August  2017, 14(4): 953-973. doi: 10.3934/mbe.2017050

A tridiagonal patch model of bacteria inhabiting a Nanofabricated landscape

1. 

Department of Mathematics, The University of Miami, Coral Gables, FL 33124, USA

2. 

Department of Public Health Division of Biostatistics, Miller School of Medicine, The University of Miami, Miami, FL 33136, USA

* Corresponding author: RSC

Received  July 2016 Accepted  January 2017 Published  February 2017

In this paper we employ a discrete-diffusion modeling framework to examine a system inspired by the nano-ecology experiments on the bacterium Escherichia coli reported upon in Keymer et al. (2006). In these experiments, the bacteria inhabit a linear array of 85" microhabitat patches (MHP's)", linked by comparatively thinner corridors through which bacteria may pass between adjacent MHP's. Each MHP is connected to its own source of nutrient substrate, which flows into the MHP at a rate that can be controlled in the experiment. Logistic dynamics are assumed within each MHP, and nutrient substrate flow determines the prediction of the within MHP dynamics in the absence of bacteria dispersal between patches. Patches where the substrate flow rate is sufficiently high sustain the bacteria in the absence of between patch movement and may be regarded as sources, while those with insufficient substrate flow lead to the extinction of the bacteria in the within patch environment and may be regarded as sinks. We examine the role of dispersal in determining the predictions of the model under source-sink dynamics.

Citation: Robert Stephen Cantrell, Brian Coomes, Yifan Sha. A tridiagonal patch model of bacteria inhabiting a Nanofabricated landscape. Mathematical Biosciences & Engineering, 2017, 14 (4) : 953-973. doi: 10.3934/mbe.2017050
References:
[1]

K.J. Brown and S.S. Lin, On the existence of positive solutions for an eigenvalue problem with an indefinite weight function, Journal of Mathematical Analysis and Applications, 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.

[3]

F. CentlerI. Fetzer and M. Thullner, Modeling population patterns of chemotactic bacteria in homogeneous porous media, Journal of Theoretical Biology, 287 (2011), 82-91.  doi: 10.1016/j.jtbi.2011.07.024.

[4]

B. Fiedler and T. Gedeon, A Lyapunov function for tridiagonal competitive-cooperative systems, SIAM Journal on Mathematical Analysis, 30 (1999), 469-478.  doi: 10.1137/S0036141097316147.

[5]

J.K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 20 (1989), 388-395.  doi: 10.1137/0520025.

[6]

J.E. KeymerP. GalajdaC. MuldoonS. Park and R.H. Austin, Bacterial metapopulations in nanofabricated landscapes, Proceedings of the National Academy of Sciences, 103 (2006), 17290-17295.  doi: 10.1073/pnas.0607971103.

[7]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Mathematische Annalen, 258 (1982), 459-470.  doi: 10.1007/BF01453979.

[8]

J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM Journal on Mathematical Analysis, 15 (1984), 530-534.  doi: 10.1137/0515040.

[9]

H.R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAM Journal on Mathematical Analysis, 24 (1993), 407-435.  doi: 10.1137/0524026.

show all references

References:
[1]

K.J. Brown and S.S. Lin, On the existence of positive solutions for an eigenvalue problem with an indefinite weight function, Journal of Mathematical Analysis and Applications, 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.

[3]

F. CentlerI. Fetzer and M. Thullner, Modeling population patterns of chemotactic bacteria in homogeneous porous media, Journal of Theoretical Biology, 287 (2011), 82-91.  doi: 10.1016/j.jtbi.2011.07.024.

[4]

B. Fiedler and T. Gedeon, A Lyapunov function for tridiagonal competitive-cooperative systems, SIAM Journal on Mathematical Analysis, 30 (1999), 469-478.  doi: 10.1137/S0036141097316147.

[5]

J.K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 20 (1989), 388-395.  doi: 10.1137/0520025.

[6]

J.E. KeymerP. GalajdaC. MuldoonS. Park and R.H. Austin, Bacterial metapopulations in nanofabricated landscapes, Proceedings of the National Academy of Sciences, 103 (2006), 17290-17295.  doi: 10.1073/pnas.0607971103.

[7]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Mathematische Annalen, 258 (1982), 459-470.  doi: 10.1007/BF01453979.

[8]

J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM Journal on Mathematical Analysis, 15 (1984), 530-534.  doi: 10.1137/0515040.

[9]

H.R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAM Journal on Mathematical Analysis, 24 (1993), 407-435.  doi: 10.1137/0524026.

Figure 1.  (Experiment 1) We illustrate the role of increasing the bacterial self-aggregation parameter $\gamma$ in Experiment 1. Here the value of $\gamma$ is: $(a)$ $0$, $(b)$ $4$, $(c)$ $10$ and $(d)$ $50$.
Table 1.  (Experiment 2) We display equilibrium values for the model (6) with $7$ MHP's. Values of all parameters except $\beta_{3}$ and $\beta_{5}$ are fixed as in the text. Values for $(\beta_{3},\beta_{5})$ for each experiment are: $(a)$ $(0.3, 0.14)$, $(b)$ $(0.26, 0.18)$, $(c)$ $(0.24, 0.2)$, $(d)$ $(0.2, 0.24)$, $(e)$ $(0.16, 0.28)$, $(f)$ $(0.15, 0.29)$, $(g)$ $(0.14, 0.3)$, $(h)$ $(0.14, 0.39)$.
Patch Number
1234567
a0.00005890.0003310.001870.01340.001390.0002080.0000305
b0.00005840.0003070.001720.01330.001480.0002190.0000324
c0.00005300.0002970.001650.01330.001530.0002260.0000333
d0.00004970.0002780.001540.01330.001640.0002420.0000356
e0.00004700.0002630.001440.01330.001780.0002590.0000382
f0.00004630.0002590.001420.01340.001810.0002640.0000389
g0.00004570.0002560.001400.01340.001850.0002690.0000397
h0.00004720.0002640.001450.01450.002290.0003260.0000478
Patch Number
1234567
a0.00005890.0003310.001870.01340.001390.0002080.0000305
b0.00005840.0003070.001720.01330.001480.0002190.0000324
c0.00005300.0002970.001650.01330.001530.0002260.0000333
d0.00004970.0002780.001540.01330.001640.0002420.0000356
e0.00004700.0002630.001440.01330.001780.0002590.0000382
f0.00004630.0002590.001420.01340.001810.0002640.0000389
g0.00004570.0002560.001400.01340.001850.0002690.0000397
h0.00004720.0002640.001450.01450.002290.0003260.0000478
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