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How the interplay of dormancy and selection affects the wave of advance of an advantageous gene

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 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.

1 Introduction and main results

Background: The F-KPP equation with dormancy
The F-KPP equation (named after Fisher [16] and Kolmogorov, Petrovski and Piscounov [24]) is the simplest and most prominent example of a non-linear reaction-diffusion system. In population genetics, it is used to describe the propagation of an 'advantegous gene' or 'beneficial allele' in a bi-allelic population under the influence of directional selection. Denoting by p(t, x) ∈ [0,1] its solution at time t ≥ 0 and spatial position x ∈ R, which we here interpret as the fraction of a beneficial allele present in a biological population at location x and time t, the corresponding initial value problem is given by the non-linear second order partial differential equation ∂ t p(t, x) = ∆ 2 p(t, x) + p(t, x)(1 − p(t, x)), (1.1) and suitable initial conditions, this systems has a monotone travelling-wave solution of speed λ, i.e. the solution takes the form where w is a decreasing function with lim x→−∞ w(x) = 1 and lim x→∞ w(x) = 0, and this solution is unique up to translations. While wave-speeds above the 'critical value' can be obtained by starting from initial conditions with suitable decay behaviour, the critical wave-speed √ 2 is realized in particular when starting in a Heaviside initial condition. These (and much finer results) can be found in the classical works [32], [10], [9], [26], and the model has been extended in many directions, including coupled systems with multiple components, see e.g. [17], [12].
Recently, the F-KPP equation has been extended to incorporate the biological concepts of dormancy and seed banks. Here, in addition to undergoing spatial dispersal and selective pressure, individuals may independently switch into a state of low metabolic activity. In this dormant state, individuals neither move nor reproduce. The dormant individuals of a population thus form a 'seed bank' which buffers genetic diversity. In biology, such a type of dormancy is often regarded as a bet-hedging strategy against unfavourable environmental conditions. The corresponding trait is wide-spread among many taxa, including many microbial species, see e.g. [27] and [28] for overviews.
For spatial systems such as the F-KPP equation, a way to incorporate a seed bank comprised of dormant individuals is to introduce a second component to the system, which describes, for each spatial position, the relative frequency of the beneficial allele in the seed bank. Active and dormant components then interact via local two-way migration ('switching'). This idea has been formalized in [6], leading to the coupled two-type system A 'common source of confusion' in the context of the F-KPP equation is the use of two different conventions regarding the sign in front of the non-linearity of the equation. Historically, Fisher was interested in the advance of the advantageous gene which lead to the introduction of Equation (1.1) with a positive sign (directional selection increases the frequency of the advantageous type). However, in order to exploit the fruitful probabilistic method of duality to analyse the system, it is necessary to switch focus from the advantageous to the deleterious gene whose frequency is given by This transformation leads to a sign change in front of the selection term, with u now solving the equation which is of course equivalent to the original system. However, note that the direction of travelling wave solutions will now be reversed (and the initial conditions need to be transformed as well). We will follow this convention, and in what follows focus on the 'transformed' F-KPP equation with dormancy (and selection strength s > 0) given by ∂ t u(t, x) = ∆ 2 u(t, x) + c(v(t, x) − u(t, x)) − su(t, x)(1 − u(t, x)), ∂ t v(t, x) =c (u(t, x) − v(t, x)), (1.4) for the deleterious allele (and with corresponding initial conditions). This is in line with notation used in a majority of the probabilistic literature concerning the classical F-KPP equation (see e.g. [19], [32] or [9]).
As mentioned above, a powerful approach to analyze solutions to the above system is via its dual Markov process (see e.g. [38] and [2] for the method of duality in spatial population models). Indeed, the dual process of the (transformed) F-KPP equation is given by branching Brownian motion (BBM; this link was pointed out by [32] and heavily exploited in [10], but is also already present in [39] and [21,22,23]). Branching Brownian motion and its generalizations have been a central object of study in modern probability theory and statistical physics for several decades, see e.g. [7] for an overview.
For the coupled two-type system including dormancy, a similar duality has recently been established in [6], where the dual is now given by so-called on/off branching Brownian motion (o/oBBM). This dual is again a branching Markov process, denoted by M = (M t ) t≥0 , formally taking values in the space (1.5) Note that for fixed k, we interpret elements of R k × {a, d} k as the spatial positions of k particles in R, each carrying a flag from {a, d}. Particles with flag a are deemed active, while particles flagged with a d are deemed as dormant. Starting from some initial value M 0 = (x 1 , σ 1 ), · · · , (x n , σ n ) ∈ R n × {a, d} n , for some n ∈ N, the process evolves according to the following rules: • Active particles, i.e. particles carrying flag a, disperse in R according to independent Brownian motions and branch into two active particles at rate s. • Independently, each active particle falls dormant at rate c, changing its flag from a to d.
• Dormant particles neither move nor branch.
• Independently, each dormant particle resuscitates at rate c changing its flag from d to a.
To keep track of the number of active and dormant individuals (at each time t ≥ 0), we denote by I = (I t ) t≥0 and J = (J t ) t≥0 the (time-dependent) index sets of active and dormant particles of M , respectively, further set K t = I t ∪ J t and let N t := |K t | be the total number of particles at time t ≥ 0. For example, if for t ≥ 0 we have Note that this brief description can be expanded and formalized using e.g. the Ulam-Harris labelling (see e.g. [4]), but for brevity we refrain from going into the details here.
With the help of the aforementioned dual processes, a probabilistic representation of the solution to (1.4) can be given as follows (see [6,Cor. 1.8] for details). Indeed, starting in a Heaviside initial condition given by we have the (analog of the classical) McKean representation for the solution u given by where (R t ) t≥0 is the position of the rightmost particle of an on/off branching Brownian motion as defined above, started with a single active particle in 0, i.e. M 0 = (0, a).
A question that can be answered for the classical F-KPP equation via the McKean representation concerns the speed of propagation of advantageous genes (e.g. when starting with reversed Heaviside initial conditions). Indeed, the position R t of the rightmost particle of the dual branching Brownian motion at time t relates to the critical wave-speed of the original equation via the a.s. equality In line with intuition from biology (see e.g. [28]), namely that seed banks should contribute to the diversity and resilience of populations, one expects that the presence of dormancy in (1.4) should at least slow down the speed of traveling wave solutions, if not preventing their emergence entirely. Following this line of thought, a first relatively crude result in [6] shows that the speed of the right-most particle is indeed reduced significantly due to the effect of dormancy. More precisely, via a simple first moment bound in combination with a many-to-one lemma, the upper bound has been obtained (for the case c = c = s = 1). However, it was left open in [6] whether this bound is actually sharp, and even whether non-trivial travelling wave solutions exist at all.
In the present paper, our first aim is to extend this result by providing the existence of travelling wave solutions, the exact value of the critical wave-speed of the F-KPP equation with dormancy and the asymptotic speed (up to first order) of the right-most particle of the on/off branching Brownian motion. In the case c = c = s = 1, we will see that the previous upper bound is not sharp, and that the correct value is given by Our second aim in this paper is to expand our modelling approach. Indeed, there is another natural way to incorporate seed-banks into an F-KPP based model, that was not considered in [6], and for which we aim to obtain similar results while also investigating their quantitative differences.

Two models for the interplay of dormancy and dispersal
Understanding the role of different forms of dormancy in population genetics and ecology is currently an active field of research, and this holds in particular for the interplay between dormancy and spatial dispersal ( [40], [11], [18] with initial conditions u 0 , v 0 ∈ B(R, [0,1]) and parameters c, c ≥ 0, where the selection term is of the form for a given probability distribution p = (p k ) k∈N ⊆ [0,1] N such that ∞ k=1 p k = 1 and ∞ k=1 p k k < ∞.
Note that the form of the selection term ensures the duality to on/off branching Brownian motion with general (not necessarily binary) branching mechanism. In fact, this branching mechanism is precisely given by the probability distribution (p k ), i.e., the probability to see k + 1 offspring in a branching event (which happen with overall rate κ) is precisely p k (cf. e.g. [2]).
The second variant is distinguished from the previous one by moving the Laplacian from the first to the second component: Definition 1.2 (F-KPP equation with dormancy, variant II: spore model). The initial value problem associated with the F-KPP equation with dormancy, variant II is given by the coupled system with initial conditionsũ 0 ,ṽ 0 ∈ B(R, [0,1]) and parametersc,c , κ, where the selection term is of the form for a given probability distributionp = (p k ) k∈N ⊆ [0,1] N such that ∞ k=1p k = 1 and ∞ k=1p k k < ∞. Remark 1.3 (Alternative interpretation of model II). Note that in our interpretation, the selective pressure is always applied to the active population (e.g. corresponding to fertility selection). An equivalent interpretation of model variant II can be reached by interchanging the roles ofũ andṽ: Now, the interpretation is that active individuals disperse, but are not subject to selective pressure, which acts only on dormant individuals. Remark 1.4 (Relation to existing theory for coupled reaction-diffusion systems). Note that both model variants I and II can be considered as coupled reaction-diffusion systems. Coupled systems have been considered e.g. by Freidlin [17] and Champneys et. al. [12]. However, there the assumption is that the Laplace-operator is present in both sub-populations, which is not the case in our set-up. Still, several of the arguments of Champneys et. al., can be employed for the analysis of our system, as we will see later in the paper. While the delay in the first model is essentially the same as in the model considered in [6], simply including a more general selection term, the delay formulation of the second variant can in general only be provided implicitly. The reason is that if the Laplace operator is moved to the second component, it is the first component that can be represented as solution of an ODE. However, since the selection term is of second order (or higher), one ends up with a Riccati-type equation, which in general cannot be solved explicitly. Since the delay-representation is not required for our results (in contrast to the situation in [6]), we refrain from going into the details here.
Both of the above model variants again have dual spatial branching processes. In order to provide the corresponding duality statements, we first formally introduce the two variants of on/off branching Brownian motions (o/oBBM, variant I and II) arising as duals to (1.8) and (1.9). We employ the notation from the preceding section. Definition 1.6 (on/off-BBM, variant I). On/off branching Brownian motion variant I corresponds to the system (1.8) and is the unique Markov process (M t ) t≥0 with state space Γ from (1.5) evolving according to the following rules: • Active particles (carrying flag a) disperse in R according to independent Brownian motions.
• Active particles branch at rate κ into k + 1 ∈ N offspring particles according to the distribution p = (p k ) k∈N . • Independently, each active particle falls dormant at rate c, changing its flag from a to d.
• Dormant particles neither move nor branch.
• Independently, each dormant particle resuscitates at rate c changing its flag from d to a. Definition 1.7 (on/off-BBM, variant II). On/off branching Brownian motion variant II corresponds to the system (1.9) and is the unique Markov process (M t ) t≥0 with state space Γ from (1.5) evolving according to the following rules: • Active particles do not move but branch at rate κ into k + 1 ∈ N offspring particles according to the distribution p = (p k ) k∈N . • Independently, each active particle falls dormant at rate c, changing its flag from a to d.
• Dormant particles (carrying flag a) disperse in R according to independent Brownian motions.
• Dormant particles do not reproduce.
• Independently, each dormant particle resuscitates at rate c changing its flag from d to a.
For each particle alive at tine t, indexed by some γ ∈ I t ∪ J t , we denote its trajectory process by (M γ t ) t≥0 for variant I, and correspondingly by (M γ t ) t≥0 for variant II. With this notation we are now in a position to provide a formal statement of the duality between o/oBBM and the F-KPP equation with seed bank in each of the variants I and II.
and the analogous statement holds for variant II with u, v, M replaced byũ,ṽ,M .
The proof for variant I is a simple extension of the corresponding result in [6], and the proof for variant II uses similar arguments and thus both are omitted.

The linearized systems and their 'wave-speed functions'
We aim to establish, for both model variants I and II, critical wave-speeds λ * ,λ * ∈ (0, ∞) and the existence of monotone travelling wave solutions (f, g) (depending on λ) such that the pair is a solution to Equation (1.8) for λ ≥ λ * , and similarly (f ,g) for (1.9) withλ ≥λ * . For variant I, the pair (f, g) then solves the system with s = κ ∞ k=1 p k (k − 1) (the equation for variant II is analogous and again omitted for brevity). Note that all information of the branching mechanism encoded in the selection term here is condensed into the real number s, which is given by the overall branching rate κ times the expected number of offspring (excluding the parent). The corresponding traveling wave solutions (f, g) then solve the linear system (1.12) It will be suitable to interpret this system (as in [19]) as a vector-valued equation given by The usual Ansatz to solve these is to choose (f, g) in dependence on a decay parameter µ ≤ 0 of the form which leads to the Eigenvalue problem Solving this for λ gives, for each µ, two possible values, namely From Perron-Frobenius-Seneta theory [37] it is not hard to see that for each µ < 0, the eigenvalue −µλ + µ in (1.15) is positive, and consequently that λ + µ is a positive function. Moreover, the corresponding eigenvector d µ can be computed explicitly, and in particular has strictly positive entries. See Section 5.2 in the Appendix for details. We now focus on λ + µ and omit the superscript + in what follows. Proposition 1.9 (Speed function for traveling wave solutions, model variant I). The differentiable map called the speed function of the linearized traveling wave equation for model variant I, has a unique global minimum at µ * on the negative half axis. The corresponding value is called critical wave-speed, and its position µ * is called critical decay rate.
Again, this is not hard to check and we refer to the Appendix for details. For variant II, we obtain for the linearized system the matrices and similar arguments as above lead to the positive solutioñ of the corresponding eigenvalue problem. We defineλ * andμ * as the quantities from Proposition 1.9 corresponding to the speed function of variant II. They satisfy analogous properties and results, and the details are again omitted for brevity.
Below we will see that the minimal values of the speed functions, λ * = λ µ * andλ * =λμ * , indeed provide the critical wave-speeds of monotone traveling wave solutions for model variants I and II. Figure  3 depicts these speed functions, and their minimizers, for parameters c,c, c ,c , s,s = 1 in models I and II, and also for the classical F-KPP equation. Their dependence on the parameter values in our models will be further discussed in Section 4.

Main results
We are now in a position to state our main results. Their proofs can be found in Sections 2 and 3. We begin with results on the speed of the right-most particle of o/oBBM, for both variants I and II. Using the McKean representation, the position of the right-most particle gives information about the solution of the dual F-KPP equations. While this already establishes a 'speed' of propagation when started in a Heaviside initial condition, the result can be significantly strengthened. Theorem 1.11 (Existence of traveling wave solutions). For each µ ∈]µ * ,0[ there exists a travelling wave solution (f µ , g µ ), increasing from 0 to 1 and of speed λ µ , to Equation (1.8). In particular, there are travelling wave solutions for all speeds λ > λ * . The analogous statement involvingμ * andλ * holds for the system (1.9).
Regarding the shape of the traveling wave solutions in this 'supercritical case', that is, for wave speeds strictly greater than the critical speed, we obtain the following asymptotic decay result.
The asymptotic shape of the traveling waves is thus described by the solutions to the linearized systems. In fact, depending on the decay rate of the initial condition, in the supercritical case, all solutions converge to the corresponding traveling wave solutions. Theorem 1.13 (Convergence of solutions in the supercritical regime). Consider system (1.8) with parameters c, c , s ≥ 0 and initial conditions u 0 , v 0 . Let λ * be the critical value of the corresponding speed function, which is attained at the critical decay rate µ * < 0. Then, for any µ ∈]µ * , 0[ and x ∈ R we have that if As for the classcial F-KPP equation, there are no monotone traveling wave-solutions with speed strictly below λ * (subcritical case).
The speed of propagation of the advantageous allele in systems with dormancy, when started in Heaviside initial conditions, falls into the critical regime. For this important case, we have the following result.
Again, we see that the beneficial allele propagates at a linear speed, which can be computed explicitly, depending on the model parameters. However, we currently lack finer results on traveling wave solutions. Some of the reasons and difficultires will be discussed in Section 4.
While both model variants so far exhibit the same qualitative behaviour, it is certainly interesting to investigate the quantitative differences. Unfortunately, due to the rather implicit description of the speed functions λ + µ andλ + µ , it is not easy to achieve general analytic results. The following is an example of what can be observed for fixed sets of parameters. However, all quantities are readily accessible via simulation.
Note that the result below already shows that for "unit parameters", dormancy slows the speed of propagation of beneficial alleles more severely in the spore model than in the seed bank model, and both significantly reduce the spread of beneficial alleles in comparison to the classical F-KPP model. The picture emerging for other parameter choices is quite rich and a more detailed discussion with concrete values using numerical methods will be provided in Section 4.
This situation is depicted in Figure 1 The proof is straightforward, see again the Appendix for details.

Organization of the paper
The proofs for the derivation of the critical wave-speed and the existence of a corresponding travelling wave solution follow well-trodden yet elegant paths and employ a convergence analysis of suitable additive and multiplicative martingales. Indeed, in Section 2, we derive solutions to the linearized wave equation (1.11) in terms of a corresponding additive martingale based on the dual on/off branching Brownian motion. The convergence properties of this martingale then provide information about the asymptotic speed of the right-most particle in the on/off branching Brownian motion.
In Section 3, we establish the existence of travelling wave solutions. This is done again following a well-known general recipe by characterizing solutions to the wave-equation (1.12) in terms of suitable multiplicative martingales.
We complete the paper by a discussion (Section 4) of the wave-speed in all model variants in dependence on the underlying parameters, by addressing open problems related in particular to the critical case, and by outlining possible future research.
Finally, the Appendix (Section 5) establishes technical results, including the existence and uniqueness of a related PDE necessary to estblish the crucial martingale property in Section 2.
2 Additive Martingales and the speed of the rightmost particle in o/oBBM In this section, we focus on model variant I. Analogous results (and proofs) can be obtained for variant II, but will be skipped for brevity. We employ the classic martingale approach dating back to Neveu [34] and McKean [32].

The additive martingale
We first characterize the solutions to the linearized system (1.11) in terms of an additive martingale, defined in terms of the dual on/off BBM.
be the on/off branching Brownian motion with offspring distribution p dual to model variant I as defined in 1.6, and let {F t } t≥0 be its canonical filtration. Then (f, g) is a solution of the linearised travelling wave equation (1.12) with speed λ > 0 iff the process X λ = (X λ t ) t≥0 given by Proof. Recall that active particles of the dual process at time t, indexed by α ∈ I t , move like Brownian motions, while dormant particles, indexed by β ∈ J t do not move. Hence we have (locally) d[M α r + λr] = dr and d(M β r + λr) = λdr, where [·] denotes quadratic variation. Further, the expected increase of active particles per time unit (and per active particle) due to branching is given by while the expected loss due to switches into dormancy is c (and the expected gain per dormant particle is c ). Hence, by Ito's formula (with jumps), using the independence of the offspring mechanism, we obtain after localization Now, replacing t with an arbitrary bounded stopping time τ we see using the optional stopping theorem that the second term on the right hand side is almost surely the zero process iff X is a local martingale. But this is equivalent to (f, g) being a solution of Equation (1.12).
Note that the local martingale property only rests on the expected increase of the number of particles during each branching event, and not on the finer properties of the offspring distribution. This corresponds to the information that is lost when switching from the original F-KPP equation to the linearised equation.
Our next goal is to show that for certain choices of λ and corresponding test-functions (f, g) we get that the local martingale (X λ ) = (X λ t ) t≥0 from Proposition 2.1 is actually a true martingale. This will then allow us to investigate its convergence behaviour by means of the martingale convergence theorem.
Proposition 2.2. Let µ < 0 and (f µ , g µ ) be given as in Equation (1.14) with decay rate µ. Let λ µ be the value of the speed function from Prop. 1.9 at µ. Finally, let {M t } t≥0 be an on/off BBM and {F t } t≥0 its canonical filtration. Then, the local martingale (X λµ ) = (X λµ t ) t≥0 given by Proof. Note first that (X λµ ) is a positive local martingale and hence by Fatou's lemma a supermartingale.
Step 1: We first show that in order to compute the expectation of X λµ t (for some t ≥ 0) we only require a single trajectory from our branching particle system, i.e.
where B = (B t ) t≥0 is an on/off Brownian motion and I, J are the random time intervals in which it is active, respectively dormant.
To show this consider the PDE given by with initial conditionsū(0, ·) = f µ andv(0, ·) = g µ . Further, by a monotone class argument we can assume w.l.o.g. thatū(0, ·),v(0, ·) ∈ C ∞ b (R) and are hence bounded. By Proposition 5.2 this PDE then has a solution which is unique by [5,Theorem I]. Then on the one hand by the Feynman Kac formula for on/off Brownian motion (see Proposition 5.7) we know that where (x, a) means that the on/off BM is started from one single particle at position x ∈ R in an active state.
On the other hand we may show that E[X λµ t ] solves the same PDE by considering for fixed T > 0 Then by Ito's formula we see after localizing that Hence, by the optional stopping theorem since (ū,v) solve Equation (2.1) we see that ζ is a local martingale. Now, since the initial conditions of the PDE are bounded we have for every t ≥ 0

But then sup
which is integrable. Hence, ζ is a true martingale and for all T > 0 and x ∈ R as desired.
Step 2: Next, we aim to show that is constant in time. Again by Ito's formula we see that after localization since −µλ µ is the Eigenvalue of 1 2 µ 2 A + Q + R. Thus, ξ is a local martingale implying that we have by non-negativity that ξ is actually a supermartingale. Now elaborating and adapting the argument from [12, p. 102] by Doob's maximal inequality we see for c > 0 and choosing someμ < µ, that and thus for any d > 0 taking c = e −μd since either |I t | ≥ 1 or |J t | ≥ 1 is integrable for each t ≥ 0. But then we have since there exists some C t,µ > 0 such that that ξ µ is also integrable and hence indeed a true martingale. This gives constant expectation for (X λµ ) as required using Step 1.
2.2 L 1 convergence on (µ * , 0) In this section we establish the convergence of the additive martingale (X λµ ) to an almost surely positive limit X λµ whenever µ * < µ < 0. where Z α t (s) is an conditional on F s independent version of (X λµ ) started from an active particle and analogously Z β t (s) is started from a dormant particle. Now, we apply Neveu's Lemma 5.9 with conditional expectations to see for any p ∈ (1,2] Thus, by taking expectations Now since the map µ → λ µ is increasing on [µ * ,0] we may find some p ∈ (1,2] such that λ µ > λ µp . But then since X λµp is a martingale. This implies that for every n ∈ N is a submartingale we get L p boundedness for (X λµ t ) t≥0 . The martingale convergence theorem then gives the desired result.
The next step is to analyze this limiting random variable. We begin by showing a 0 − 1 law.  Proof. This follows from the fact that E[X λµ 0 ] > 0 and the L 1 -convergence.

Almost sure convergence to 0 on (−∞, µ * )
In this section we show that the additive martingale (X λµ t ) t≥0 converges almost surely to zero if µ < µ * < 0. This section follows along the lines of Lemma 4.11 in [12]. We begin by examining the negativity of the diagonal entries of the matrix 1 2 µ 2 A + Q + R + µλI 2 . Lemma 2.6 (Negativity Lemma). Let F a (µ) be the top left diagonal entry and F d (µ) be the bottom right diagonal entry of 1 2 µ 2 A + Q + R + µλI 2 and µ ∈ R (cf. 1.15). Then we have F y < 0 for each y ∈ {a, d} in a suitable neighbourhood of µ.
Proof. Let d be a Frobenius-Perron-Eigenvector of the Matrix 1 2 µ 2 A + Q + R with corresponding Eigenvalue −µλ. By (a variant of) the Frobenius-Perron-Theorem [37, Theorem 1.1], we have that d > 0. The assertion then follows from ( 1 2 µ 2 A + Q + R + µλI 2 ) d = 0 and c, c > 0 and the continuity of F y . Next we consider an upper bound on the expected value of the limiting random variable.
Proof. Note that by the branching property of the on/off branching Brownian motion, we have that where λ = λ µ and T 1 is the first time where the initial particle of the on/off branching Brownian motion either wakes up or goes to sleep, T 2 is the first time that the initial particle branches, X λµ,0 is an independent copy of the limiting random variable X λµ started in the active/dormant state of the initial particle at the time T 1 and X µ,1 and X µ,2 are independent copies of the limiting random variable started in the active/dormant state of the first particle at time T 2 . Note that from this we can derive, that If the on/off branching Brownian motion starts in the awake state, then we can evaluate the expected values as in [12, p. 106]. If the on/off branching Brownian motion starts in the dormant state, we have T 1 < T 2 almost surely and therefore get via the moment-generating function of the exponentially distributed random variable T 1 , that since the initial particle does not move and first has to wake up before it can branch. For the other expected value we have, that {T 2 < T 1 } is a nullset.
Proof. Since the additive martingale is non-negative and converges almost surely, we have 0 ≤ θ a , θ d . From the previous lemma we have .
Due to the negativity lemma we have −F a (pµ) + 2s > 0 and −F d (pµ) > 0 for p close enough to 1.
Rewriting the above inequalities yields where λ := λ µ . Inverting the matrix 1 2 (µp) 2 A + Q + R + µpλI 2 by using it's adjunct we get Since the determinant in the above expression is a polynomial of degree 3 (in µp), with positive leading coefficient we have that it is strictly postive between its two smallest zeroes 1 . Therefore the above adjunct is a matrix with strictly negative entries and we get θ = ( 1 2 (µp) 2 A + Q + R + µpλI 2 ) −1 ( 1 2 (µp) 2 A + Q + R + µpλI 2 )θ ≤ 0 and in total θ = 0.
1 Since µ * is the unique local minimum of the speed function there are at most two values µ 1 , µ 2 (one of them being µ itself) on the negative half axis with λµ 1 = λµ 2 = λµ which correspond to the two smallest zeros. The last zero is obtained at the unique µ 3 on the positive half axis with λ − µ 3 = λµ. This value must exist since lim µ→0 λ − µ = ∞ and limµ→∞ λ − µ = 0.

The speed of the rightmost particle of o/oBBM
In this section we derive the exact asymptotic speed of the rightmost particle of an on/off BBM. Let the position of the rightmost, respectively the leftmost, particle of an on/off BBM with arbitrary starting point. We can now use the almost sure convergence of the additive martingale (X µ t ) to the trivial limit zero to show an upper bound for the asymptotic speed.
Proof. Let > 0 and λ := λ * + and µ < µ * < 0 corresponding to λ. From we get L t + λt t→∞ −→ +∞ almost surely. In other words almost surely there is an N ≥ 0 such that for all t > N we have L t + λt > 0. From this we get Since > 0 was arbitrary the assertion follows.
Next, we use the a.s. convergence to a positive random variable to obtain the corresponding lower bound.
Proof. This follows along the lines of [12, Theorem 1.44]. Fix µ * < µ < 0. Since X λµ is an almost surely positive random variable of expectation one (when the corresponding on/off BM is started from a single dormant particle at zero otherwise rescaling with a constant is necessary) we may define an equivalent probability measure via dQ µ = X λµ dP.
Moreover, by exchanging expectation and derivative we notice that any derivative with respect to µ of (X λµ t ) t≥0 is still a martingale. Hence, we notice that S γ t > 0 for each γ ∈ K t and t ≥ 0 and γ∈Kt S γ t = 1.
Hence, by Jensen's inequality we see that Moreover, where we used that N 2 is a martingale under Q µ . But then Doob's maximal inequality yields for every > 0 and n ∈ N Q µ sup Since the right hand side is summable over n ∈ N we get Since the preceding holds true for every µ ∈]µ * ,0[ and the right hand side is continuous we may infer lim sup Now, recall that µ * was the minimizer of the map µ → λ µ on the negative half axis which yields that ∂ µ µλ µ | µ=µ * = λ * + µ∂ µ (λ µ )| µ=µ * = λ * and thus lim sup Using that the on/off BBM is sign invariant we finally get lim inf The preceding statement allows us to show that when started from Heaviside initial conditions the solution of our PDE is an approximate wave form of speed λ * .
Corollary 2.11. Consider the solution to Equation (1.8) with initial conditions u 0 = v 0 = 1 R + . Then for any λ > λ * u(t, x + λt) → 1 and v(t, x + λt) → 1, Proof. This follows directly from the duality relations Remark 2.12. In simulations one may see the emergence of approximate travelling waves when the system is started from Heaviside initial conditions (see Figure 2.12 below). The preceding corollary establishes that the bulk of the mass of the approximate waves cannot move faster or slower than λ * t for large t which is why we speak of approximate travelling waves of speed λ * .

Multiplicative martingales and travelling wave solutions
The convergence results in the preceding section contain all the necessary tools to turn back to our original PDE (1.8). We again reduce solutions of the corresponding wave equation (1.10) to martingales of a specific form. In this case we will look for multiplicative martingales rather than additive martingales. As before, we focus on model variant I. Analogous results hold for variant II.

The multiplicative martingale (Y λ t )
Proposition 3.1. Let M = (M t ) t≥0 be an on/off BBM. Then (f, g) is a solution to Equation (1.10) iff (Y λ ) given by is a local martingale.
Proof. By Ito's formula and adding and subtracting the compensator of the jumps we have Hence, by the optional stopping theorem as before we get the desired result.

The supercritical case λ > λ *
We now use the limiting random variable X λµ from Theorem 2.3 to construct a suitable multiplicative martingale. In order to do so we define for µ * < µ < 0 We claim that the corresponding multiplicative quantity is a martingale.
Theorem 3.2. Let µ * < µ < 0, M = (M t ) t≥0 be an on/off BBM and Then is a martingale and f µ , g µ are increasing from 0 to 1.
Proof. First note that by taking limits in Equation (2.2) we obtain the following representation for X λµ for each t ≥ 0 where Z α µ and Z β µ are i.i.d. copies of X λµ started at an active particle, respectively a dormant particle. Then we have by the Markov property and that X λµ under P x is the same as X λµ e µx under P 0 where in the penultimate equality we used Equation (3.1). For the last part of the statement note that we have which immediately yields that both f µ and g µ are increasing from 0 to 1.
Next, we use our representation of the travelling wave solution to establish the asymptotic behaviour of our waves at infinity.
Proof. On the one hand we have using the branching property that Rearranging yields On the other hand we also obtain This gives by monotone convergence as desired. Replacing our representation for f µ by the corresponding representation for g µ we obtain the second part of the statement.
Finally, we show that for initial conditions with this decay behaviour the corresponding solution to Equation (1.8) will converge into the previously established wave form.
Theorem 3.4. Consider Equation (1.8) with initial conditions u 0 , v 0 . Then for any µ * < µ < 0 and x ∈ R we have that if Proof. We are guided by [12,Theorem 1.41]. We note first that by our assumptions for ε > 0 small enough and x large enough For the upper bound pick x large enough such that For the lower bound pick x large enough such that Now, note that for ε small enough we then have where we possibly have to pick x even larger. Rearranging then yields the lower bound. Analogously, Now, since L t + λ µ t → ∞ as t → ∞ we get pathwise for large enough t recalling Taking limits and applying expectations yields when started from an active particle at x ∈ R: where we used the preceding inequality and the duality relation from Proposition 1.8. Since ε > 0 was arbitrarily small we see as t → ∞ and by a similar calculation starting the dual process from a dormant particle at x ∈ R we also get v(t, x + λ µ t) → g µ (x) as desired.
3.3 Non-existence of travelling waves in the subcritical case λ < λ * Proposition 3.5. There does not exist a travelling wave solution to Equation (1.9) increasing from 0 to 1 of speed λ < λ * .
Proof. The proof follows along the lines of [19,Section 3.4]. Assume that there exist a travelling wave solution (f λ , g λ ) to Equation (1.8) of speed λ < λ * . Then by Proposition 3.1 is a local martingale bounded in [0,1] and hence a true martingale. Thus, starting our on/off-BBM from a single active particle we get and when started from a dormant particle Since (Y λ t ) t≥0 is a positive and bounded martingale there exists some random variable Y λ such that almost surely and in L 1 as t → ∞. Thus, on the one hand On the other hand since f λ , g λ take values in [0,1] Recalling from Section 2.4 that L t t → −λ * almost surely we see that L t + λt → −∞ and deduce from the fact that (f λ , g λ ) are travelling waves from 0 to 1 Hence, Y λ = 0 almost surely. But by Equation (3.2) this implies that a contradiction to the assumption that f λ , g λ are travelling waves from 0 to 1.

Discussion and outlook
In what follows we aim to analyze the impact of the introduction of dormancy in both variants of the F-KPP equation with seed bank on the wave speed. We thus set c =c, c =c , s =s for the remainder of this section.

Comparing the models
For the natural parameter choices c = c = s = 1, Proposition 1.16 shows that the critical wave-speed of the classical F-KPP model dominates the one of model variant I ("seed bank model"), which in turns dominates the critical wave-speed of model variant II ("spore model"). Interestingly, this monotonicity seems to hold also with regard to the entire graphs of the respective speed functions. However, Figure  3 shows that a similar monotonicity does not hold for the position of the minima. Indeed, while the position of the minimum of the speed function for the classical F-KPP equation (remarkably) coincides with the position of the minimum for variant II, this is not the case for variant I. For the above unit parameters, the reduction of the speed of advance of an advantageous gene due to dormancy is more severe in the spore model than in the seed bank model. On an intuitive level, this can be understood from their corresponding dual processes through Theorem 1.10. In o/oBBM of model variant I, individuals can both move and reproduce while being active. In particular, newborn particles can contribute to the spread of the population immediately after birth and are only slowed down due to switches into dormancy later on. In contrast, in the o/oBBM of model variant II, newly created actively reproducing individuals are initially non-moving and need to switch their type from a to d before they can be subject to dispersal, in turn preventing them from branching.
However, the picture of the impact of dormancy over the whole parameter space is relatively complex, as we will see below. Indeed, although the values of the critical wave-speeds λ * andλ * of model variants I and II can be explicitly characterized, they are not just simple functions of the parameters c, c and s, and in practice will often have to be computed numerically. To get a feeling for the effects of the different parameters on the critical wave-speed in the various models, we highlight the following concrete scenarios.
Fixed transition rates c, c = 1, selection varying from 0 to ∞: This situation is depicted in the upper left panel in Figure 4.1. The critical wave-speeds in all three models grow as (apparently) concave functions from 0 to ∞. However, the relative impact of dormancy in model I seems to become smaller and smaller in comparison to the classical F-KPP model. Interestingly, this is not the case for variant II, where the critical wave-speed is always precisely half the speed of the classical model. In fact, we have that λ * ,classical (s) = √ 2s = 2λ * (s) for all s ∈ (0, ∞). It would be interesting to find a quantitative intuitive argument for this precise relationship. One hint in this direction seems to be that if c = c , then the asymptotic fraction of time that an individual spends in the active resp. dormant state is precisely 1/2 (but this is of course also true for model I, which shows different behaviour).
Fixed selection rate s = 1, transition rates c = c simultaneously varying from 0 to ∞: This situation is depicted in the lower right panel in Figure 4.1. The fixed relationship c = c ensures that the active and the dormant pool of individuals are of the same relative size. Again, the classical critical wave-speed is precisely twice as large as the one of variant II, and they agree with the case s = 1 = c = c . This means in particular that the critical wave-speed for the spore-model is independent of the overall transitioning rate c = c in this case. But the dependence of model I on c = c is non-trivial: For small overall transition rates, transitions from the active population into the seed bank are rare. For the dual o/oBBM process of variant I this means that newly born particles will spend a long time in an actively moving and reproducing state. It is thus intuitively clear that the speed of the right-most particle approaches the classical case as c = c → 0. In contrast, for very large switching rates, particles will almost immediately enter (and leave) the seed bank, with on average about half of the particles (including the newly created ones) being in a dormant state at any given time. These rapid fluctuations between active and dormant state slow the action of the Laplacian and the selection term by a factor of 1/2, explaining the reduced critical wave-speed, which converges to the one of variant II.
Fixed s = c = 1 while c varies from 0 to ∞: This situation is depicted in the upper right panel in Figure 4.1, where the critical wave-speed is represented as a function of c. Small values of c correspond to a 'small seed bank', that is, particles spend most of their time in an active state. It is thus not surprising that the behaviour of variant I is close to the classical case for small c, in contrast to variant II, where the dual process needs particles to be in the d state to engage in spatial motion. Surprisingly, the critical wave-speed as a function of c seems to be uni-modal in variant II, hinting at a trade-off between selection and switching effects. Again this seems to be due to the mutually exclusive branching vs motion character of the dual of variant II. More precisely, on the one hand diffusion is needed to colonize new areas and initiate traveling waves, but this only takes place while in the dormant non-reproducing state. On the other hand, the branching (selection) term in the active component contributes most to the linear speed of the right-most particle, but it can only be effective if particles switch quickly into dormancy to find their way into new environments. It is thus reasonable to expect an equilibrium of sorts for these two effects that leads to a maximal wave speed. Our simulations suggest that this equilibrium is attained at c = c , i.e. whenever switching to and from the dormant state happens at the same rate, which is again remarkable.
Finally, as c → ∞, the critical wave-speeds of both variants I and II approach 0. In this case, the effect of the dormant component dominates, and hence the effect of the selection becomes smaller and smaller.
Fixed s = c = 1 while c varies from 0 to ∞: This situation is depicted in the lower left panel in Figure 4.1. Note that large values of c mean that dormant particles in the dual processes 'wake up quickly'. For variant I the critical wave speed decreases as c decreases, since in the dual process, dormant particles cannot produce offspring. However, it is an interesting question whether 0 can actually be reached, and in fact a phase-transition seems to emerge: If c is close to 0, switching into dormancy (at rate c) amounts to an effective 'killing', and branching can only happen during the initial active phase. The expected time in this initial phase is 1/c, so that iff the branching process will be (effectively) 'super-critical', and the limit of the critical wave-speed should stay above 0 as c → 0, whereas in the opposite case, one should see convergence to 0. This is consistent with the simulations underlying Figure 4.1, for the values c = 1 and s = 3/2 (super-critical) and s = 1/2 (sub-critical). This is another example where arguments via the probabilistic dual process shed light on the behaviour of the original analytic system. The critical wave-speed of variant II again exhibits a uni-modal shape as the result of a trade-off that can be understood from the dual process with similar arguments as in the previous scenario.

Related models
Reaction-diffusion models similar to variant I also appear in theoretical neuroscience in the form of nerve-axon equations, see e.g. [15]. There, a rather general multi-component reaction diffusion system is  investigated, where, similar to our systems, a spatial diffusion term is only present in a single component. However, the author is specifically interested in traveling (nerve-)pulses rather than traveling waves, meaning that the 'wave-shape w' is supposed to go to zero on both ends (i.e. w(x) → 0 for x → +∞ and x → −∞). The very general assumptions on the coupling terms in these papers have provided a framework for a rather large body of literature of related works, and many further variants of coupling terms have been discussed. The corresponding papers are typically focused on analytic and numerical aspects and indeed often do not exhibit duality relations to stochastic processes such as branching Brownian motion. Formulas for the speed of traveling waves of certain variants of the nerve-axon equations can for example be found in [20] (see p.4) on contaminant transport.
The related paper [41] considers a model for a cholera epidemic, where the dispersing individuals are interpreted as infectious bacteria and the non-dispersing individuals are considered as humans, acting as 'incubators' for the reproduction of the bacteria whose mobility is negligible. This model essentially looks the same as the spore-model, however the coupling term in the second line of the spore-model does not fulfill the required assumptions in [41] on its second derivative 2 .
In [42] the authors use so-called speed index functions to investigate the wave-speed. These are essentially Laplace-type transforms of an integral kernel. The wave-speed in [42] then appears as a solution to an equation involving the speed-index-function and the model parameters. This is somehow reminiscent of the speed-functions considered in the present paper. According to figures 5-7 and 11-17 of [42] the dependence on the model parameters of the wave-speed is monotone in contrast to the spore model, where we find a non-monotone dependence.

Open questions and future research
Given the large body of work on the classical F-KPP equation and its variants, our results seem far from complete. In this section we briefly touch upon several open questions including the related technical difficulties, and outline some aspects for future research.
Convergence to the critical wave and its speed and shape. Note that our convergence theorem involving speed and shape of the wave (Theorem 1.13) only covers the super-critical regime, and that our results for the critical case (Theorem 1.15) neither provide an asymptotic shape nor any convergence behaviour into the critical travelling wave. This is stark contrast to the classical F-KPP equation, where one has convergence of the solution, when started from suitably decaying initial conditions, into the critical travelling wave (see e.g. [9]). More precisely, there exists a process (m t ) t≥0 such that where u is the solution to the classical F-KPP equation started from e.g. Heaviside initial conditions and w its travelling wave solution of corresponding to the critical speed √ 2. Here, m t is chosen as the unique value such that u(t, m t ) = 1 2 for each t ≥ 0. The asymptotics of m t have been studied in detail by e.g. McKean, Bramson and Roberts in [32,10,9,36], who obtained It would be desirable to reach similar results for the seed bank versions of the models, starting with the question whether one has the existence of a process (m t ) t≥0 , or rather two potentially different processes (m 1 t ) t≥0 and (m 2 t ) t≥0 , such that as t → ∞, alongside with the finer asymptotic results for their speeds. For coupled two-component systems as in our case this seems to have been an open question for the last couple of decades (see e.g. [14, p.2,5]). Unfortunately, this also implies that many further results are currently inaccessible for higher order system like ours, including e.g. the probabilistic representation of the travelling wave of Lalley and Selke and related results regarding the shape of the critical wave such as [26,Theorem 1].
Convergence of the additive martingale and existence of travelling waves in the critical case. Related to the above issues is the problem that our results regarding the convergence of additive martingales in Section 2 cover everything except the critical case µ * . In the context of the classical F-KPP equation this can be covered through a spine argument using the Girsanov Theorem and that the quadratic variation of the Brownian motion is deterministic (see e.g. [25]). However, in the case of an on/off Brownian motion (without branching) the quadratic variation is truly probabilistic making an application of the Girsanov Theorem difficult: For instance if c = c = 1, the on/off Brownian Motion can be expressed by where N is a Poisson process with rate one and B is a standard Brownian Motion. Its quadratic variation is then given by the process which is a random piece-wise linear function.
This also leads to a lack of existence results for travelling waves in the critical case λ * . An alternative approach through adoption of the stopping line theory by Chauvin [13] along the lines of [19] turns out to also be challenging for the multi type case.
Uniqueness of travelling waves modulo translation. For the classical F-KPP equation it is well known that the monotone travelling wave solutions from 0 to 1 are unique up to translations. A probabilistic approach may be found in [19] but relies on the convergence of the additive martingale for the critical case µ * . We do however believe that the methods outlined in [12,Theorem 1.41] in combination with the analytic methods from [12, Section 3 (f)] may also yield the corresponding result for Variant I of the F-KPP equation with seed bank. Note however that small modifications will be necessary. For example, the phase plane formulation of the equation will involve the 3-dimensional vector (u, v, u ) instead of the 4-dimensional vector (u, v, u , v ) and the proof of Lemma 3.2 in [12] will only yield exponential decay for the active part (u − 1, (u − 1) ) instead of the whole system (u − 1, v − 1, (u − 1) , (v − 1) ). These difficulties can be overcome, for example, by considering a delay reformulation of the system as in [6]. Since we however want to focus on probabilistic methods for the analysis of our models in this paper and the aforementioned methods require heavy use of phase plane analysis and differential equation theory we refrain from providing the technical details here.
Properties of on/off branching Brownian motions. Branching Brownian motion has been a classical object of study in probability theory for more than 50 years, apparently beginning with [33]. In the last decade, it has experienced increased interest due to e.g. the construction and analysis of its extremal process ( [1], [3]).
It appears to be an interesting task to investigate related extremal properties of our two new variants of this model in the form of on/off branching Brownian motions. This seems to extend the recent line of research on time-inhomogeneous or variable speed BBM ( [8], [30]) or in variable environments, e.g. [31] into a novel direction.
Proof. We provide a rough outline of the necessary steps of the proof. For the missing details we refer the reader to [35,Section 3]. Let T > 0. We begin by noting that the PDE (1.9) with initial conditionū(0, ·) ≡ 1 has the constant 1 function as a solution. And similarly using the initial condition u(0, ·) ≡ 0 the equation has the constant 0 function as a solution. Next, define the maps which are Lipschitz with some common constant K > 0 on the interval [0,1] 2 . Then we consider the following reformulation of Equation (1.9): for which we will simplify the notation to L 1ũ =f (ũ,ṽ) and L 2ṽ =g(ũ,ṽ).
We then define iteratively the sequence (ū k ,v k ) k∈N viā , y),v(s, y)) ds dx (5.4) andū 0 =v 0 ≡ 1, where G 1 and G 2 are the Green's function of the operator L 1 and L 2 , respectively. Similarly, we define a corresponding sequence (u k , v k ) with u 0 = v 0 ≡ 0. The proof can then be finished in the following steps: 1. Check by induction whether (ū k ,v k ) exhibits enough regularity such such that the mild formulation from Equation (5.4) implies that (ū k ,v k ) satisfies the corresponding differential equation L 1ū k+1 =f (ū,v), and L 2v k+1 =g(ū,v) for all k ∈ N. Repeat the procedure for (u k , v k ).
Proof. Consider the space L ∞ ([0, T ], C 1 b (R)) and proceed via Picard iteration. Note first that the linear elliptic operator given by Rigorously, we define v as above and consider the recursive sequence given by for n ∈ N and t ≥ 0. Now, by induction if u n ∈ L ∞ ([0, T ], C 1 b (R)) we have that u n+1 (t) ∈ C 1 b (R) for every n ∈ N and t ≥ 0. Moreover, for t ≥ 0 we have since (e tA ) T ≥t≥0 ∈ C ∞ ([0, T ], L(C 1 b (R))) u n+1 (t) ∞ + ∇u n+1 (t) ∞ ≤ sup By a similar calculation using the linearity of the semigroup operators we see that Hence, we get existence of a solution u ∈ L ∞ ([0, T ], C 1 b (R)) through a limiting argument for the integral equation u(t) = e tA f + c t 0 e (t−s)A v(s) ds (5.8) where v is given by Equation (5.7). Hence, since u ∈ L ∞ ([0, T ], C 1 b (R)) we can infer that v actually solves the equation and v ∈ L ∞ ([0, T ], C 1 b (R)). Next, by [29, Proposition 4.2.

Feynman-Kac representation
As in the non seed bank case one is able to obtain Feynman-Kac representations for specific PDEs in terms of (on/off) Brownian motions. For details we refer to future work by one of the present authors (F.N.).
Proposition 5.7 (on/off Feynman-Kac for Variant I). Let T, λ, c, c , s > 0 and u, v ∈ C 1,2 b ([0, T ] × R) be solutions to the PDE with initial conditions u 0 , v 0 ∈ C b (R). Then for an on/off Brownian motion of Variant I (B t ) t≥0 started in the active state at x ∈ R we have the probabilistic representation u(t, x) = E (x,a) e t 0 s1 I (r) dr (u 0 (B t + λt)1 I (t) + v 0 (B t + λt)1 J (t)) where I, J ⊆ [0, T ] are the random time intervals in which the on/off Brownian motion is active and dormant, respectively.
with initial conditions u 0 , v 0 ∈ C b (R). Then for an on/off Brownian motion of Variant II (B t ) t≥0 started in the active state at x ∈ R we have the probabilistic representation u(t, x) = E (x,a) e t 0 s1 I (r) dr (u 0 (B t + λt)1 I (t) + v 0 (B t + λt)1 J (t)) where I, J ⊆ [0, T ] are the random time intervals in which the on/off Brownian motion is active and dormant, respectively.