Predator-Prey Models with Diffusion and Time Delay

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1 Predaor-Prey Models wih Diffsion and Time Delay Irina Kozloa 1, Manmohan Singh 1 and Alan Eason 1,2 1 School of Mahemaical Sciences, Swinbrne Uniersiy of Technology, Hawhorn, Vicoria, Asralia. 2 Presen address: School of Naral and Physical Sciences, Uniersiy of PNG, Naional Capial Disric, Papa New Ginea. Absrac: - Lckinbill [1] demonsraed in a laboraory ha poplaions of Paramecim arelia as a prey and Didinim nasm as a predaor cold ehibi ssained oscillaory behaior. Harrison [2] modelled his daa for esing predaor-prey models and sccessflly demonsraed he general feares of he eperimen inclding he ssained oscillaions. We hae sdied he aboe models in order o deermine he effec of diffsion and ime delay. I has been shown ha he inclsion of hese erms gies rise o differen growh paerns. I has been obsered ha while diffsion acceleraes he process of growh or eincion of a poplaion, he ime delay modifies he process of growh or eincion of he poplaion and he properies of he cyclic seady sae. These resls can be sed o obain a beer fi o field daa and improe predicions in he behaior of predaor-prey sysems. Key-Words: Paramecim, Didinim, diffsion, Loka-Volerra, response fncions and ime delay. 1 Inrodcion This paper is deoed o predaor-prey models based on Lckinbill s [1] eperimen wih Paramecim arelia as a prey ogeher wih Didinim nasm as a predaor. Paramecim arelia is a small nicelllar organism of he ciliae gens. I is elongaed, ranges in size from 12 o 3 µm and lies mosly in freshwaer ponds, where i feeds on arios baceria, small proozoans, algae and yeass. Didinim nasm is also a wellknown proozoan ciliae, which lies in freshwaer habias and feeds on oher ciliaes, especially Paramecim, which is seeral imes larger. 1 I is acally differen in shape from Paramecim, hicker and less oblong, egg shaped wih size 8 17 µm in lengh and 6 8 µm in diameer. Lckinbill s [1] eperimen wih Paramecim arelia and Didinim nasm was sccessfl in mainaining oscillaory behaior becase i was condced in 6 ml of cerophyl medim hickened wih mehyl celllose o sabilise he sysem. This eperimen was modelled by Harrison [2] by fiing he parameers in Loka-Volerra models in order o find he bes fi beween he seleced model and he obsered poplaion densiies. He sed differen kinds of fncional responses, inclding re-

2 sponses wih he predaor mal inerference, raio-dependency, Leslie and sigmoid ypes. He also sed a delay componen in he predaor nmerical response. Howeer, he was no compleely saisfied wih accracy of he resls becase he solions were ery dependen on he iniial condiions and he poplaion densiies fell oo low beween he cycles. Jos and Ardii [3] noed ha differen parameers were fond for he basic growh and carrying capaciy parameers and caion ha he approach may no idenify he appropriae feares of he model. Haefner [4] eplains in some deail he reasons for sccess of he Lckinbill s eperimen. He also nderlines he imporance of ndersanding he feares of he models seleced before he alidaion process is aemped. This paper inesigaes he models sed by Harrison wih eensions o inclde spaial dependence. The main poin of his paper is o eamine he role of he ime delay in he nmerical response of he predaor. 2 Eqaions Le (, ) be he prey poplaion densiy and (, ) be he predaor poplaion densiy. The logisic predaor-prey model wih inclsion of a predaor ime delay effec and diffsion consiss of he eqaions ( = ρ 1 ) 2 ω f 1 (, ) + d 1 K 2,(1) = γ + σ f 2 2(, ) + d 2 2, (2) where f 1 (, ) = f((, ), (, )), f 2 (, ) = f((, τ), (, )), ρ is he specific growh rae of prey in he absence of predaor, K is he carrying capaciy of he prey, ω is he maimm nmber of prey ha can be eaen by a predaor per ni ime, γ is he moraliy rae of predaor in he absence of prey, σ is a conersion efficiency o predaor biomass, τ is he ime delay consan and d 1 and d 2 are diffsiiy consans. The delay is inrodced in he nmerical response erm in he eqaion (2) o consider he effec of a ime delay on he poplaion densiies of he prey and predaor. As seen, he poplaion densiy (, ) is replaced by (, τ), where τ is delay erm. All of he compaions are carried o oer he domain [ 2, 2] so ha he effec of diffsion can be obsered. The bondary condiions applied for his domain are ( 2, ) (2, ) = = ( 2, ) (2, ) =, =. In each nmerical eperimen, we se he iniial condiion = 15 sech 2 (1), 2 2,, 2 <.5, = 6,.5.5,,.5 < 2, 15, Figre 1: Iniial condiions for he prey and predaor. This is chosen wih iniially concenraed a he origin in he cenre of he domain niformly occpied by. I is graphed a inerals =.1 in Figre 1. 2

3 3 Response Fncions and Daa Vales In his paper, we se he response fncions and associaed daa ales sed by [2]. The responses are 1. f(, ) = φ +, 2. f(, ) = (φ + ) (1 + β), 3. f(, ) = φ + (1 (1 + µ) e µ ). In hese eqaions, φ is a saraion consan, β is he predaor inerference consan and µ is a consan. Response fncion 1 is he sandard Holling ype 2 response for which φ is he half saraion consan, he leel of prey a which half of he maimm consmpion rae occrs. There are en parameers (ρ, K, ω, d 1, γ, σ, d 2, φ, β and µ) o be chosen in he model eqaions inclding he response fncions. In he calclaions below he response parameers µ =.17 and β =.728 and he carrying capaciy parameer K = 898 are no changed. Diffsion parameers are aken in wo ses, d 1 = d 2 = and d 1 =.2, d 2 =.1, o presen cases wiho and wih diffsion accordingly. The oher fie parameers are considered in grops called Daa 1, Daa 2 and Daa 3 ρ ω φ σ γ Daa , Daa , Daa All of he response fncions hae he general feares of haing ale zero when = and increasing oward 1 as he poplaion densiy of he prey increases. Response fncions 1 and 2 behae in a similar manner een hogh he poplaion densiy in 2 depends on he predaor as well as he prey. Howeer, for 2 he response is significanly redced for large ales of he predaor densiy. 4 Nmerical Calclaions The model eqaions (1) and (2) are soled nmerically sing he operaor spliing mehod [5] which has been sccessflly applied preiosly by he ahors. The differenial eqaion sysem is spli ino a pair of non linear reacion eqaions 1 (1 2 = ρ ) ω f 1 (, ), (3) K 1 2 = γ + σ f 2(, ), (4) which are sed for he firs half of he ime sep, and a pair of linear diffsion eqaions 1 2 = d 2 1, 2 (5) 1 2 = d 2 2, 2 (6) which are sed for he second half of he ime sep. The nmerical mehod sed for he reacion eqaions and he diffsion eqaions is he forward Eler scheme. Then eqaions (3) and (4) become ū n+1/2 = n + ( ( ) ) ρ 1 n n K ω n f 1 ( n, n ), (7) n+1/2 = n + ( γ n + σ n f 2 ( n, n ) ), (8) where n and n indicae he approimae ales of and a he posiions = 2 +, =, 1, and ime n = n, n =, 1,, and ū n+1/2 and n+1/2 indicae he represenaie ales a he half ime sep. Similarly, eqaions (5) and (6) become n+1 n+1 = ū n+1/2 + d 1 2 ( ū n+1/2 1 2ū n+1/2 = n+1/2 + d 2 2 ( n+1/2 1 2 n+1/2 ) + ū n+1/2 +1, (9) ) + n+1/2 +1. (1) 3

4 The nmerical schemes (9) and (1) for he diffsion eqaions gie sable solions proided d i.5, i = 1, 2. ( ) 2 These condiions are saisfied for he calclaions of all cases of his paper sing he incremen ale =.1 and he ime sep =.5 days. The ale of has been chosen afer eamining he conergence of calclaed ales of and we hae soled eqaions (1) and (2) wih response fncion 1, Daa 1 for arios ales of from.1 o.5. The solion consiss of a seqence of he prey and predaor plses wih increasing amplide oward he seady sae limi cycle. As he ime sep is decreased, he peak ales become smaller approaching he conerged solion nil he ime sep =.5. We choose a ime sep =.5 o ensre ha he solion has conerged. 5 Resls and Discssion he ransien ops and he ime seqences of he solion for he poplaion densiies and for each of he hree ses of response fncions and daa ses for no diffsion and wih diffsion, no delay and wih delay τ = 2 hors. Each figre presens picres in 3 rows and 2 colmns. Row order for each case incldes he ransien ops, ime seqences wiho delay and ime seqences wih delay τ = 2. The order of he colmns is wiho diffsion and wih diffsion. The ransien ops are shown as spaial cres for and a imes =.5, 1., 1.5,, 1 days hrogh he domain [ 2, 2]. Time seqences for he prey and predaor are shown as densiy disribions in ime a he posiion =, i.e. a he middle poin of he space ineral [ 2, 2]. In each of he Figres 2-4 he case wih no diffsion in row (b) is he resl from Harrison s model [2]. To illsrae delay effec from a wider poin of iew, we conclde wih a series compaions wih delays of, 1, 3 and 5 hors presened as phase planes in Figre 5. In all of he cases illsraed, he ime seqences conine p o 6 days. (a), d =, d = -2-1, d =.2, d = (a), d =, d = -2-1, d =.2, d = (b), d =, d =, d =.2, d =.1 (b), d =, d =, d =.2, d =.1 (c), d =, d =, τ =2, d =.2, d =.1, τ =2 (c), d =, d =, τ =2, d =.2, d =.1, τ =2 Figre 2: Predaor-prey solions for response fncion 1 and Daa 1. (a) Transien op of and, (b) op wiho and wih diffsion wih no delay, (c) op wiho and wih diffsion wih ime delay. In all he graphs in Figres 2-4 we hae gien Figre 3: Predaor-prey solions for response fncion 2 and Daa 2. (a) Transien op of and, (b) op wiho and wih diffsion wih no delay, (c) op wiho and wih diffsion wih ime delay. Figre 2 shows he ops sing he response 4

5 (a), d =, d =, d =.2, d =.1 (a) d =, d =, τ = d =.2, d =.1, τ = (b), d =, d =, d =.2, d =.1 (b) d =, d =, τ =1 d =.2, d =.1, τ = (c), d =, d =, τ =2, d =.2, d =.1, τ =2 (c) d =, d =, τ =3 d =.2, d =.1, τ = (d) d =, d =, τ =5 d =.2, d =.1, τ =5 Figre 4: Predaor-prey solions for response fncion 3 and Daa 3. (a) Transien op of and, (b) op wiho and wih diffsion wih no delay, (c) op wiho and wih diffsion wih ime delay. fncion 1 and Daa 1. Solions ehibi he oscillaions wih some growing and some decaying. Wih no delay, he cases wih and wiho diffsion show he iniial growh of he prey is limied by he growh of he predaor nil he predaor poplaion densiy decreases de o he small poplaion densiy of he prey. While he prey is a a sbsisance leel he predaor poplaion densiy conines o redce nil i reaches sch a small leel ha an obreak of he prey can occr. This growh is finally limied by he predaor and he procedre is repeaed. The poplaion densiies of and grow o a seady paern of seady cycles of periods of almos zero poplaion densiy followed by plses wih amplides close o 76 and 3 for he prey and predaor respeciely. The disances beween he sccessie plses also increase close o a period of 1-days and he maimm amplides of he plses increase oward heir seady sae ales. For he no diffsion cases firs plses reach smaller ales han wih diffsion, and i also akes a longer ime o reach he seady cyclic sae. I is clear ha diffsion acceleraes he process Figre 5: Phase-plane behaior of he predaorprey solions for he Daa 2, he response fncion (c), and he iniial condiion gien, (a) wiho ime delay componen, (b) ime delay τ = 1 hor, (c) τ = 3 hors, and (d) τ = 5 hors. Wih ime delay, he seady sae prey and predaor peak ales close o 89 and 52 are approached more qickly, and he ime ineral beween sccessie cycles is increased o approimaely 2 days. Diffsion again acceleraes he process of reaching seady sae ales. Figre 3 presens he ops sing he response fncion 2 and Daa 2. Wih no delay, he seady sae solions consis of reglar oscillaions of prey and predaor plses wih magnides approimaely 52 and 18 accordingly. The firs prey and predaor peak plse ales are mch smaller han wiho diffsion, and ery close in ales o ohers for diffsion one. Time periods beween peaks are also greaer. The final paern is reached more qickly wih diffsion and in fewer oscillaions. Wih ime delay, he cyclic seady sae is achieed sooner and is reached afer only hree plses. Figre 4 presens he ops sing he response fncion 3 and Daa 3. Here he seady 5

6 paern is achieed ery qickly in hree or for of he cycles. Wih diffsion he firs peaks are mch higher han wiho i. Wih ime delay he maimm ales are larger and he ime beween peaks is greaer. There are more plses for he ime period here compared wih Figres 2 and 3. This is a resl of he response fncion which is mch smaller han he oher responses for small prey ales. Hence he growh of he predaor redces more qickly and he prey ales increase more qickly. In order o eamine more closely he inflence of delay on he poplaion growh process, he resls for response fncion 2 and Daa 2 hae been calclaed wih delay coefficiens τ =, 1, 3 and 5 for comparison wih he solion for poin of iew, we inclde τ = 2 shown in Figre 3. The resls are presened in Figre 5 as phase-plane plos. The raecories are raersed aniclockwise wih increasing ales oward he limi cycles. The graphs show ha increasing he delay coefficien τ significanly increases he peak ales of he plses and increases he ime ineral beween he plses. Wih he inclsion of diffsion he raecories grow o heir seady cyclic paern more qickly. 6 Conclsion Each he graphs from Figres 2-4 show ha here is a cyclic seady sae. I is ineresing o noe ha his is no necessarily he case if he response fnciona are sed wih he oher daa ses. Then seady saes may be consan ales. The inclsion of diffsion acceleraes he process of reaching he final growh paern. Increasing he ime delay leads o increased ales of local erema for boh he prey and predaor and an increased ime beween heir occrrence. This was bes seen in he phase-plane plos. These resls nderline he imporance of he choice of appropriae feares of predaor-prey ineracions in mahemaical biology. Time delay in he nmerical response has he physiological meaning ha he conersion of prey biomass o predaor biomass is delayed. I cold occr in naral siaions. Inclding delay when modelling biological processes cold significanly improe he realism of he model as well as he accracy of he predicions for he chosen daa. References [1] Lckinbill, L. S., Coeisence in laboraory poplaion of Paramecim arelia and is predaor Didinim nasm, Ecology, 54,1973, pp [2] Harrison, G. W., Comparing predaorprey models o Lckinbill s eperimen wih Didinim and Paramecim,Ecology, 76, 1995, pp [3] Jos, C. and Ardii, R., em Idenifying predaor prey processes from ime series, Theoreical Poplaion Biology, 57,, pp [4] Haefner, J. W., Modelling Biological Sysems: Principles and Applicaions, Chapman and Hall, New York, [5] Yanenko, N. N., The Mehod of Fracional Seps, Springer Verlag, New York, [6] Eason. A., Singh, M. and Ci, G., Solions of wo species reacion-diffsion sysems, Canadian Applied Mahemaics Qarerley, 5, 1997, pp [7] Singh, M., Eason. A., Ci, G. and Kozloa, I., Nmerical sdy of he wodimensionmal sprce-bdworm reaciondiffsion eqaion wih densiy dependen diffsion, Naral Resorce Modeling, 11, 1998, pp