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May  2017, 22(3): 859-875. doi: 10.3934/dcdsb.2017043

Persistence in phage-bacteria communities with nested and one-to-one infection networks

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA

Received  August 2015 Revised  January 2016 Published  January 2017

We show that a bacteria and bacteriophage system with either a perfectly nested or a one-to-one infection network is permanent, a.k.a uniformly persistent, provided that bacteria that are superior competitors for nutrient devote the least to defence against infection and the virus that are the most efficient at infectinghost have the smallest host range.By ensuring that the density-dependent reduction in bacterial growth rates are independent of bacterial strain, we are able to arrive at the permanence conclusion sought by Jover et al [3].The same permanence results hold for the one-to-one infection network considered by Thingstad [9] but without virus efficiency ordering.In some special cases, we show the global stability for the nested infection network, and obtain restrictions on the global dynamics for the one-to-one network.

Citation: Dan A. Korytowski, Hal L. Smith. Persistence in phage-bacteria communities with nested and one-to-one infection networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 859-875. doi: 10.3934/dcdsb.2017043
References:
[1]

J. Hale, Ordinary Differential Equations Robert E. Krieger Publishing Co. , Malabar, Fl, 1980. Google Scholar

[2]

J. Hofbauer and K. Sigmund, Evolutionary Games Cambridge Univ. Press, 1998. Google Scholar

[3]

L. F. JoverM. H. Cortez and J. S. Weitz, Mechanisms of multi-strain coexistence in host-phage systems with nested infection networks, Journal of Theoretical Biology, 332 (2013), 65-77.  doi: 10.1016/j.jtbi.2013.04.011.  Google Scholar

[4]

D. Korytowski and H. L. Smith, How nested and monogamous infection networks in host-phage communities come to be, Theoretical Ecology, 8 (2015), 111-120.  doi: 10.1007/s12080-014-0236-6.  Google Scholar

[5]

D. Korytowski and H. L. Smith, Persistence in Phage-Bacteria Communities with Nested and One-to-One Infection Networks arXiv: 1505.03827 [q-bio. PE]. Google Scholar

[6]

H. Smith and H. Thieme, Dynamical Systems and Population Persistence GSM 118, Amer. Math. Soc. , Providence R. I. , 2011. Google Scholar

[7]

H. Smith and P. Waltman, The Theory of the Chemostat Cambridge Univ. Press, 1995. Google Scholar

[8]

H. R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.  Google Scholar

[9]

T. F. Thingstad, Elements of a theory for the mechanisms controlling abundance, diversity, and biogeochemical role of lytic bacterial viruses in aquatic systems, Limnol. Oceanogr., 45 (2000), 1320-1328.  doi: 10.4319/lo.2000.45.6.1320.  Google Scholar

show all references

References:
[1]

J. Hale, Ordinary Differential Equations Robert E. Krieger Publishing Co. , Malabar, Fl, 1980. Google Scholar

[2]

J. Hofbauer and K. Sigmund, Evolutionary Games Cambridge Univ. Press, 1998. Google Scholar

[3]

L. F. JoverM. H. Cortez and J. S. Weitz, Mechanisms of multi-strain coexistence in host-phage systems with nested infection networks, Journal of Theoretical Biology, 332 (2013), 65-77.  doi: 10.1016/j.jtbi.2013.04.011.  Google Scholar

[4]

D. Korytowski and H. L. Smith, How nested and monogamous infection networks in host-phage communities come to be, Theoretical Ecology, 8 (2015), 111-120.  doi: 10.1007/s12080-014-0236-6.  Google Scholar

[5]

D. Korytowski and H. L. Smith, Persistence in Phage-Bacteria Communities with Nested and One-to-One Infection Networks arXiv: 1505.03827 [q-bio. PE]. Google Scholar

[6]

H. Smith and H. Thieme, Dynamical Systems and Population Persistence GSM 118, Amer. Math. Soc. , Providence R. I. , 2011. Google Scholar

[7]

H. Smith and P. Waltman, The Theory of the Chemostat Cambridge Univ. Press, 1995. Google Scholar

[8]

H. R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026.  Google Scholar

[9]

T. F. Thingstad, Elements of a theory for the mechanisms controlling abundance, diversity, and biogeochemical role of lytic bacterial viruses in aquatic systems, Limnol. Oceanogr., 45 (2000), 1320-1328.  doi: 10.4319/lo.2000.45.6.1320.  Google Scholar

Figure 5.1.  NIN simulation with $n=3$ and parameters: $r_1=6,r_2=5,r_3=3,e_1=3,e_2=2,e_3=1,n_1=n_2=n_3=1,a_1=1,a_2=3,a_3=0.5$
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