Effects of indirect facilitation through grazing on dryland ecosystems

In this slideshow, I present some preliminary results of a vegetation dynamic model.

This model is an extension of two precedent models (Kéfi et al., 2007; Schneider & Kéfi, 2016, full citation in the slides).

I notably show that indirect facilitation through grazing can (i) maintain stable coexistence between two species in competition and that (ii) it may shift the transition type of the beneficiary species from catastrophic to linear.

Transcript

1. Introduction Model Coexistence Stability Conclusion Effects of indirect facilitation through grazing on dryland ecosystems Evidences from a pair approximation model Alain Danet, Florian Dirk Schneider & Sonia K´ efi 1th June, 2018 Models in Ecology & Evolution, Montpellier Licence Creative Commons cbea https://alaindanet.github.io/ 1
2. Introduction
3. Introduction Model Coexistence Stability Conclusion 40% of the terrestrial surface 1/3 of the world population 2
4. Introduction Model Coexistence Stability Conclusion Catastrophic shifts Vegetation cover Environmental stress / grazing K´ efi et al. (2007a) 3
5. Introduction Model Coexistence Stability Conclusion Catastrophic shifts Vegetation cover Environmental stress / grazing K´ efi et al. (2007a) 3
6. Introduction Model Coexistence Stability Conclusion Catastrophic shifts Vegetation cover Environmental stress / grazing K´ efi et al. (2007a) 3
7. Introduction Model Coexistence Stability Conclusion Catastrophic shifts Vegetation cover Environmental stress / grazing Bistability K´ efi et al. (2007a) 3
8. Introduction Model Coexistence Stability Conclusion Direct facilitation Improving local conditions Resources Erosion Evapotranspiration Rietkerk et al. (2004); K´ efi et al. (2007b); Callaway (2007) 4 Feedback loop Vegetation Water + +
9. Introduction Model Coexistence Stability Conclusion Indirect facilitation through grazing Protegee Callaway (2007); Brooker et al. (2008); Smit et al. (2009); Lortie et al. (2016) 5
10. Introduction Model Coexistence Stability Conclusion Indirect facilitation through grazing - Protegee Callaway (2007); Brooker et al. (2008); Smit et al. (2009); Lortie et al. (2016) 5
11. Introduction Model Coexistence Stability Conclusion Indirect facilitation through grazing - - Nurse Protegee Callaway (2007); Brooker et al. (2008); Smit et al. (2009); Lortie et al. (2016) 5
12. Introduction Model Coexistence Stability Conclusion Indirect facilitation through grazing - - + Nurse Protegee Callaway (2007); Brooker et al. (2008); Smit et al. (2009); Lortie et al. (2016) 5
13. Introduction Model Coexistence Stability Conclusion Research question What are the effects of indirect facilitation on dryland ecosystems ? Verwijmeren et al. (2013); Schneider and K´ efi (2016) 6
14. Model
15. Introduction Model Coexistence Stability Conclusion A dryland ecosystem into a lattice Cellular automata K´ efi et al. (2007b); Schneider and K´ efi (2016) 7
16. Introduction Model Coexistence Stability Conclusion Cell states Empty : 0 Nurse : +N Degraded : − Protegee : +P 8
17. Introduction Model Coexistence Stability Conclusion Cell states Empty : 0 Nurse : +N Degraded : − Protegee : +P Degradation Regeneration 8
18. Introduction Model Coexistence Stability Conclusion Cell states Empty : 0 Nurse : +N Degraded : − Protegee : +P Degradation Mortality Regeneration Mortality 8
19. Introduction Model Coexistence Stability Conclusion Cell states Empty : 0 Nurse : +N Degraded : − Protegee : +P Colonization Colonization Degradation Mortality Regeneration Mortality 8
20. Introduction Model Coexistence Stability Conclusion Transition equations • Mortality : • Degradation : • Regeneration : 9
21. Introduction Model Coexistence Stability Conclusion Transition equations • Mortality : wn,o =m wp,o =m • Degradation : • Regeneration : 9
22. Introduction Model Coexistence Stability Conclusion Transition equations • Mortality : wn,o =m wp,o =m • Degradation : wo,− =d • Regeneration : 9
23. Introduction Model Coexistence Stability Conclusion Transition equations • Mortality : wn,o =m wp,o =m • Degradation : wo,− =d • Regeneration : w−,o =r + fq+|− qi|j : proportion of i cells around j cells (local neighborhood) 9
24. Introduction Model Coexistence Stability Conclusion Colonization equations - Nurse colonization : wo,n =(δρn + (1 − δ)qn|o )(b − cρ+ − γ) - Protegee colonization : wo,p =(δρp + (1 − δ)qp|o )(b − cρ+ − g(1 − p)) Global dispersal 10
25. Introduction Model Coexistence Stability Conclusion Colonization equations - Nurse colonization : wo,n =(δρn + (1 − δ)qn|o )(b − cρ+ − γ) - Protegee colonization : wo,p =(δρp + (1 − δ)qp|o )(b − cρ+ − g(1 − p)) Local dispersal 10
26. Introduction Model Coexistence Stability Conclusion Colonization equations - Nurse colonization : wo,n =(δρn + (1 − δ)qn|o )(b − cρ+ − γ) - Protegee colonization : wo,p =(δρp + (1 − δ)qp|o )(b − cρ+ − g(1 − p)) Maximum survival & germination 10
27. Introduction Model Coexistence Stability Conclusion Colonization equations - Nurse colonization : wo,n =(δρn + (1 − δ)qn|o )(b − cρ+ − γ) - Protegee colonization : wo,p =(δρp + (1 − δ)qp|o )(b − cρ+ − g(1 − p)) Global competition for resources 10
28. Introduction Model Coexistence Stability Conclusion Colonization equations - Nurse colonization : wo,n =(δρn + (1 − δ)qn|o )(b − cρ+ − γ) - Protegee colonization : wo,p =(δρp + (1 − δ)qp|o )(b − cρ+ − g(1 − p)) Cost of defense against grazing 10
29. Introduction Model Coexistence Stability Conclusion Colonization equations - Nurse colonization : wo,n =(δρn + (1 − δ)qn|o )(b − cρ+ − γ) - Protegee colonization : wo,p =(δρp + (1 − δ)qp|o )(b − cρ+ − g(1 − p)) Grazing and indirect facilitation p(γ, qn|o ) 10
30. Introduction Model Coexistence Stability Conclusion Pair approximation modeling Approximation of spatial clustering 11
31. Introduction Model Coexistence Stability Conclusion Pair approximation modeling Approximation of spatial clustering ... Densities of cell pairs 11
32. Coexistence
33. Introduction Model Coexistence Stability Conclusion Indirect facilitation increases coexistence coexistence nurse protégée extinct Without indirect facilitation 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 12
34. Introduction Model Coexistence Stability Conclusion Indirect facilitation increases coexistence coexistence nurse protégée extinct 0.5 0.6 0.7 0.8 0.9 1.0 Environmental quality With indirect facilitation Without indirect facilitation 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 12
35. Introduction Model Coexistence Stability Conclusion Indirect facilitation increases coexistence coexistence nurse protégée extinct 0.5 0.6 0.7 0.8 0.9 1.0 Environmental quality With indirect facilitation Without indirect facilitation 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality But it is costly for the nurse 12
36. Stability
37. Introduction Model Coexistence Stability Conclusion Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 13
38. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 13
39. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
40. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
41. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
42. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
43. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
44. Introduction Model Coexistence Stability Conclusion Without indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
45. Introduction Model Coexistence Stability Conclusion With indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 13
46. Introduction Model Coexistence Stability Conclusion With indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
47. Introduction Model Coexistence Stability Conclusion With indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 13
48. Introduction Model Coexistence Stability Conclusion With indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 1. Bistability area decreases 13
49. Introduction Model Coexistence Stability Conclusion With indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 1. Bistability area decreases 2. Transition shifts from catastrophic to linear 3. Nurse transition is more sudden 13
50. Introduction Model Coexistence Stability Conclusion With indirect facilitation Environmental quality 0.00 0.25 0.50 0.75 1.00 0.2 0.4 0.6 0.8 1.0 rho 0.0 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 Grazing Environmental quality 0.5 0.6 0.7 0.8 0.9 1.0 Nurse Protegee Starting cover: Low High 1. Bistability area decreases 2. Transition shifts from catastrophic to linear 3. Nurse transition is more sudden 13
51. Conclusion
52. Introduction Model Coexistence Stability Conclusion Take home message Indirect facilitation . . . 14
53. Introduction Model Coexistence Stability Conclusion Take home message Indirect facilitation . . . 1. increases coexistence area between two species in competition 14
54. Introduction Model Coexistence Stability Conclusion Take home message Indirect facilitation . . . 1. increases coexistence area between two species in competition • smooth the transition between the two species • is costly for the nurse 14
55. Introduction Model Coexistence Stability Conclusion Take home message Indirect facilitation . . . 1. increases coexistence area between two species in competition • smooth the transition between the two species • is costly for the nurse 2. alters stability of a dryland ecosystem 14
56. Introduction Model Coexistence Stability Conclusion Take home message Indirect facilitation . . . 1. increases coexistence area between two species in competition • smooth the transition between the two species • is costly for the nurse 2. alters stability of a dryland ecosystem • may modify the transition type of the beneficiary • may induce a more sudden shift for the nurse 14
57. Introduction Model Coexistence Stability Conclusion Perspectives Effects of species interactions on ecosystem properties 15
58. Introduction Model Coexistence Stability Conclusion Thank you for listening! Coauthors The Florian D. Schneider Sonia Kéfi & 16
59. R´ ef´ erences Brooker, R. W., Maestre, F. T., Callaway, R. M., Lortie, C. L., Cavieres, L. A., Kunstler, G., Liancourt, P., Tielb¨ orger, K., Travis, J. M. J., Anthelme, F., Armas, C., Coll, L., Corcket, E., Delzon, S., Forey, E., Kikvidze, Z., Olofsson, J., Pugnaire, F., Quiroz, C. L., Saccone, P., Schiffers, K., Seifan, M., Touzard, B., and Michalet, R. (2008). Facilitation in plant communities : the past, the present, and the future. Journal of Ecology, 96(1) :18–34. Callaway, R. M. (2007). Positive interactions and interdependence in plant communities. Springer, Dordrecht. K´ efi, S., Rietkerk, M., Alados, C. L., Pueyo, Y., Papanastasis, V. P., ElAich, A., and de Ruiter, P. C. (2007a). Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems. Nature, 449(7159) :213–217. K´ efi, S., Rietkerk, M., van Baalen, M., and Loreau, M. (2007b). Local facilitation, bistability and transitions in arid ecosystems. Theoretical Population Biology, 71(3) :367–379. Lortie, C. J., Filazzola, A., and Sotomayor, D. A. (2016). Functional assessment of animal interactions with shrub-facilitation complexes : a formal synthesis and conceptual framework. Functional Ecology, 30(1) :41–51. Rietkerk, M., Dekker, S. C., Ruiter, P. C. d., and Koppel, J. v. d. (2004). Self-Organized Patchiness and Catastrophic Shifts in Ecosystems. Science, 305(5692) :1926–1929. Schneider, F. D. and K´ efi, S. (2016). Spatially heterogeneous pressure raises risk of catastrophic shifts. Theoretical Ecology, 9(2) :207–217. Smit, C., Rietkerk, M., and Wassen, M. J. (2009). Inclusion of biotic stress (consumer pressure) alters predictions from the stress gradient hypothesis. Journal of Ecology, 97(6) :1215–1219. Verwijmeren, M., Rietkerk, M., Wassen, M. J., and Smit, C. (2013). Interspecific facilitation and critical transitions in arid ecosystems. Oikos, 122(3) :341–347.
60. textwidth in cm : 10.79846cm textheight in cm : 8.2662cm paperwidth in cm : 12.79817cm paperheight in cm : 9.59863cm
61. Les annexes
62. Equations i dρnn dt =2ρnowo,n − 2ρnnm dρn− dt =ρn0d + ρ0−(δρn + (1 − δ) z − 1 z ρn0 ρ0 )(b − cρ+ − γ) − ρn−m − ρn−(r + f ( 1 z + z − 1 z ( ρn0 ρ0 + ρp0 ρ0 )))
63. Equations ii dρnp dt =ρn0(δρp + (1 − δ) z − 1 z ρp0 ρ0 )(b − cρ+ − g(1 − p)) + ρp0(δρn + (1 − δ) z − 1 z ρn0 ρ0 )(b − cρ+ − γ) − 2ρnpm dρpp dt =ρp0(δρp + 1 − δ z + (1 − δ) z − 1 z ρp0 ρ0 )(b − cρ+ − g(1 − p)) − 2ρppm
64. Equations iii dρp− dt =ρp0d + ρ0−(δρp + (1 − δ) z − 1 z ρp0 ρ0 )(b − cρ+ − g(1 − p)) − ρp−m − ρp−(r + f ( 1 z + z − 1 z ( ρn0 ρ0 + ρp0 ρ0 ))) dρ−− dt =2ρ0−d − 2ρ−−(r + f z − 1 z ( ρn0 ρ0 + ρp0 ρ0 ))
65. Equations iv dρp dt =ρ0(δρp + (1 − δ) ρp0 ρ0 )(b − cρ+ − g(1 − p)) − ρpm dρn dt =ρ0(δρn + (1 − δ) ρn0 ρ0 )(b − cρ+ − γ) − ρnm dρ− dt =ρ0d − ρ−(r + f ( ρn0 ρ0 + ρp0 ρ0 ))
66. Pair approximations • Neighborhood of a cell : qi|j = ρij ρj • Neighborhood of a cell pair : qi|ij = ρiji ρij Depends on triplet densities ! (And etc. . . ) • Pair approximation : qi|ij ≈ qi|j = ρij ρj
67. Credits Dryland map by FAO (2014) : Link Dryland pictures by Florian Schneider Goat by Simon Li from the Noun Project Flower by Marek Polakovic from the Noun Project Shrub by Laymik from the Noun Project