2 Faculty of Mechanics and Mathematics, Moscow State University.

Transcription

1 th World IMACS / MODSIM Congress, Cairns, Astralia 3-7 Jl 9 Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle. Dnn, J.M., T.D. Wentzel, S. Schreider and L. McArthr School of Mathematics and Geospatial Science, RMIT Uniersit, Melborne, Astralia. jessica.dnn@rmit.ed.a Faclt of Mechanics and Mathematics, Moscow State Uniersit. Abstract: Understanding the spatial behaior of poplations is a crcial element for gaining a cohesie pictre of the oerall dnamics of a model and shold be eplored in conjnction with analtical techniqes. As an eample, when considering egetation process modelling of ecosstems, the inclsion of spatial components is critical for accrate modelling reslts (Jörgensen, ). Howeer, throgh the inclsion of additional components, models often tend to be comple and it becomes increasingl more difficlt to analse the oerall sstem. Techniqes then trn to nmerical methods (Mickens, 3); with applications in (Jesse, 999). The problem considered in this paper originates from poplation growth modeling of seeral grops of marine phtoplankton, and algae species in Astralian coastal lagoons. A simplified ersion of a diffsion, growth, competition/predator sstem of P.D.E. s is considered in order to model the poplation dnamics. More specificall, this work is concerned with the maimm-minimm theorem and the one-sided maimm principle for the diffsie Lotka-Volterra tpe eqations, with poplations of N = and species ( and ), nder imposed on Nemann bondar conditions. The predator-pre and competition sstems with two-dimensional diffsion are eplored. The analticall proen theorem indicates that in the case of eqal diffsion coefficients a certain fnction of and has no maimm inside the bonding rectangle, < < l and < t < T, and on the eternal bondar t T and therefore attains its maimm on the base t = or on the ertical sides = and = l. It is also shown that proportionall larger initial poplations with higher growth rates will maintain a competitie adantage oer their conterparts in the competition eqations. Nmerical comptations hae been implemented to eamine the sstem s behaior for the case when N = with simplified lagoon geometr when two-dimensional diffsion is considered. The reslts of nmerical eperiments illstrate that the sstem s behaior are consistent with the analtic conclsions obtained for the one-dimensional case. Kewords: Competition, Diffsion, Lotka-Volterra, Poplation Modelling, Predation

2 Dnn et. al., Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle.. INTRODUCTION The Lotka-Volterra eqations for modelling species interactions, also termed the predator-pre and competition eqations, hae been widel analsed. The competition-diffsion sstem has been the focs of seeral analtic works: for two species (Leng et al., ), for three species (Mimra and Fife, 9), for n-species (Rothe, 97). In this std, analtical techniqes hae been implemented to gie general reslts for the maimm-minimm theorem for the Lotka-Volterra predator-pre eqations, and the one-sided maimm principle for the Lotka- Volterra competition eqations, both with one-dimensional diffsion. The analtical reslts are then etended and compared to nmerical simlations for two-dimensional diffsion. The sstems are eamined nder aring diffsie rates and different initial conditions in order to proide an oerall pictre of the sstem dnamics. The application sed for the nmerical simlations originates from poplation growth modeling of seeral grops of marine phtoplankton and algae species in Astralian coastal lagoons. To aoid compleities the lagoon geometr is simplified to a dimensionless bonding sqare with on Nemann bondar conditions. Nmerical simlations are carried ot sing a finite element approach.. PREDATOR-PREY INTERACTIONS Introdcing one-dimensional diffsion terms to the D Lotka-Volterra Predator Pre eqations and assming diffsion coefficients are eqal to one, the sstem of eqations is gien b: L = t = ( β) in (, l) [, T ] L = t = (α ) () = = = = =l = =l =. For this case, we can proe the following statement (the proof is omitted here, bt proided in the fll paper which is in preparation): Lemma.: In ma Lφ and φ t and φ there is no maimm inside the bonding rectangle, < < l and < t T, and the maimm can onl be assmed on the base t = or on the ertical sides = and = l.. Nmerical Simlations: Nmericall, we wish to eamine the predator-pre eqations with two-dimensional diffsion to see Lemma. holds for a bonding rectangle. The initial conditions for the predator () and the Pre () are gien in Figre (a) and (b) with initial reslts gien in Figre (c). Densit Predator () 5 Densit Pre () a=., b=. a=.5, b=. a=., b=. a=.5, b=.5 Predator 5 3 Pre 5 3 Pre () Predator () (a) Initial Condition: Predator (b) Initial Condition: Pre (c) Predator-Pre nmerical reslts. Figre : The predator-pre eqations: IC and reslts. It is hard to describe the maimm in terms of two dimensional space when diffsion, d, is considered at sch a large rate (d = ). Oer this time step (t = ), diffsie rates as large as these tend to reslt in total diffsion within a short space of time and so diffsion in the FEM is onl considered in the initial time step. The final reslts, then become completel dependant on the Lotka-Volterra interaction terms. The maimm is therefore 9

3 Dnn et. al., Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle. dependant on the initial conditions at time (t = ), where if diffsion is large, the maimm will be within the initial condition or dependant on the initial poplations (also at t = ) and the scale of the interaction coefficients. Figre (c) demonstrates that proportionall larger α leads to initial decline in predator nmbers with dominance of pre. Large β tends to an initial dominance of pre before the species leels ot de to no predation. Eqal α and β shows an initial die-off of pre which reslts in the decline of predators de to a shortage of resorces. This howeer is linked to the initial condition of the predator which is significantl larger than the pre species. In all simlations it is noted that diffsie terms dominate the initial time-steps. 3. COMPETITION INTERACTIONS The competitie Lotka-Volterra eqations for species who contend for the same resorce, withot diffsion are: d dt d dt = (β ) = (α ) () In a manner similar to the operations in Section., the trajectories are leel lines of the fnction and are gien b the eqation ψ = C, where C is some constant. ψ = + βln() + αln() (3) The point (α,β) corresponds to ψ = and ψ = bt is not a minimm bt a montain pass point for ψ. The eqation for the leel line passing throgh (α, β) is αln() + βln() = α αln(α) β + βln(β) () For Sstem () with diffsion and diffsie rates eqal to, we can obtain an eqation for ψ. An attempt is now made to obtain a single eqation for the fnction ψ = f(ψ). The eqation for ψ t is ψ = ψ t + ψ (5) If the R.H.S. of Sstem () are mltiplied b ψ and ψ correspondingl and smmed, the following eqalit arises: It follows that ψ (β ) + ψ (α ) = () Lψ = (ψ + ψ + ψ ) (7) for an fnction ψ(, ). Consider ψ = f(ψ), where ψ is gien b (3), the qestion is, can ψ be cone locall? To answer this, consider ψ = f (ψ)ψ ψ = f (ψ)ψ ψ = f (ψ)ψ + f (ψ)ψ ψ = f (ψ)ψ + f (ψ)ψ ψ = ψ = f (ψ)ψ ψ + f (ψ)ψ () Sbstittion of the eqations from Sstem () into the conditions for coneit: ψ ψ ψ = (f (ψ)ψ + f (ψ)ψ )(f (ψ)ψ + f (ψ)ψ ) (f (ψ)ψ ψ +f (ψ)ψ ) = f (ψ)(ψ ψ ψ ) + f (ψ)f (ψ)(ψ ψ ψ ψ ψ + ψ ψ ) +(f (ψ ψ ψ ψ )

4 Dnn et. al., Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle. Defining f(ψ) b letting f(ψ) = e ψ and ψ = + βln() + αln() gies ψ = α, ψ = α, ψ = + β, ψ = β (9) and Epression () is positie when ( αβ + α( β) ψ ψ ψ = e ψ β( α) ) () α( β) β( α) > αβ () The points (α, β + β), (α, β β) lie on the bondar of the domain determined b ineqalit (): α( β) β( α) = αβ () The domain (), is sch that in it either < β β (lower part) or < β + β (pper part). The lower part has intersection with { >, > } onl for β >. Sbstitting the eqations from Sstem (9) into the conditions for coneit gies: ψ = e ψ ( ( α ) + α ) ψ = e ψ ( ( β ) β ) > > = e ψ ( β β)( β + β) > which holds for either the pper or lower ale of, < β β or > β + β. This implies that ψ is cone in domain (), and a one-sided maimm principle is alid. Hence, in the domain () Lψ < and ψ has no maimm in [,l] (,T ] as long as, remain in the domain (). The same is tre for ψ since ψ is monotonicall increasing with respect to ψ. Ths if (, ) ɛ (), ψ ma ɛ[,l] ψ( (), ()) = C (3) Figre () illstrates the cres ψ = C. Refer to Eqation () for the eqation of the cre containing (α, β) which corresponds to C = C o = α αln(α) β + βln(β) () The left and right branch cres correspond to C > C. The cres with pper and lower branches correspond to C < C. In the case when C < C the domain ψ C is shaded. (a) Cres ψ = C. (b) The bordering cre of the coneit domain. Figre : ψ = C and the bordering cre of the coneit domain. Figre ((b)) illstrates the bordering cre of the coneit domain, a hperbola with asmptotes: β β α = ± α (5)

5 Dnn et. al., Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle. The qestion now is: Does the ineqalit (3) make the ales of (, ) remain in the domain ()? To answer this, consider now the pper parts of the domains gien b () and (3) sing notations () and (3) for those pper parts. Consider the right hand and pper branch of the cre ψ = C, that is, C < C gien b: and βln() + αln() = C () For > α, >, we can estimate d d from below: d d = α β (7) α >, β < () and therefore d d > for > α, > (9) This means that as, and Eqation () gies as. Indeed: For the hperbola (Eqation ()): ( β ln() ) ( = α ln() ) C ln() ( α = ) ( β ln() ) = C ( β ln() ) = ( + ε()), ε() as () = β α ( + ε ()), ε () as () For α > β and for large, the cre () lies higher than (). The rest of the cre () lies higher than () for sfficientl large C. The cre () intersects the line = at = β + β( + α) and the whole line for < α lies lower than β + β( + α). On the cre the minimal ale of is assmed for = α and is defined b and if > βln() = α αln(α) C () > C + α ln(α) = (3) It follows that for negatie C large in absolte ale, is also large and the whole cre () is sitated higher than (). Therefore Eqation (3) implies Eqation (). The reslt is as epected, gien that α and β are the reprodctie coefficients for and in the absence of other species. If has a large adantage for t = and reprodces qicker than, it is natral that it keeps the adantage. Corollar: If for the hperbola, Figre ((b)), <, the cre ψ = C will be sitated higher than the domain or the set which is bonded within ψ = C. Therefore ψ is cone in this set and hence, ψ does not hae a maimm and ψ ma t= ψ. Becase ψ is monotonicall increasing then ψ ma t= ψ and will be aboe the cre. For large C, the domain, ψ C: βln() + αln() C belongs to the domain of large, positie (becase it also eists in the lower part of closer to zero). Indeed the cre:

6 Dnn et. al., Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle. has a minimm for βln() + αln() = C () α = (5) i.e. for = α. Then βln() = C = C + α αln(α) () For large C, C is also large and the graph of the L.H.S. of () has a shape indicated in Figre ((b)) and horizontal straight lines g() = C intersecting () at two points. For the cre ψ = C lies higher than the corresponding branch of the hperbola (). Sppose that this is tre for >. Now is it possible to choose C so large that the rest if the cre ψ = C lies higher than the remaining part of the hperbola. The fnction (), defined b ψ = C has a minimm at = α, and this minimm tends to infinit as C. If C is so large that >, > β + β( + α), and the whole cre ψ = C lies oer the hperbola (where β is defined in (3). Indeed for (ε), ( ε) for the hperbola α, where β α <. For < < (ε) and (), and all of the cre () is located as far as necessar aboe for large enogh C. These conclsions onl relate to the pper part of the domain, ψ = C. Therefore the initial conditions hae to belong to the pper domain () β + β. 3. Nmerical Simlations: The most interesting qestion is how keeps its adantage mentioned in the paragraph aboe: does it eliminate or does keep non-zero eqilibrim less than ales of. Figre illstrates the different simlations. Inclded in the simlations of the diffsie competition eqations are the intraspecific competition terms for each species. Hence the eqations for simlation in D are: t d ( + ) = (r α α ) t d ( + ) = (r α α ) in (, l) [, T ] (7) = = = = = = = = (,, ) = (,, ) =. In Sstem (7) d i are the coefficients of diffsion and α ii and α ij represent intraspecific and interspecific competition respectiel with i =, and j =,. The simlations are initialized with a mch larger for t = and the sstem is obsered nder different diffsie and growth rates. The initial conditions for and are presented in Figres 3(a) and 3(b). The graphical reslts are also presented in Figre 3. If has a large adantage for t = and reprodces qicker than, it is natral that it keeps the adantage. The reslt is as epected, gien that α and β are the reprodctie coefficients for and in the absence of other species. This is gien sbject to the condition that the intraspecific competition rates are reasonable, i.e. there are abndant resorces to sstain species. When the intraspecific rates are zero with eqal growth we see a dominance of species as is also the case with a larger growth rate for, Figres 3(i and j). The diffsie rates maintain the same propert as that of the predator-pre sstem.. CONCLUSIONS AND FUTURE WORK We proide analtical qantification of poplation dnamics described b predator-pre eqations with diffsion for one the dimensional case, l and t T. We also implemented nmerical simlations for the two spatial dimensional case. The simlation was implemented for a rectanglar spatial domain with on Nemann bondar conditions. The reslts of the simlation confirmed the conclsion obtained for the one dimensional analtical case. Ftre work incldes writing a program for the nmerical soltion of the on Nemann problem for the diffsie Lotka-Volterra eqations when the spatial domain has a sophisticated geometr reflecting the real shape of estarine lakes. 3

7 Dnn et. al., Nmerical eamination of competitie and predator behaior for the Lotka-Volterra eqations with diffsion based on the maimm-minimm theorem and the one-sided maimm principle. Densit Densit 7 5 d=.,, d=. d=., d=. d=., d= d=., d=. d=., d= d=. d=.5 d=. d=.5 d=. d=.5 d= (a) Initial Condition: Competitie (b) Initial Condition: Competitie (c) Var.Diff.: α = α = α = α =., r = r =. (d) Eq.Diff.: α = α = α = α =., r =., r =. d=., d=.,. 5, d=., d=. d=., d=. d=., d=..35 d=., d=.5 d=., d=.5 d=., d=. d=., d=. d=., d=. d=., d=..3 d=..5 d=.5 d=. d=. d=.5. d=.5 d=. d=. d=.5 d=.5.5 d=. d= d=.5 d= (e) Var.Diff.: α = α = α = α =., r =., r =. (f) Eq.Diff.: α = α = α = α =., r = r =. (g) Var.Diff.: α = α = α = α =., r = r =. (h) Eq.Diff.: α = α = α = α =., r = r = d=.,, d=. d=., d= d=., d=.5 d=., d=. d=., d= d=.. d=.5.3 d=. d=.5. d=. d=.5. d= d=.,, d=. d=., d=.. d=., d=.5 d=., d=..7 d=., d=. d=., d= (i) Var.Diff.: α = α =, α =., α =., r = r =. (j) Eq.Diff.: α = α =, α =., α = r =., r =. (k) Var.Diff.: α = α =, α =., α = r =., r =. Figre 3: Competition eqations: IC and reslts. REFERENCES Jörgensen, S.E. (). Oeriew of the model tpes aailable for deelopment of ecological models. Ecological Modelling 5: 3-9. Jesse, K.J Modelling of a diffsie Lotka-Volterra-Sstem: the climate-indced shifting of tndra and forest realms in North-America. Ecological Modelling 3: 53-. Leng, A. W., Ho, X., Li, Y.,. Eclsie traeling waes for competitie reaction-diffsion sstems and their stabilities. J. Math. Anal. Appl., 33, 9-9. Mickens, R.E. 3. A nonstandard finite-difference scheme for the Lotka-Volterra sstem. Applied Nmerical Mathematics 5: Mimra, M., Fife, P. C., 9. A 3-component sstem of competition and diffsion. Hiroshima Math. J.,, 9-7. Rothe, F., 97. Conergence to the Eqilibrim State in the Volterra-Lotka Diffsion Eqations. J. Math. Bio. 3,39-3.