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Chemostat Theory with Several Nutrients Today’s starting point: dN i i ( R1 , R2 ,..., Rm ) N i DN i dt dR D S j R j i N i qi dt i i 1,..., n j 1,..., m i i 0, 0 i, j; R j strict inequality for at least one j for every i Mass balance constraints = conservation principle: T j R j Ni qij i dNi qij D S j R j i Ni qij i Ni qij DNi qij dt dt dt i i i DS j DT j dT j dR j As t → ∞, Tj → Sj, and asymptotically S j R j qij Ni i Alfred Lotka first noted the significance of such multiple constraints. For two resources and two competitors, a 2d graphical approach is sufficient to represent dynamics (Tilman, others in the 1970’s). Nutritionally Substitutable Resources Zi Nutrient 1 Concentration, R1 Nutrient 2 Concentration, R2 Nutrient 2 Concentration, R2 Nutritionally Essential (Complementary) Resources Zi Nutrient 1 Concentration, R1 Zero-Net-Growth-Isoclines display nutrient concentrations where a single population is at steady state (dNi / dt = 0). Nutrient 2 Concentration, R2 Z1 Z2 Nutrient 1 Concentration, R1 Superimposing Zero-Net-Growth-Isoclines for two species determines whether steady state coexistence is possible. There must be a tradeoff between competitive abilities for the two resources. Nutrient 2 Concentration, R2 M2 (S1,S2) M1 Z1 I2 I1 Z2 Nutrient 1 Concentration, R1 Adding mass conservation constraints determines feasibility and stability of coexistence at equilibrium, given nutrient supplies. Experimental Test: Competition between two diatoms, Synedra filiformis (species 1) and Asterionella formosa (species 2), for phosphorus (resource 1) and silicon (resource 2) in laboratory cultures (drawn from the data of Tilman, 1981). Z1 10 Z2 (S1,S2) 40 6 4 Silicon (M) Silicon (M) 8 I2 I1 M2 20 M1 10 2 0 0.000 30 Z1 0.002 0.004 0.006 Phosphorus (M) 0.008 0 0.0 Z2 0.1 0.2 0.3 0.4 0.5 0.6 Phosphorus (M) Resource Ratio Hypothesis: Outcome depends on supply ratio of the resources. Synedra filiformis (sp. 1) wins when Si:P supply ratio is high; Asterionella formosa (sp. 2) wins when Si:P supply ratio is low. They coexist for intermediate ratios. Variable-internal-stores model of competition for several resources: dNi i (Qi1 , Qi 2 ,..., Qim ) Ni DNi dt dQij ij ( R j , Qij ) i Qij dt dR j D S j R j Ni ij dt i For analysis of equilibria, set dQij / dt = 0 and obtain Q*ij. Then derive an equivalent steady state model where growth depends on Rj. For two species and two resources the same graphical approach then applies. Parameterized models for algae suggest that coexistence can turn to bistability at high dilution rates. Commensalism – Graphical Theory Nutrient 2 Concentration, R2 Z1 M1 (S1,S2) I1 Z2 M2 I2 Nutrient 1 Concentration, R1 Two species compete for one nutrient resource. The inferior competitor releases another resource that is essential to the superior competitor. Commensalism and Planktonic Ecosystems Organic Carbon Algae Onshore / inland Bacteria Offshore Inorganic Nutrient Vertical Distribution of Light in a Well-Mixed Water Column Two competing species (j = 1,2) absorb light in proportion to density Nj and water and dissolved substances absorb light: j k j N j z Irradiance (I) Iin Depth (z) I ( z ) I in e kbg zm Iout Competition for light, following Huisman et al… Light-dependent growth rate at depth z: j ( I ( z )) D max I ( z) j KI , j I ( z) D Depth-averaged growth rate: dN j 1 dt zm zm 0 max I ( z) j D N j dz KI , j I ( z) Only I(z) depends on z, so: K I , j I in ln K I dt I , j out kbg k j N j zm j dN j max j N j DN j Competition for light follows something like the R*-rule. Consider each species growing alone, and let it go to equilibrium to get I*out,j The species with lowest I*out,j wins competition, others go extinct. Competition for light and a nutrient is similar to competition for two essential nutrients. Relationship between supply and outcome is a little more complicated… For intermediate supply ratios of light-to-nutrient, bistability is experimentally more likely than coexistence (for two nutrients, coexistence is common). Competition for more than two resources (work is almost all numerical): Competition for one or two resources in a chemostat always goes to equilibrium. Moreover, no. persisting species is always ≤ no. resources. For three resources oscillations can occur among three species, if each species is the best competitor for one resource and the worst for another, and if each species consumes the most of the resource for which it is the intermediate competitor (Huisman & Weissing 2001): R11* R12* R13* q12 q13 q11 * 21 R R R q23 q21 q22 * * * R33 R31 R32 q31 q32 q33 * 22 * 23 Oscillations can be limit cycles or heteroclinic cycles: Huisman & Weissing, 2001 Oscillations can then allow invasion and persistence of more species than resources (“supersaturated coexistence”): Huisman & Weissing, 2001 With 5 species and 5 resources, the oscillations can be chaotic (“competitive chaos”): Huisman & Weissing, 1999 How important are “supersaturated coexistence” and “competitive chaos” in the natural world? High amplitude cycles → close approach to extinction (even when uniformly persistent). Heteroclinic cycles are not uniformly persistent (close to extinction for long periods of time). “Supersaturated coexistence” and “competitive chaos” are highly parameter dependent (ordering relations and values). Schippers et al. (2001): Randomly parameterized species usually not supersaturated: Also, adding a randomly parameterized species to a supersaturated assemblage often reduces no. coexisting species to ≤ no. resources. Huisman et al. (2001): Supersaturated coexistence more likely when certain tradeoffs restrict randomized parameterization of species: Scenario 1 – completely random. Scenario 5 – ordering relations for R* and q from original work. Scenario 3 – ordering relation where low R* ↔ low q. Huisman et al. argued that scenario 3 was theoretically plausible and had some empirical support. Roelke & Eldridge (2008): Supersaturated coexistence is less likely in spatially variable systems (gradostat-like), and is altered by periodic forcing: Multiple nutrients and microbial interactions other than competition, e.g. predation (Grover 2003) Z C zooflagellate B bacteria N P dissolved nutrients recycling fluxes omitted bacteria that have been ingested Photo by T.H. Chrzanowski Governing population dynamic equations: dB B B DB mZ aBZ dt dZ Z Z DZ mZ Z dt B Z B, Z m B, m Z D a = bacteria density = zooflagellate density = growth functions = mortality rates (constant) = dilution rate (constant) = attack rate (constant) Growth functions combine Droop’s Equation with Liebig’s Law of the Minimum: min Q j ,i max i i 1 max j Q j ,i i = B, Z j = C, N, P Bacterial quota dynamics (from Thingstad 1987): dQ j , B dt V j ,B V j ,B BQ j ,B R j ,B uptake "use in growth" for j C , N , P release [ j ] Q max j ,i Q j ,i V K [ j ] Q max Q min j ,B j ,i j ,i maximal max j ,B uptake R j ,B Michaelis-Menten saturation term for j C , N , P decrease with quota 0 for j N , P g B QC , B m QC , B QCmin ,B growth term maintenance term Note: [j] denotes dissolved concentration of element j (=C,N,P). Note: Release of carbon (RC,B) is respiration and disappears from the system. There is mass conservation for N and P, but not C. Zooflagellate quota dynamics : dQ j ,Z dt aBQ j , B Z Q j ,Z R j ,Z ingestion R j ,Z "use in growth" for j C , N , P release Q max j ,Z Q j ,Z aBQ j , B ae j BQ j , B max Q Q min j ,Z j ,Z ingestion assimilation, decreases with quota Q max j ,Z Q j ,Z aBQ j , B 1 e j max Q Q min j ,Z j ,Z Note: For j =N,P assimilation efficiency ej = 1. Note: Release of carbon (RC,Z) is respiration and disappears from the system. There is mass conservation for N and P, but not C. Dissolved nutrient dynamics : dC D Cin C BVC , B mZ ZQC ,Z mB BQC , B dt dN D N in N BVN , B ZRN ,Z mZ ZQN ,Z mB BQN , B dt dP D Pin P BVP , B ZRP ,Z mZ ZQP ,Z mB BQP , B dt Note: For carbon (j = C), only mortality is recycled. Note: For nitrogen and phosphorus (j = N, P), mortality is recycled and so is excess nutrient that is ingested but not assimilated. Topics to investigate with multi-nutrient microbial predator-prey model: 1. Paradox of enrichment 2. Relation of carbon decomposition to supply of other nutrients 1a. Paradox of enrichment is much like one-nutrient models when ratios of nutrient supplies are “balanced” for prey and predator: 1b. Paradox is modified or eliminated when enrichment of one nutrient occurs at “unbalanced” ratios: 1c. Limit cycles in the paradox of enrichment arise only with sufficient balance of nutrients: 2. Decomposition of dissolved organic carbon is reduced by limit cycles and enhanced by balanced nutrients: Adding a competitor (Grover 2004): Z A C zooflagellate B N bacteria, two species P dissolved nutrients recycling fluxes omitted Paramaterized so that species A is a better competitor for C and N, species B is a better competitor for P. Relation of asymptotic outcomes to nutrient supplies when species B is preferred prey: Multiple attractors (1): coexistence in a limit cycle versus coexistence at equilibrium Multiple attractors (2): exclusion in a limit cycle versus coexistence at an equilibrium Relation of asymptotic outcomes to relative preference of predator for prey: < 1 means species B is preferred; > 1 means species A is preferred. Modeling the dynamics of mixotrophs (Ph.D. student, Ken Crane): Mixotroph – a microorganism that can grow phototrophically or heterotrophically. Phototrophic growth – uses photosynthesis to produce complex organic matter from simple constituents while absorbing light. Heterotrophic growth – consumes complex organic matter (“food”), either in dissolved form or in the form of prey or particulate matter. Mixotrophs that prey on bacteria are very common in oceans and lakes, and there is a great variety of species. Flagellates that prey on bacteria to obtain nutrients and that also use photosynthesis to make complex organic matter often dominate total microbial biomass in large parts of the ocean in many lakes, especially those with low nutrient supply. chloroplasts for photosynthesis An organism we study in the lab is an example (Ochromonas danica). Without light it grows purely as a predator of bacteria. bacteria that have been ingested Photo by T.H. Chrzanowski Mixotrophs typically coexist with microorganisms pursuing other nutritional strategies A = “algae” that use only photosynthesis (“autotrophs” or “phototrophs”) Z = “zooflagellates” that can only eat bacteria B = bacteria that are prey both mixotrophs and zooflagellates There is a complex web of resource-mediated interactions I = light, used by algae and mixotrophs C = dissolved organic carbon, produced by algae and mixotrophs, consumed by bacteria P = inorganic phosphorus, consumed by algae, mixotrophs, bacteria There is also nutrient recycling from various processes. A M Z B I C P Interesting questions: Under what circumstances do species of these different nutritional strategies coexist? Under what circumstances to mixotrophs persist and become abundant? What kind of mixotrophs do best? There is a spectrum, from mixotrophs that are almost purely autotrophic to those that are almost purely heterotrophic. No published mathematical models explicitly address all the resources involved. A M Z B I C P Formulation of bacterial dynamics: dB ( B D mB aBZ Z aBM M ) B dt dQPB VPB B QPB dt B min PB , CB PB min Q max PB B 1 Q PB VPB max Q P max PB QPB VPB max min k P Q Q PB PB PB CB max B C k C BC Note: Bacterial quota for carbon is constant, and carbon-dependent growth depends directly on the dissolved organic carbon concentration (C). Formulation of algal dynamics: dA ( A D mA ) A dt dQPA VPA A QPA dt A min PA , CA PA min Q max PA A 1 Q PA VPA max Q P max PA QPA VPA max min k P Q Q PA PA PA CA Amax kCA I in ln k I CA out I out I in e ( kbg k B BQCB k A AQCA Note: Algal quota for carbon is constant, and carbon-dependent growth is equivalent to light-dependent growth. A well-mixed water column of depth z is assumed. k M M QCM k Z ZQCZ ) z Formulation of zooflagellate dynamics: dZ ( Z D mZ ) Z dt max dQPZ QPZ QPZ aBZ BQPB max Z QPZ min dt QPZ QPZ Z min PZ , CZ PZ max Z min QPZ 1 Q PZ CZ aBZ eZ B QCB QCZ Note: Zooflagellate quota for carbon is constant. Formulation of mixotroph dynamics combines characteristics of algae and zooflagellates: dQPM dt dM ( M D mM ) M dt max QPM QPM VPM (1 ) aBM BQPB max M QPM min QPM QPM M min PM , CM PM min Q max PM M 1 Q PM VPM V max PM CM kCM I in ln kCM I out max P QPM QPM max min k PM P QPM QPM Mmax QCB (1 ) a e B BM M QCM I out I in e ( kbg k B BQCB k A AQCA Note: Mixotrophy parameter ranges 0 to 1 and defines the mixture of predation and photosynthesis. k M M QCM k Z ZQCZ ) z Formulation of dissolved organic carbon dynamics: dC D (Cin C ) B BQCB eA max(0, CA PA ) AQCA eM max(0, CM PM ) M QCM dt Term 1 Term 2 Term 3 Term 4 mB BQCB mM M QCM mA AQCA mZ Z QCZ Terms 58 Term 1 – Chemostat supply Term 2 – Excretion of organic carbon when algae are nutrient limited Term 3 – Excretion of organic carbon when mixotrophs are nutrient limited Terms 5-8 – Recycling of organic carbon from dead organisms Carbon fluxes from organisms are regarded as respiration and are dissipated from the system. There is no mass conservation constraint for carbon. Formulation of dissolved phosphorus dynamics: dP D ( Pin P ) BVPB AVPA M VPM dt Terms 2 4 Term 1 mB B QPB mM M QPM mA AQPA mZ Z QPZ Terms 58 (1 ) aBM min min QPM QPM QPZ QPZ B M QPB max a B Z Q BZ PB min max min Q Q Q Q PM PM PZ PZ Term 9 Term 10 Term 1 – Chemostat supply Terms 2-4 – Uptake Terms 5-8 – Recycling of phosphorus from dead organisms Term 9 – Recycling of phosphorus ingested by mixotrophs but not assimilated Term 9 – Recycling of phosphorus ingested by algae but not assimilated All fluxes of phosphorus from organisms appear as dissolved phosphorus. There is a mass conservation constraint for phosphorus. Numerical analysis strategy Examine bifurcations as mixotrophy parameter varies from 0 to 1. Fix all other biological parameters, which are assigned so that generalist mixotrophs are poorer competitors than specialist algae and zooflagellates, for the specialists’ resources. Mixotrophs are poorer competitors for P and light than algae, and poorer competitors for bacterial prey than zooflagellates. Examine various combinations of resource supply parameters (Cin, Pin, Iin, z, kbg). System is very stiff -- use Matlab ode23s solver (modified Rosenbrock triple, one step implicit method with new Jacobian every time step). Four groups of coexistence sequences for 0 < < 1: 1. 2. 3. 4. Persisting species Resource supply conditions BAZ – BAM – BAMZ – BAZ High P supply BAZ – BAM – BM – BMZ – BAZ Low P supply (Pin) and Low C supply (Cin) BAZ – BM – BAM – BM – BMZ - BAZ Low P supply (Pin) and High C supply (Cin) Various others Dark (low Iin with high kbg and high z) Mixotrophs persist under most resource supply conditions unless they are too specialized ( near 0 or 1). Mixotrophs often outcompete either algae or zooflagellates when P supply is low. All four nutritional types typically coexist when P supply is high and for mixotrophs is intermediate. The most abundant mixotrophs are usually those with relatively high (mostly photosynthetic, getting supplemental nutrition from prey). Microbial ecologists think this is the most common and abundant mixotrophic strategy. “Eutrophic” Lake – High P Supply A. Bacteria B. Algae 1012 1013 1012 1011 1010 109 108 107 106 105 104 103 102 3 Density (cells / m ) 3 Density (cells / m ) 1014 1011 1010 109 108 107 0.0 0.2 0.4 0.6 0.8 1.0 0.0 C. Mixotrophs 0.4 0.6 0.8 1.0 0.6 0.8 1.0 D. Zooflagellates 1011 3 3 Density (cells / m ) 1011 Density (cells / m ) 0.2 1010 109 108 1010 109 108 107 106 105 0.0 0.2 0.4 0.6 Parameter 0.8 1.0 0.0 0.2 0.4 Parameter P-limited growth C-limited growth Alternation during limit cycle “Oligotrophic” Lake – Low P Supply B. Algae 1012 3 Density (cells / m ) 3 Density (cells / m ) A. Bacteria 1014 1013 1012 1011 1010 109 108 107 106 105 104 103 102 1011 1010 109 108 107 0.0 0.2 0.4 0.6 0.8 1.0 0.0 C. Mixotrophs 0.4 0.6 0.8 1.0 0.6 0.8 1.0 D. Zooflagellates 1011 3 3 Density (cells / m ) 1011 Density (cells / m ) 0.2 1010 109 108 1010 109 108 107 106 105 0.0 0.2 0.4 0.6 0.8 Parameter 1.0 0.0 0.2 0.4 Parameter P-limited growth C-limited growth Alternation during limit cycle Conclusions: 1. Multiple resource models needed for realism in many applications. 2. Complex dynamics can arise (multiple attractors, chaos, heteroclinic cycles, supersaturated coexistence, long transients). 3. Practical implications of complex dynamics unclear. With realistic parameters dynamics are often simpler (equilibria and limit cycles). 4. Classical ecological principles often emerge (resource partitioning, predator-mediated coexistence). 5. Challenges include incorporating still more complexity (numbers of species & resources, food web structure), spatially variable habitats, numerical analysis problems.