We study a minimal stochastic individual-based model for a microbial population challenged by a persistent (lytic) virus epidemic. We focus on the situation in which the resident microbial host population and the virus population are in stable coexistence upon arrival of a single new “mutant” host individual. We assume that this mutant is capable of switching to a reversible state of dormancy upon contact with virions as a means of avoiding infection by the virus. At the same time, we assume that this new dormancy trait comes with a cost, namely a reduced individual reproduction rate. We prove that there is a non-trivial range of parameters where the mutants can nevertheless invade the resident population with strictly positive probability (bounded away from 0) in the large population limit. Given the reduced reproductive rate, such an invasion would be impossible in the absence of either the dormancy trait or the virus epidemic. We explicitly characterize the parameter regime where this emergence of a host dormancy trait is possible, determine the success probability of a single invader and the typical amount of time it takes the successful mutants to reach a macroscopic population size. We conclude this study by an investigation of the fate of the population after the successful emergence of a dormancy trait. Heuristic arguments and simulations suggest that after successful invasion, either both host types and the virus will reach coexistence, or the mutants will drive the resident hosts to extinction while the virus will stay in the system.

In this paper we propose a Moran model that describes the population dynamics of two types: While the first type has a selective advantage during reproduction, the second type can avoid replacement during reproduction with some positive probability by switching temporarily into a dormant state. We investigate the interplay of both evolutionary strategies by studying the invasion dynamics of the dormant type into the resident (selectively advantageous) population in the large population limit of the system. It turns out that the dormancy trait can not only invade and subsequently fixate under suitable parameter assumptions despite its selective disadvantage (a phenomenon that has already been observed in a related context in Blath and Tóbiás (2020)), but that there is also a novel regime of stable coexistence of both types due to a frequency-dependent balancing effect that did not arise in the previous setup with Lotka–Volterra type symmetric competition. The emergence of a coexistence regime here rests in part on specific properties of the Moran modelling framework, in particular its fixed overall population size that enforces instant re-colonization after death events, as well as on the (positive) mortality and resuscitation rates of the dormant state. We provide heuristic explanations for the observed types of behaviour and the corresponding proofs, which involve comparisons to suitable branching processes, approximations by dynamical systems, and an analysis of asymptotic behaviour of the latter.

In this paper we investigate the spread of advantageous genes in two variants of the F-KPP model with dormancy. The first variant, in which dormant individuals do not move in space and instead form "localized seed banks", has recently been introduced in Blath, Hammer and Nie (2020). However, there, only a relatively crude upper bound for the critical speed of potential travelling wave solutions has been provided. The second model variant is new and describes a situation in which the dormant forms of individuals are subject to motion, while the "active" individuals remain spatially static instead. This can be motivated e.g. by spore dispersal of fungi, where the "dormant" spores are distributed by wind, water or insects, while the "active" fungi are locally fixed. For both models, we establish the existence of monotone travelling wave solutions, determine the corresponding critical wave speed in terms of the model parameters, and characterize aspects of the asymptotic shape of the waves depending on the decay properties of the initial condition. Interestingly, the slow-down effect of dormancy on the speed of propagation of beneficial alleles is often more serious in model variant II (the "spore model") than in variant I (the "seed bank model"), and this can be understood mathematically via probabilistic representations of solutions in terms of (two variants of) "on/off branching Brownian motions". Our proofs make rather heavy use of probabilistic tools in the tradition of Watanabe (1967), McKean (1975), Bramson (1978), Neveu (1987), Lalley and Sellke (1987), Champneys et al (1995) and others. However, the two-compartment nature of the model and the special forms of dormancy also pose obstacles to the classical formalism, giving rise to a variety of open research questions that we briefly discuss at the end of the paper.

Therapy evasion - and subsequent disease progression - is a major challenge in current oncology. An important role in this context seems to be played by various forms of cancer cell dormancy. For example, therapy-induced dormancy, over short timescales, can create serious obstacles to aggressive treatment approaches such as chemotherapy, and long-term dormancy may lead to relapses and metastases even many years after an initially successful treatment. In this paper, we focus on individual cancer cells switching into and out of a dormant state both spontaneously as well as in response to treatment. We introduce an idealized mathematical model, based on stochastic agent-based interactions, for the dynamics of cancer cell populations involving individual short-term dormancy, and allow fora range of (multi-drug) therapy protocols. Our analysis - based on simulations of the many-particle limit - shows that in our model, depending on the specific underlying dormancy mechanism, even a small initial population (of explicitly quantifiable size) of dormant cells can lead to therapy failure under classical single-drug treatments that would successfully eradicate the tumour in the absence of dormancy. We further investigate and quantify the effectiveness of several multi-drug regimes (manipulating dormant cancer cells in specific ways, including increasing or decreasing resuscitation rates or targeting dormant cells directly). Relying on quantitative results for concrete simulation parameters, we provide some general basic rules for the design of (multi-)drug treatment protocols depending on the types and processes of dormancy mechanisms present in the population.

In this paper, we investigate the consequences of dormancy in the ‘rare mutation’ and ‘large population’ regime of stochastic adaptive dynamics. Starting from an individual-based micro-model, we first derive the Polymorphic Evolution Sequence of the population, based on a previous work by Baar and Bovier (2018). After passing to a second ‘small mutations’ limit, we arrive at the Canonical Equation of Adaptive Dynamics, and state a corresponding criterion for evolutionary branching, extending a previous result of Champagnat and Méléard (2011).The criterion allows a quantitative and qualitative analysis of the effects of dormancy in the well-known model of Dieckmann and Doebeli (1999) for sympatric speciation. In fact, quite an intuitive picture emerges: Dormancy enlarges the parameter range for evolutionary branching, increases the carrying capacity and niche width of the post-branching sub-populations, and, depending on the model parameters, can either increase or decrease the ‘speed of adaptation’ of populations. Finally, dormancy increases diversity by increasing the genetic distance between subpopulations.

In this paper, we introduce a type switching mechanism for the Contact Process on the lattice $\mathbb{Z}^d$. That is, we allow the individual particles/sites to switch between two (or more) types independently of one another, and the different types may exhibit specific infection and recovery dynamics. Such type switches can eg.\ be motivated from biology, where 'phenotypic switching' is common among micro-organisms. Our framework includes as special cases systems with switches between 'active' and 'dormant' states (the Contact Process with dormancy, CPD), and the Contact Process in a randomly evolving environment (CPREE) introduced by Broman (2007). The 'standard' multi-type Contact Process (without type-switching) can also be recovered as a limiting case. After constructing the process from a graphical representation, we establish several basic properties that are mostly analogous to the classical Contact Process. We then provide couplings between several variants of the system, which in particular yield the existence of a phase transition. Further, we investigate the effect of the switching parameters on the critical value of the system by providing rigorous bounds obtained from the coupling arguments as well as numerical and heuristic results. Finally, we investigate scaling limits for the process as the switching parameters tend to 0 (slow switching regime) resp.\ $\infty$ (fast switching regime). We conclude with a brief discussion of further model variants and questions for future research.

This paper introduces a stochastic adaptive dynamics model for the interplay of several crucial traits and mechanisms in bacterial evolution, namely dormancy, horizontal gene transfer (HGT), mutation and competition. In particular, it combines the recent model of Champagnat, M\'el\'eard and Tran (2021) involving HGT with the model for competition-induced dormancy of Blath and T\'obi\'as (2020). Our main result is a convergence theorem which describes the evolution of the different traits in the population on a `doubly logarithmic scale' as piece-wise affine functions. Interestingly, even for a relatively small trait space, the limiting process exhibits a non-monotone dependence of the success of the dormancy trait on the dormancy initiation probability. Further, the model establishes a new `approximate coexistence regime' for multiple traits that has not been observed in previous literature.

We introduce and construct on/off super-Brownian motion (on/off SBM) as a measure-valued scaling limit of critical on/off branching Brownian motions. The distinguishing feature of this process is that its infinitesimal particles can switch individually into and out of a state of dormancy, in which they neither move nor reproduce. Related dormancy traits have received interest in mathematical population biology recently, introducing memory and delay (in the form of a seed bank) into the corresponding processes. It turns out that the properties of on/off SBM differ significantly from those of classical super-Brownian motion. In particular, the process does not die out in finite time with probability one despite criticality of reproduction. However, the size of the active subpopulation does hit 0 in finite time with positive probability, a result which can be shown using methods from polynomial diffusion theory. We expect distinct behaviour in various other qualitative and quantitative respects, for which we provide heuristics, and conclude with a brief discussion of several directions for future research.

We investigate a stochastic individual-based model for the population dynamics of host-virus systems where the microbial hosts may transition into a dormant state upon contact with virions, thus evading infection. Such a contact-mediated defence mechanism was described in Bautista et al (2015) for an archaeal host, while Jackson and Fineran (2019) and Meeske et al (2019) describe a related, CRISPR-Cas induced, dormancy defense of bacterial hosts to curb phage epidemics. We first analyse the effect of the dormancy-related model parameters on the probability and time of invasion of a newly arriving virus into a resident host population. Given successful invasion in the stochastic system, we then show that the emergence (with high probability) of a persistent virus infection ('epidemic') in a large host population can be determined by the existence of a coexistence equilibrium for the dynamical system arising as the deterministic many-particle limit of our model. This is an extension of a dynamical system considered by Beretta and Kuang (1998) that is known to exhibit a Hopf bifurcation, giving rise to a 'paradox of enrichment'. In our system, we verify that the additional dormancy component can, at least for certain parameter ranges, prevent the associated loss of stability. Finally, we show that the presence of contact-mediated dormancy enables the host population to attain higher equilibrium sizes - while still being able to avoid a persistent epidemic - than host populations without this trait.

We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of “on/off”-moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an “on/off-branching Brownian motion”. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be “switched off” for an exponential amount of time after which they get “switched on” again. On/off branching Brownian motion shows qualitatively different behaviour to classical branching Brownian motion and is an interesting object for study in itself. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed √(2) , is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by √(√(5)-1)≈ 1.111 under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.

We review recent progress in the understanding of the role of multiple- and simultaneous multiple merger coalescents as models for the genealogy in idealised and real populations with exceptional reproductive behaviour. In particular, we discuss models with `skewed offspring distribution' (or under other non-classical evolutionary forces) which lead in the single locus haploid case to multiple merger coalescents, and in the multi-locus diploid case to simultaneous multiple merger coalescents. Further, we discuss inference methods under the infinitely-many sites model which allow both model selection and estimation of model parameters under these coalescents.

In the present article, we investigate the effects of dormancy on an abstract population genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in general, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.

We investigate a stochastic individual-based model for the population dynamics of host–virus systems where the hosts may transition into a dormant state upon contact with virions, thus evading infection. Such a dormancy-based defence mechanism was described in Bautista et al (2015). We first analyse the effect of the dormancy-related model parameters on the probability of invasion of a newly arriving virus into a resident host population. It turns out that the probability of dormancy initiation upon virus contact plays a crucial role, while the lengths of the dormancy periods or the death rate during dormancy are largely irrelevant. Given successful invasion, we then show that the emergence of a persistent virus infection (‘epidemic’) in the host population corresponds to the existence of a coexistence equilibrium for the deterministic many-particle limit of our model. In this context, all dormancy-related parameters have a significant impact. Indeed, while related systems without dormancy may exhibit a Hopf bifurcation, giving rise to a variant of the ‘paradox of enrichment’, we argue that the inclusion of dormancy can prevent this loss of stability. Finally, we show that the presence of contact-mediated dormancy enables the host population to maintain higher equilibrium sizes (resp. fitness values) – while still being able to avoid a persistent epidemic – than host populations without this trait, for which high fitness values would imply a high risk for the emergence of a persistent epidemic. This adds a twist to the relevance of ‘reproductive trade-offs’ usually associated with costly dormancy traits. MSC 2010. 92D25, 60J85, 34D05, 37G15.

Across the tree of life, populations have evolved the capacity to contend with suboptimal conditions by engaging in dormancy, whereby individuals enter a reversible state of reduced metabolic activity. The resulting seed banks are complex, storing information and imparting memory that gives rise to multi-scale structures and networks spanning collections of cells to entire ecosystems. We outline the fundamental attributes and emergent phenomena associated with dormancy and seed banks, with the vision for a unifying and mathematically based framework that can address problems in the life sciences, ranging from global change to cancer biology.

The goal of this article is to contribute towards the conceptual and quantitative understanding of the evolutionary benefits for (microbial) populations to maintain a seed bank consisting of dormant individuals when facing fluctuating environmental conditions. To this end, we discuss a class of ‘2-type’ branching processes describing populations of individuals that may switch between ‘active’ and ‘dormant’ states in a random environment oscillating between a ‘healthy’ and a ‘harsh’ state. We incorporate different switching strategies and suggest a method of ‘fair comparison’ to incorporate potentially varying reproductive costs. We then use this concept to compare the fitness of the different strategies in terms of maximal Lyapunov exponents. This gives rise to a ‘fitness map’ depicting the environmental regimes where certain switching strategies are uniquely supercritical.