14. Martina Sassone-Corsi M, Nuccio S-P, Liu H, Hernandez D, Vu CT, Takahashi AA,
Edwards RA, Raffatellu M (2016) Microcins mediate competition among Enterobacteriaceae
in the inflamed gut. Nature 540:280–283
15. Petrovskii S, Shigesada N (2001) Some exact solutions of a generalized Fisher equation related
to the problem of biological invasion. Math Biosci 172:73–94
16. Sherratt JA, Lewis MA, Fowler AC (1995) Ecological chaos in the wake of invasion. Proc Natl
Acad Sci U S A 92:2524–2528
17. Shigesada N, Kawasaki K, Teramoto E (1986) Traveling periodic waves in heterogeneous
environments. Theor Popul Biol 30:143–160
18. Perotti JI, Billoni OV, Tamarit FA, Chialvo DR, Cannas SA (2009) Emergent self-organized
complex network topology out of stability constraints. Phys Rev Lett 103:108701
19. Hutchinson GE (1961) The paradox of the plankton. Am Nat 95:137–145
20. Brooker RW, Maestre FT, Callaway RM, Lortie CL, Cavieres LA, Kunstler G et al (2008)
Facilitation in plant communities: the past, the present, and the future. J Ecol 96:18–34
21. van der Heijden MGA, Horton TR (2009) Socialism in soil? The importance of mycorrhizal
fungal networks for facilitation in natural ecosystems. J Ecol 97:1139–1150
22. Pascual-Garcia A, Bastolla U (2017) Mutualism supports biodiversity when the direct competition is weak. Nat Commun 8:14326
23. James A, Pitchford JW, Plank MJ (2012) Disentangling nestedness from models of ecological
complexity. Nature 487:227–230
24. Fassoni AC, Martins ML (2014) Mathematical analysis of a model for plant invasion mediated
by allelopathy. Ecol Complex 18:49–58
25. Strogatz SH (2000) Nonlinear dynamics and chaos: with applications to physics, biology,
chemistry, and engineering. Westview Press, Cambridge, MA
26. Britton N (2003) Essential mathematical biology. Springer, London
27. Fox MD, Fox BD (1986) The susceptibility of natural communities to invasion. In: Groves RH,
Burdon JJ (eds) Ecology of biological invasions: an Australian perspective. Australian Academy of Science, Canberra, pp 97–105
28. Carvalho SA, Martins ML (2018) Invasion waves in the biochemical warfare between living
organisms. Phys Rev E 97:042403
29. Gordon DM, O’Brien CL (2006) Bacteriocin diversity and the frequency of multiple bacteriocin
production in Escherichia coli. Microbiology 152:3239–3244
30. Melchionda D, Pastacaldi E, Perri C, Banerjee M, Venturino E (2018) Social behavior-induced
multistability in minimal competitive ecosystems. J Theor Biol 439:24–38
31. May RM (1973) Stability and complexity in model ecosystems. Princeton University Press,
Princeton
32. Reichenbach T, Mobilia M, Frey E (2007) Mobility promotes and jeopardizes biodiversity in
rock-paper-scissors games. Nature 448:1046–1049
33. Cheng H, Yao N, Huang Z-G, Park J, Do Y, La Y-C (2014) Mesoscopic interactions and species
coexistence in evolutionary game dynamics of cyclic competitions. Sci Rep 4:7486
34. Szabó P, Czárán T, Szabó G (2007) Competing associations in bacterial warfare with two toxins.
J Theor Biol 248:736–744
35. Carvalho SA, Martins ML (2018) Community structures in allelopathic interaction networks: an
eco-evolutionary approach. https://arxiv.org/submit/2491687. Submitted to arXiv 30 Nov 2018
36. Edwards KF et al (2018) Evolutionary stable communities: a framework for understanding the
role of trait evolution in the maintenance of diversity. Ecol Lett. https://doi.org/10.1111/
ele.13142
37. Barabási A-L (2016) Network science. Cambridge University Press, Cambridge, MA
38. May RM (1972) Will a large complex system stable? Nature 238:413
39. Inderjit, Wardle DA, Karban R, Callaway R (2011) The ecosystem and evolutionary contexts of
allelopathy. Tends Ecol Evol 26:655–662
40. Glimm J, Sharp DH (1997) Multiscale science. A challenge for the twenty-first century. SIAM
News 30(4):17–19
20 Biochemical Warfare Between Living Organisms for Survival: Mathematical. . .
503
Edwards RA, Raffatellu M (2016) Microcins mediate competition among Enterobacteriaceae
in the inflamed gut. Nature 540:280–283
15. Petrovskii S, Shigesada N (2001) Some exact solutions of a generalized Fisher equation related
to the problem of biological invasion. Math Biosci 172:73–94
16. Sherratt JA, Lewis MA, Fowler AC (1995) Ecological chaos in the wake of invasion. Proc Natl
Acad Sci U S A 92:2524–2528
17. Shigesada N, Kawasaki K, Teramoto E (1986) Traveling periodic waves in heterogeneous
environments. Theor Popul Biol 30:143–160
18. Perotti JI, Billoni OV, Tamarit FA, Chialvo DR, Cannas SA (2009) Emergent self-organized
complex network topology out of stability constraints. Phys Rev Lett 103:108701
19. Hutchinson GE (1961) The paradox of the plankton. Am Nat 95:137–145
20. Brooker RW, Maestre FT, Callaway RM, Lortie CL, Cavieres LA, Kunstler G et al (2008)
Facilitation in plant communities: the past, the present, and the future. J Ecol 96:18–34
21. van der Heijden MGA, Horton TR (2009) Socialism in soil? The importance of mycorrhizal
fungal networks for facilitation in natural ecosystems. J Ecol 97:1139–1150
22. Pascual-Garcia A, Bastolla U (2017) Mutualism supports biodiversity when the direct competition is weak. Nat Commun 8:14326
23. James A, Pitchford JW, Plank MJ (2012) Disentangling nestedness from models of ecological
complexity. Nature 487:227–230
24. Fassoni AC, Martins ML (2014) Mathematical analysis of a model for plant invasion mediated
by allelopathy. Ecol Complex 18:49–58
25. Strogatz SH (2000) Nonlinear dynamics and chaos: with applications to physics, biology,
chemistry, and engineering. Westview Press, Cambridge, MA
26. Britton N (2003) Essential mathematical biology. Springer, London
27. Fox MD, Fox BD (1986) The susceptibility of natural communities to invasion. In: Groves RH,
Burdon JJ (eds) Ecology of biological invasions: an Australian perspective. Australian Academy of Science, Canberra, pp 97–105
28. Carvalho SA, Martins ML (2018) Invasion waves in the biochemical warfare between living
organisms. Phys Rev E 97:042403
29. Gordon DM, O’Brien CL (2006) Bacteriocin diversity and the frequency of multiple bacteriocin
production in Escherichia coli. Microbiology 152:3239–3244
30. Melchionda D, Pastacaldi E, Perri C, Banerjee M, Venturino E (2018) Social behavior-induced
multistability in minimal competitive ecosystems. J Theor Biol 439:24–38
31. May RM (1973) Stability and complexity in model ecosystems. Princeton University Press,
Princeton
32. Reichenbach T, Mobilia M, Frey E (2007) Mobility promotes and jeopardizes biodiversity in
rock-paper-scissors games. Nature 448:1046–1049
33. Cheng H, Yao N, Huang Z-G, Park J, Do Y, La Y-C (2014) Mesoscopic interactions and species
coexistence in evolutionary game dynamics of cyclic competitions. Sci Rep 4:7486
34. Szabó P, Czárán T, Szabó G (2007) Competing associations in bacterial warfare with two toxins.
J Theor Biol 248:736–744
35. Carvalho SA, Martins ML (2018) Community structures in allelopathic interaction networks: an
eco-evolutionary approach. https://arxiv.org/submit/2491687. Submitted to arXiv 30 Nov 2018
36. Edwards KF et al (2018) Evolutionary stable communities: a framework for understanding the
role of trait evolution in the maintenance of diversity. Ecol Lett. https://doi.org/10.1111/
ele.13142
37. Barabási A-L (2016) Network science. Cambridge University Press, Cambridge, MA
38. May RM (1972) Will a large complex system stable? Nature 238:413
39. Inderjit, Wardle DA, Karban R, Callaway R (2011) The ecosystem and evolutionary contexts of
allelopathy. Tends Ecol Evol 26:655–662
40. Glimm J, Sharp DH (1997) Multiscale science. A challenge for the twenty-first century. SIAM
News 30(4):17–19
20 Biochemical Warfare Between Living Organisms for Survival: Mathematical. . .
503