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Cari 2020: A minimalistic model of spatial structuration of humid savanna vegetation

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Tega II Simon Rodrigue

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Cari 2020: A minimalistic model of spatial structuration of humid savanna vegetation

  1. 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. 2. CARI 2020 Tega II Simon Rodrigue Introduction Model construction Mathematical analysis Numerical illustration Conclusion and Upcoming work Outline 1) Introduction
  3. 3. Why study spatial structuring of humid savanna vegatation.
  4. 4. Mathematical modelling of Savanna Dynamic with Deterministic models.
  5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 :
  17. 17. highlight the fire resistance strategies of trees.
  18. 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. 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 :
  20. 20. highlight the fire resistance strategies of trees.
  21. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 41. 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
  42. 42. CARI 2020 Tega II Simon Rodrigue Introduction Model construction Mathematical analysis Numerical illustration 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. 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. 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. 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. 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. 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. 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. 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. 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. 51. CARI 2020 Tega II Simon Rodrigue Introduction Model construction Mathematical analysis Numerical illustration Conclusion and Upcoming work 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. 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. 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. 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. 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. 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. 57. 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 0.05 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. 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. 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

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