If you cant beat them, eat them'

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If you cant beat them, eat them.

• Mathematical models of mixotrophy
• Jon Pitchford and Astrid Hammer
• Departments of Maths and Biology
• University of York.

Dinoflagellate Prorocentrum minimum
Chrysophyte Dynobrion
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Typical annual plankton population dynamics
Phytoplankton
Population µg N/l
Zooplankton
0
50
100
150
200
250
300
350
Truscott and Brindley 1994, Bull. Math. Biol. 56
Time (days)
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The plan

• Mixotrophy and cannibalism
• General framework
• Excitable medium models equilibria and blooms
• A Baltic estuary ugly facts getting in the way
  of elegant maths

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1. Mixotrophy in the real world Ashes 2005
Bowlers who can bat a bit. N.B. No batsman took
wickets
A genuine all-rounder. Rare. Expensive.
Inconsistent.
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Mixotrophy in the real world (Stoecker 1998)
TYPE I
"Ideal Mixotrophy"
Balanced phototrophy and phagotrophy
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Secondary Endosymbiosis
Cryptophyte
2 µm
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Cannibalism in the real world Hands up!
Real cannibalism we all do it Intratrophic
predation we cant ignore it
GRASS
SHEEP
PLANTS
MAMMALS
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What are we trying to do? To understand how to
incorporate mixotrophy into models To quantify
its consequences To consider DYNAMICS, not
STATICS How? Simple maths (2-D nonlinear
dynamical systems) Question the dogma of
predator-prey and Lotka-Volterra
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2. Intratrophic predation a general model
Pitchford and Brindley, Bull. Math. Biol. 1998
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3. Plankton populations as excitable media Wake
et al. (1990?), Truscott and Brindley (1994)
Ingredients Nonlinearity (x2/1x2)
P fast, Z slow (g ltlt 1)
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Type III Mixotrophs
Photosynthesising zooplankton/protozoa typically
photosynthesise when prey is limiting.
Euglenoid (hired plastids)
Poterioochromonas (own plastids)
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For our excitable system
with null clines
O(e) change
O(e/g) change
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Changes in null cline structure Stability is
increased. Or is it?
dZ/dt nonmixo 0
dP/dt 0
Zooplankton µg N/l
.
PM P eB B ..-1/? (big!)
0
20
40
60
80
Phytoplankton µg N/l
Hammer Pitchford (2005) ICES J. Mar. Sc.
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What triggers blooms in an excitable system?
How do you cause a Hopf bifurcation? Do you
flush a toilet in the same way? Spring bloom
iff i) (1/r)(dr/dt) gt A and ii) r(t1)
r(t2) gt B Singular limit analysis g 0,
asymptotics. J. Truscott, J. Plank. Res. (1995)
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Same equilibrium, same forcing
No mixotrophy
1 mixotrophy
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Response of Systems to forcing ( increase of
growth rate at equilibrium)
100
80
nonmixo
60
Response, Pmax
40
20
0
1
2
3
4
5
6
dr/dt
-3
x 10
Hammer Pitchford (2005) ICES J. Mar. Sc.
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Type II mixotrophy in an excitable system?
b is an encounter rate (mixotrophic P eating
P). We can calculate this (to order of
magnitude). RESULT Type II Mixotrophy is
negligible in marine systems
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Conclusions Mixotrophy in excitable systems

• Type III mixotrophy
• Small mixotrophy O(e) can have large effects
  O(e/g)
• Stabilising effect i) moves equilibrium away from
  bifurcation
• Stabilising effect ii) even with same
  equilibrium, reduces susceptibility to blooms
• Type II mixotrophy
• Negligible effects on equilibria and dynamics

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Meanwhile. Strange things are happening in
Germany Darss Zingst Bodden Chain (muddy
estuaries) Hammer and Pitchford, Mar. Ecol.
Prog. Ser., December 2006
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Type II Mixotrophs - Occurrence
Acidic mining lakes
Eutrophic lakes, Estuaries
Prorocentrum minimum brown harmful blooms
Antarctic lakes
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4. Seasonal dynamics in the DZBC

• Summer blooms of algae (as usual)
• Excess of nutrients, all year round
• High population of bacteria, all year round
• Well-mixed water column
• All standard stuff for estuaries, but
• Winter blooms of Type II mixotrophs
  (cryptophytes) associated with periods of ice
  cover

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So lets try to..

• demonstrate the succession of cryptophyte blooms
  and coexistence of species in phytoplankton
  communities
• explore the plausibility of the mixotrophy
  hypothesis as an explanation of winter bloom
  forming
• put sensible constraints on the possible
  mechanisms governing phytoplankton coexistence
  and dynamics
• We include data from detailed laboratory and
  field studies.

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Can mixotrophy explain coexistence? Does
mixotrophy explain bloom dynamics? Simple
questions, detailed data. Consider algae A and
cryptophytes C growing in competition for light.
p(T,L) is the cryptophyte preference for
photosynthesis a function of temperature T and
light L.
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Table 1. Biological variables and parameters used
in numerical simulation
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Experiment 1 Can mixotrophy explain
coexistence? No! Were still in the world of
Lotka-Volterra competitive exclusion.
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Experiment 2 Can environmental seasonality
and/or stochasticity explain coexistence? In
theory? Yes. (Tilman 1977, Levins 1979, Namba
1984, Namba and Takahashi 1993) Within bounds
of real data? No chance.
Experiment 3 Can asynchrony explain
coexistence? e.g. Algae adapt more slowly to
increased light levels In theory? Yes.
(Wangersky and Cunningham 1957, Ruan 1995, Zhang
and Fan 2004) Within bounds of real data? No
chance.

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So were in bother. Allelopathy (confirmed 2004,
in Southern Baltic) the release of non-toxic
extracellular compounds that inhibit the growth
of other microorganisms Cryptophytes are
particularly susceptible.
Modify model p p(T,L) p (A) where
so that cryptophytes dont photosynthesise at
high algal densities (there is allegedly a
biological basis for this).
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Experiment 4 Allelopathy
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Summer coexistence
Winter C dominance
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So we can have (static) coexistence. How about
seasonal dynamics, species succession, and ice
winters? Numerical simulations daily light and
temperature data from DZBC over 3 years,
speculative allelopathy formulation, other
parameters from Table 1.
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Cryptophytes data
Cryptophytes model
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Algae data
Algae model
Details of life history and/or nutrient could
account for the discrepancies, as could a more
careful formulation of allelopathy.
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• Modelling mixotrophy Conclusions
• Question two dogmas of theoretical ecology
• Predator prey
• Competitive exclusion
• Modified models to account for biological
  reality
• O(e) perturbations can have amplified affects
• Or not

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• The future
• 1. Individual-based evolutionary models (with
  Calvin Dytham, York) how should I allocate my
  resources in response to
• My environment (static/changing)?
• My competitors (evolving)?
• 2. Immortality! (Max Planck Institute for
  Demography, Rostock, Germany)