Luận án tiến sĩ: Mathematical analysis of trophic interactions: From bacteria competition to lemming cycles

The sto-ichiometric modeling of the bacteria-algae lake system is relatively new, whilethe lemming population cycle has attracted the attention of several generations oftheoretical and e

BACTERIA COMPETITION TO LEMMING CYCLES

byHao Wang

A Dissertation Presented in Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

ARIZONA STATE UNIVERSITY

May 2007

Copyright 2007 by Wang, Hao

All rights reserved.

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BACTERIA COMPETITION TO LEMMING CYCLES

byHao Wang

has been approvedNovember 2006

Mechanistic and phenomenological models and careful parameter tions are presented through both aquatic and terrestrial ecosystems The sto-ichiometric modeling of the bacteria-algae lake system is relatively new, whilethe lemming population cycle has attracted the attention of several generations oftheoretical and experimental biologists and continues to be an issue of controversy.

estima-Bacteria-algae interaction in epilimnion is modeled with explicit ation of carbon (energy) and phosphorus (nutrient) Global qualitative analysisand bifurcation diagrams of this model are presented Competition of bacterialstrains are modeled to examine Nishimura’s hypothesis that in severely P-limitedenvironments, such as Lake Biwa, P limitation exerts more severe constraints onthe growth of bacterial groups with higher nucleic acid contents, which allows low

consider-nucleic acid bacteria to be competitive.

Through a series of carefully derived models of the well documented amplitude, large-period fluctuations of lemming populations at Point Barrow, onecan argue that, when appropriately formulated, autonomous differential equationsmay capture much of the desirable rich dynamics such as the existence of a periodicsolution with period and amplitude close to that of approximately periodic solu-tions produced by the more natural but mathematically daunting nonautonomousmodels This, together with the bifurcation analysis, indicates that neither sea-sonal factors, nor the moss growth rate and lemming death rate, are the maindeterminants of the observed multi-year lemming cycles

high-What ecological factors control population cycles? For some species —

ili

tors and the functional response of predation appear to be the primary nants Maturation delay almost completely determines the cycle period, whereasthe functional response greatly affects its amplitude and even its existence Thisresult is obtained from sensitivity analysis of all the parameters and comparison

determi-of the lemming-stoat and hare-lynx systems

iv

First of all I would like to thank my advisor Dr Yang Kuang for offering

me precious chances to implement experiments and work with biologists, also hisencouragement during my Ph.D study at Arizona State University I would like tothank my co-advisor Dr Hal Smith for his numerous discussions on mathematicalanalysis and invaluable help on all aspects, and Dr James J Elser, Dr John Nagyfor their many insightful discussions in biology I also thank Dr Olivier Gilg,

Dr Sebastian Diehl, Dr James Grover, Dr Val Smith, Dr Irakli Loladze and

Dr Jiaxu Li for helpful discussions and suggestions In addition, thanks to

Dr Horst Thieme and Dr Sharon Crook for their interesting courses I am blessed

to work with these excellent mathematicians and biologists

Last but not the least, I am forever indebted to my wife, Qiangying Cao,

for her pure love and inspiration.

Hao Wang

December 25, 2006

Arizona State University

vi

LIST OF TABLES 0.0.0 0000200040,

LIST OF FIGURES 2 00.0000 eee,

CHAPTER 1 BIOLOGICAL BACKGROUND

3 Persistence and Invasion of Bacteria

4, Competing Bacterial Strains

CHAPTER 4 LEMMING-PREDATOR SYSTEMS IN GREENLAND 67

1 Simple Lemming-Stoat Model Ốc 681.1 Parameter Estimation 0.0 0.004 701.2 Numerical Solution and Limitation 72

2 Lemming-Stoat Delay Model 0000.4 732.1 Mathematical Rationality ch So 732.2 Empirical Data Fitting 200 752.3 Sensitivity Analysis 2 00.0.0 0 0004, 76

3 Snowshoe Hare-Lynx Delay Model 0 80

4 Comparison and Interpretation ch 0200.4 85

5 Summary and Discussion 0 00.0.0 0 05 eee 89

CHAPTER 5 CONCLUDING REMARKS 91

1 Key Points of Thesis 0 2 ee 91

APPENDIX A MATLAB PROGRAMS co 108

vill

Table Page

Variables in bacteria-algae system (2.l) 16

Parameters in bacteria-algae system (2.l) 17

Parameters in Barrow model (3.1) (Turchin and Batzli 2001) 47

Parameters in moss-lemming models 53

Comparison of all four lemming models 65

Parameters in lemming-stoat systems 70

Empirical lemming data from Olivier Gilg , 76

Parameters in hare-lynx system (4.4) 84

ix

proof of Theorem 3 0 ho

A bifurcation diagram for system (2.2), illustrating that the lower the mixed layer the better for algae in system (2.2) Thisbifurcation diagram confirms our mathematical findings When

shal-Ry > 1, the algae extinction equilibrium is unstable, and the onlypositive equilibrium appears to be globally attractive The branch-ing point occurs at Ro = 1 When Rp < 1, there is no positiveequilibrium and the algae extinction equilibrium is globally attract-ing This numerical result is generated by the continuation software

‘MatCont’ in MATLAB 0 2 Q Q Q he

14

25

Phase plane when Ry > 1 for system (2.2) The algae extinction

equilibrium Ep = (0,Q, Pn) is globally attracting on the subspace

9 = {2 € | A = 0} but a uniform weak repeller for Q, = {x €Q| AO}, and A is persistent in this case 2 2.2.0 Algae dynamics without bacteria (system (2.2)) with respect todifferent depths of the mixed layer: All the variables approach thepositive equilibrium in about two months The first two figuressuggest that with deeper mixing depths, P increases, but A de-creases On the other hand, average light intensity should decreasewith larger depths, because of shading effect Hence, algal growth

is more limited by energy than nutrient in a lake with a deepermixed layer This numerical simulation is generated by ‘ode23s’ inMATLAB using the initial conditions: A = 20,P = 0.1,Q = 0.004

xi

27

27

12.

13.

An abstract phase plane diagram for system (2.1) when Ro > 1 and

R, > 1 Q,P are placed on one axis (say x-axis), A,C are placed

on another axis (say y-axis) and B is on the vertical axis (z-axis)

Extinction equilibrium eo = (0,Q, Pin, 0,0) is globally attracting

on the subspace {x € 9 | A = B = C =0}, but a repeller for

Q, = {ce € 2 | B = 0) Bacteria extinction only equilibrium

e, = (A,Q, P,0,C) is globally attracting on the subspace M2, but

a repeller for Q) = {x €2| BO} B is persistent, and at least

one coexistence equilibrium exists 6 0 eee ee es

Regions of P;, versus In, for survival and extinction of bacteria andalgae Both algae and bacteria go extinct (Ry < 1) in the greyregion Both algae and bacteria survive (Ro > 1,R1 > 1) in thewhite region Algae survive but bacteria go extinct (Ro > 1, Ri <1) in the red/dark region We run simulations of system (2.1) foreach pair of (Jin, Pin), plot the point in grey if both A and B go tozero, in white if both persist, and in red/dark if A persists but B

A quantitative relationship between bacteria and algae at the low Plevel (Pi = 30), illustrating that B: A ratio is decreasing in solarenergy input They also confirm the qualitative result (Figure 12).These two figures are generated by ‘MatCont’ in MATLAB

xil

34

36

37

in MATLAB with the initial conditions: A = 20,Q = 0.004, P =0.1,B)=1,B,=1,C=100 0 0.00005When the mixed layer is shallow with z„ = 1, system (2.1) exhibitscomplex dynamics These simulations are generated by ‘ode23s’ inMATLAB with the initial conditions: A = 350,Q@ = 0.004, P =

Bifurcation diagrams of system (2.1) for depth of epilimnion Theseare generated by 'MatCont'in MATLAB .Numerical simulation of Barrow model (3.1) with the median val-ues of parameters shown in Table 3 This numerical simulation

is generated by ‘ode23s’ in MATLAB with the initial conditions:

V=100,M@=1000,H=20 0,000.00 00.Numerical simulation of nonautonomous moss-lemming model (3.2)with the median values of parameters shown in Table 3 and Z =0.01 This numerical simulation is generated by ‘ode23s’ in MAT-LAB with the initial condition: M = 1000, H = 20

21.

22.

u(r(t)) vs u(t) and d(r(t)) vs d(t) u(t) and d(t) are

rep-resented by the red continuous curves Using mean values of

Us, ds,dy, we compare u(t),d(t) with u(r), d(r) statistically The

standard/average error of u(t) with respect to u(r(t)) is 1.538 and

the relative error is 1.538/u,; = 0.128 The standard/average

er-ror of d(t) with respect to d(r(t)) is 0.401 and the relative erer-ror

is 0.401/(d, — d;) = 0.141 These two continuous functions areconstructed manually Then we plot each of them with the cor-responding discontinuous function in one figure and also calculate

CITOTS oo

A typical solution of the nonautonomous moss-lemming model (3.4)

with the median values of parameters shown in Table 3 and / = 0.01

This numerical simulation is generated by ‘ode23s’ in MATLAB

with the initial conditions: z = 1000,=20

A typical solution of the autonomous moss-lemming model (3.5)

with the median values of parameters shown in Table 3 and / = 0.01

This numerical simulation is generated by ‘ode23s’ in MATLAB

with the initial conditions: x = 1000,y=20

XIV

53

56

Bifurcation diagrams and limit cycles for the autonomous system(3.5) These bifurcation diagrams are generated by plotting maxi-mums and minimums of all the eventually stabilized oscillations .Period diagrams for the autonomous system (3.5) These period bi-furcation diagrams are generated by modifying the ‘ode23’ program

in MATLAB with Jiaxu Li’s great help The location in NE Greenland where field experiments were carriedout by Olivier Gilg and colleagues © 2 eee.Stoat functional response to lemmings (Gilg et al 2003)

A typical solution of the simple lemming-stoat system (4.1) withparameter values in Table 6 This numerical solution is generated

by ‘ode23s’ in MATLAB with the initial conditions: z = 0.1,y =0.1 We use the command ‘plotyy’ to plot this multi-scale graph

72

30.

ỏ1.

32.

Lemming-stoat dynamics predicted by the delay system (4.3)

com-pared to empirical data (points) The empirical data is in

Ta-ble 7 Stoat density is collected in winters; hence, stoat data

points are shifted to the left half a year The numerical

predic-tions are generated by ‘dde23’ in MATLAB with the initial tions: x = 0.1, = 0.1 The command ‘plotyy’ is applied to plot multi-scale graphs of both model predictions and empirical data Sensitivity analysis on the period of the lemming cycle predicted by

condi-system (4.3) for each parameter and different functional responses

We run simulations by increasing or decreasing each parameter,

or by changing functional response types, then all the periods of

eventually stabilized oscillations are saved as outputs In the end, these saved outputs and default period are imported into ‘Adobe

Illustrator CS’, and then we get these histograms

Sensitivity analysis on the amplitude of the lemming cycle

pre-dicted by system (4.3) for each parameter and different functional

responses We use the similar method to plot these histograms as

Period bifurcations for key parameters in system (4.3) These

pe-riod bifurcation diagrams are generated by modifying the ’dde23’

program in MATLAB with Jiaxu Li’s help -.-

Xvi

77

78

78

A numerical solution of the hare-lynx delay system (4.4) with rameter values in Table 8 mimics the 10-year snowshoe hare cycle.This numerical solution is generated by the commands ‘dde23’ and

pa-‘plotyy’ in MATLAB with the initial conditions: z = 3,y = 0.5

We run the simulation for 60 years, but only choose the latter 40years (20-60) where the oscillations are stabilized .The details of the effect of maturation delay in system (4.3) Thesenumerical simulations are generated by ‘dde23’ and ‘plotyy’ inMATLAB with the initial conditions: r=O.1l,y=01 .Competitions of two daphnia species under different light intensi-ties These figures are generated by ‘Microsoft Excel’ Photos of our experiment in Elserslab -Moss dispersal and growth This figure is from Wikipedia

Evolution direction with “nutrient hypothesis” which is modifiedfrom the bifurcation diagrams of system (3.5), Figure 24 and 25

97

BIOLOGICAL BACKGROUND

1 Ecological Stoichiometry

“Ecological stoichiometry” (Sterner and Elser 2002), a newly emergingbranch of ecology, deals with the balance of energy and chemical elements inecological systems Examples have shown that stoichiometric reasoning can pre-dict macroscopic phenomena using its microscopic principles (Anderson 1997, Lo-ladze et al 2000) A typical biological process is in essence a set of chemicalreactions In an ordinary chemical reaction, matter and component elements areneither created nor destroyed (Conservation Law of Matter) Carbon, nitrogen,and phosphorus are three of the main constituents in biomass However, C, N,and P are not particularly abundant on Earth or in the universe as a whole andthus it seems that living things made a very discriminating selection of elementsfrom the environment (Sterner and Elser 2002) Many biological processes areeffectively studied and modeled with the applications of some simple yet powerfulstoichiometric constraints This suggests that the ubiquitous and natural stoichio-metric constraints may also be useful for modeling species growth and interactions(Grover 2002, Klausmeier et al 2004, Loladze et al 2000, Loladze et al 2004).Nitrogen and phosphorus are two elements commonly considered to be lim-iting for autotrophs and heterotrophs Often, there is a mismatch in the elementalcomposition of food and heterotrophic consumers For example, the P:C ratio oflemmings is about 0.06gP/gC (Batzli et al 1980) while the median P:C ratio of

reproduction and growth of consumers In general, consumers have high ent contents whereas autotrophs have low and highly variable nutrient contents.C:N:P ratios of autotrophs are highly variable because of physiological plasticity

nutri-of plant biomass in relation to environmental conditions (light, nutrient supply,COx, etc.) Although these ratios can vary somewhat in heterotrophs, they typ-ically vary much less than in autotrophs (Elser et al 2000a) Thus, it has beencustomary to assume that consumers have strict homeostasis; that is, consumershave fixed elemental composition in biomass (Andersen and Hessen 1991)

It is observed that plant quality can dramatically affect the growth rate ofherbivores and may even lead to their extinction Specifically, if the concentration

of an essential element in plant biomass is lower than the minimum threshold forits consumer, then the consumer’s growth may suffer This has been shown forboth aquatic (Nelson et al 2001, Sterner et al 2002), and terrestrial systems(Newman et al 1999)

While consumers have C:N:P ratios that are relatively constant for a givenspecies, considerable variation does exist when different species are compared

To explain this, Elser et al (1996, 2000b) have proposed the “Growth RateHypothesis” (GRH) to explain variation among organisms in biomass C:P andN:P ratios Their argument reasons that rapid growth rate requires increasedallocation to P-rich ribosomal RNA to meet the protein synthesis demands ofrapid growth (Figure 1) Thus, fast-growing organisms have biomass with low C:Pand N:P ratios, raising their requirements for P from their environment or dietand making them poor competitors for this potentially key resource Considerable

natural selection

on growth rate

\

cellular investment (ribosome content)

biochemical investment (RNA:protein)

Figure 1 A flow chart for GRH (Elser et al 1996)

data supporting the GRH have begun to accumulate (Gorokhova et al 2002, Elser

et al 2003)

Stoichiometric models deal with both food quantity and food quality

ex-plicitly, and most are, in fact, related to nutrient ratios The threshold model of

only one nutrient limitation, has been known at least since Liebig (1840, 1842).

A piecewise linear function is used by Andersen (1997) for herbivore’s growth

un-der stoichiometric and energetic constraints Loladze et al (2000) modified the framework of Andersen (1997) to include light intensity through the autotroph

carrying capacity Both conceptual (Sterner et al 1997) and quantitative

(Lo-ladze et al 2000) work in ecological stoichiometry has called attention to the issue

on how light and nutrients together affect trophic interactions Thus, both solarenergy and the nutrient element (phosphorus) are mechanistically modeled in theaquatic bacteria-algae system in this thesis Stoichiometric impacts exist not only

(Sterner et al 1998, Urabe et al 2002a,b) Thus, stoichiometric theory can prove both qualitative and quantitative insights in population ecology Many newmathematical and biological predictions can be gained by adding stoichiometricdimensions to classic population dynamics.

im-2 Population Cycles

Population fluctuations in small mammals such as voles, snowshoe haresand lemmings have been a constant inspiration to numerous influential andthought-provoking articles since the pioneering work of Elton (Elton 1924, Hanski

et al 2001) Lemmings are mouse-like arctic rodents characterized by small, shortbodies about 13 cm (about 5 in) long, with very short tails Lemmings live inextensive burrows near the water, feed on vegetation, and build nests out of hair,grass, moss, and lichen The female produces several broods a year, each of whichcontains about four or five young All key reproductive events for lemmings takeplace during winter Empirical research is difficult to perform on organisms thatlive under snow due to the extreme low temperature and the fact that you cannot

see them.

In order to find plausible mechanisms for generating population cycles,biologists often perform short-term field observational studies of population pro-cesses that might cause cycles (Krebs 1996) However, it is almost impossible

to determine which ones cause cyclic behavior from verbal population models,since multiple dynamic factors may make contributions Mathematical popula-

cyclic dynamics (Kendall et al 1999).

Over the years, a variety of ecological mechanisms have been proposed ascauses of population cycles (Krebs 1994) Some researchers have argued that the

“cycles” are nothing more than stochastic fluctuations, but this view is mined by obvious synchrony across broad, even continental, geographic regions(Krebs et al 2002) Ecological dispersal, cyclic weather patterns, parasitic andother diseases and even the sun spot cycle have all joined food supply, predationand stochasticity as potential causes Recently, however, ecologists tend to believethat the cause of such oscillations is either an interaction between lemmings andtheir food supply, so-called “bottom-up” regulation (Turchin and Batzli 2001), or

under-an interaction between lemmings under-and their munder-any predators, so-called “top-down”regulation, (Hanski et al 2001, Gilg et al 2003) These trophic mechanismsare still open for debate This uncertainty is reflected in three questions posed byHudson and Björnstad (2003) First, what precise ecological mechanisms generatepopulation cycles? Second, do these mechanisms apply to all cyclic populations?Finally, do these mechanisms explain why some populations are cyclic whereas

others are not?

For example, cycles in brown lemming (Lemmus trimucronatus) tions at Point Barrow, Alaska, appear to be driven by bottom-up regulation(Turchin 2003, Turchin et al 2000) Turchin and coworkers find evidence forthis conclusion in the shape of population density curve at its peaks If cycleswere driven primarily by predators, then prey peaks should be “blunt,” because

popula-by the time predator density increases sufficiently to cause the prey population to

ever, lemming populations exhibit very sharp peaks, with rarely more than oneobservation period at the peak (Turchin 2003) Therefore, predators are probablynot causing cycles in the Point Barrow brown lemming population.

In contrast, evidence suggests that cycles in collared lemming (Discrostonyzgroenlandicus) populations in Northeast Greenland are driven by predation Thissystem is well studied and astoundingly simple, with this single prey specieshunted essentially by only four predator species (Gilg et al 2006) A mathe-matical model studied by Gilg et al (2003) predicted cycles with a periodicitythat matched field data very well over a 15 year time span, a remarkable resultprimarily because the model was parameterized with independent field data in-stead of being fit to the field data with a statistical procedure In the model,cycles were driven by stoat or short-tailed weasel (Mustela ermina) predation

To make matters more complicated, recent evidence has forced ecologists

to examine hypotheses that include both bottom-up and top-down forces

act-ing together These ideas are mainly introduced by Oksanen et al (1999) and

summarized nicely by Korpimaki et al (2004) They conclude that, at least forvoles, lemmings and snowshoe hares, the increase phase of the cycle occurs largelybecause individuals are more likely to survive, not because females increase re-productive output Population density plateaus when food becomes scarce, and

is driven into the decrease phase as predator populations become so dense thatthe prey population can no longer sustain predation losses

Pioneering work on resource-consumer dynamics includes the well knownwork of Lotka (1925) and Volterra (1926), which introduced the classical Lotka-

ical ecology One of the most frequently used resource-consumer models is theRosenzweig-MacArthur (1963) model, which produces two generic asymptotic be-haviors — equilibria and limit cycles Bazykin (1974) added a self-limitation term

to the Rosenzweig-MacArthur model to account for the rather ubiquitous densitydependent mortality rate (see also Bazykin et al 1998) All these models produceoscillatory solutions that seem to mimic the fluctuating populations observed innature.

Because ecologists have been collecting empirical data for long time series ofpopulation abundances, it is the right time now to bring empirical time-series dataand mechanistic population modeling together We construct a few mathematicalmodels that embody the biologically plausible hypotheses, use these models togenerate numerical simulations, and then compare the numerical time series to theempirical data quantitatively Our models are capable of resembling the 4-yearlemming cycle qualitatively and nearly quantitatively with biologically reasonableparameter values Mathematical and numerical results of these models may giveinsight into the major factors controlling population cycles and their features

Most parameters in Chapter 3 are estimated by Turchin and Batzli (2001),while those in Chapter 4 are estimated by Gilg, and they are consistent from thesame field — Northeast Greenland Because of the constraints induced by popu-lation structure and the chosen function forms, the model may fit the empiricaldata poorly in period and amplitude even with the “correct” parameter values.However, our delay differential equation model in Chapter 4 resembles the fielddata reasonably well in period and amplitude

In Chapter 2, solar energy (organic carbon) and nutrients (inorganic phorus) are mechanistically modeled for the interaction between algae and bacte-ria in the epilimnion of a lake system In order to get a “realistic” stoichiometric

phos-model, all the close-to-truth assumptions are explicitly posed, and function forms

of most terms are biologically tested and carefully generated We perform a globalqualitative analysis and present bifurcation diagrams of our models In addition,competing bacterial strains are modeled to test some hypotheses of Nishimura et

al (2005).

The “bottom-up” trophic regulation is modeled for the brown lemmings in

predator-prey competitionbottom-up top-down algae dynamics

brown lemming) collared lemming | [lake bacteria,

population cycle stoichiometry

biologyFigure 3 The structure of this thesis

Bacterium

Alaska Greenland

Brown Lemming Collared Lemming

Figure 4 A checklist of all the locations and species analyzed in the thesis

Alaska following the Barrow model of Turchin and Batzli (2001) in Chapter 3.

We compare four different systems with data and suggest that the autonomoussystem can indeed be a good approximation to the moss-lemming interaction

at Point Barrow Through bifurcation diagrams of the autonomous system, theseasonal factor may not be the main culprit of the observed multi-year lemmingcycles.

In Chapter 4, the “top-down” trophic regulation is modeled with delaydifferential equations for the collared lemmings in NE Greenland The modelstructure is the same as the historic predator-prey model, but all the terms usemore realistic formulations The functional response of stoats to collared lemmingsresembles the Holling Type III (Gilg et al 2003) The lemming growth takes themodified logistic growth term that is recently published by Sibly et al (2005)following the GPDD database Stoat maturation delay and death are explicitlyinvolved in the stoat growth term Further, the stoat death rate takes a function

of the lemming density (Gilg et al 2003) All these painful modifications makeour model more specific for the collared lemmings in NE Greenland Many peopleargued that one model could be applied to many different biological situations, but

to identify key differences is much more important and difficult in a modelingprocess Based on this parameterized model, we perform sensitivity analysis,bifurcation diagrams, and compare the 4-year lemming cycle with the 10-yearsnowshoe hare cycle Through these technical processes, we obtain some newqualitative and quantitative insights, that can help us figure out key factors aswell as guide future field tests of plausible hypotheses

The last chapter discusses philosophy of mathematical modeling and all the

key points of the dissertation Additionally, we propose future extensive ical and experimental work beyond this dissertation.

theoret-STOICHIOMETRY OF LAKE BACTERIA AND ALGAE

Solar energy (which is captured as organic carbon in photosynthesis) andnutrients (phosphorus, nitrogen, etc.) are main factors regulating ecosystem char-acteristics and species density Phosphorus is often a limiting nutrient for algalproduction in lakes (Edmondson 1991) In Lake Biwa, phosphorus is an extremelylimiting element for bacterial growth (Nishimura et al 2005) Lake Biwa is a large(surface area, 674km?) and deep (maximum depth, 104m) lake located in the cen-tral part of Honshu Island, Japan Nishimura et al (2005) used flow cytometry

to examine seasonal variations in basin-scale distributions of bacterioplankton inLake Biwa They hypothesized that, in severely P-limited environments such asLake Biwa, P limitation exerts more severe constraints on the growth of bacterialgroups with higher nucleic acid (HNA) contents, which allows low nucleic acid(LNA) bacteria to be competitive and become an important component of the mi-crobial community A main purpose of this chapter is to examine this hypothesis

The interaction between bacteria and algae in pelagic ecosystems is complex(Cotner et al 2002) Bacteria are nutrient-rich organisms, whose growth is easilylimited by nutrient supply and organic matter produced by plants, which havevery flexible stoichiometry compared to bacteria Suspended algae, also calledphytoplankton, live in almost all types of aquatic environments Algae grow inopen water by taking up nutrients such as phosphorus and nitrogen from the wa-ter and capturing energy from sunlight Extra fixed energy is exuded in the form

of organic carbon from algae during photosynthesis Bacteria require previouslyfixed organic carbon (OC) as energy source Hence, algae is an important source

of OC to bacteria Since bacteria remineralize nutrient elements during sition, bacteria and algae have often been considered to have a loose mutualistic

decompo-or commensal relationship However, bacteria and algae may also compete witheach other if bacteria are limited by phosphorus (Hessen et al 1994) To simplifythe model, we only consider the freely living bacteria, excluding particle-attacheddecomposers, and assume there is no exchange between them

A lake can be separated by a thermocline into two parts: the epilimnionand the hypolimnion The epilimnion is the upper warmer layer overlying thethermocline It is usually well-mixed The hypolimnion is the bottom colder layer.The absorption and attenuation of sunlight by the water column and algae aremajor factors controlling photosynthetic potential and temperature Solar energy,essential for algae, decreases exponentially with water column depth Nutrientsare redistributed from epilimnion to hypolimnion as the plankton-derived detritusgradually sinks to lower depths and decomposes; the redistribution is partiallyoffset by the active vertical migration of some plankton (Horne et al 1994)

Figure 6 The cartoon lake system for our mathematical models.

In lakes, algal OC exudation and environmental input are two primary energysources for bacterial growth To simplify the study of algal stimulation on bacterialgrowth, we assume below algal OC exudation is the only source for bacterialsubsistence.

Algae dynamics in a lake system have been modeled by many researchers(Huisman et al 1994, Huisman et al 1995, Klausmeier et al 2001, Diehl 2002,Diehl et al 2005, Berger et al 2006) Chemostat theory and experiments havebeen applied to nutrient competition between bacteria (Grover 1997, Smith et al

1994, Codeco 2001, Smith et al 2003) Bacteria-algae interaction was modeled

by Bratbak and Thingstad (1985) Their work provides us a useful framework

to develop a more realistic model In recent years, stoichiometric modeling offood web systems has gained much attention (Andersen 1997, Diehl et al 2005,Hessen et al 1997, Loladze et al 2000, Kuang 2004, Kuijper 2004, Logan et

al 2004, Grover 2002) However, these models are not directly applicable tophytoplankton-bacteria interaction Our models, motivated by the experimentsand hypotheses of Nishimura et al (2005) can be viewed as an extension as well

as a variation of the work of Diehl et al (2005) where they modeled algal growth(without bacteria) experiments subject to varying light and nutrient availability

We will model the stoichiometry of bacteria and algae in epilimnion underthe “well mixed” assumption (Berger et al 2006, Huisman et al 1994, Huisman

et al 1995) We will perform a global qualitative analysis and present bifurcationdiagrams of the model We will discuss the implications of these bifurcationdiagrams and the basic reproductive numbers of bacteria and algae Competing

bacterial strains are modeled to test the hypotheses of Nishimura et al (2005).

A brief discussion section concludes this chapter

1 Model Formulation

All the variables and parameters are defined in Table 1 and Table 2 cording to Lambert-Beer’s law, the light intensity at depth s of a water columnwith algal abundance A is (Huisman et al 1994)

Ac-I(s, A) = la exp[—(kA + Kog) 8}

I(s, A)Algal carbon fixation function takes the Monod form Ts, A)+H (Diehl et al.2005), where H is the half-saturation constant

The epilimnion is well-mixed (Diehl et al 2005, Huisman et al 1994).

Algal average growth function for carbon is (Huisman et al 1994, Berger et al

Table 1 Variables in bacteria-algae system (2.1)

Var | Meaning Unit

A Algal carbon density mgC/m?

Q Algal cell quota (P:C) gP/gC (= mgP/mgC)

P Dissolved mineral phosphorus concentration | mgP/m®

B Heterotrophic bacterial abundance mgŒ/m

C | OC concentration mụŒ/m3

2006)

ape 1SA) ao 1 n(_ đẻ là

tmJo I(s,A)+H ĐC tm(kA+ Koy) \H+1(zm.A)/

Algal growth function for phosphorus takes the Droop model form 1 — Qn where

Qm is the minimum algal cell quota and Q is the actual algal cell quota

Algal sinking takes place at the interface between epilimnion and polimnion, and its rate is negatively related to the depth of epilimnion, becausewith deeper epilimnion, there is proportionately less of the total species abun-dances or element concentrations available for sinking For convenience, we as-sume that algal sinking rate is inversely proportional to the mixed-layer depth2m (Diehl et al 2005, Berger et al 2006) D is the water exchange rate acrossthe interface between epilimnion and hypolimnion and between the epilimnionand in- and outflowing streams We assume that there is a constant phosphorusconcentration, P,,, in the hypolimnion and in the inflow As for algal sinking, weassume the water exchange is also inversely proportional to z,, We assume thatbacteria have a fixed stoichiometry, since compared to algae, it is relatively con-

hy-stant (Sterner and Elser 2002) We assume bacterial growth functions of carbon

_ C

7 Ko +C’and phosphorus take the Monod form: f(P) and g(C)~ Kp+P

Table 2 Parameters in bacteria-algae system (2.1)

Par | Meaning Value Ref

Ii, | Light intensity at surface 300umol(photons)/(m? - s) | [20]

k Specific light attenuation co- | 0.0003 — 0.0004m?/mgC | 20,17]eff of algal biomass

Kyg | Background light attenua- | 0.3 — 0.9/m (7],(20]tion coefficient

H | Hs‹c for light-dependent al- | 120mol(photơns)/(m2 - s) | [20]

gal production

2m | Depth of epilimnion > 0m, 30m in Lake Biwa | [85]

Qm | Algal cell quota at which | 0.004gP/9C [20]

growth ceases

Qu | Algal cell quota at which nu- | 0.04gP/gC [20)

trient uptake ceases

Øm | Maximum specific algal nu- | 0.2— 1gP/gC/day [20], [7]trient uptake rate

M | H.s.c for algal nutrient up- | 1.5mgP/m* [20)

6 Bacterial fixed cell quota 0.0063 — 0.1585mgP/mgC | [16], [39]

uy | Bacterial respiration loss 0.1 — 2.5/day [15], [34]

fg | Grazing mortality rate of | 0.06 — 0.36/daw [85]

respectively, where Kp, Kc are half-saturation constants.

The exudation rate of OC by algae is the difference between the potential

growth rate attained when growth is not phosphorus-limited,

HA ~ [5 I(s, - Tay atsand actual growth rate,

This deviation actually assumes that algae always fix carbon at rate

and then have to get rid of excessive carbon As in Diehl et al (2005), we assumethat the algal phosphorus uptake rate is

_ Qu — Q P

AQ, P) = Pm tan M+P

At minimum cell quota, specific phosphorus uptake rate is just a saturation

func-tion of P At maximum cell quota, there is no uptake The algal cell quota

dilution rate is proportional to algal growth rate (Berger et al 2006).

All the assumptions above yield the following bacteria-algae interaction

system:

P consumption by algae P consumption by bacteria

P input and exchange

In the rest of this chapter, we assume that (with units and sources given

in Table 2): Iin = 300; k = 0.0004; Ky, = 0.3; H = 120; 2m = 30; Qm = 0.004;

Qu = 0.04; pm = 0.2; M = 1.5; nạ = 1s lạ = 0.1; v = 0.25; D = 0.02; Pin = 120;

Kp = 0.06; Kc = 100; pp = 3; 8 = 0.1; t„ = 0.2; uạ = 0.1; r = 0.5

2 Algae Dynamics

In order to have a comprehensive understanding of model (2.1), we study

first the algae dynamics without bacteria (B = 0):

( dA Qm\ 1 [7 I(s,A) v+D““= ` - ` = AU(A,Q), = =HAA ụ 5 ) — f To AC nd “TA = Avis.)

Qu, P(0) > 0 We analyze this system on the positively invariant set

9={(A,@,P) ER) |A>0,Q„ <Q < Qu, P > 0}.

Obviously the set where A = 0 is invariant for the system It is easy to see that

Qm <Q < Quy whenever Qm < Q(0) < Quy; that is, the cell quota stays within

the biologically confined interval.

Our first theorem states that all the variables are bounded and solutions

eventually enter a bounded region.

Theorem 1 The algae system (2.2) is dissipative and bounded

Proof Let S = AQ+ P, which is the total phosphorus of system (2.2)

dS — D

<

as 7 tần — 8).

There can be two types of steady state solutions for system (2.2): the algae

extinction steady state Ey = (0, QO, Pin), Where

_ Pm P

_ Qu -QmM+P

Q= >0 with 6P)

and positive steady state(s) E* = (4,Q,P) with 0(4,đ) =0

Standard computation shows that the basic reproductive number for algae

is the potential average light intensity in epilimnion without algal shading Indeed,

Rp is calculated from (0, Ô) so that Ry > 1 © W(0,Ô) > 0 ñ is the average

amount algae produced by one unit of existing algae (measured in carbon biomass)over the average algal life span in epilimnion It is an indicator of algal viability.Theorem 2 states that Ro is an indicator for the local stability of Ep

Theorem 2 When Ry < 1, Eo ts locally asymptotically stable and when Ro > 1,

Where A; and Az are negative numbers It is easy to see that the eigenvalues of

J(Ep) are (0, Ô),À¡ and Ag Ro < 1 implies W(0, Q) <0 Hence Ep is locally

asymptotically Ro > 1 implies ữ(0,Ô) > 0, which implies that Eo is unstable.

L]

We observe that increasing solar input or phosphorus input enhances algalviability, since

Olin ôh(0) Olin > 0 and an ~ 98(Pn) 0P, > 0.

Weakening water exchange enhances algal viability, since

Theorem 3 is our main mathematical result When Rp < 1, we establishthe global stability of Hy, which is equivalent to saying that algae will die out

It can be shown that there is no positive equilibrium #* when Roy < 1, in whichcase the existing results of general competitive systems can be applied to provethe global stability of Eg When Ro > 1, we prove algae is uniformly persistentand there is a unique positive steady state ETM.