Progress towards Modeling Red Tides and Algal Blooms

Transcription

1 Pogess towads Modeling Red Tides and Algal Blooms Maxim Lowe College of the Atlantic Ba Habo, ME Novembe 23, 2014 Abstact Conditions in the ocean sometimes allow specific species to populate so quickly that these species fom dense aggegations of individuals. Many species of micoscopic algae in paticula ae known to fom in these dense aggegations, o blooms. Some species that fom blooms ae toxic to maine oganisms o humans. In this pape, one method of pedicting whethe o not blooms will occu involves exploing the impact of gazing zooplankton on algae populations, and how the toxin poduced by the phytoplankton a ects those zooplankton populations. Results show an incease in toxin poduction yields a decease in the gaph s peiod and an incease in atio between zooplankton and phytoplankton populations. Anothe model exploes how the elationship between di usion and patch size influences algal blooms. This pape exploes these two simple models of pedicting blooms and how di eent paametes a ect the esults, and which model gives moe consistent outcomes. A final model involves conveting the one-dimensional di usion model to a two-dimensional model, and yields inteesting changes egading the elationship of the patch size to whethe o not thee will be subsequent algal bloom (o lack theeof). 1 Intoduction Algal blooms ae apid inceases in the population of micoscopic phytoplankton within a ma- Coesponding Autho ine o lake [1] envionment. So-called Red Tides ae algal blooms of dinoflagellates o diatoms that poduce toxins o acids that cause tempoay envionmental ham (a ecting wate quality and killing fish, bids and mammals) in the ecosystems in which they ae pesent. Some species that cause ed tides give the suounding wate a ed discoloation, hence thei name [2]. Gazing zooplankton have been shown to have an impact on species that cause algal blooms. Di eent zooplankton ae a ected in di eent ways. Fo some, dinoflagellates can disupt the mechanical and chemical sensoy system [3] of gazes. In othe species of zooplankton, the gazes show no appaent side e ects fom the toxins, but etain high concentations of toxins, causing a cascading e ect as the toxicity inceases on the tophic pathway [4] [5]. The e ect of gazing on toxic dinoflagellate blooms vaies fom species to species, and depends on the species composition of the gazing community [5]. Algal blooms and ed tides have been happening befoe ecoded histoy, but in ecent times blooms have been happening moe fequently and moe intensely. Anthopogenic influences (such as nutient un-o inducing ed tide blooms and the tanspot of dinoflagellate cysts in ballast wate of ships) have been suggested as possible causes [6]. In lake envionments, whee algal blooms ae nealy exclusively caused by an excess of nutients as a esult of dainage, wate un o fom agicultual field [1], models have been ceated to simulate how nutients impact algal blooms [1]. 1

2 Di usion is often used in modeling algal blooms [7] because spatial and tempoal distibution of phytoplankton is fundamentally govened by the movement of wate, because they ae lacking in mobility o have only weak mobility [8]. Consideing that ed tides bing with them toxins that significantly a ect the envionment they e in, attempting to model the theshold that dictates whethe o not a bloom will occu could, if applied, help pedict when di eent algal blooms occu and why. As the esults of a ed tide can a ect the ecology of an envionment, local fisheies and beachgoes [9], and in some cases dinking wate[10], modeling blooms can help lead to counteacting the negative envionmental, social, and economic e ects they cause. In this pape I exploe di eent models with the intent of finding the best with which to pedict an algal bloom. The main contibution of this study involves expanding a one-dimensional di usion model to two dimensions, and exploing the subsequent changes to elationships between paametes that comes with adding a dimension. The est of this pape is oganized as follows: In Sec. 2 I intoduce thee models fo this study and descibe the techniques used to analyze them. In Sec. 3 I pesent analytic esults, and in Sec. 4 I discuss my esults and how I plan to move fowad with eseach. 2 Methods This pape pactices thee methods of attempting to pedict algal blooms. One involves the inteaction of gazing zooplankton on algae, while two othes attempt to pedict the occuence of a bloom based on di usion. Fo all models, compute-geneated gaphs and animations ae plotted using Python Lotka-Voltea The fist, a Lotka-Voltea model pai of equations [11]: dp dt = P 1 P K P Z, (1) dz dt = PZ µz P Z. (2) + P This model exploes the elationship between phytoplankton and gazing zooplankton. Hee P and Z epesent the density of phytoplankton and zooplankton population espectively. (>0) is the specific pedation ate, (> 0) epesents the atio of biomass consumed pe zooplankton fo the poduction of new zooplankton. µ (> 0) is the motality ate of zooplankton. (> 0) is the ate of toxin poduction pe phytoplankton species and (> 0) is the half satuation constant [11]. The model was coded in Python. The following paametes wee used: P = 3, Z = 0, =.175, = 1, =.35, µ =.0825, = 0.2, K = 160, =.06. The values of,,, µ, and wee chosen by aveaging each of thei Repoted anges in Table 1 fom [11]. The population atios between P and Z wee also taken fom [11]. K was chosen to be 160, and the values fo wee vaied in the models. Values fo wee inceased by a facto of 10 fo fou figues, stating at 0.01 and inceasing, to see the di eences in esults Dimensional = u + u The above model [7] is a simple one dimensional di usion model with gowth. If the function u, of x and t, is split up into two functions, each dependent upon one vaiable, then x and t as vaiables can be isolated fom one anothe: u(x, t) =a(x)b(t) (4) Plugging the above equation into Eq. = ab + ab Simplify equation so that a and b ae isolated on eithe side and divide though by D: 2

3 ( b D D 2 x a (6) By setting each side equal to a common constant, the left and ight side can be sepaated. Since the vaiables ae now totally independent fom one anothe, the patial deivatives become odinay deivatives, and we can now solve these equations. This pocess of sepaating vaiables and solving thei coesponding functions makes it possible to find key elationships between paametes. An impotant inequality in the model aises: L D (7) To see in moe detail the steps that lead up to this inequality, see pages of Edwad Beltami s Mathematical Models in the Social and Biological Sciences. What this inequality states is that fo any given di usivity constant and ate of algal epoduction, thee is a coesponding minimum patch width at which a population of algae can sustain itself without di using out of the system into the suounding wate. Using di eent patch widths, Eq. 3 was plotted fo multiple time values using Python, and the esulting plots can be viewed in Section 3. Animations fo each model wee posted, accessible though a link mentioned in the desciption of each figue. 3 Results Thee ae thee main esults fom this study: one fo each of the models employed. Limiting cases: In Eq s. 1) and 2, setting Z = 0 and K = 160 yields the following expected gaph in Fig. 1: Figue 1: P (blue line) = 3, Z (geen line) = 0 Similaly, setting P = 0 yields a similaly expected esult in Fig. 2, in which the Zooplankton gadually die o since thee is nothing to gaze. 2.3 Two-Dimensional = D u (8) This model is a modification to Eq. 3, and simulates the di usion of algae acoss two dimensions to povide a moe ealistic simulation. The model in Eq. 8 was also coded using Python, and the esulting plotted images wee sequenced to fom a video, accessible though a link mentioned in the desciption of each figue. The last fame of each video is used in the figues fo this pape. Figue 2: P (blue line) = 0, Z (geen line) = 1, = Lotka-Voltea Results Choosing a vey small value, =0.01, the esult in Fig. 3, yields a peiod between oscillations of appoximately 150 time units. The population 3

4 of phytoplankton peaks between 4.5 and 5, and the population of zooplankton peaks at about 1.8. Choosing a lage value, = 1, the esult in Fig. 5, yields a peiod between oscillations between 16 and 25 time units. The population of phytoplankton peaks at just above 7, and the zooplankton at about 1. The phytoplankton population dips just below 1 at its initial lowest tough. Figue 3: P (blue line) = 3, Z (geen line) = 1, =0.01 Choosing the value =0.1, the esult in Fig. 4 yields a peiod between oscillations of appoximately 125 time units. The population of phytoplankton peaks slightly highe than in Fig. 3, and the population of zooplankton slightly lowe. Figue 5: P (blue line) = 3, Z (geen line) = 1, =1 Choosing a lage value, = 10, the esult in Fig. 6, yields an initial spike in phytoplankton population that peaks at between 55 and 60. The phytoplankton population density does not dip below 10 at its lowest tough, and the zooplankton population density emains below one. It can be obseved that equilibium is eached at a population density appoximate to 29. Figue 4: P (blue line) = 3, Z (geen line) = 1, =0.1 Figue 6: P (blue line) = 3, Z (geen line) = 1, = 10 It was found that as values of incease, the peiod of the gaph deceases, and the atios between the population peaks fo the phytoplankton and zooplankton inceases. All in all, this model is useful fo obseving the e ects of gazing zooplankton upon ed tide algae, but its 4

5 esults ae less significant fo pedicting algal blooms. The impact of gazes on algal blooms is negligible when it comes to attempting to pedict a bloom, and in many cases, only specific species ae a ected by the toxins Dimensional Di usion Results Fo the following esults, these paametes wee used: a stating population density of P = 50, a gowth ate of =.3, a di usivity ate of 30, and patch width L was vaied. Using these paametes, the minimum patch width to yield a bloom is calculated to be 10. The following occus in Fig. 7 when the patch width is at 25, below the minimum fo a bloom to occu. Figue 8: L = 32. To see the animation fo this simulation, go to WKg02KqG8dA mum. As expected, the bloom occus. The inequality holds tue fo all values tested. Figue 7: L = 25. To see the animation fo fo the simulation unde these paametes, go to The algae quickly di use out of the system. Thee is no sign of gowth - this is in accodance with the inequality. The following occus in Fig. 8 when the patch width is at 32, just above the minimum fo a bloom to occu. As expected, the algae do not di use out of the system. In fact, the algae begin to gow afte the minimum population is eached, howeve it takes a long time fo the bloom to occu. The following occus in Fig. 9 when the patch width is at 50, a numbe well above the mini- Figue 9: L = 50. To see the animation fo this simulation, go to 5Pcp5l Dimensional Di usion Results The same paametes that wee used in the 1- Dimensional Di usion model wee used in this one. Fig. 10 occus when the patch width is at 25, again, below the minimum stated by the inequality. As expected, much like in the one dimensional equivalent the algae di use out of the system befoe being able to bloom. 5

6 about 1.413, o appoximately p 2. Shown in Fig. 12 is the model with a patch width of Much like the one dimensional model at a patch width of 32, the population density seems to stagnate at its lowest point - it gows, but vey slowly, and in eality would not cause a lage bloom. Figue 10: L = 25, time = 7.9. To see the animation fo this simulation, go to be/dyzxjhz1fwm Fig. 11 occus when the patch width is at 32, a numbe just above the minimum accoding to the inequality. Figue 12: L = 44.5, time = 7.9. To see the animation fo this simulation, go to http: //youtu.be/rmzzfxzza2a To bette undestand the easons fo this new minimum, p 2 is plugged into the inequality: L p 2 D Reaanging the tems in ode to isolate D: (9) Figue 11: L = 32, time = 7.9. To see the animation fo this simulation, go to be/umzicavtb6y This yields an unexpected esult. Afte 7.9 time units, the algae have nealy completely diffused out of the system. Running the model longe shows that the algae have completely diffused. This equies a eevaluation of the inequality. Though tial and eo, I found that the minimum patch width fo this two dimensional model was appoximately The atio between the two-dimensional and one-dimensional models is D 2 apple L2 2 (10) Reaanging the tems in ode to isolate : 2 D 2 L 2 (11) What this new inequality assets is that fo a two dimensional di usion model, the di usivity ate of the system must be one half of that of the one dimensional di usion model. Altenatively, the ate of population gowth must be twice that of the one dimensional di usion model. Since in actuality the di usivity and the ate of gowth would emain elatively constant, the patch width would have to be p 2 times that of the one dimensional model. These findings 6

7 make sense, because the algae is now di using acoss two dimensions, and the new paametes must make up fo the faste ate at which the algae is lost. Fig. 13 occus with a patch width of 75, well above the minimum accoding to the new inequality. This final un of the model veifies that the new inequality is tue fo numbes lage than the minimum. di using acoss a second dimension it must epoduce at twice the ate to sustain a bloom. Futue wok will include adding moe paametes to the equation, including algae death, dependence upon nutients, adding vaiance in diffusivity, and potentially incopoating the Lotka- Voltea elationship into the impoved di usion model. Also among my goals ae modeling diffusion acoss thee dimensions, as well as finding the new elationship coesponding between patch size and the occuence of blooms in thee dimensions. Acknowledgments I would like to thank Dave Feldman fo his help with code and undestanding models, and to Chis Petesen fo his advice and help with my pesentation. Figue 13: L = 75, time = 7.9. To see the animation fo this simulation, go to be/9dyvt96aiy 4 Conclusion and Futue Reseach The Lotka-Voltea model was helpful as fa as exploing the impact of zooplankton on phytoplankton. It is not ideal fo pedicting the occuence of blooms, but is useful fo potentially contolling ed tides though the intoduction of gazing zooplankton. This model could be vey useful fo modeling vey specific conditions of ed tide; how to contol specific ed tide species with specific gazes. Both the second and thid models ae simple, but ae good ways of detemining whethe o not blooms will occu based on the algae population acoss a cetain patch aea. The biggest supise in gaphing these models was the unpecedented change in the inequality. It tuned out to be logical, since consideing the algae population is Refeences [1] J.B. Shukla, A.K. Misa, and Peeyush Chanda. Modeling and analysis of the algal bloom in a lake caused by dischage of nutients. Applied Mathematics and Computation, 196: , [2] Yong-Yeng Lin, Matin Risk, Sammy M. Ray, Donna Van Engen, Jon Clady, Jezy Golik, John C. James, and Koji Nakanishi. solation and stuctue of bevetoxin b fom the ed tide dinoflagellate ptychodiscus bevis (gymnodinium beve). Jounal of Ameican Chemical Society, 103(22): , Novembe [3] M. Delgado and M. Alcaaz. Inteactions between ed tide micoalgae and hebivoous zooplankton : the noxious e ects of gyodinium cosicum ( dinophyceae ) on acatia gani ( copepoda : Calanoida ). Jounal of Plankton Reseach, 21(12): , [4] Alan W. White. Maine zooplankton can accumulate and etain dinoflagellate tox- 7

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