Russian Academy of Sciences A.M. Obukhov Institute of Atmospheric Physics Laboratory of Mathematical Ecology Nikolay N. Zavalishin Dynamics of the carbon cycle functioning in Russian peatlands under the climate change and human perturbations Supported by the program “Hydrospheric and atmospheric processes: forming, changing and regulating the Earth Climate” of the Russian Academy of Sciences
General problem of a dynamic model design by a given «storage-flow» diagram f k i = 8.5 a set of compartment schemes for time moments t 0 , t 1 ,…, collected from the q k = 0.5 q i = 8 x k =3.5 x i =70 f k i = 15 field studies y i = 3 y k = 0.8 f ik (x i , x k ) ? x 1 = 180 q k ( t ) q i ( t ) x k ( t ) x i ( t ) q 1 = 30 y 1 = 10 f ki ( x k , x i ) y k ( x k ) y i ( x i ) dynamic model for storages in x 1 ( t ) reservoires q 1 ( t ) y 1 ( x 1 ) dx n = − + − ∑ The main problem: how to make dynamic model i ( ) q y f f i i ki ik from one flow-balanced diagram? dt = ≠ k 1 , k i d x = − + q(x) y(x) f(x) Dynamic equations in general form: dt q - vector of input flows from the environment; y - vector of output flows to the environment; f ki - intercompartment flow from i to j
Modified method for dynamic system design Main assumptions (1-5c) for given stationary schemes: 1) q * + f * = y * - at least one of the given diagrams is a dynamic equilibrium; 2) f ki = f ki (x k , x i ); f ik = f ik (x i , x k ) – intercompartment flows depend only on participating storages; 3) q i = const – input flows can have only constant form; 4) y i = m i x i – output flows are linear; 5) intercompartment flow control types : Coefficients of flow functions are α ij x donor ; a) f ki = calculated from the given scheme: i β ij x b) f ki = recipient; * f * y f * * γ f j β γ α x x ki = ki = i = ki = ki ki ; ; ki i ; m ; c) f ki = Lotka-Volterra * * ij i j * * * x x x x x k i i k i 5d) additional control types with saturation : Coefficients of flow functions with K x x saturation are calculated from = K x x K x x ki k i = g = ki k i ki k i g , g , + + ki several given schemes or by special ( L x )( N x ) + + ki ki L x L x ki k ki i calibration procedures ki k ki i − + β − = ∑ α ( ), i s ; m i ik ki i Dynamic compartment model: i,s = 1 ,…n ≠ b k i = s − β ≠ α , i s . n dx ∑ n n si is = − + + + − ∑ ∑ i i q ( g g ) m x b x x c x si − γ γ . i i s s i is s c is i si is = dt is = = s 1 s 1 = s 1 = + + + x / x ( ,..., ) x ( x ) d dt q B diag C G x x 1 n
Open compartment model: typical carbon cycle in the bog ecosystem Data from (Alexandrov et al., 1994): Tajozhny Log mesotrophic bog, Novgorod Region, Russia q 3 =0.0 q 1 =987.55 Flow control selection: x * x * * =337.4 1 =8490.3 3 =35.0 f 14 f 14 = α 14 x 1 f 24 = α 24 x 2 f 34 = α 34 x 3 y * * =279.9 1 =612.1 y 3 f 12 = γ 12 x 1 x 2 f 43 = γ 43 x 3 x 4 f 42 = γ 42 x 2 x 4 f * * =305 12 =38.05 f 34 * =584.9 f 43 Storages (in g/m 2 of dry weight): f * 42 =36.9 x 1 - plant biomass; x 2 - animal x * q 2 =0.0 q 4 =35.73 x * biomass; x 3 - fungi and bacteria 2 =1.246 4 =8835.6 biomass; x 4 - dead trees, dead * =58.205 f 24 * =114.73 * =16.745 y 4 y 2 roots and litter 5 Intercompartment flows (in g/m 2 ·year of Input and output flows (in g/m 2 ·year of dry weight) : dry weight) : f 12 – consumption by animals, f 14 – litterfall q 1 – carbon assimilation by plants, q 4 – from plants, f 42 , f 43 – consumption of dead input from neighboring ecosystems, y 1 , y 2 , organics by animals, fungi and bacteria, f 24 , y 3 - respiration, y 4 – surface runoff + peat f 34 – death of animals, fungi and bacteria formation + abiotic oxidation 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.
Local bifurcation curves for the carbon cycle steady states Equilibrium 2: Equilibrium 1: γ α γ α q q = + = + 43 1 14 43 1 14 m ( q ) m ( q ) + α + α + α + α 4 4 4 4 m m m m 3 34 1 14 3 34 1 14 α γ = + 14 43 γ α + + α m ( q q ) - Hopf ( q q ( m )) + α α 4 1 4 = m 42 14 1 4 1 14 m 1 14 34 + α + α − γ + α + α 4 m m ( m )( m ) q = + α − γ 1 14 2 24 12 1 3 34 1 14 q ( m ) γ γ 1 2 24 42 43 12 Equilibrium 4: Equilibrium 3: γ α + + α + α + α m m ( q q ( m )) = = + α − γ 42 14 1 4 1 14 3 34 1 14 m q ( m ) γ γ + α + α − γ 4 1 2 24 42 ( m )( m ) q 43 12 1 14 2 24 12 1 γ α γ − γ + α γ γ + α − γ α + α γ ( ( m )) q m ( m ) m m = + + 43 24 43 42 3 34 43 4 1 42 3 34 43 24 1 2 14 43 ( ) m q q + α + α γ + α 4 1 1 K ( m ) m ( m ) 1 3 34 3 34 12 3 34 TC 31 – saddle-node H 4 - Hopf H 3 - Hopf
Periodic and chaotic dynamics after CO 2 atmospheric increase Limit cycles Γ 2 (4) and Γ 4 (4) as a result of Limit cycle Γ 1 (4) as a result of Hopf period doubling bifurcation for the cycle bifurcation for the given equilibrium. Γ 1 (4) . q 1 =1018 g/(m 2 year) and q 1 =1025.25 q 1 =1010 g/(m 2 year), m 4 =0.012986. Initial g/(g 2 year). conditions: x 0 =[8550 1.8 36 8900]. Strange attractor after the period-doubling bifurcation. q 1 =1027.6 g/(m 2 year).
Periodic regime Γ 1 (4) of carbon cycle functioning in the bog Chaotic regime S 1 after the period-doubling bifurcation
Compartment model: typical carbon cycle in a bog ecosystem Data from (Alexandrov et al., 1994): Tajozhny Log bogs, Novgorod Region, Russia Storages (in g/m 2 of dry weight): q 2 =0.0 f * q 1 =987.55 12 =38.05 x * x * 2 =36.246 1 =8490.3 x 1 - plant biomass; x 2 – animal, fungi and bacteria * =363.21 y * * =296.645 f 23 1 =612.1 y 2 biomass; x 3 - dead trees, dead roots and litter * =337.4 * =621.8 f 13 f 43 Input and output flows (in g/m 2 ·year of dry weight) : q 3 =35.73 x * 3 =8835.6 q 1 – carbon assimilation by plants, q 3 – input from * =114.735 y 3 neighboring ecosystems, y 1 , y 2 , y 3 – surface runoff 5 + peat formation + abiotic oxidation Flow control selection: f 13 = α 13 x 1 f 23 = α 23 x 2 f 12 = γ 12 x 1 x 2 f 23 = γ 23 x 3 x 2 f 32 = γ 32 x 3 x 4 Intercompartment flows (in g/m 2 ·year of dry weight) : Mass balance dynamic equations : f 12 – consumption by phytophages and animals, f 13 – litterfall from plants, f 32 – consumption of dead = − − − γ α d / dt q x m x x x x 1 1 1 13 1 1 2 12 1 organics by animals, fungi and bacteria, f 23 – = − − + + γ γ α d / dt ( ) x m x x x x x mortality of animals, fungi and bacteria 2 2 2 23 2 3 1 2 32 12 = − + − γ α d / dt q x m x x x x 3 3 3 13 1 2 3 3 32 The model form is valid for oligtrophic ( q 3 =0), mesotrophic and q 1 - carbon assimilation from atmosphere by vegetation, g/m 2 year eutrophic bogs m 4 – specific intensity of run-off and peat formation, 1/year
Steady states of the carbon cycle model and interpretation α q 1 q = + ( 1 ) 1 1 13 a fen or raised bog x [ ; 0 ; ( q )] + α + α 3 m m ( m ) 1 13 4 1 13 ± = sphagnum pine forest or a meadow ± ± ± ( 3 ) x [ ; ; ]; x x x 1 2 3 and a mesotrophic bog − = ( 3 ) x [ 8830 . 5 ; 36 . 246 ; 8835 . 6 ] a measured mesotrophic bog Multistability of of steady regimes: convergence of time plots to steady states x (2) and x (3+) .
Stability domains of the carbon cycle in a bog and perturbations Specific intensity of run-off and peat formation peat mining or atmospheric CO 2 melioration increase due to the climate change Organics decomposition decrease Carbon assimilation by plants, g/m 2 year dry weight Stationary dynamic regimes of carbon cycle functioning : 1 –transitional bog is stable; 2 – a eutrophic fen; 3 – raised bog ecosystem is stable under q 3 =0 or a eutrophic fen is stable; 4 – multistability of raised bog and a meadow; 5 – sphagnum pine forest is stable; 6 – a fen or a flowing water reservoir
Different types of bog ecosystems in the parameter space Specific intensity of run-off and peat formation Carbon assimilation by plants, g/m 2 year dry weight Mesotrophic bogs: Oligotrophic bogs: North-West of European Russia Western Siberia North of European Russia
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 CO 2 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 !
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