he consumer populaion is equal o zero. The unimodal curve is he vegeaion nullcline, where he rae of change over ime also is equal o zero. Figure 3.3 shows a periodic orbi equilibrium creaed by a sligh decrease in H in he defaul parameric values. We reduced he half sauraion deermining consan, H, by one hundredh which changed an inward spiraling sable equilibrium ino a periodic orbi. In Figure 3.4, we crashed he consumer and vegeaion populaions (,)=(0,0) by furher reducing H o Applicaion of he Model: The Effecs of Harvesing In his secion we examine hrough simulaions he possible impac of various managemen pracices. Specifically, we will consider vegeaion and consumer harvesing, as well as he impac of ferilizaion. We begin he applicaions of our model by sudying he harvesing of vegeaion. The following sequence of figures shows paired phase plane and ime series plos of sysem simulaions. 4a. egeaion Harvesing We have kep H a 8 and model he harvesing of vegeaion, by increasing h 1. Based on he algebraic form of he vegeaion nullcline, increasing h 1 ends o lower he nullcline. arying h 1 has no effec on he (verical) consumer nullcline. We find ha low vegeaion harvesing will no preven he sysem from crashing a h 1 = ( 3), Figure 4.1a. In he corresponding ime series plo, Figure 4.1b, an immediae crash occurs for he vegeaion and soil nurien populaions and here is a momenary increase in he consumer populaion before i also crashes. When he harvesing increases o h 1 =4 he consumer vegeaion populaions approach a periodic orbi, Figure 4.2a. There is a larger populaion flucuaion in he vegeaion versus he consumer and soil nurien over ime, Figure 4.2b, shown by he periodic orbi. The ampliude of he periodic orbi becomes smaller as he harvesing increases, unil a h 1 =6, 14

18 Figure 4.3a, an inward spiral o an equilibrium poin occurs. The inward sable spiral coninues hrough Figure 4.4a, h 1 =5. As h 1 has increased i has he effec of lowering he coexisence equilibrium for he consumer coordinae. Afer iniial flucuaions in he corresponding ime series plos a sable populaion is reached for he vegeaion consumer soil ecosysem as shown in Figures 4.3b and 4.4b. In Figure 4.5a, h 1 =6, and Figure 4.6a, h 1 =9, he populaions of consumer and vegeaion boh crash o zero. This illusraes ha he increased harvesing of vegeaion can cause a susainable sysem o become suddenly unsusainable, Figures 4.5b and 4.6b. As he harvesing of vegeaion coninues he equilibrium values of he vegeaion mee a a single poin on he vegeaion axis for h 1 > 5. For even larger values of h 1 he vegeaion nullcline and he vegeaion axis do no inersec, Figure 4.6a, where h 1 =9. Since nullclines no longer inersec when > 0, i is no possible for he populaions o coexis as we winessed in he preceding Figures 4.1a 4.5a. Thus, under exreme harvesing he vegeaion ends o zero populaion, as does he consumer. All foregoing phase plane graphs have a verical consumer nurien coordinae and a horizonal vegeaion nurien coordinae. 0 HredL, HgreenL,.004*S HblueL Figure 4.1a: H=8; h 1 =( 3); ( 0, 0 )=(0.70,) Figure 4.1b 15

19 0 HredL, HgreenL,.004*S HblueL Figure 4.2a:h 1 =4; ( 0, 0 )=(0.70,) Figure 4.2b HredL, HgreenL,.004*S HblueL Figure 4.3a: h 1 =6; ( 0, 0 )=(0.70,) HredL, HgreenL,.004*S HblueL Figure 4.3b Figure 4.4a: h 1 =5; ( 0, 0 )=(0.70,) Figure 4.4b 16

20 0 HredL, HgreenL,.004*S HblueL Figure 4.5a: h 1 =6; ( 0, 0 )=(0.70,) Figure 4.5b 0 HredL, HgreenL,.004*S HblueL Figure 4.6a: h 1 = 9; ( 0, 0 )=(0.70,) Figure 4.6b 4b. onsumer Harvesing In he nex sequence of figures we are modeling he harvesing of he consumer by increasing h 2. The verical nullcline of he consumer will move in he posiive direcion on he vegeaion axis as he consumer experiences an increase in harvesing. The vegeaion nullcline is no affeced by h 2. An increased vegeaion equilibrium value will occur since he rae of growh of he consumer is zero bu he consumer populaion is declining. Figure 4.7a illusraes he parameric defaul values where H=0.8 and h 2 =0. As we iniiae consumer harves, h 2 =1, he consumer vegeaion populaions lock ono a large periodic orbi around 17

21 he inersecion of he nullclines. In he ime series plo of Figure 4.8b he vegeaion plo has a greaer flucuaion in populaion over ime versus he consumer and soil nurien and is similar o he plo of Figure 4.2b. When h 2 =4, Figure 4.9a, here is a igh inward spiral o he sable equilibrium poin where he wo nullclines inersec. The periodic orbi has been los due o a reverse Hopf bifurcaion a he coexisence equilibrium. The consumer nullcline has moved o he righ on he vegeaion axis. Our ime series plo shows a ransien flucuaion in he populaions of he hree eniies over ime, approaching sable equilibrium values as > +, Figure 4.9b. oninuing he harvesing of he consumer we noice he vegeaion consumer populaion curve goes oward he vegeaion nullcline and follows i unil reaching he equilibrium poin where he wo nullclines inersec, Figures 4.10a and 4.11a. The soil nurien equilibrium poin is a lesser value han he consumer equilibrium poin, Figure 4.10b. Each of he ime series plos, Figure 4.10b, h 2 =8, Figure 4.11b, h 2 =3, and Figure 4.12b, h 2 =4 shows an immediae movemen of he vegeaion consumer populaion rajecory oward an equilibrium sae of he hree eniies in he ecosysem. In Figure 4.12a, he consumer nullcline has moved o he righ on he vegeaion axis so he nullclines no longer can inersec. Recall ha he vegeaion nullcline is no affeced by quaniies of he consumer harvesing coefficien, h 2. The equilibrium poin ends up becoming he larger valued *, which is he carrying capaciy of he consumerfree sysem. 18

22 0 HredL, HgreenL,.004*S HblueL H0.70, L h2= Figure 4.7a: H=0.8; h 2 =; ( 0, 0 )=(0.7,) Figure 4.7b 0 HredL, HgreenL,.004*S HblueL Figure 4.8a: h 2 = 1; ( 0, 0 )=(0.70,) Figure 4.8b HredL, HgreenL,.004*S HblueL Figure 4.9a: h 2 = 4; ( 0, 0 =(0.70,) Figure 1.9b 19

23 0 HredL, HgreenL,.004*S HblueL Figure 4.10a: h 2 =8; ( 0, 0 )=(0.70,) Figure 4.10b 0 HredL, HgreenL,.004*S HblueL Figure 4.11a: h 2 =3; ( 0, 0 )=(0.70,) HredL, HgreenL,.004*S HblueL Figure 4.11b Figure 4.12a: h 2 =4); ( 0, 0 )=(0.70,) Figure 4.12b

24 4c. Soil Ferilizaion The nex sequence of figures will help o reveal he impac of adding nuriens o he soil, as modeled by increasing F > 0. A he defaul parameer values wih H=8 and F=0, he vegeaionconsumer populaion rajecory crashed o zero. By adding ferilizaion o he soil, F=1, he soluion locks ono a periodic orbi around he inersecion of he nullclines, Figure 4.13a. Noice ha he vegeaion nullcline no longer inersecs he posiive vegeaion axis wice. So ferilizaion appears o have removed he Allee effec from he sysem. In he corresponding ime series plo he soil is now flucuaing in populaion more han he consumer and somewha less han he vegeaion. Figure 4.14a, he soil nurien addiion is 100 imes greaer han in Figure 4.13a, F=1.00. A periodic orbi is found similar o he previous Figure 4.13a bu he ime series plo (he exended verical graph) shows he change in quaniy of soil nurien added did no significanly change he equilibrium of he ecosysem. Oher experimenaion wih he simulaor for larger F showed lile significan change by addiional ferilizaion. 0 HredL, HgreenL,.004*S HblueL Figure 4.13a: H=8; F=1; ( 0, 0 )=(0.70,) Figure 4.13b 21

25 0 HredL, HgreenL,.004*S HblueL Figure 4.14a: F=1.00; ( 0, 0 )=(0.70,) 0 0 Figure 4.14b 5. Implicaions of he Model and Topics for Furher Sudy Alhough we have considered hree harvesing scenarios in he previous secion, he mehods of his paper allow various hybrid cases as well. For example, in he remaining four figures we se F=1, bu now we will harves he vegeaion and inerpre wha kind of effec he ferilizaion will have on he susainabiliy of ecosysem. A h 1 =3 he vegeaion consumer curve crashed, refer o Figure 4.1a. However, in Figure 5.1a we have se h 1 =3 and he addiion of soil nurien has forced he ecosysem ino a periodic orbi. The ime series plo has lesser flucuaions of he hree eniies han in Figure 4.13b bu he comparisons in flucuaions are similar. Refer o Figure 4.4a o see he graph for h 1 =6, a loose inward spiral. In Figure 5.2a, he inward spiral is igher and he equilibrium poin has a higher consumer value. We need o noice he nullcline for he vegeaion is larger han in Figure 4.4a, which allows for he higher consumer value a he sable equilibrium poin. By Figure 4.5a, h 1 =6, and Figure 4.6a, h 1 =9, we see he vegeaion and consumer populaions crashing o zero. However, afer adding soil nurien, F=1, Figures 5.1a 5.4a show ha he sable equilibrium poins are values oher han (0,0). In Figure 5.3a he equilibrium is a he 22

26 inersecion of he nullclines and in Figure 5.4a he equilibrium is he higher consumer free equilibrium value, *, since he nullclines do no inersec a > 0. The ime series plos, Figures 5.3b and 5.4b depic an iniial flucuaion in he populaions of he hree eniies, followed by an approach o posiive equilibrium poins. Thus, he addiion of ferilizaion appears o eliminae he populaion crash observed in 4.1a, where F=0. 0 HredL, HgreenL,.004*S HblueL 0.8 Figure 5.1a: h 1 =3; F=1; ( 0, 0 )=(0.70,) Figure 5.1b 0 HredL, HgreenL,.004*S HblueL 0.8 Figure 5.2a: h 1 =6; F=1; ( 0, 0 )=(0.70,) Figure 5.2b 23

27 0 HredL, HgreenL,.004*S HblueL 0.8 Figure 5.3a: h 1 =6; F=1; ( 0, 0 )=(0.70,) 0 HredL, HgreenL,.004*S HblueL Figure 5.3b Figure 5.4a: h 1 =9; F=1; ( 0, 0 )=(0.70,) Figure 5.4b 0.8 The mahemaical model we have presened furhers our undersanding of he dynamics involved in he soil vegeaion consumer ecosysem. We found ha he differenial equaions we used were sensiive o he parameric values incorporaed in our Mahemaica simulaions. Sudden changes occurred in he approached equilibrium of he ecosysem when we changed harvesing and ferilizaion values. The ime series plos helped us o undersand he feriliy of he soil, even hough our graphs 24

28 were based on a wo dimensional sysem. Our sysem was observed o be sensiive o he adding of nuriens. We concluded ha ferilizaion dampened he effecs of harvesing vegeaion and reduced he likelihood of he vegeaion ending o zero populaion. An Allee effec was quie noiceable in some of our simulaions, wih he model supporing simulaneous sable no life sae ==0 and sable coexisence soluions. The Allee effec was eliminaed when ferilizaion was inroduced due o he increased dimensions of he vegeaion nullcline. Our goal has been o derive and parameerize a basic model. Finding he various sabiliy properies for he equilibrium poins would be a logical nex sep for his mahemaical model. This is accomplished by compuing he eigenvalues for he equilibrium poins. Finally, i should also be observed ha our work has involved simulaions of he wodimensional quasi equilibrium sysem where i is assumed S = 0. I would be ineresing o see how hese resuls compare o simulaions of he full hree dimensional sysem. 25

29 6. Appendix: Mahemaica odes Manipulae[soln1=NDSolve[{'[]((d*((F+a*L*[])/(α/(1+β*[])))/(b+((F+a*L*[])/(α/(1+β*[]))) ))*(1-[]/k)-(Pmax*[])/(Pmax*H+[])-L-h1)*[],'[][] *((γ*pmax*[])/(pmax*h+[])-mh2),[0]1,[0]1},{[],[]},{,0,max},maxseps ,MaxSepSize 2]; Show[onourPlo[ {0==(d*((F+a*L*)/(α/(1+β*)))/(b+((F+a*L*)/(α/(1+β*)))))*(1-/k)- (Pmax*)/(Pmax*H+)-L-h1,0,0==((γ*Pmax*)/(Pmax*H+)-mh2),[0]1,[0]1},{,min,max},{,min,max}, PloPoins 40,onourSyle {Darker[Blue],Black,Red}],ParamericPlo[Evaluae[{[],[]}/.soln1],{,0, max},axes True,PloSyle Black,AxesLabel {"",""},PloRange {{min,max},{min,max}},pl opoins 50],Graphics[{PoinSize[.01],Red,Poin[{1,1}]}]], Syle["Model Parameers",Bold],{{k,.95},0,5,.001},{{b,.04},0,3,.01},{{d,1},0,3,.01},{{F,0},0,10,.01},{{a,.093},0,.1,.00 1},{{L,.4},0,1,.01},{{α,1},0,1.2,.01},{{β,100},.01,200,.01},{{h1,0},0,1,.01},{{h2,0},0,1,.01},{{Pmax, 1},0, 10},{{H,.06},0,1,.01},{{γ,.9},0.,1,.01},{{m,.68},0.,1,.01}, Delimier,Syle["Iniial ondiions / Simulaion Lengh",Bold],{{1,.7},0,max,.01},{{1,.05},0,max,.01},{{max,300},1,500,1},Delimier,Syle["Win dow Size:",Bold],{{max,.85},.1,1.,.01},{{max,.2},.01,.5,.001},{{min,.0001},0,30,.1},{{min,-.01},- 1,1,.01},onrolPlacemen Righ] Model Parameers k 0.95 b d 1 F 0 a 4 L 0 a 1 b 100 h1 h2 0 0 Pmax 1 H 0.8 g 0.88 m 4 Iniial ondiions ê Simulaion Lengh 1 1 max Window Size: max max min min 26

30 Manipulae[soln2=NDSolve[{'[]((d*((F+a*L*[])/(α/(1+β*[])))/(b+((F+a*L*[])/(α/(1+β*[])))))*(1- []/k)-(pmax*[])/(pmax*h+[])-l-h1)*[],'[][] *((γ*pmax*[])/(pmax*h+[])-mh2),[0]1,[0]1},{[],[]},{,0,max},maxseps ,MaxSepSize 2]; Plo[Evaluae[{[],[],.004*((F+a*L*[])/(α/(1+β*[])))}/.soln2],{,min,max},Axes True,PloSyle {Darke r[green],darker[red], Darker[Blue]},AxesLabel {""," (red), (green),.004*s (blue)"},plorange {{min,max},{ymin,ymax}},plopoins 50], Syle["Model Parameers",Bold],{{k,.95},0,5,.001},{{b,.04},0,3,.01},{{d,1},0,3,.01},{{F,0},0,10,.01},{{a,.093},0,.1,.001},{{L,. 4},0,1,.01},{{α,1},0,1.2,.01},{{β,100},.01,200,.01},{{h1,0},0,1,.01},{{h2,0},0,1,.01},{{Pmax,1},0, 10},{{H,.06},0,1,.01},{{γ,.9},0.,1,.01},{{m,.68},0.,1,.01}, Delimier,Syle["Iniial ondiions / Simulaion Lengh",Bold],{{1,.7},0,3,.01},{{1,.05},0,1,.01},{{min,0},0,500,1},{{max,150},1,500,1},Delimier,Syle["Wi ndow Size:",Bold],{{ymax,.6},.01,2.0,.01},{{ymin,-.01},-1,1,.01},onrolPlacemen Righ] Model Parameers k b 0.95 d F 1 a 0 L 4 a 1 HredL, HgreenL,.004*S HblueL b h1 h Pmax 1 H g 0.88 m 4 Iniial ondiions ê Simulaion Lengh 1 1 min max Window Size: ymax ymin 27

31 7. References (1)Ibanez, J., Marinez, J., and Schnabel S., Deserificaion due o overgrazing in a dynamic commercial livesock grass soil sysem. Ecological Modelling 205: (2)Koppel, J., and Riekerk, M., Herbivore regulaion and irreversible change in semi arid grazing sysems. OIKOS 90: (3)Mahemaica 7.0. Wolfram Associaes (4)Mesa, E., Bhaya, A., Kaszkurewicz, M., and osa, M., On off policy and hyseresis onoff policy conrol of he herbivore vegeaion dynamics in a semi arid sysem. Ecological Engineering 28: (5)Pasor, J., Mahemaical Ecology of Populaions and Ecosysems. Wiley and Blackwell. (6)Rosenzweig, M.L., and MacArhur,R.H.,1963. Graphical Represenaion and Sabiliy ondiions of Predaor Prey Ineracions. American Nauralis 97: (7)Sephens, P.A., Suherland, W.J., Freckleon, R.P., Wha is he Allee Effec? Oikos 87: (8)Wang, Z., Huang, G., Wang, G., and Gao, J., Modeling of egeaion Erosion Dynamics in Waershed Sysems. Journal of Environmenal Engineering 130: