Dynamic model of vegetation pattern in Nothern Euro-Asia based on - PowerPoint PPT Presentation

Dynamic model of vegetation pattern in Nothern Euro-Asia based on probabilistic plant types interaction scheme Nikolay N.Zavalishin Laboratory of mathematical ecology, A.M.Obukhov Institute of atmospheric physics RAS, 3, Pyzhevsky Lane, Moscow, Russia, 119017, nickolos@ifaran.ru

Global Vegetation Pattern is a mosaic of terrestrial plant functional types varying under the Climate Change and anthropogenic perturbations Bioclimatic schemes Global cycles functioning models Climatic parameters best define the Methods based on the dependence of distribution of plant communities around physiological growth processes on climate the Earth surface initiating Static Global and carbon cycle functioning Vegetation Models (Foley et al., 1996; Hybrid v.3.0 – Friend et al., 1997; LPJ: Sitch et al., 2003) (Holdridge, 1947; Prentice, 1990) Mathematical ecology approach Dynamic equations for fractions of competing vegetation types with coefficients induced by climatic variables, soil properties, carbon, nitrogen and water cycles etc. (Svirezhev, Ecological Modelling , v.124, 1999) Two spatial scales are introduced at the terrestrial point ( x, y ) : micro-unit as the small area occupied by the only vegetation type, macro-unit as a set of micro-units.

Simplest probabilistic scheme with two vegetation types in a spatial “macro-unit” – the primitive case Let N f ( x,y,t ) – the number of forest “micro-units” at point ( x, y ) at time t, N g ( x,y,t ) – the number of grass “micro-units”, N = N f + N g – the total number of micro-units. f – forest particle, g – grass particle π – probability of forest particle to win the grass one Forest fraction: = p ( x , y , t ) N ( x , y , t ) / N f grass fraction: = q ( x , y , t ) N ( x , y , t ) / N g + = + π ∆ = π − → π < 2 p ( t 1 ) p ( t ) 2 p ( t ) q ( t ), p ( 2 1 ) pq , p ( t ) 0 if 1 / 2 ∆ = − π − → π > + = − π q ( 2 1 ) pq 2 p ( t ) 1 if 1 / 2 q ( t 1 ) q ( t ) + 2 ( 1 ) p ( t ) q ( t ),

Simplest probabilistic scheme with two vegetation types with primitive forest age structure f – forest micro-unit, g – grass micro-unit π – probability of forest to win grass, s – probability for forest to survive, 1- s – forest mortality Equilibrium points: π − 2 s 1 ∗ ∗ = = p 0 and p π − 1 2 s ( 2 1 ) Discrete-time dynamic system : [ ] + = + π Stability conditions: 2 p t ( 1 ) s p t ( ) 2 p t q t ( ) ( ) , [ ] < ∗ ≤ ≤ π > + = − π + − + π 2 2 0 p 1 , if s 1 and 2 s 1 q t ( 1 ) q t ( ) + 2 1 ( ) ( ) ( ) p t q t ( 1 s ) p t ( ) 2 p t q t ( ) ( ) 2

Probabilistic scheme of three vegetation type interactions with simple forest age structure Let N f ( x,y,t ) – the number of forest micro-units at point ( x, y ) at time t, N g ( x,y,t ) – the number of grass micro-units, N d ( x,y,t ) – the number of desert micro-units, N = N f + N g +N d – the total number of micro-units. v d f – forest micro-unit, Forest fraction: 1 -v g g 1 - v g g – grass micro-unit, v d q 2 = p ( x , y , t ) N ( x , y , t ) / N s d – desert micro-unit f f 1 -s f d 1 - s f s - desert fraction - grass fraction: p 2 f s f 1 -s f 1 - s d = = β f r ( x , y , t ) N ( x , y , t ) / N q ( x , y , t ) N ( x , y , t ) / N s f g d f δ g 1 - s d g 2 q p f s γ f Factors and probabilities of vegetation types g te rre s tria l g s m a c ro -u n it interaction: f 2 r p 1 -s μ f d 1 - s - competition forest – grass ( β , γ , δ ); f s f η f d 1 -s - interaction forest – desert ( μ , η , ω ); d f s ω 2 q r f d - interaction grass – desert ( u, w, v ); d d - natural mortality of forest ( m= 1 – s ). u g r 2 g g w g d d v d d Discrete-time dynamic system: d d t t + 1 T im e [ ] + = + σ + µ p ( t 1 ) sp ( t ) p ( t ) 2 q ( t ) 2 r ( t ) , + = ω − σ ω + − µ ω + ω 2 q ( t 1 ) q ( t ) + 2 ( 1 ) p ( t ) q ( t ) 2 ( 1 ) p ( t ) r ( t ) 2 r ( t ) q ( t ), [ ] [ ] + = + − + − ω + σ − + − σ − ω + µ − + − µ − ω 2 2 2 r ( t 1 ) r ( t ) ( 1 s ) p ( t ) ( 1 ) q ( t ) 2 p ( t ) q ( t ) ( 1 s ) ( 1 )( 1 ) 2 p ( t ) r ( t ) ( 1 s ) ( 1 )( 1 )

Probabilistic scheme of vegetation type interactions with complex forest age structure s i f i – forest of age i, i= 1,…,n f i+1 1 -s i f i f 1 σ ij 1 f 0 s i f i+1 Factors and probabilities of vegetation α ij 1 -s i f i f i f 1 1 -s j interaction: f j f j s j σ ji f j+1 - competition in n forest age groups ( σ ij , s j f j 1 -s j α ij ) f 1 1 f 0 s i p i p j - competition forest of group i – grass ( β i , f i+1 1 -s i f i β i γ i , δ i ); 1 f 1 f 0 - interaction forest of group i – desert ( μ i , s i f i+1 f i δ i 1 -s i f i 2qp i g η i , ω i ); 1 f 1 g γ i terrestrial 1 g - interaction grass – desert ( u, w, v ); g g macro-unit 1 g g q 2 - natural mortality of forest ( m i = 1 – s i ). 1 g g 1 s i g g f i+1 1 -s i f i f 1 μ i 1 f 0 2rp i s i f i+1 f i η i 1 -s i f i f 1 d d 1 ω i d 1 d d 1 d d 2qr u g g 1 -u d d 1- u r 2 u g 1 d d 1 d t d t +1

Dynamic model of vegetation pattern with continuous forest age structure and continuous time Dynamic system for vegetation types with a continuous forest age τ : p i ( t ) → p ( t , τ ): ∂ τ ∂ τ +∞ p ( t , ) p ( t , ) + ∫ = − τ τ + − τ γ τ + ω τ + σ ξ τ ξ ξ p ( t , )[ m ( ) ( 1 m ( ))( ( ) q ( t ) ( ) r ( t ) ( , ) p ( t , ) d ) )] ∂ ∂ τ t 0 ( τ +∞ dq t , ) ∫ = − + γ τ − β τ τ τ ( )[ ( )( ) ( ( ) ( )) ( , ) ] q t r t u v p t d dt 0 ( τ +∞ dr t , ) ∫ = − + ω τ − µ τ τ τ r ( t )[ q ( t )( v u ) ( ( ) ( )) p ( t , ) d ] dt 0 Renewal equation for p ( τ =0): +∞ ( ) ∫ = − − + τ β τ + τ − γ τ τ + 2 ( , 0 ) ( 1 ) ( ) ( , ) ( ) ( )( 1 ( )) p t q r q t p t m d 0 +∞ +∞ +∞ ( ) ( ) ∫ ∫ ∫ τ µ τ + τ − ω τ τ + − τ τ ξ − σ ξ τ ξ τ r ( t ) p ( t , ) ( ) m ( )( 1 ( )) d ( 1 m ( )) p ( t , ) p ( t , ) 1 ( , ) d d 0 0 0 New model variables - fractions of young and mature trees and total forest fraction: τ ∞ τ m = τ m (T,P) - average maturity age of trees m ∫ ∫ = τ τ = τ τ = + p ( t ) p ( , t ) d p ( t ) p ( , t ) d p p p 1 2 1 2 τ l = τ l (T,P) - a life span (maximal age) of trees τ 0 m

Simplification of a continuous-time model for two-age classes forest Simplifying hypothesis on competition coefficients - they are constants for each of two forest age-groups: Ω Ω γ τ ≤ τ τ ≤ τ β τ ≤ τ    , if , m 1 if , , , if , γ τ = τ = β τ = 1 m m 1 m    ( ) m ( ) ( ) γ τ > τ τ > τ β τ > τ    ,if . m , if ,if . 2 m 2 m 2 m µ ≤ τ < τ ω ≤ τ < τ   , if 0 , if 0 µ τ = ω τ = 1 m 1 m   ( ) ( ) µ τ ≤ τ < +∞ ω τ ≤ τ < +∞   , if , if 2 m 2 m β = γ = µ = σ = σ ξ τ ∈ Ω  0 Ω Ω , if , , 1 2 2 12 11 1  σ ξ τ ∈ Ω , if , , σ ξ τ =  ( , ) 12 2 p σ ξ τ ∈ Ω τ = , if , , 1 p ( , t )  21 3 τ m σ ξ τ ∈ Ω  , if , , m 22 4 Final Vegetation Pattern model for a one spatial macro-unit dp p dp = − + ω − σ + ω + 2 = ω − − − + γ − 2 ( A a ) p a p a ( p q ) ( 1 p q )( k p p ) q ( k p p ) τ 2 2 22 2 2 1 2 2 1 1 2 dt dt m − β µ 1 [ ] dq u v 1 1 = − = − = + =γ − − + − = + = + a 1 2 2 k 1 k 1 k 1 A ( 1 )( 1 ) ) q q k k p k p τ − τ γ ω γ τ τ − τ 1 2 3 1 3 3 1 2 dt 1 1 1 m l m l m

Steady states, stability and calibration of the simplified model The model has steady states corresponding to pure desert, grass, forest, forest- desert, forest-grass, and fully mixed vegetation in the macro-unit. Stability boundaries in the { k 1 , k 2 , k 3 } space acquire the particular form with linking competition coefficients to the climatic parameters – average temperature and annual precipitation. Exponent-polynomial form of boundaries between biomes on the Lieth diagram: = 0 . 05 T P ( T ) 1235 . 3 e , = + AB k (T,P) b (T) b (T)P 0 1 1 = − − + 2 P ( T ) 4 . 73 T 69 . 78 T 144 . 51 CD = 0 . 05 T P ( T ) 431.5 e , EF = + = k (T,P) c (T) c (T)P 0 . 05 T P ( T ) 255.4 e . 0 1 2 GH Approximation of mean forest ages: 178 k τ =const for coniferous and τ = τ = τ ( T ) ; k ( T ) τ − m l m 0 . 85 ( T T ) deciduous trees cr