# American Institute of Mathematical Sciences

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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 and 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. [2] J. Hofbauer and K. Sigmund, Evolutionary Games Cambridge Univ. Press, 1998. [3] L. F. Jover, M. 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. [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. [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]. [6] H. Smith and H. Thieme, Dynamical Systems and Population Persistence GSM 118, Amer. Math. Soc. , Providence R. I. , 2011. [7] H. Smith and P. Waltman, The Theory of the Chemostat Cambridge Univ. Press, 1995. [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. [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.

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##### References:
 [1] J. Hale, Ordinary Differential Equations Robert E. Krieger Publishing Co. , Malabar, Fl, 1980. [2] J. Hofbauer and K. Sigmund, Evolutionary Games Cambridge Univ. Press, 1998. [3] L. F. Jover, M. 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. [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. [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]. [6] H. Smith and H. Thieme, Dynamical Systems and Population Persistence GSM 118, Amer. Math. Soc. , Providence R. I. , 2011. [7] H. Smith and P. Waltman, The Theory of the Chemostat Cambridge Univ. Press, 1995. [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. [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.
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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