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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 - - 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,
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), ( ) ( ) 1 ( 2 + ) ( ) 1 ( ), ( ) ( 2 ) ( ) 1 (
2 2
t q t p t q t q t q t p t p t p π π − = + + = +
pq q pq p ) 1 2 ( , ) 1 2 ( − − = ∆ − = ∆ π π 2 / 1 if 1 ) ( 2 / 1 if ) ( > → < → π π t p t p
Simplest probabilistic scheme with two vegetation types in a spatial “macro-unit” – the primitive case
Forest fraction: grass fraction: N t y x N t y x p
f
/ ) , , ( ) , , ( = N t y x N t y x q
g
/ ) , , ( ) , , ( = Let Nf(x,y,t) – the number of forest “micro-units” at point (x, y) at time t, Ng(x,y,t) – the number of grass “micro-units”, N = Nf + Ng – the total number of micro-units. f – forest particle, g – grass particle π – probability of forest particle to win the grass one
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Simplest probabilistic scheme with two vegetation types with primitive forest age structure
Discrete-time dynamic system : 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:
) 1 2 ( 1 2 and
2 1
− − = =
∗ ∗
π π s s p p
[ ] [ ]
p t s p t p t q t q t q t p t q t s p t p t q t ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + + = − + − + 1 2 1 2 1 1 2
2 2 2
π π π +
Stability conditions: 1 2 and 1 s if , 1
2
> ≤ ≤ <
∗
s p π
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Probabilistic scheme of three vegetation type interactions with simple forest age structure
[ ] [ ] [ ]
) 1 )( 1 ( ) 1 ( ) ( ) ( 2 ) 1 )( 1 ( ) 1 ( ) ( ) ( 2 ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ), ( ) ( 2 ) ( ) ( ) 1 ( 2 ) ( ) ( ) 1 ( 2 + ) ( ) 1 ( , ) ( 2 ) ( 2 ) ( ) ( ) 1 (
2 2 2 2
ω µ µ ω σ σ ω ω ω µ ω σ ω µ σ − − + − + − − + − + − + − + = + + − + − = + + + = + s t r t p s t q t p t q t p s t r t r t q t r t r t p t q t p t q t q t r t q t p t sp t p
Discrete-time dynamic system:
N t y x N t y x p
f
/ ) , , ( ) , , ( =
N t y x N t y x q
g
/ ) , , ( ) , , ( = N t y x N t y x r
d
/ ) , , ( ) , , ( =
- desert fraction
q 2 p 2 2 r p 2 q r β γ μ ω r 2
t t+ 1
s 1 -s 1 -s s s 1 -s 1 -s s 1 -s s s 1 -s 1 -s s u v δ 1 -s s η w v 1 -v 1 -v v
T im e
te rre s tria l m a c ro -u n it 2 q p f d f g f f f f g g f f d d d d f d f f d f g d f d f d f g g d d g f d f d f g d d g d g g
Let Nf (x,y,t) – the number of forest micro-units at point (x, y) at time t, Ng (x,y,t) – the number of grass micro-units, Nd (x,y,t) – the number of desert micro-units, N = Nf + Ng+Nd – the total number of micro-units. Forest fraction:
- grass fraction:
Factors and probabilities of vegetation types interaction:
- competition forest – grass (β, γ, δ );
- interaction forest – desert (μ, η, ω );
- interaction grass – desert (u, w, v );
- natural mortality of forest (m=1 – s ).
f – forest micro-unit, g – grass micro-unit, d – desert micro-unit
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Probabilistic scheme of vegetation type interactions with complex forest age structure
fi – forest of age i, i=1,…,n
q 2 p i p j 2rp i 2qr σ ij σ ji α ij β i γi δ i μ i ω i η i 1 1 1 1 r 2 s i 1-si 1 s i 1-s i 1 1 1 1 s i 1-si s i 1-si 1 s i 1-s i 1 s i 1-s i 1-sj s j s j 1-sj 1 1 1
t t+1
u 1-u 1-u u terrestrial macro-unit 2qp i g d g g fi d fi g fi fj fi f0 fi fj fj f0 fi f0 fi g g g fi f0 fi d d d d d d d d d d f1 fi+1 f1 fi+1 g g g f1 fi+1 fi+1 f1 fi+1 f1 fi+1 f1 fj+1 f1 g g g d g
Factors and probabilities of vegetation interaction:
- competition in n forest age groups (σij,
αij )
- competition forest of group i – grass (βi,
γi, δi );
- interaction forest of group i – desert (μi,
ηi, ωi );
- interaction grass – desert (u, w, v );
- natural mortality of forest (mi = 1 – si ).
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Dynamic model of vegetation pattern with continuous forest age structure and continuous time
)] ) ) , ( ) , ( ) ( ) ( ) ( ) ( ))( ( 1 ( ) ( )[ , (
∫
+∞
+ + − + − = ξ ξ τ ξ σ τ ω τ γ τ τ τ d t p t r t q m m t p
] ) , ( )) ( ) ( ( ) )( ( )[ (
∫
+∞
− + − = τ τ τ β τ γ d t p v u t r t q
Renewal equation for p (τ=0):
τ τ τ ∂ ∂ + ∂ ∂ ) , ( ) , ( t p t t p dt t dq ) , ( τ dt t dr ) , ( τ
] ) , ( )) ( ) ( ( ) )( ( )[ (
∫
+∞
− + − = τ τ τ µ τ ω d t p u v t q t r
( )
∫
+∞
+ − + + − − =
2
)) ( 1 )( ( ) ( ) , ( ) ( ) 1 ( ) , ( τ τ γ τ τ β τ d m t p t q r q t p
( ) ( )
τ ξ τ ξ σ ξ τ τ τ τ ω τ τ µ τ d d t p t p m d m t p t r ) , ( 1 ) , ( ) , ( )) ( 1 ( )) ( 1 )( ( ) ( ) , ( ) ( − − + − +
∫ ∫ ∫
+∞ +∞ +∞
Dynamic system for vegetation types with a continuous forest age τ:
pi(t) → p( t,τ ): τm = τm(T,P) - average maturity age of trees τl = τl(T,P) - a life span (maximal age) of trees
2 1 2 1
) , ( ) ( ) , ( ) ( p p p d t p t p d t p t p
m m
+ = = =
∫ ∫
∞ τ τ
τ τ τ τ New model variables - fractions of young and mature trees and total forest fraction:
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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:
m m m
m m
( ) , , τ τ τ τ τ = ≤ >
1 if
, if
2
β τ β τ τ β τ τ ( ) , , . = ≤ >
1 2
if ,if
m m
γ τ γ τ τ γ τ τ ( ) , , . = ≤ >
1 2
if ,if
m m
Ω ∈ Ω ∈ Ω ∈ Ω ∈ = , , if , , , if , , , if , , , if , ) , (
4 22 3 21 2 12 1 11
τ ξ σ τ ξ σ τ ξ σ τ ξ σ τ ξ σ
+∞ < ≤ < ≤ = τ τ µ τ τ µ τ µ
m m
if , if , ) (
2 1
+∞ < ≤ < ≤ = τ τ ω τ τ ω τ ω
m m
if , if , ) (
2 1
m m
p t p τ τ
1
) , ( =
12 2 2 1
= = = = σ µ γ β
Ω Ω Ω Ω
Final Vegetation Pattern model for a one spatial macro-unit
) ( ) (
2 2 2 22 2 2 2
q p a p a p a A p dt dp
m
+ + − + − = ω σ ω τ
) ( ) )( 1 (
2 1 1 2 2 1
p p k q p p k q p dt dp − + − − − = γ ω
[ ]
) ) 1 )( 1 (
2 1 3 3 1
p k p k k q q dt dq − + − − =γ
m l m
A τ τ τ − + = 1 1
m l
a τ τ − − = 1 1
1 2 1
1 γ β + = k
1 2 2
1 ω µ + = k
1 3
1 γ v u k − − =
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Steady states, stability and calibration of the simplified model
. 255.4 ) ( , 431.5 ) ( 51 . 144 78 . 69 73 . 4 ) ( , 3 . 1235 ) (
05 . 05 . 2 05 . T GH T EF CD T AB
e T P e T P T T T P e T P = = + − − = =
Exponent-polynomial form of boundaries between biomes on the Lieth diagram:
) ( ; ) ( 178 ) (
85 .
T k T T T
m l cr m
τ τ τ
τ
= − =
(T)P b (T) b (T,P) k 1
1
+ =
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 {k1, k2, k3} space acquire the particular form with linking competition coefficients to the climatic parameters – average temperature and annual precipitation.
(T)P c (T) c (T,P) k 1
2
+ =
Approximation of mean forest ages:
kτ=const for coniferous and deciduous trees
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Numerical simulation of the Vegetation Pattern in Nothern Euro-Asia under the Climate Change
Any explicit climate scenario as a sequence of temperature – precipitation values, generates dynamics of the vegetation cover as a spatial mosaic with initial distribution calculated for the modern climate in Northern Euro-Asia. Using scenarios generated by some climatic models and based on A2 and B1 carbon emission scenarios by IPCC, we estimate the vegetation dynamics for Northern Euro-Asia over 120-year time interval and pay attention to the evolution of regions with transient dynamics. Initial state of vegetation cover in 1980s ( a - forest fraction, b - grass fraction)
a b
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Numerical simulation of the Vegetation Pattern in Nothern Euro-Asia under the Climate Change
a
Final state of vegetation pattern in 2100 under scenario SRES B1 ( a - forest fraction, b - grass fraction )
b a
Final state of vegetation pattern in 2100 under scenario SRES A2 ( a - forest fraction, b - grass fraction)
b
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Conclusions
1) The vegetation pattern model satisfactorily reproduces the initial state of vegetation cover in terms of a forest function (Brovkin et al., Ecological Modelling, 1997); 2) Steady states of the model have clear biogeographic interpretation, and their stability conditions can be connected with current biome boundaries in the climatic parameter space
- f temperature-precipitation;