Nikolay N. Zavalishin Dynamics of the carbon cycle functioning in - - PowerPoint PPT Presentation

SLIDE 1

Nikolay N. Zavalishin

Russian Academy of Sciences A.M. Obukhov Institute of Atmospheric Physics Laboratory of Mathematical Ecology

Supported by the program “Hydrospheric and atmospheric processes: forming, changing and regulating the Earth Climate” of the Russian Academy of Sciences

Dynamics of the carbon cycle functioning in Russian peatlands under the climate change and human perturbations

SLIDE 2

fki ( xk , xi )

x k ( t ) x i ( t ) x 1 ( t )

yk ( xk ) qk ( t ) qi ( t )

yi ( xi ) q1( t )

y1( x1 )

fik (xi , xk )

xk=3.5

xi=70 x1= 180

yk = 0.8 qk = 0.5

f k i = 15

q i = 8

yi = 3

q1 = 30 y1 = 10

f k i = 8.5

?

General problem of a dynamic model design by a given «storage-flow» diagram

a set of compartment schemes for time moments t0, t1,…, collected from the field studies dynamic model for storages in reservoires

Dynamic equations in general form:

∑ − + − =

≠ = n i k k ik ki i i i

f f y q dt dx

, 1

) (

f(x) y(x) q(x) x + − = dt d

q - vector of input flows from the environment; y - vector of output flows to the environment; fki - intercompartment flow from i to j

The main problem: how to make dynamic model from one flow-balanced diagram?

SLIDE 3

5) intercompartment flow control types: a) fki = donor; b) fki = recipient; c) fki = Lotka-Volterra

j i ij

x x γ

i ijx

α

j ij x

β

Modified method for dynamic system design

Main assumptions (1-5c) for given stationary schemes:

;

* *

x y m

i i i =

;

* *

x f

i ki ki =

β

;

* *

x f

k ki ki =

α

;

* * *

x x f

k i ki ki =

γ

Coefficients of flow functions are calculated from the given scheme: cis

. γ γ

is si −

= =

b

i s

. , ; ), ( s i s i m

is si ik i k ki i

≠ − = − ∑ + −

≠

β α α β

1) q* + f* = y* - at least one of the given diagrams is a dynamic equilibrium; 2) fki = fki (xk, xi); fik = fik(xi, xk) – intercompartment flows depend only on participating storages; 3) qi = const – input flows can have only constant form; 4) yi = mixi – output flows are linear; 5d) additional control types with saturation:

,

i ki i k ki ki

x L x x K g + =

,

k ki i k ki ki

x L x x K g + = ) )( (

i ki k ki i k ki ki

x N x L x x K g + + =

Coefficients of flow functions with saturation are calculated from several given schemes or by special calibration procedures

Dynamic compartment model:

∑

= = =

− + ∑ + ∑ + − =

n s is si n s s is i n s s i s i i i i

g g x c x x b x m q dt dx

1 1 1

) (

) ( ) ,..., ( /

1

x x x q x G C x x diag B dt d

n

+ + + =

i,s = 1,…n

SLIDE 4

x*

1=8490.3

x*

3=35.0

x*

2=1.246

x*

4=8835.6

y*

1=612.1

f*

12=38.05

f14

*=337.4

q3=0.0 y3

*=279.9

f43

*=584.9

q4=35.73 y4

*=114.73

5 f24

*=58.205

f*

42=36.9

y2

*=16.745

q2=0.0 f34

*=305

q1=987.55

Storages (in g/m2 of dry weight): x1 - plant biomass; x2 - animal biomass; x3 - fungi and bacteria biomass; x4 - dead trees, dead roots and litter

Open compartment model: typical carbon cycle in the bog ecosystem

Input and output flows (in g/m2·year of dry weight): q1 – carbon assimilation by plants, q4 – input from neighboring ecosystems, y1, y2 , y3 - respiration, y4 – surface runoff + peat formation + abiotic oxidation

Data from (Alexandrov et al., 1994): Tajozhny Log mesotrophic bog, Novgorod Region, Russia

Intercompartment flows (in g/m2·year of dry weight): f12 – consumption by animals, f14 – litterfall from plants, f42, f43 – consumption of dead

• rganics by animals, fungi and bacteria, f24,

f34 – death of animals, fungi and bacteria The main purpose: to investigate stability and bifurcations of steady states as a reaction of the carbon cycle to climatic and human perturbations by means of parametric system portrait analysis. Flow control selection: f14=α14x1 f24=α24x2 f34=α34x3 f12=γ12x1 x2 f43=γ43x3 x4 f42=γ42x2x4

SLIDE 5

Local bifurcation curves for the carbon cycle steady states

Equilibrium 1:

1 12 24 2 14 1 14 1 4 1 14 42 4

) )( ( )) ( ( q m m m q q m γ α α α α γ − + + + + = ) (

4 14 1 14 1 34 3 43 4

q m q m m + + + = α α α γ

12 14 1 43 34 3 42 24 2 1

) ( γ α γ α γ α + + − + = m m m q

34 43 4 1 14 1 14 4

) ( α γ α α q q m m + + = ) ( ) ( ) ( )) ( ( ) (

34 3 12 43 14 2 1 24 43 34 3 42 1 34 3 4 43 1 34 3 1 34 3 42 43 24 43 1 4

α γ γ α α γ α γ α γ α α γ γ α γ + + − + + + + + + − = m m m m m m q q m K m q m Equilibrium 2:

) (

4 14 1 14 1 34 3 43 4

q m q m m + + + = α α α γ

Equilibrium 3: Equilibrium 4:

• Hopf

1 12 24 2 14 1 14 1 4 1 14 42 4

) )( ( )) ( ( q m m m q q m γ α α α α γ − + + + + =

12 14 1 43 34 3 42 24 2 1

) ( γ α γ α γ α + + − + = m m m q

H3 - Hopf H4 - Hopf TC31 – saddle-node

SLIDE 6

Periodic and chaotic dynamics after CO2 atmospheric increase

Limit cycle Γ1

(4) as a result of Hopf

bifurcation for the given equilibrium. q1=1010 g/(m2year), m4=0.012986. Initial conditions: x0=[8550 1.8 36 8900]. Limit cycles Γ2

(4) and Γ4 (4) as a result of

period doubling bifurcation for the cycle Γ1

(4). q1=1018 g/(m2 year) and q1=1025.25

g/(g2 year). Strange attractor after the period-doubling bifurcation. q1=1027.6 g/(m2 year).

SLIDE 7

Periodic regime Γ1

(4) of carbon cycle functioning in the bog

Chaotic regime S1 after the period-doubling bifurcation

SLIDE 8

x*

1=8490.3

x*

2=36.246

x*

3=8835.6

y*

1=612.1

f*

12=38.05

f13

*=337.4

q2=0.0 y2

*=296.645

f43

*=621.8

y3

*=114.735

5 q3=35.73 f23

*=363.21

q1=987.55

Storages (in g/m2 of dry weight):

x1 - plant biomass; x2 – animal, fungi and bacteria biomass; x3 - dead trees, dead roots and litter

Input and output flows (in g/m2·year of dry weight):

q1 – carbon assimilation by plants, q3 – input from neighboring ecosystems, y1, y2, y3 – surface runoff + peat formation + abiotic oxidation

Intercompartment flows (in g/m2·year of dry weight):

f12 – consumption by phytophages and animals, f13 – litterfall from plants, f32 – consumption of dead

• rganics by animals, fungi and bacteria, f23 –

mortality of animals, fungi and bacteria

Flow control selection: f13=α13x1 f23=α23x2 f12=γ12x1 x2 f23=γ23x3 x2 f32=γ32x3x4

Compartment model: typical carbon cycle in a bog ecosystem

Data from (Alexandrov et al., 1994): Tajozhny Log bogs, Novgorod Region, Russia

Mass balance dynamic equations :      − + − = + + − − = − − − = x x x x m q dt x d x x x x x m dt x d x x x x m q dt x d

3 2 32 1 13 3 3 3 3 2 1 12 3 32 2 23 2 2 2 2 1 12 1 13 1 1 1 1

/ ) ( / / γ α γ γ α γ α

The model form is valid for

• ligtrophic (q3=0), mesotrophic and

eutrophic bogs q1 - carbon assimilation from atmosphere by vegetation, g/m2 year m4 – specific intensity of run-off and peat formation, 1/year

SLIDE 9

Steady states of the carbon cycle model and interpretation

] 6 . 8835 ; 246 . 36 ; 5 . 8830 [

) 3 (

=

−

x

Multistability of of steady regimes: convergence of time plots to steady states x(2) and x(3+).

a fen or raised bog a measured mesotrophic bog sphagnum pine forest or a meadow and a mesotrophic bog )] ) ( ( 1 ; ; [

13 1 13 1 3 4 13 1 1 ) 1 (

α α α + + + = m q q m m q x

]; ; ; [

3 2 1 ) 3 ( ± ± ± ± =

x x x x

SLIDE 10

Stability domains of the carbon cycle in a bog and perturbations

Carbon assimilation by plants, g/m2 year dry weight

Specific intensity of run-off and peat formation

peat mining or melioration atmospheric CO2 increase due to the climate change Organics decomposition decrease

5 – sphagnum pine forest is stable; 6 – a fen or a flowing water reservoir Stationary dynamic regimes of carbon cycle functioning : 1 –transitional bog is stable; 2 – a eutrophic fen; 3 – raised bog ecosystem is stable under q3=0 or a eutrophic fen is stable; 4 – multistability of raised bog and a meadow;

SLIDE 11

Different types of bog ecosystems in the parameter space

Specific intensity of run-off and peat formation Carbon assimilation by plants, g/m2 year dry weight Mesotrophic bogs: Oligotrophic bogs: North-West of European Russia Western Siberia North of European Russia

SLIDE 12

Conclusions

1) method of dynamic compartment model design by static “storage-flow” schemes can be applied both to open and closed ecosystems allowing to find steady states and analyze their stability properties; 2) current equilibrium of carbon cycle in the bog ecosystem can lose stability under the climate change induced CO2 atmospheric concentration increase, the critical stability value can be calculated using the bifurcation theory; 3) parametric portrait, analytical and numeric investigations show complex dynamic behavior of the carbon cycle in the open mesotrophic bog ecosystem with attractors sensitive to the climate dependent parameters.

Thank you for attention !

SLIDE 13

Bogs and carbon cycle: types and dynamics

x*

1=8490.3

x*

3=35.0

x*

2=1.246

x*

4=8835.6

y*

1=612.1

f*

12=38.05

f14

*=337.4

q3=0.0 y3

*=279.9

f43

*=584.9

q4=35.73 y4

*=114.73

5 f24

*=58.205

f*

42=36.9

y2

*=16.745

q2=0.0 f34

*=305

q1=987.55

Carbon cycle in Tajozhny Log mesotrophic bog, Novgorod Region, Russia, Alexandrov et al., 1994

Oligotrophic Mesotrophic Eutrophic Permafrost Storages (in g/m2 of dry weight): x1 - plant biomass; x2 - animal biomass; x3 - fungi and bacteria biomass; x4 - dead trees, dead roots and litter (Vompersky et al., 2005)