Shifts in evolutionary balance of microbial phenotypes under environmental changes

Environmental conditions shape entire communities by driving microbial interactions. These interactions then find their reflection in the evolutionary outcome of microbial competition. In static, homogeneous environments a robust, or evolutionary stable, outcome in microbial communities is reachable, if it exists. However, introducing heterogeneity and time-dependence in microbial ecology leads to stochastic evolutionary outcomes determined by specific environmental changes. We utilise evolutionary game theory to provide insight into phenotypic competition in dynamic environments. We capture these effects in a perturbed evolutionary game describing microbial interactions at a phenotypic level. We show that under regular periodic environmental fluctuations a stable state that preserves dominant phenotypes is reached. However, rapid environmental shifts, especially in a cyclic interactions, can lead to critical shifts in the evolutionary balance among phenotypes. Our analysis suggests that an understanding of the robustness of the systems current state is necessary to understand when system will shift to the new equilibrium. This can be done by understanding the systems overall margin of safety, that is, what level of perturbations it can take before its equilibrium changes. In particular, the extent to which an environmental shift affects the system’s behaviour.

In our framework, we consider behavioural flexibility to be the potential expression of different phe-85 notypes when organisms are exposed to new, unfamiliar environmental conditions. As this flexibility 86 occurs due to the imperfect realisation of the bacteria's 'expected' phenotype, it is referred to as in- where I is the identity matrix and τ is the environmental time scale.
where Q(λ(τ )) T is the transpose of Q(λ(τ )). This can be interpreted in the following way. Firstly, 110 consider pairwise interactions in a given population subjected to new environmental conditions. These 111 organisms have a finite number, n, of available strategies, or phenotypes. Hence, interacting individuals 112 compete using their chosen strategies, receiving a payoff according to the fitness matrix R. However, we 113 also assume that individuals may be imperfect in their strategy execution, and may execute a different 114 strategy according to the probabilities given in matrix Q(λ(τ )). Thus, the errors made by incompetent 115 individuals during their interactions lead us to a perturbed payoff r ij (λ(τ )) from R(λ(τ )). The (i, j)th 116 entry of the latter is merely the expected payoff that incorporates all possible mistakes made by the 117 two interacting phenotypes i and j. From the evolutionary perspective, behavioural mistakes perturb 118 population fitness over time as bacteria respond to new conditions. When the environment changes all 119 bacteria are incompetent, except a small portion that were incompetent in the previous environment. That 120 is, they had deficits in the old environment that are competencies in the new environment. The fitness 121 defines reproductive success of each phenotype or behavioural strategy as where R(λ(τ )) denotes an incompetent fitness matrix (3) with a time-varying incompetent matrix

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(2), and the mean fitness of the entire population is defined as follows We exploit the well-studied replicator equation This is a widely studied game (e.g. [33]), and it has been shown that the game possesses an evolution- Rock P aper Scissors For all simulations, from (2), we define the limiting incompetence matrix, S, to be 159 P henotype 1 P henotype 2 P henotype 3 Such probability distributions provide us an opportunity to compare three abstract phenotypes which dis-160 play different forms of flexibility. From (6), Phenotype 1 may exhibit Phenotype 2. Phenotype 3 is most 161 likely to exhibit Phenotype 2 but may also express Phenotype 1, whereas Phenotype 2 is evenly random 162 in its switching. These probabilities are taken to be high enough to reflect critical environmental shifts 163 i.e. those which impose significant stress on the system. We chose such probabilities in order to demon- A periodic form of λ(τ ) reflects seasonal or daily regular fluctuations, and may be defined by 172 and then α is interpreted as a frequency of fluctuations in the environmental conditions. For instance, 173 small α reflects a longer period after which the perturbation cycle begins again.

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However, in the case of rapid environmental shifts, the system moves to a new, fixed state. We where a determines the steepness of the change curve and b is the inflection point. This function begins 178 with a slow change rate, followed by a steep decrease before levelling of in its approach to the limiting 179 state.

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Results and discussion 181 We show that such an assumption might break existing periodic relationships or lead to formation of  We incorporated periodic environmental changes, described in (7), into the static model of a Hawk- not only determines the rate of response required from organisms, but also their potential to capitalise in 215 their interactions. Firstly, the system loses its stability at a critical time t c defined as where λ c is the bifurcation value of λ, where the dynamical system (4) changes its qualitative behaviour 217 [31]. We show that if the environment shifts back to its original conditions, the system is also able where the system balances out itself due to the relation between phenotypes, sometimes referred to as predict which phenotype will assume the dominating position.

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As can be seen in Figure 3 panel d, the system has a limit of perturbations it can take: in the begin-277 ning of the environmental shift, the system evolves in a similar cyclic manner as in the old environment.

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The capacity to buffer the shift is not unlimited. Once the system hits a critical threshold, the cyclic rela- phenotype may change to one that did not initially appear evolutionarily competitive.

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The inherently unstable rock-paper-scissors model is particularly appropriate for bacteria because of The non-technical discussion presented in the main body of the text of this manuscript is based on 315 rather detailed mathematical analyses. For the sake of completeness, in this Appendix, we outline these 316 analyses. For more details the reader is referred to [42].

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Let us first introduce some notation. Consider the replicator dynamics for the case with full compe-318 tence, that is, the original evolutionary game. We shall define such a game as Γ 1 , the fully competent 319 game 1 : where g(x) is a replicator equation given by Next, we define a game with the fixed incompetence parameter as a λ-fixed incompetent game Γ λ : where g(x, λ) is a replicator equation given by Further, we shall call a λ(t)-varying game Γ λ(t) with λ being time-dependent as follows.

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Theorem 2. Letx be an ESS of Γ 1 and x 1 (t) be a resulting solution for some being a periodic function of period T = 2π α from (7) and a sufficiently small δ, if ||S − I|| < δ, then 339 there exists a periodic solution x 2 (t) with x 2 (0) = x 0 of Γ λ(t) that stays close to x 1 (t).

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Proof: In order to apply Theorem 1 we need to estimate |g(x, t) − g(x)|, |g x (x, t) − g (x)|, and 341 |g t (x, t)|, and show that they are sufficiently small. The first distance |g(x, t) − g(x)| is sufficiently 342 small due to the Lipschitz continuity of the replicator dynamics (see p. 141 [45]), that is, where M is a Lipschitz constant.

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Next, we can rewrite the requirement ||S − I|| < δ as S = I + E(δ), where E(δ) is a matrix with 345 entries ij (δ) that are of magnitude less than δ. For the simplicity of notation we omit the dependence 346 on δ and simply use E. From (2) we have Substituting (13) into the replicator dynamics and setting X to be a diagonal matrix with entries of 349 the vector x on the diagonal, we obtain Hence, and therefore |g x (x, t) − g (x)| → 0 as δ → 0.

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Let us now define the critical time of the learning adaptation process as the first time when incom-406 petence function, λ(t), attains the maximal critical value, λ u , of Theorems 3-4. Let λ u be the maximal 407 bifurcation value of the incompetence parameter forx, then the critical time is given by which follows immediately from the functional form of λ(t) given by the sigmoid function. Namely, we 409 solve for t u the equation