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Cari 2020: A minimalistic model of spatial structuration of humid savanna vegetation
1. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
A minimalistic model of spatial structuration of
humid savanna vegetation
By :
Tega II Simon Rodrigue
co-authors
Yatat Valaire, Tewa Jean Jules, and Couteron Pierre
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 1/27
2. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Outline
1) Introduction
5. Problematic.
2) Model construction.
3) Mathematical analysis.
4) Numerical illsutration.
5) Conclusion and upcoming work.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 2/27
6. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Illustration of spatial structuration of vegetation in
arid environemental context
FIGURE: Tiger Bush in Somalia and Niger (Lefever and Lejeune (1997)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 3/27
7. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Illustration of spatial structuration of vegetation in
humid environemental context
FIGURE: Forest-grassland mosaic in Lopé (Gabon)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 4/27
8. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Why Study spatial structuring of humid savanna
vegetation
Why study humid savanna ?
Biodiversity conservation in face of climatic change
Economic importance
The most widespread biome in Central Africa.
Four main directions have been undertaken by the researchers.
Influence of climate (precipitations) on vegetation
physiognomies.
Fires disturbance.
Herbivory.
Tree-grass interaction in Savanna.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 5/27
9. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Why Study spatial structuring of humid savanna
vegetation
Why study humid savanna ?
Biodiversity conservation in face of climatic change
Economic importance
The most widespread biome in Central Africa.
Four main directions have been undertaken by the researchers.
Influence of climate (precipitations) on vegetation
physiognomies.
Fires disturbance.
Herbivory.
Tree-grass interaction in Savanna.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 5/27
10. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Why Study spatial structuring of humid savanna
vegetation
Why study humid savanna ?
Biodiversity conservation in face of climatic change
Economic importance
The most widespread biome in Central Africa.
Four main directions have been undertaken by the researchers.
Influence of climate (precipitations) on vegetation
physiognomies.
Fires disturbance.
Herbivory.
Tree-grass interaction in Savanna.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 5/27
11. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Mathematical modelling of Savanna Dynamic
Deterministic models based on :
Ordinary Differential Equation.
Impusilve Differential Equation.
Partial differential Equation.
PDE models : Pattern formation
Pattern formation models are based on :
Symetric breaking instability.
Stabilization of pattern by non linear terms.
PDE models for pattern formation : Classification
Turing-Like instability models
Differential flow models
Kernels based models
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 6/27
12. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Mathematical modelling of Savanna Dynamic
Deterministic models based on :
Ordinary Differential Equation.
Impusilve Differential Equation.
Partial differential Equation.
PDE models : Pattern formation
Pattern formation models are based on :
Symetric breaking instability.
Stabilization of pattern by non linear terms.
PDE models for pattern formation : Classification
Turing-Like instability models
Differential flow models
Kernels based models
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 6/27
13. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Mathematical modelling of Savanna Dynamic
Deterministic models based on :
Ordinary Differential Equation.
Impusilve Differential Equation.
Partial differential Equation.
PDE models : Pattern formation
Pattern formation models are based on :
Symetric breaking instability.
Stabilization of pattern by non linear terms.
PDE models for pattern formation : Classification
Turing-Like instability models
Differential flow models
Kernels based models
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 6/27
14. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Mathematical modelling of Savanna Dynamic
Kernels based models
This type of models are generally used in arid or semi-arid context
and :
describe the dynamics of a single species (Tree dynamic).
symetric-breaking induced by non local interactions.
Short range of cooperation among plants.
Long range of competition.
Example of structuration of kernels based models
∂u
∂t
= h(u) + ∫ ω(r,r′
)[u(r′
,t) − u0]dr′
(1)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 7/27
15. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Mathematical modelling of Savanna Dynamic
Kernels based models
This type of models are generally used in arid or semi-arid context
and :
describe the dynamics of a single species (Tree dynamic).
symetric-breaking induced by non local interactions.
Short range of cooperation among plants.
Long range of competition.
Example of structuration of kernels based models
∂u
∂t
= h(u) + ∫ ω(r,r′
)[u(r′
,t) − u0]dr′
(1)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 7/27
16. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Problematic
Aim
Build a tractable spatio-temporal model allowing, to illustrate the
spatial structuration of vegetation in wet savanna zone and
specifically in Cameroon.
Specifically :
18. illustrate the emergence of spatial pattern.
FIGURE: Forest-savanna (grassland) mosaic in Ayos,Cameroon
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 8/27
19. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Problematic
Aim
Build a tractable spatio-temporal model allowing, to illustrate the
spatial structuration of vegetation in wet savanna zone and
specifically in Cameroon.
Specifically :
21. illustrate the emergence of spatial pattern.
FIGURE: Forest-savanna (grassland) mosaic in Ayos,Cameroon
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 8/27
22. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Our model follows Yatat et al. (2017) and Tchuinté et al. (2017).
ODE version of Tchuinté et al. (2017)
⎧⎪⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎪⎩
dG
dt
= γGG(1 − G
KG
) − δGG − γTGTG − λfGfG
dT
dt
= γTT(1 − T
KT
) − δTT − λfTfω(G)exp(−pT)T
(2)
Where
G and T stand for grass and tree biomass in t.ha−1
.
ω(G) has the form of a Holling type III function.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 9/27
23. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Starting from this, we incoporate a spatial component on state
variables.
Model assumptions
G(x,t) and T(x,t) are normalized by their capacities of
charge respectively.
Trees and grasses biomasses, have a logistic growth with a
non local intra-specific competition.
There is a factor of cooperation Ω ∈ R between trees
promoting regrowth and growth of young trees.
Trees exert competition on grasses in a non-local way.
We insert a probability of fire induced-tree mortality at a
space point x. This probability is therefore, a decreasing
function of tree density.
G and T experience local isotropic biomass propagation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 10/27
24. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Starting from this, we incoporate a spatial component on state
variables.
Model assumptions
G(x,t) and T(x,t) are normalized by their capacities of
charge respectively.
Trees and grasses biomasses, have a logistic growth with a
non local intra-specific competition.
There is a factor of cooperation Ω ∈ R between trees
promoting regrowth and growth of young trees.
Trees exert competition on grasses in a non-local way.
We insert a probability of fire induced-tree mortality at a
space point x. This probability is therefore, a decreasing
function of tree density.
G and T experience local isotropic biomass propagation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 10/27
25. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Starting from this, we incoporate a spatial component on state
variables.
Model assumptions
G(x,t) and T(x,t) are normalized by their capacities of
charge respectively.
Trees and grasses biomasses, have a logistic growth with a
non local intra-specific competition.
There is a factor of cooperation Ω ∈ R between trees
promoting regrowth and growth of young trees.
Trees exert competition on grasses in a non-local way.
We insert a probability of fire induced-tree mortality at a
space point x. This probability is therefore, a decreasing
function of tree density.
G and T experience local isotropic biomass propagation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 10/27
26. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Starting from this, we incoporate a spatial component on state
variables.
Model assumptions
G(x,t) and T(x,t) are normalized by their capacities of
charge respectively.
Trees and grasses biomasses, have a logistic growth with a
non local intra-specific competition.
There is a factor of cooperation Ω ∈ R between trees
promoting regrowth and growth of young trees.
Trees exert competition on grasses in a non-local way.
We insert a probability of fire induced-tree mortality at a
space point x. This probability is therefore, a decreasing
function of tree density.
G and T experience local isotropic biomass propagation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 10/27
27. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Starting from this, we incoporate a spatial component on state
variables.
Model assumptions
G(x,t) and T(x,t) are normalized by their capacities of
charge respectively.
Trees and grasses biomasses, have a logistic growth with a
non local intra-specific competition.
There is a factor of cooperation Ω ∈ R between trees
promoting regrowth and growth of young trees.
Trees exert competition on grasses in a non-local way.
We insert a probability of fire induced-tree mortality at a
space point x. This probability is therefore, a decreasing
function of tree density.
G and T experience local isotropic biomass propagation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 10/27
28. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
Starting from this, we incoporate a spatial component on state
variables.
Model assumptions
G(x,t) and T(x,t) are normalized by their capacities of
charge respectively.
Trees and grasses biomasses, have a logistic growth with a
non local intra-specific competition.
There is a factor of cooperation Ω ∈ R between trees
promoting regrowth and growth of young trees.
Trees exert competition on grasses in a non-local way.
We insert a probability of fire induced-tree mortality at a
space point x. This probability is therefore, a decreasing
function of tree density.
G and T experience local isotropic biomass propagation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 10/27
29. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
All of this lead to the following model
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂G
∂t
= DG
∂2
G
∂x2
+ γGG(1 − ∫
+∞
−∞
φM1 (x − y)G(y,t)dy) − δGG
−γTGG(∫
+∞
−∞
φM2 (x − y)T(y,t)dy) − λfGfG,
∂T
∂t
= DT
∂2
T
∂x2
+ γTT(1 + ΩT)(1 − ∫
+∞
−∞
φM2 (x − y)T(y,t)dy)
−δTT − λfTfω(G)exp(−p∫
+∞
−∞
φM2 (x − y)T(y)dy)T.
(3)
x ∈ I = [−L,L], L > 0, t > 0 with homogeneous Neumann
boundary condition and for 0 < M ≤ L we choose :
φMi (x) =
⎧⎪⎪⎪
⎨
⎪⎪⎪⎩
1
2Mi
, x ≤ Mi
0 , x > Mi
with :φ0(x) = 1.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 11/27
30. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Model construction
All of this lead to the following model
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂G
∂t
= DG
∂2
G
∂x2
+ γGG(1 − ∫
+∞
−∞
φM1 (x − y)G(y,t)dy) − δGG
−γTGG(∫
+∞
−∞
φM2 (x − y)T(y,t)dy) − λfGfG,
∂T
∂t
= DT
∂2
T
∂x2
+ γTT(1 + ΩT)(1 − ∫
+∞
−∞
φM2 (x − y)T(y,t)dy)
−δTT − λfTfω(G)exp(−p∫
+∞
−∞
φM2 (x − y)T(y)dy)T.
(3)
x ∈ I = [−L,L], L > 0, t > 0 with homogeneous Neumann
boundary condition and for 0 < M ≤ L we choose :
φMi (x) =
⎧⎪⎪⎪
⎨
⎪⎪⎪⎩
1
2Mi
, x ≤ Mi
0 , x > Mi
with :φ0(x) = 1.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 11/27
31. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Our aim in this section is to derived a condition on spatial
convolution such that, savanna space homegeneous steady state of
(3) is stable in the case M1 = M2 = 0 but unstable for some Mi > 0,
i = 1 ; 2.
Space homogeneous solution of (3)
{
γGG(1 − G) − δGG − γTGTG − λfGfG = 0,
γTT(1 + ΩT)(1 − T) − δTT − λfTfω(G)exp(−pT) = 0.
(4)
It is assumed that :
γT − δT > 0 and γG − δG > 0. (5)
Set :
RG =
γG
δG + fλfG
and RF =
γG
δG + λfGf + γTGTi
. (6)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 12/27
32. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Our aim in this section is to derived a condition on spatial
convolution such that, savanna space homegeneous steady state of
(3) is stable in the case M1 = M2 = 0 but unstable for some Mi > 0,
i = 1 ; 2.
Space homogeneous solution of (3)
{
γGG(1 − G) − δGG − γTGTG − λfGfG = 0,
γTT(1 + ΩT)(1 − T) − δTT − λfTfω(G)exp(−pT) = 0.
(4)
It is assumed that :
γT − δT > 0 and γG − δG > 0. (5)
Set :
RG =
γG
δG + fλfG
and RF =
γG
δG + λfGf + γTGTi
. (6)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 12/27
33. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Our aim in this section is to derived a condition on spatial
convolution such that, savanna space homegeneous steady state of
(3) is stable in the case M1 = M2 = 0 but unstable for some Mi > 0,
i = 1 ; 2.
Space homogeneous solution of (3)
{
γGG(1 − G) − δGG − γTGTG − λfGfG = 0,
γTT(1 + ΩT)(1 − T) − δTT − λfTfω(G)exp(−pT) = 0.
(4)
It is assumed that :
γT − δT > 0 and γG − δG > 0. (5)
Set :
RG =
γG
δG + fλfG
and RF =
γG
δG + λfGf + γTGTi
. (6)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 12/27
34. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Our aim in this section is to derived a condition on spatial
convolution such that, savanna space homegeneous steady state of
(3) is stable in the case M1 = M2 = 0 but unstable for some Mi > 0,
i = 1 ; 2.
Space homogeneous solution of (3)
{
γGG(1 − G) − δGG − γTGTG − λfGfG = 0,
γTT(1 + ΩT)(1 − T) − δTT − λfTfω(G)exp(−pT) = 0.
(4)
It is assumed that :
γT − δT > 0 and γG − δG > 0. (5)
Set :
RG =
γG
δG + fλfG
and RF =
γG
δG + λfGf + γTGTi
. (6)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 12/27
35. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Proposition 3.1
If RG ≤ 1, then system (3) admits two homogeneous stationary
solutions :
a) desert equilibrium E0 = (0,0).
b) forest equilibrium :
∗ if Ω = 0, then ET1
= (0,
γT − δT
γT
)
∗ if Ω > 0, then
ET2
=
⎛
⎜
⎜
⎜
⎝
0,
√
(1 − Ω)
2
+ 4Ω(1 − δT
γT
) − (1 − Ω)
2Ω
⎞
⎟
⎟
⎟
⎠
If RG > 1 then we have, E0, ETi , (i = 1,2) and a grassland
equilibrium : EGe = (Ge,0) = (1 −
1
RG
,0).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 13/27
36. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Proposition 3.1
If RG ≤ 1, then system (3) admits two homogeneous stationary
solutions :
a) desert equilibrium E0 = (0,0).
b) forest equilibrium :
∗ if Ω = 0, then ET1
= (0,
γT − δT
γT
)
∗ if Ω > 0, then
ET2
=
⎛
⎜
⎜
⎜
⎝
0,
√
(1 − Ω)
2
+ 4Ω(1 − δT
γT
) − (1 − Ω)
2Ω
⎞
⎟
⎟
⎟
⎠
If RG > 1 then we have, E0, ETi , (i = 1,2) and a grassland
equilibrium : EGe = (Ge,0) = (1 −
1
RG
,0).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 13/27
37. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Proposition 3.1
If RG ≤ 1, then system (3) admits two homogeneous stationary
solutions :
a) desert equilibrium E0 = (0,0).
b) forest equilibrium :
∗ if Ω = 0, then ET1
= (0,
γT − δT
γT
)
∗ if Ω > 0, then
ET2
=
⎛
⎜
⎜
⎜
⎝
0,
√
(1 − Ω)
2
+ 4Ω(1 − δT
γT
) − (1 − Ω)
2Ω
⎞
⎟
⎟
⎟
⎠
If RG > 1 then we have, E0, ETi , (i = 1,2) and a grassland
equilibrium : EGe = (Ge,0) = (1 −
1
RG
,0).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 13/27
38. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Proposition 3.1
If RG ≤ 1, then system (3) admits two homogeneous stationary
solutions :
a) desert equilibrium E0 = (0,0).
b) forest equilibrium :
∗ if Ω = 0, then ET1
= (0,
γT − δT
γT
)
∗ if Ω > 0, then
ET2
=
⎛
⎜
⎜
⎜
⎝
0,
√
(1 − Ω)
2
+ 4Ω(1 − δT
γT
) − (1 − Ω)
2Ω
⎞
⎟
⎟
⎟
⎠
If RG > 1 then we have, E0, ETi , (i = 1,2) and a grassland
equilibrium : EGe = (Ge,0) = (1 −
1
RG
,0).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 13/27
39. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Set
a = −
λfGf + δG
γTG
,
b =
γG
γTG
,
θ = 2(a + b)bΩγT + γT(1 − Ω)b,
α = ΩγTb2
,
q = (γT − δT) + γT(Ω − 1)(a + b) − ΩγT(a + b)2
,
m = λfTf exp(−p(a + b)),
θ∗
=
24α + mpb((pb)2
+ 6(pb) + 6)exp(pb)
6
(7)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 14/27
40. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Space homogeneous steady states
Proposition 3.2 (Savanna equilibrium)
If RF,f=0 > 1, then we have the unique savanna equilibrium
Es = (G∗
,T∗
) such that
G∗
= 1 −
1
RF,f=0
and T∗
= Ti,i = 1,2. (8)
If f > 0 and RG > 1, then a savanna equilibrium Es = (G∗
,T∗
)
must satisfy these two relations :
−α(G∗
)4
+θ(G∗
)3
−mexp(pbG∗
)(G∗
)2
+(q−αg2
0)(G∗
)2
+θg2
0G∗
+qg2
0 = 0,
(9)
T∗
= (a + b) − bG∗
and max{Ge −
γTG
γG
;0} < G∗
< Ge. (10)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 15/27
41. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
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Conclusion
and Upcoming
work
Space homogeneous steady states
Proposition 3.2 (Savanna equilibrium)
If RF,f=0 > 1, then we have the unique savanna equilibrium
Es = (G∗
,T∗
) such that
G∗
= 1 −
1
RF,f=0
and T∗
= Ti,i = 1,2. (8)
If f > 0 and RG > 1, then a savanna equilibrium Es = (G∗
,T∗
)
must satisfy these two relations :
−α(G∗
)4
+θ(G∗
)3
−mexp(pbG∗
)(G∗
)2
+(q−αg2
0)(G∗
)2
+θg2
0G∗
+qg2
0 = 0,
(9)
T∗
= (a + b) − bG∗
and max{Ge −
γTG
γG
;0} < G∗
< Ge. (10)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 15/27
42. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
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Conclusion
and Upcoming
work
Space homogeneous steady states
Case 1 : θ < mpb
Condition q < m + αg2
0 q > m + αg2
0
Maximal number of savanna equilibria 2 3
TABLE: Maximal number of savanna equilibria of (3) with
θ < mpb
Case 2 : θ > mpb,
Condition θ < θ∗
θ > θ∗
Maximal number on savanna equilibria 4 3
TABLE: Maximal number of savanna equilibria of (3) with
θ > mpb
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 16/27
43. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
we define the following thresholds :
R∗
1 =
γT [(1 − Ω) + 2ΩT∗
]
pλfTfω(G∗)exp(−pT∗)
and R∗
2 =
γTGω′
(G∗
)
pγGω(G∗)
, (11)
Proposition 3.3
case 1 : Assume that f = 0, then Es = (G∗
,T∗
) is locally
asymptotically stable (LAS) when it exist.
case 2 :Assume that f > 0, then if :
R∗
1 − R∗
2 > 1 (12)
then Es = (G∗
,T∗
) is locally asymptotically stable.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 17/27
44. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
we define the following thresholds :
R∗
1 =
γT [(1 − Ω) + 2ΩT∗
]
pλfTfω(G∗)exp(−pT∗)
and R∗
2 =
γTGω′
(G∗
)
pγGω(G∗)
, (11)
Proposition 3.3
case 1 : Assume that f = 0, then Es = (G∗
,T∗
) is locally
asymptotically stable (LAS) when it exist.
case 2 :Assume that f > 0, then if :
R∗
1 − R∗
2 > 1 (12)
then Es = (G∗
,T∗
) is locally asymptotically stable.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 17/27
45. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
Our aim now is to derive a condition on spatial convolution such
that savanna homogeneous steady state Es = (G∗
;T∗
) is locally
asymptotically stable in the case M1 = M2 = 0, but unstable for
some Mi > 0 ; i = 1 ; 2. we set :
a11 = −γGG∗
,
a12 = −γTGG∗
,
a21 = λfTfω′
(G∗
)exp(−pT∗
)T∗
,
a22 = −γT [(1 − Ω)T∗
+ 2Ω(T∗
)2
] + pλfTfω(G∗
)exp(−pT∗
)T∗
,
c = γTΩT∗
(1 − T∗
).
(13)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 18/27
46. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
Our aim now is to derive a condition on spatial convolution such
that savanna homogeneous steady state Es = (G∗
;T∗
) is locally
asymptotically stable in the case M1 = M2 = 0, but unstable for
some Mi > 0 ; i = 1 ; 2. we set :
a11 = −γGG∗
,
a12 = −γTGG∗
,
a21 = λfTfω′
(G∗
)exp(−pT∗
)T∗
,
a22 = −γT [(1 − Ω)T∗
+ 2Ω(T∗
)2
] + pλfTfω(G∗
)exp(−pT∗
)T∗
,
c = γTΩT∗
(1 − T∗
).
(13)
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 18/27
47. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
Proposition 3.4 (linearized system)
Set g(x,t) = G(x,t) − G∗
and h(x,t) = T(x,t) − T∗
two
pertubations around the savanna homogeneous steady state
⎧⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂g
∂t
= DG
∂2
g
∂x2
+ a11 ∫
+∞
−∞
φM1
(x − y)g(y,t)dy+
a12 ∫
+∞
−∞
φM2
(x − y)h(y,t)dy,
∂h
∂t
= DT
∂2
h
∂x2
+ (a22 − c)∫
+∞
−∞
φM2
(x − y)h(y,t)dy + ch + a21g.
(14)
{
λg(k) = −DGk2
g(k) + a11φM1 (k)g(k) + a12φM2 (k)h(k),
λh(k) = −DTk2
h(k) + ch(k) + (a22 − c)φM2 (k)h(k) + a21g(k),
(15)
where k is the wavenumber (k ∈ R).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 19/27
48. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
Proposition 3.4 (linearized system)
Set g(x,t) = G(x,t) − G∗
and h(x,t) = T(x,t) − T∗
two
pertubations around the savanna homogeneous steady state
⎧⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂g
∂t
= DG
∂2
g
∂x2
+ a11 ∫
+∞
−∞
φM1
(x − y)g(y,t)dy+
a12 ∫
+∞
−∞
φM2
(x − y)h(y,t)dy,
∂h
∂t
= DT
∂2
h
∂x2
+ (a22 − c)∫
+∞
−∞
φM2
(x − y)h(y,t)dy + ch + a21g.
(14)
{
λg(k) = −DGk2
g(k) + a11φM1 (k)g(k) + a12φM2 (k)h(k),
λh(k) = −DTk2
h(k) + ch(k) + (a22 − c)φM2 (k)h(k) + a21g(k),
(15)
where k is the wavenumber (k ∈ R).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 19/27
49. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
The characteristic equation of system (15) is :
λ2
− Tr(k,M1,M2)λ + Det(k,M1,M2) = 0, (16)
where :
Tr(k,M1,M2) = −(DG+DT)k2
+a11φM1 (k)+a22φM2 (k)+(1−φM2 (k))c
Det(k,M1,M2) = DGDTk4
− [a22DGφM2 (k)+
a11DTφM1 (k) + cDG(1 − φM2 (k))]k2
+
a11(a22 − c)φM1 (k)φM2 (k) + ca11φM1 (k)−
a12a21φM2 (k).
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 20/27
50. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Linear analysis stability
Proposition 3.5 (Stationary pattern condition)
Consider z1 and z2, (z1 < z2) two positive solutions of the equation
tan(z) = z, such that : µj =
sinzj
zj
< 0, j = 1,2. Then, suppose that :
a11(c − a22)µ1µ2
ca11µ1 − a12a21µ2
< 1. (17)
If :
Mj > MT
j =zj (
DGDT
(a11a22 − ca11)µ1µ2 + ca11µ1 − a12a21µ2
)
1/4
, j = 1,2
(18)
and then we have the appearance of periodic solutions in space in
the neighborhood of savanna equilibrium.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 21/27
51. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
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Conclusion
and Upcoming
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Numerical illustration
Parameter Value of parameter Source
γG 3.1 Tchuinté et al. (2017)
δG 0.1 Tchuinté et al. (2017)
γTG 0.04 Tchuinté et al. (2017)
λfG 0.5 Tchuinté et al. (2017)
γT 1.5 Tchuinté et al. (2017)
δT 0.015 Tchuinté et al. (2017)
λfT 0.7 Tchuinté et al. (2017)
f 0.6 Tchuinté et al. (2017)
DG 1 assumed
DT 1 assumed
p 0.15 Tchuinté et al. (2017)
g0 2 Tchuinté et al. (2017)
TABLE: Value of parameter
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 22/27
52. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Numerical illustration
Our numerical simulation will be made according to Ω values.
If Ω = 5, Es = (0.03;0.9983) then the Turing condition for
instability is
M1 > 12.14m and M2 > 29.47m (19)
We choose for illustration M1 = 15m and M2 = 35m.
If Ω = 0.8, Es = (0.04;0.9944) then :
M1 > 15.97m and M2 > 38.97m (20)
and we choose M1 = 16m and M2 = 45m
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 23/27
53. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Numerical illustration
Our numerical simulation will be made according to Ω values.
If Ω = 5, Es = (0.03;0.9983) then the Turing condition for
instability is
M1 > 12.14m and M2 > 29.47m (19)
We choose for illustration M1 = 15m and M2 = 35m.
If Ω = 0.8, Es = (0.04;0.9944) then :
M1 > 15.97m and M2 > 38.97m (20)
and we choose M1 = 16m and M2 = 45m
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 23/27
54. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Numerical illustration
Our numerical simulation will be made according to Ω values.
If Ω = 5, Es = (0.03;0.9983) then the Turing condition for
instability is
M1 > 12.14m and M2 > 29.47m (19)
We choose for illustration M1 = 15m and M2 = 35m.
If Ω = 0.8, Es = (0.04;0.9944) then :
M1 > 15.97m and M2 > 38.97m (20)
and we choose M1 = 16m and M2 = 45m
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 23/27
55. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Numerical illustration
Our numerical simulation will be made according to Ω values.
If Ω = 5, Es = (0.03;0.9983) then the Turing condition for
instability is
M1 > 12.14m and M2 > 29.47m (19)
We choose for illustration M1 = 15m and M2 = 35m.
If Ω = 0.8, Es = (0.04;0.9944) then :
M1 > 15.97m and M2 > 38.97m (20)
and we choose M1 = 16m and M2 = 45m
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 23/27
56. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Numerical illustration
0 50 100 150 200 250 300
Space
0
0.02
0.04
0.06
0.08
0.1
0.12
G
Profil in final time of Grass in the non local PDE with =5
FINAL TIME GRASS =5000
(a) Profil of grass in space with
M1 = 15m.
0 50 100 150 200 250 300
Space
0
5
10
15
20
25
T
Profil in final time of Trees in the non local PDE with =5
FINAL TIME TREE =5000
(b) Profile of tree in space with M2 = 35m
FIGURE: Illustration of Grass and Tree distribution in space with Ω = 5.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 24/27
57. CARI 2020
Tega II Simon
Rodrigue
Introduction
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Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
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Numerical illustration
0 50 100 150 200 250 300
Space
0
0.05
0.1
0.15
0.2
0.25
0.3
G
Profil in final time of Grass in the non local PDE with =0.8
FINAL TIME GRASS=5000
(a) Profil of grass in space with
M1 = 16m.
0 50 100 150 200 250 300
Space
0
5
10
15
20
25
T
Profil in final time of Trees in the non local PDE with =0.8
FINAL TIME TREE=5000
(b) Profile of tree distribution with
M2 = 45m.
0 50 100 150 200 250 300
Space
0
0.02
0.04
0.06
0.08
0.1
0.12
G
Profil in final time of Grass in the non local PDE with =0.8
FINAL TIME GRASS =50000
(c) Profil of grass in space with
M1 = 16m.
0 50 100 150 200 250 300
Space
0
5
10
15
T
Profil in final time of Trees in the non local PDE with =0.8
FINAL TIME TREE =50000
(d) Profile of tree distribution with
M2 = 45m.
FIGURE: Grass and Tree distribution in space with Ω = 0.8.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 25/27
58. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
Conclusion and Upcoming work
conclusion
In this work we have been able to :
Find a condition on non-local interaction for the apperarence
of steady periodic solution in space.
Illustrate numerically this condition.
Upcoming work
Focus on the appearence of localized structures in space
Explicit introduction in the model of factors linked to
precipation.
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 26/27
59. CARI 2020
Tega II Simon
Rodrigue
Introduction
Model
construction
Mathematical
analysis
Numerical
illustration
Conclusion
and Upcoming
work
END ! ! !
Thanks
For Your Kind Attention
Tega II Simon Rodrigue CARI 2020, Thiès (Sénégal) 27/27