exploring spatial pattern formation using a simple individual-based model
TRANSCRIPT
![Page 1: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/1.jpg)
Exploring microbial patterns formation using a simple IBM
Exploring microbial patterns formation using asimple IBM
Nabil Mabrouk
www.cemagref.fr
15 decembre, 2009
![Page 2: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/2.jpg)
Exploring microbial patterns formation using a simple IBM
Introduction
Introduction
Microscopic observation of microbial systems reveals adiversity of spatial patterns
![Page 3: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/3.jpg)
Exploring microbial patterns formation using a simple IBM
Introduction
Introduction
Microscopic observation of microbial systems reveals adiversity of spatial patterns
![Page 4: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/4.jpg)
Exploring microbial patterns formation using a simple IBM
Introduction
Introduction
Our aim: investigate how these large-scale patterns emerge
Our approach: individual-based modeling
Represent the individuals explicitlySimulate the pattern formation under different conditions
![Page 5: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/5.jpg)
Exploring microbial patterns formation using a simple IBM
Introduction
Introduction
Our aim: investigate how these large-scale patterns emerge
Our approach: individual-based modeling
Represent the individuals explicitlySimulate the pattern formation under different conditions
![Page 6: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/6.jpg)
Exploring microbial patterns formation using a simple IBM
Introduction
Introduction
Our aim: investigate how these large-scale patterns emerge
Our approach: individual-based modeling
Represent the individuals explicitly
Simulate the pattern formation under different conditions
![Page 7: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/7.jpg)
Exploring microbial patterns formation using a simple IBM
Introduction
Introduction
Our aim: investigate how these large-scale patterns emerge
Our approach: individual-based modeling
Represent the individuals explicitlySimulate the pattern formation under different conditions
![Page 8: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/8.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
Model description
Simple is beautiful, and necessary (Deffuant et al., 2003)
![Page 9: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/9.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 10: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/10.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 11: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/11.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability d
birth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 12: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/12.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability d
birth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 13: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/13.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 14: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/14.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 15: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/15.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birth
b = d = constant
mean-field limit (for large N):
![Page 16: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/16.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):
![Page 17: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/17.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
2D domain with individualsrepresented as point particles
Two processes:
death with a probability dbirth with a probability b
We are interested in the case:
wb << L : local birthb = d = constant
mean-field limit (for large N):dNdt = (b − d)N
![Page 18: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/18.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
Simulation with wb/L = 0.015
Figure: t = 0
![Page 19: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/19.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
Simulation with wb/L = 0.015
Figure: t = 400
![Page 20: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/20.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
Simulation with wb/L = 0.1
Figure: t = 400
![Page 21: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/21.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
Two processes:
death with a probability di ,i = 1..Nbirth with a probability b
We are interested in the case:
wb << L : local birthbirth probability b isconstant
death probabilities dependon the neighborhood (thepattern)
di = d1 + d2∑
j Kd
(||xi−xj ||
wb
)wb << wd , b > d1 and d2 > 0
![Page 22: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/22.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
Two processes:
death with a probability di ,i = 1..Nbirth with a probability b
We are interested in the case:
wb << L : local birthbirth probability b isconstantdeath probabilities dependon the neighborhood (thepattern)
di = d1 + d2∑
j Kd
(||xi−xj ||
wb
)wb << wd , b > d1 and d2 > 0
![Page 23: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/23.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
Two processes:
death with a probability di ,i = 1..Nbirth with a probability b
We are interested in the case:
wb << L : local birthbirth probability b isconstantdeath probabilities dependon the neighborhood (thepattern)
di = d1 + d2∑
j Kd
(||xi−xj ||
wb
)
wb << wd , b > d1 and d2 > 0
![Page 24: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/24.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
A simple birth-death model
Overview:
Two processes:
death with a probability di ,i = 1..Nbirth with a probability b
We are interested in the case:
wb << L : local birthbirth probability b isconstantdeath probabilities dependon the neighborhood (thepattern)
di = d1 + d2∑
j Kd
(||xi−xj ||
wb
)wb << wd , b > d1 and d2 > 0
![Page 25: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/25.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
Simulation with wb/L = 0.015 and wd >> wb
Figure: t = 0
![Page 26: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/26.jpg)
Exploring microbial patterns formation using a simple IBM
A simple birth-death model
Simulation with wb/L = 0.015 and wd >> wb
Figure: t = 800
![Page 27: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/27.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
A birth-death model with motility
Overview:
Three processes:
death with a probability di ,i = 1..Nbirth with a probability bmotility with a probabilitymi , i = 1..N
We are interested in the case:
motility probabilities dependon the neighborhood
mi = m1−m2∑
j Kv
(||xi−xj ||
wv
)
![Page 28: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/28.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
A birth-death model with motility
Overview:
Three processes:
death with a probability di ,i = 1..Nbirth with a probability bmotility with a probabilitymi , i = 1..N
We are interested in the case:
motility probabilities dependon the neighborhood
mi = m1−m2∑
j Kv
(||xi−xj ||
wv
)
![Page 29: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/29.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
A birth-death model with motility
Overview:
Three processes:
death with a probability di ,i = 1..Nbirth with a probability bmotility with a probabilitymi , i = 1..N
We are interested in the case:
motility probabilities dependon the neighborhood
mi = m1−m2∑
j Kv
(||xi−xj ||
wv
)
![Page 30: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/30.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
A birth-death model with motility
Overview:
Three processes:
death with a probability di ,i = 1..Nbirth with a probability bmotility with a probabilitymi , i = 1..N
We are interested in the case:
motility probabilities dependon the neighborhood
mi = m1−m2∑
j Kv
(||xi−xj ||
wv
)
![Page 31: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/31.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
A birth-death model with motility
Overview:
Three processes:
death with a probability di ,i = 1..Nbirth with a probability bmotility with a probabilitymi , i = 1..N
We are interested in the case:
motility probabilities dependon the neighborhood
mi = m1−m2∑
j Kv
(||xi−xj ||
wv
)
![Page 32: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/32.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
Parameters
9 parameters:
wb, wd , wm, wv
b, d1, d2, m1 and m2
Additional assumptions:
wb (birth) << wd (death)wm (mobility) >> wb (birth)wv (”viscosity’) > wd (death)b >> d1 m1 = 1.0 and d2, m2 > 0
![Page 33: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/33.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
Simulation results
Figure: t = 0
![Page 34: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/34.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
Simulation results
Figure: t = 800
![Page 35: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/35.jpg)
Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
Are these patterns realistic?
Figure: (Xavier et al., 2009) Fluorescent microscopy of yellow[U+FB02]uorescent protein-labeled biofilm shows cells in spatial patternswith holes, labyrinths, and wormlike shapes.
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Exploring microbial patterns formation using a simple IBM
Birth-death model with motility
Are these patterns realistic?
Figure: (Xavier et al., 2009) Continuous variation of spatial patternsacross the surface of the coverslip is produced by the systematic variationof nutrient concentration. This image is a montage of four contiguousphase-contrast microscopy images.
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Exploring microbial patterns formation using a simple IBM
Conclusion
”A change without pattern is beyond Science” (Zeide, 1991)
Experimental data contains: meaningful pattern andmisleading noise
IBM (modeling) can help in extracting patterns andunderstanding how they form and impact the population
Perspectives ...
![Page 38: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/38.jpg)
Exploring microbial patterns formation using a simple IBM
Conclusion
”A change without pattern is beyond Science” (Zeide, 1991)
Experimental data contains: meaningful pattern andmisleading noise
IBM (modeling) can help in extracting patterns andunderstanding how they form and impact the population
Perspectives ...
![Page 39: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/39.jpg)
Exploring microbial patterns formation using a simple IBM
Conclusion
”A change without pattern is beyond Science” (Zeide, 1991)
Experimental data contains: meaningful pattern andmisleading noise
IBM (modeling) can help in extracting patterns andunderstanding how they form and impact the population
Perspectives ...
![Page 40: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/40.jpg)
Exploring microbial patterns formation using a simple IBM
Conclusion
”A change without pattern is beyond Science” (Zeide, 1991)
Experimental data contains: meaningful pattern andmisleading noise
IBM (modeling) can help in extracting patterns andunderstanding how they form and impact the population
Perspectives ...
![Page 41: Exploring spatial pattern formation using a simple individual-based model](https://reader031.vdocuments.site/reader031/viewer/2022022202/587d02f01a28ab1e7e8b6cf3/html5/thumbnails/41.jpg)
Exploring microbial patterns formation using a simple IBM
Conclusion
The end!