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B1 Exclusion or bistability in a two-strain community. a The isoclines for the sensitive (blue) and producer (red) strains are shown in the p-s plane. The arrows give the flow of a point describing the densities of the two strains. When the producer is insufficiently toxic (g < g c ), the isoclines do not cross. From nearly all starting positions, the "community point" moves to the equilibrium on the s axis (given by the blue sphere). That is, the sensitive strain displaces the producer. b Here, we see the same dynamics expressed as the frequency of the sensitive strain over time. Despite the starting conditions, the sensitive type fixes (that is, it approaches a frequency of 1). c A symbolic representation of the community dynamics (see 2). The arrow pointing from the producer node to the sensitive node indicates that the sensitive strain will outcompete the producer under any starting conditions. d When the producer is sufficiently toxic (g > g c ), the isoclines cross and a new internal equilibrium (the gray sphere) is introduced. This new equilibrium is unstable. In this community, the initial strain densities become important-if the producer is sufficiently abundant relative to the sensitive strain, then the producer will displace the sensitive strain and vice versa. This is a bistable system where both edge equilibria (the red and blue spheres) are locally stable. e Now, the sensitive strain fixes only if frequent enoughotherwise, it goes extinct (and the producer fixes). f The symbolic representation shows arrows pointing to each node with an unstable internal node in between

B1 Exclusion or bistability in a two-strain community. a The isoclines for the sensitive (blue) and producer (red) strains are shown in the p-s plane. The arrows give the flow of a point describing the densities of the two strains. When the producer is insufficiently toxic (g < g c ), the isoclines do not cross. From nearly all starting positions, the "community point" moves to the equilibrium on the s axis (given by the blue sphere). That is, the sensitive strain displaces the producer. b Here, we see the same dynamics expressed as the frequency of the sensitive strain over time. Despite the starting conditions, the sensitive type fixes (that is, it approaches a frequency of 1). c A symbolic representation of the community dynamics (see 2). The arrow pointing from the producer node to the sensitive node indicates that the sensitive strain will outcompete the producer under any starting conditions. d When the producer is sufficiently toxic (g > g c ), the isoclines cross and a new internal equilibrium (the gray sphere) is introduced. This new equilibrium is unstable. In this community, the initial strain densities become important-if the producer is sufficiently abundant relative to the sensitive strain, then the producer will displace the sensitive strain and vice versa. This is a bistable system where both edge equilibria (the red and blue spheres) are locally stable. e Now, the sensitive strain fixes only if frequent enoughotherwise, it goes extinct (and the producer fixes). f The symbolic representation shows arrows pointing to each node with an unstable internal node in between

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Summary The use of model laboratory communities, model organisms, and mathematical models has deeply enriched our understanding of the causes and consequences of toxin production in bacteria. In particular, such models have provided much insight into the dynamics of microbial communities with toxin producers. Both experimental and theoretical appro...

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... current state of the two-strain community can be expressed as a point (p, s) on the two-dimensional p-s plane (see Fig. 6 The arrows give the flow of a point describing the densities of the two strains. When the producer is insufficiently toxic (g < g c ), the isoclines do not cross. From nearly all starting positions, the "community point" moves to the equilibrium on the s axis (given by the blue sphere). That is, the sensi- tive strain displaces the ...
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... axis (given by the blue sphere). That is, the sensi- tive strain displaces the producer. b Here, we see the same dynamics expressed as the fre- quency of the sensitive strain over time. Despite the starting conditions, the sensitive type fixes (that is, it approaches a frequency of 1). c A symbolic representation of the commu- nity dynamics (see Fig. 6.2). The arrow pointing from the producer node to the sensitive node indicates that the sensitive strain will outcompete the producer under any starting conditions. d When the producer is sufficiently toxic (g > g c ), the isoclines cross and a new internal equilibrium (the gray sphere) is introduced. This new equilibrium is unstable. ...
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... assumptions (B1.3), (B1.4) and (B1.5), g c > 0. If g < g c (that is, the producer is not very toxic), then the isoclines do not cross in the positive quadrant of the p-s plane (see Fig. 6.B1a, where the sensitive isocline is in blue and the producer isocline is in red). The arrows in Fig. 6.B1a trace out the potential movement of a point giving the strain densities. Note that the arrows cut the blue line horizontally (because vertical movement of the point corresponds to changes in the sensitive strain, and the ...
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... assumptions (B1.3), (B1.4) and (B1.5), g c > 0. If g < g c (that is, the producer is not very toxic), then the isoclines do not cross in the positive quadrant of the p-s plane (see Fig. 6.B1a, where the sensitive isocline is in blue and the producer isocline is in red). The arrows in Fig. 6.B1a trace out the potential movement of a point giving the strain densities. Note that the arrows cut the blue line horizontally (because vertical movement of the point corresponds to changes in the sensitive strain, and the sensitive strain does not change its density on its isocline), and the arrows cut the red line vertically ...
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... the elevated deaths of sensitive cells) is disproportionately available to toxin-producing cells. In this way, small clumps of producers can "toxically clear-cut" sensi- tive cells at their periphery and radiate outward into a sea of sensitivity (see capacity would lead to the exclusion of producers by the invading sensi- tive strain). In Fig. 6.B1b, we see that the frequency of the sensitive strain approaches unity despite starting conditions ( Fig. 6.B1c shows this behav- ior schematically). Thus, without sufficient toxicity, the producer always goes extinct in head-to-head ...
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... this way, small clumps of producers can "toxically clear-cut" sensi- tive cells at their periphery and radiate outward into a sea of sensitivity (see capacity would lead to the exclusion of producers by the invading sensi- tive strain). In Fig. 6.B1b, we see that the frequency of the sensitive strain approaches unity despite starting conditions ( Fig. 6.B1c shows this behav- ior schematically). Thus, without sufficient toxicity, the producer always goes extinct in head-to-head ...
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... point is an unstable equilibrium (this can be shown locally using linear stability analysis; see the Appendix). From most starting posi- tions, either the sensitive strain displaces the producer or vice versa (see Fig. 6.B1d). Thus, initial community composition becomes important in determining which strain dominates. Generally, if sufficiently abundant, the producer displaces the sensitive strain, otherwise it goes extinct. This is shown in Fig. 6.B1e (and schematically in Fig. 6.B1f). This bistability was demonstrated in vitro with E. coli ( Adams et ...
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... the Appendix). From most starting posi- tions, either the sensitive strain displaces the producer or vice versa (see Fig. 6.B1d). Thus, initial community composition becomes important in determining which strain dominates. Generally, if sufficiently abundant, the producer displaces the sensitive strain, otherwise it goes extinct. This is shown in Fig. 6.B1e (and schematically in Fig. 6.B1f). This bistability was demonstrated in vitro with E. coli ( Adams et al. 1979;Chao and Levin ...
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... posi- tions, either the sensitive strain displaces the producer or vice versa (see Fig. 6.B1d). Thus, initial community composition becomes important in determining which strain dominates. Generally, if sufficiently abundant, the producer displaces the sensitive strain, otherwise it goes extinct. This is shown in Fig. 6.B1e (and schematically in Fig. 6.B1f). This bistability was demonstrated in vitro with E. coli ( Adams et al. 1979;Chao and Levin ...
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... Fig. 6.2a, we give the "boundary dynamics" on a de Finetti diagram for the rock-paper-scissors game (Frean and Abraham 2001;Czárán et al. 2002). We see that any community comprised of only "rock" and "paper" fixes for paper (since paper beats rock), any community of only "paper" and "scis- sors" fixes for scissors, and any community of only ...
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... on a de Finetti diagram for the rock-paper-scissors game (Frean and Abraham 2001;Czárán et al. 2002). We see that any community comprised of only "rock" and "paper" fixes for paper (since paper beats rock), any community of only "paper" and "scis- sors" fixes for scissors, and any community of only "scissors" and "rock" fixes for rock. In Fig. 6.2b, we show the dynamics when all three players are present -and we see continued cycles. In Fig. 6.2c, we give the boundary dynamics for the resistant-producer-sensitive game (when the producer is fairly toxic). Here, we see that we do not have a simple flow from one vertex to the next on the outside of the triangle. Rather, on the ...
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... 2002). We see that any community comprised of only "rock" and "paper" fixes for paper (since paper beats rock), any community of only "paper" and "scis- sors" fixes for scissors, and any community of only "scissors" and "rock" fixes for rock. In Fig. 6.2b, we show the dynamics when all three players are present -and we see continued cycles. In Fig. 6.2c, we give the boundary dynamics for the resistant-producer-sensitive game (when the producer is fairly toxic). Here, we see that we do not have a simple flow from one vertex to the next on the outside of the triangle. Rather, on the "sensitive-producer" edge we have flow going in both directions -both the sensitive pool and pro- ducer pool ...
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... producer is fairly toxic). Here, we see that we do not have a simple flow from one vertex to the next on the outside of the triangle. Rather, on the "sensitive-producer" edge we have flow going in both directions -both the sensitive pool and pro- ducer pool will exclude the other when they are sufficiently frequent in a two- strain community (see Fig. 6.B1f in Box 1). This is another appearance by the bistability described above. The hump (represented as a small gray point on the sensitive-producer edge in Fig. 6.2c) has reemerged. In Fig. 6.2d, we see that the dynamics are quite different than in the strict rock-paper-scissors game -the sensitive strain excludes the others from ...
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... edge we have flow going in both directions -both the sensitive pool and pro- ducer pool will exclude the other when they are sufficiently frequent in a two- strain community (see Fig. 6.B1f in Box 1). This is another appearance by the bistability described above. The hump (represented as a small gray point on the sensitive-producer edge in Fig. 6.2c) has reemerged. In Fig. 6.2d, we see that the dynamics are quite different than in the strict rock-paper-scissors game -the sensitive strain excludes the others from nearly every starting condition (see Nakamaru and Iwasa 2000, and the Appendix). Do note that these models assume infinite population sizes, and often the dynamical tra- ...
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... in both directions -both the sensitive pool and pro- ducer pool will exclude the other when they are sufficiently frequent in a two- strain community (see Fig. 6.B1f in Box 1). This is another appearance by the bistability described above. The hump (represented as a small gray point on the sensitive-producer edge in Fig. 6.2c) has reemerged. In Fig. 6.2d, we see that the dynamics are quite different than in the strict rock-paper-scissors game -the sensitive strain excludes the others from nearly every starting condition (see Nakamaru and Iwasa 2000, and the Appendix). Do note that these models assume infinite population sizes, and often the dynamical tra- jectory can come very close to ...
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... (the surface of an agar plate), Kerr et al. (2002) experimentally demonstrated that spatial structure can promote the maintenance of diversity in a bacte- riocin community. In a sense, spatial structure in these cases obliterates the "hump" on the sensitive-producer edge of the de Finetti diagram (Durrett and Levin 1997;Iwasa et al. 1998). In Fig. 6.2f, we see simulated dynamics from the lattice-based model described in Box 2. This behavior is much closer to the rock-paper-scissors game of Three-strain community dynamics. In this figure, the de Finetti diagram is used to rep- resent changes in community composition. Each vertex of the triangle is labeled with one of the three ...
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... this three-strain community is simulated using a Moore neigh- borhood, all three strains coexist under many different parameter settings. Because dispersal is local, clumps of the three strains form and these clumps chase one another at their boundaries -S clumps chase R clumps, R clumps chase P clumps, and P clumps chase S clumps (see Fig. 6.B2a, b). However, when a Global neighborhood is used, diversity is rapidly lost ...
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... Ecological and Evolutionary Dynamics of Model Bacteriocin Communities 125 (see Fig. 6.B2c, d). In the Global neighborhood, the toxic effects of pro- ducers are distributed globally. This can drive the sensitive strain to very low levels (unless the producer is not very toxic). Indeed, because our lat- tice is finite, the sensitive strain often goes extinct. Once one member of a non-transitive triplet is lost, the final ...
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... simulations with a Global neighborhood correspond closely to the dynamics given by the set of mean-field ordinary differential equations (see the Appendix and Fig. 6.2c, d). However, because such mathematical models assume infinite populations (and thus one can have an arbitrarily small density of sensitive cells), the sensitive strain is expected to "hang on" as the resistant displaces the producer, and eventually dominates the community. However, the outcome for the maintenance of diversity is the ...
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... is simulated without other competing strains, it evolves to minimize the cost of resistance (average ∆ R evolves to the minimum value in the range allowed). However, when evolution occurs in a three-strain community with local dispersal and interaction (using a Moore neighborhood), the cost of resistance does not evolve to its lowest level (see Fig. 6.3). It pays off to exercise competitive restraint in this non-hierarchical community because such restraint aids the enemy of your enemy (which, in turn, harms your enemy and thus aids you). An extremely interesting direction for future experimental work involves exploration of these counterintuitive spatial evolutionary dynamics within ...
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... Fig. 6.3, we see the maintenance of a non-minimal cost of resistance in a spatially structured three-strain community. On the other hand, if the resistant strain evolves alone in a spatially structured habitat, it does evolve to minimize its cost ( Fig. 6.3). In a structured non-transitive community, a higher cost of resistance retards ...
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... Fig. 6.3, we see the maintenance of a non-minimal cost of resistance in a spatially structured three-strain community. On the other hand, if the resistant strain evolves alone in a spatially structured habitat, it does evolve to minimize its cost ( Fig. 6.3). In a structured non-transitive community, a higher cost of resistance retards replacement of producers by resistant cells. is the average death rate of the evolving resistant strain. The minimum value that CR can obtain is n. A proxy for cost of resist- ance is CR′ = CR-n. All else being equal, the resistant strain is expected to ...
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... face a dynamical hump to get over ( Adams et al. 1979;Chao and Levin 1981;Levin 1988;Durrett and Levin 1997;Iwasa et al. 1998). In unstructured habitats, the signature of this hump is present in the resistant-producer-sensitive community, changing the dynamics from a straightforward "rock-paper- scissors" to a "one-winner" outcome (compare Fig. 6.2a, b to c, d). Population structure (e.g., spatial structure) can effectively eliminate the hump ( Chao and Levin 1981;Durrett and Levin 1997;Iwasa et al. 1998) and restore the game of rock-paper-scissors. In this spatial game, players stably chase each other around a structured arena as clumps, with balanced gains and losses occurring at the ...

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