Mathematical Modeling of Plankton Dynamics

Workshop ● Lorentz Center, Leiden NL ● December 8-12, 2014
Mathematical Modeling of Plankton Dynamics
Introductory Remarks
Horst Malchow
Institute of Environmental Systems Research
School of Mathematics & Computer Science
Osnabrück University, Germany
University of Toronto Press
1989
17th century
Anton van Leeuwenhoek
animalcules
First statistical models in epidemiology John Graunt 1662
18th century
Léonard Euler 1760
Thomas R. Malthus 1798
19th century
Benjamin Gompertz 1825
Pierre F. Verhulst 1838
Victor Hensen
plankton 1889
(Greek planktos = made to wander)
Models for fisheries
Early
20th century
Alfred J. Lotka 1910, 1925
Vito Volterra 1926
R. H. Fleming 1939
V. S. Ivlev 1945
Later
20th century
G. A. Riley 1946
H. T. Odum 1956
Later
20th century
Ilya Prigogine 1947
Thermodynamics of irreversible processes
1968 Dissipative self-organization
Alan M. Turing 1952
On the chemical basis of morphogenesis
C. S. Holling 1959
Some characteristics of simple types of predation and parasitism
M. L. Rosenzweig and R. H. MacArthur 1963
Graphical representation and stability conditions of predator-prey interactions
Hermann Haken 1971
Synergetik – Die Lehre vom Zusammenwirken (Theory of cooperativity)
Manfred Eigen 1971
Selforganization of matter and the evolution of biological macromolecules
Lee A. Segel & Julius L. Jackson 1972
Dissipative structure: an explanation and an ecological example
Alfred Gierer & Hans Meinhardt 1972
A theory of biological pattern formation
Daniel M. Dubois 1975
A model of patchiness for prey-predator plankton populations
J. S. Wroblewski & James J. O'Brien & Trevor Platt 1975
On the physical and biological scales of phytoplankton patchiness in the ocean
Simon A. Levin & Lee A. Segel 1976
Hypothesis for origin of planktonic patchiness
Later
20th century
Masayasu Mimura & James D. Murray 1978
On a diffusive prey-predator model which exhibits patchiness
Michael J. R. Fasham 1978
The statistical and mathematical analysis of plankton patchiness
Akira Okubo 1980
Diffusion and ecological problems: Mathematical models
John H. Steele & Eric W. Henderson 1981
A simple plankton model
Marten Scheffer 1991
Fish and nutrients interplay determines algal biomass: a minimal model
John H. Steele & Eric W. Henderson 1992
A simple plankton model for plankton patchiness
Mercedes Pascual 1993
Diffusion-induced chaos in a spatial predator-prey system
Horst Malchow 1993
Spatio-temporal pattern formation in nonlinear nonequilibrium plankton dynamics
J. E. Truscott & John Brindley 1994
Ocean plankton populations as excitable media
1992
Theoretical Ecology Group
Research Center Jülich, Germany
From Fasham, M. J. R. et al. 1990,
Journal of Marine Research 48: 591-639.
P - phytoplankton;
Z - zooplankton;
B - bacteria;
D - detritus;
Nn - nitrate;
Nr - ammonium;
Nd - dissolved organic nitrogen
Climate Change
Interactions
Regime Shifts
Movement
Growth
Excitability
(Bio-)Mathematics
(Rapid) Evolution
Biodiversity
Networking
Numerics
Taxis
Economy
Statistics
Iterated Maps
Patchiness
Biofilms
Cellular Automata
Coupled Map Lattices
Control
Dynamic Energy Budget Modeling
(Stochastic, Partial, Integro-) Differential Equations
Individual-based Modeling
Epidemics
Invasions
Sociology
Hybrid Modeling
Heterogeneity
Data Assimilation
Anomalous Diffusion
Hydrodynamics
(Marine) Bacteriology & Virology
Infections
Light
Computer Science
Particle Modeling
Noise
Plankton Blooms
Temperature
Early Warning Signals
Periodicity
Physiology
Spatio-temporal Pattern Formation
Stochastic reaction-diffusion-advection equations
r
n
r
r
r 
r
r
∂X i (r , t ) r  r
+ ∇ ⋅ vi X i (r , t ) − ∑ d ij ∇X j (r , t ) = f i [X (r , t ), p(r , t , ζ i )] + ξi
∂t
j =1


i = 1,2,..., n;
X = {X i ; i = 1, K , n} state variables
p = {p j ; j = 1, K , m} parameters
f = { f i ; i = 1, K , n}
r
r = ( x, y )
+ random localized rule − based even ts.
reaction functions
d ij = diffusivities,
position
r2
∂2
∂2
∇ =∆= 2 + 2
∂x
∂y
r
ξ i = ξ i (X, r , t )
r
ζ i = ζ i (r , t )
external spatiotemporal stochastic forcing
parametric stochastic forcing
r
vi
= velocities
Predation, top predation, diffusion, advection, and noise
(
)
(
)
n
n
r
r
r
∂X 1
a X1
+ ∇ ⋅ v1 − d1∇ X 1 = r X 1 (1 − X 1 ) −
X 2 + ξ1
n
n
∂t
1 + b X1
k
k
r
∂X 2 r r
a n X 1n
g
X
q
q
2
+ ∇ ⋅ v2 − d 2 ∇ X 2 =
X
−
m
X
−
f + ξ2
2
2
2
n
n
k
k
∂t
1+ b X1
1+ h X 2
Applications to plankton dynamics:
Steele & Henderson 1981-
n=1,2 q=1,2
k=2
f=0
Bistability, oscillations, internal noiseinduced patchiness
Bistability, oscillations, catastrophic shifts
Scheffer et al. 1991-
1
1
≥0
Brindley et al. 1994-
2
1
0
Excitability, external forcing, oscillations
Pascual et al. 1993-
1
1
0
Nutrient gradient, diffusion-induced chaos
Malchow et al. 1993-
1,2
1
≥0
External forcing, differential-flow and/or
noise-induced pattern formation, viruses
Medvinsky et al. 1997-
1,2
1
≥0
External forcing, rule-based fish,
population waves
Petrovskii et al. 1999-
1
1
0
Chaotic population waves, patchy invasion
Patchiness in Plankton Populations
Plankton populations are patchy on temporal and all spatial scales.
There are apparently many reasons for this, including
Growth, competition, grazing, disease, propagation
Reaction, Diffusion
Lag between build up of phytoplankton and zooplankton Delay
Patchiness of nutrients for algae
Heterogeneity
Seasonal effects
Forcing
Effects of water movement
Flow
Demographic and environmental variability
Noise
Which mechanisms can induce spatial
or spatio-temporal pattern formation?
• Diffusive (Turing) instabilities
• Differential-flow-induced instabilities (DIFII)
• Local excitability and propagation of pulses
• Time delays
• Environmental variability, incl. noise, physics and chemistry
•…
Turing structure in the Scheffer model: The 5-holes pattern
No-flux boundaries
Biologists: “Turing structures are too symmetric!”
Physicists: “Ok, let’s apply some noise … .”
What physicists view as noise is music to the ecologist (Simberloff, 1980).
ω=0.00
ω=0.01
ω=0.05
ω=0.10
ω=0.20
H.M. & F.M. Hilker & S.V. Petrovskii: Discrete and Continuous Dynamical Systems B (2004)
Instabilities by Shear Diffusion
(Evans, Kullenberg & Steele 1976, Evans 1977, 1980)
X1i
di
v
r
X2i
Dynamics:
r r
∂X 1i
= f i (X1 ) − v ⋅ ∇X 1i + d i ( X 2i − X 1i )
∂t
i=1,2,…,N.
∂X 2i
= f i (X 2 ) − d i ( X 2i − X 1i )
∂t
Instabilities of the homogeneous distribution for sufficiently different di
Time
Sherratt et al. (1995), S.V.P. & H.M. (2001). Chaos in the wake of invasion:
Irregular behaviour behind diffusive prey-predator fronts.
Space
Medvinsky et al. (2002). Pattern formation in oscillating systems with
uniform parameters and perturbed initial conditions.
Biological turbulence
Pascual (1993).
Diffusion-induced chaos
along a nutrient gradient.
Simplified Rules of Fish School Motion
Reflecting boundary
Periodic boundary
Memory and preference of previous direction
Random choice of the new direction within a
certain angle of “vision” after feeding on zooplankton down to its protective threshold
density or after some maximum residence time
Reflecting boundary
H.M. et al.: Nonlinear Analysis RWA (2000)
Nature 437, 356-361 (2005)
Hewson, I., Chow, C. and Fuhrman, J. A. 2010.
Ecological Role of Viruses in Aquatic Ecosystems.
Encyclopedia of Life Sciences.
Published Online: 15 DEC 2010
DOI: 10.1002/9780470015902.a0022546
Figure 1. Conceptualisation of the microbial loop incorporating
viruses. Viruses cause mortality of all trophic levels, regenerating
dissolved organic matter from phytoplankton, bacterioplankton
and grazers.
Figure 2. SYBR Green I stained plankton from Otisco
Lake, upstate New York, prepared following the
protocols of Noble and Fuhrman (1998). The larger
green dots are Bacteria and Archaea, whereas the
smaller dots are viruses.
Fluorescence imaging
of marine viruses
© J. Fuhrman, 1999
Viral replication cycles in host cells:
Lytic cycle without replication of the host cell:
Virus takes over operation of host cell immediately
upon entering it and then destroys it.
Lysogenic cycle with replication of the host cell:
Viruses integrate their genome into the hosts genome.
As the host reproduces and duplicates its genome, the
viral genome reproduces too. Certain stress might activate the virus to enter the lytic cycle.
Model structure
S
Noise
I
P
Diffusion
Z
F
N
j=0: Mass Action Transmission
j=1: Frequency-dependent Transmission
Susceptible Phytoplankton X1
n=2: Excitability
Infected Phytoplankton X2
Zooplankton X3
q=2: Intraspecific X3 Competition
b=0, n=1: Lotka-Volterra Dynamics
m=1,2: Harmful Phytoplankton
k=1,2: Top (Fish) Predation
∂X 1
an X1(X1 + X 2 )
X1 X 2
= r1 X 1 (1 − X 1 − X 2 ) −
X3 − λ
+ d∆X 1 + ξ1
n
j
n
∂t
(X1 + X 2 )
1+ b (X1 + X 2 )
n −1
∂X 2
an X 2 (X1 + X 2 )
X1 X 2
= r2 X 2 (1 − X 1 − X 2 ) −
X 3 − m2 X 2 + λ
+ d∆X 2 + ξ 2
n
j
n
∂t
(X1 + X 2 )
1+ b (X1 + X 2 )
n −1
∂X 3
g k X 3k
an (X1 + X 2 )
p m (X1 + X 2 )
q
q
=
X 3 − m3 X 3 −
X3 −
f + d∆X 3 + ξ 3
n
m
k
k
n
m
∂t
1+ h X3
1 + b (X1 + X 2 )
1 + c (X1 + X 2 )
n
m
Here: j=n=q=1 ; p=g=0
Switch from lysogenic to lytic replication cycle
•
No further replication of infected phytoplankton
•
Effective mortality of infected = natural mortality + virulence
•
Nonsymmetric competition : impact of infected on susceptibles but not vice versa
•
Increase of transmission rate
F.M. Hilker & H.M. & M. Langlais & S.V. Petrovskii: Ecological Complexity (2006)
Deterministic switch from lysogenic to lytic replication cycle
r1 ≥ r2 > 0
r1 > 0 ≥ r2
r2 = 0.4
r2 = 0.0
m2 = 0.2
m2 = 0.3
λ = 0.6
λ = 0.9
ω = 0.1
Switch
ω = 0.25
{
(
E X S (i ) = P4S , i, X 34S P4S , i
34
)}
X 10 = 0.10948, X 20 = 0.22385, X 30 = 0.14066
r1 = 1, a = b = 5, m3 = 0.625
Predators eventually go extinct !
Noise-induced switch from lysogenic to lytic replication cycle
Introduction of a local auxiliary quantity r3 with noisy bistable kinetics:
r
r
dr3 (r , t )
∗
(
)
(
)
(
= r3 − rmin r3 − r rmax − r3 + ωr3ξ r , t )
dt
(
)
If r3 > r ∗ then r2 = r2max
If r3 ≤ r ∗ then r2 = r2min
Noise effects
• Noise will blur distinct patterns in space and time. It can
induce transitions between and formation of temporal,
spatial, and spatio-temporal structures.
• It can generate new structures like local oscillations, global
synchronization or stationary patterns.
M. Sieber & H.M. & L. Schimansky-Geier: Ecological Complexity (2007)
• Noise can suppress periodic travelling waves and prevent
the formation of chaotic waves.
S.V. Petrovskii & A.Yu. Morozov & H.M. & M. Sieber: European Physical Journal B (2010)
M. Sieber & H.M.: European Physical Journal Special Topics (2010)
M. Sieber & H.M. & S.V. Petrovskii: Proceedings of the Royal Society A (2010)
Thanks for Cooperation
Alex James & Richard Brown, Christchurch
Michel Langlais & Jean-Baptiste Burie, Bordeaux
Diomar C. Mistro & Luiz A. Rodrigues, Santa Maria RS
Sergei V. Petrovskii & Andrei Yu. Morozov, Leicester
Jean-Christophe Poggiale, Marseille
Lutz Schimansky-Geier, Berlin
Hiromi Seno, Sendai
Nanako Shigesada, Kyoto
Michael Sieber, Potsdam
Michael Bengfort
Ivo Siekmann, Melbourne
Frank M. Hilker
Ezio Venturino, Torino
Funding
Institute of Environmental Systems Research
Osnabrück University
DAAD JSPS DFG EU
FAPERGS / CAPES Erskine UC