Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection

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Model 5G

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Theoretical Population Biology xx (xxxx) xxx–xxx

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Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb

Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection Lukas Heinrich a , Johannes Müller a,c, *, Aurélien Tellier b , Daniel Živković b a b c

Center for Mathematics, Technische Universität München, 85748 Garching, Germany Section of Population Genetics, Center of Life and Food Sciences Weihenstephan, Technische Universität München, 85354 Freising, Germany Institute for Computational Biology, Helmholtz Center Munich, 85764 Neuherberg, Germany

article

info

Article history: Received 13 October 2017 Available online xxxx MSC: 92D10 39A50 60H10 Keywords: Diffusion Moran model Seed bank Selection Site-frequency spectrum

a b s t r a c t Population genetics models typically consider a fixed population size and a unique selection coefficient. However, population dynamics inherently generate fluctuations in numbers of individuals and selection acts on various components of the individuals’ fitness. In plant species with seed banks, the size of both the above- and below-ground compartments induce fluctuations depending on seed production and the state of the seed bank. We investigate if this fluctuation has consequences on (1) the rate of genetic drift, and (2) the efficacy of selection. We consider four variants of two-allele Moran-type models defined by combinations of presence and absence of fluctuations in the population size in above-ground and seed bank compartments. Time scale analysis and dimension reduction methods allow us to reduce the corresponding Fokker–Planck equations to one-dimensional diffusion approximations of a Moran model. We first show that if the fluctuations of above-ground population size classically affect the rate of genetic drift, fluctuations of below-ground population size reduce the diversity storage effect of the seed bank. Second, we consider that selection can act on four different components of the plant fitness: plant or seed death rate, seed production or seed germination. Our striking result is that the efficacy of selection for seed death rate or germination rate is reduced by fluctuations in the seed bank size, whereas selection occurring on plant death rate or seed production is not affected. We derive the expected site-frequency spectrum reflecting this heterogeneity in selection efficacy between genes underpinning different plant fitness components. Our results highlight the importance to consider the effect of ecological noise to predict the impact of seed banks on neutral and selective evolution. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Genetic drift and natural selection are prominent forces shaping the amount of genetic diversity in populations. In diploid dioecious organisms, natural selection can be decomposed in different components: (1) viability selection as the differential survival of the genotypes from zygotes to adults, (2) fecundity (or fertility) selection as the differential zygote production, (3) sexual selection as the differential success of the genotypes at mating, and (4) gametic selection as the distorted segregation in heterozygotes (Bundgaard and Christiansen, 1972; Clegg et al., 1978). In effect, population genetic models with discrete generations or with Malthusian parameter ignoring agestructure lump these components into one unique selection parameter. Thus, experimental or genomic population studies describing changes in allele frequencies often fail to describe and dissect the respective effects of these selective modes. Several theoretical studies on fertility (Bodmer, 1965) or on sexual selection (Karlin and Scudo, 1969) as well as experimental work on animal (Prout, 1971b, a; Christiansen and Frydenberg, 1973) and plant (Clegg et al., 1978) populations have attempted to disentangle the respective influence of these selection components. Note that in plants for which seed germination depends on environmental cues, fecundity selection takes place above-ground as defined by seed production. In contrast, viability selection occurs above-ground with selection on plant survival, but also below-ground with selection on seed mortality and the ability of seeds to germinate. In age-structured populations, however, genetic drift and selection can act differently than predicted by models without age structure (overview in the book by Charlesworth, 1994). The first type of age-structured models are simply obtained by individuals’ life span and reproduction overlapping several generations. Selection for fecundity and viability can show different outcomes, such as time to allele

* Corresponding author at: Center for Mathematics, Technische Universität München, 85748 Garching, Germany. E-mail address: [email protected] (J. Müller).

https://doi.org/10.1016/j.tpb.2018.05.003 0040-5809/© 2018 Elsevier Inc. All rights reserved.

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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fixation and/or maintenance of alleles, compared to the Wright–Fisher and Moran models. This occurs if fecundity or longevity act at different ages of the structured population and under a logistic population size dynamic (Anderson and King, 1970; King and Anderson, 1971; Charlesworth and Giesel, 1972a, b). An interesting question arising from the current increasing availability of genomic data is whether in age-structured populations selection for fecundity can be disentangled from that of viability using population genomics statistics (such as the site-frequency spectrum, SFS). A second type of age-structured model is obtained when considering that individuals may remain as dormant/quiescent structures spanning several generations. Quiescence in reproductive structure is in fact wide-spread, such that plant seeds or invertebrate eggs can be persistent states that allow to buffer a variable environment (Evans and Dennehy, 2005). The time at which seeds germinate (or eggs hatch) can be variable, such that only some of the offspring live in detrimental periods, and most likely at least some in a beneficial environment. Bacterial spores or lysogenic states of temperate phages are also examples of such a bet-hedging strategy. Seed banks represent thus a storage of genetic diversity decreasing the probability of population extinction (Brown and Kodric-Brown, 1977), diminishing the effect of genetic drift (Nunney, 2002), slowing down the action of natural selection (Templeton and Levin, 1979; Koopmann et al., 2017) and favouring balancing selection (Tellier and Brown, 2009). The strength of the seed bank effect clearly depends on the organism under consideration. The delay in the germination of seeds respectively in the hedging of eggs often is in the range of several generations only. This time scale is much shorter than the evolutionary time scale, given by the average coalescent time. We call these seed banks ‘‘weak’’. Dormant states of bacteria, however, can last a longer time (many generations) even longer than the average coalescent time. These seed banks are called ‘‘strong’’, and modelled in a similar way as weak selection: the time scale of the quiescent state is scaled by the inverse of the population size. In particular Blath et al. (2015, 2016) investigated in a series of papers strong seed banks, and found mathematically appealing results as deviations from the Kingman coalescent. In the present paper, we focus on weak seed banks aiming at applications to plant or invertebrate species. In a seminal paper Kaj et al. (2001) investigated the effect of weak seed banks on the coalescent. The key parameter here is G that denotes the average number of plant generations a seed rests in the soil. Kaj et al. (2001) did show that for a large population size the coalescent with seed bank is equivalent to the Kingman coalescent, running on a slower time scale (namely G −2 , where we use the notation of the present paper). It was subsequently shown that G can be estimated per population using information on the census size and polymorphism data, the latter being summarized as the nucleotide diversity and the SFS (Tellier et al., 2011). Furthermore, neglecting seed banks may yield distorted results for the inference of past demography using for example the SFS (Živković and Tellier, 2012). Interestingly, the effect of (weak) selection is enhanced by the slow-down of the time scale due to seed banks (Blath et al., 2013, 2016; Koopmann et al., 2017). The effect of genetic drift and weak natural selection on allele frequencies can be computed in a diffusion framework in a Moran model with deterministic seed bank (Koopmann et al., 2017). While the diffusion term (defining genetic drift) is scaled by G −2 , matching the backward coalescent result of Kaj et al. (2001), the coefficient of natural selection in the drift term is multiplied only by G −1 . In biological terms, this means that the strength of selection, as defined by a unique selective coefficient, is enhanced by the seed bank compared to the effect of genetic drift, even though the time to reach fixation for an allele is increased (Koopmann et al., 2017). We investigate two additions to this current body of theoretical literature on weak seed banks (Koopmann et al., 2017). First, we compute the effect of realistic models relaxing the hypotheses of a fixed size for the population above ground, and of a deterministically large one for the seed bank compartment. By doing so, we generate fluctuations in the population size above and/or below ground. This extends the classic Moran or Wright–Fisher models, following the recent recognition that fluctuations in the population dynamics affect the rate of genetic drift or selection (Huang et al., 2015). However, this approach is faced with an additional, technical difficulty: if two alleles are present in a population of a constant size, it is sufficient to keep track of the number of individuals for one allele only, while in the case of a fluctuating population size with logistic dynamics, the state space becomes essentially two-dimensional. This more general case requires a time scale analysis and dimension reduction by singular perturbation approaches. These methods are applicable in the case where the population size becomes large (Parsons and Quince, 2007; Parsons et al., 2008; Kogan et al., 2014). Though in those papers (as in the present one) the arguments are used in a formal way, the validity of this approach is proven ( Kuehn (2015, chapter 15.5 and quotations therein), Givon et al. (2004, section 6), Majda et al. (2001)). This latter analysis reveals that the dynamics is well described by appropriate scaled diffusion approximation of a Moran model. Second, we dissect the fitness of plants into four components, which can be possibly affected by genetic drift occurring above ground and in the seed bank. We compute classic population genetics results for neutral and selected alleles and derive the expected SFS for the alleles under different fitness components. The analysis reveals the effect of fluctuations of the population size in the above-ground population (plants) as well as in the below-ground population (seeds). In particular, we find that genetic drift and selection are differently influenced by fluctuations in the above and below-ground population size. Additionally, the selection coefficient of alleles involved in seed death rate or germination rate is reduced by fluctuations of the total number of seeds, but not by the total number of plants, while the two others are not affected (plant death rate and seed production). Below-ground fluctuations also reduce the seed bank storage effect of neutral genetic diversity. This paper extends the study performed in Koopmann et al. (2017). In that paper particularly a deterministic seed bank and constant above-ground population have been investigated, with an arbitrary survival rate of seeds (non-Markovian case) and weak selection solely for seed production above-ground (selection on fecundity). In the present paper we only consider the Markov case (seeds loose their ability to germinate at constant rate). The novelties of the present study are to (1) explicitly take into account the four different fitness components of the plant life cycle above- (plant fertility and survival) and below-ground (seed mortality and the ability of seeds to germinate), and (2) assess the efficacy and signature of weak selection for these selection components. In this way we are able to go beyond the paper of Koopmann et al. (2017) as we can study the effect of different sources of stochastic fluctuations on the effect of neutral evolution and the different components of weak selection.

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2. Methods

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In previous coalescent models of seed banks, the plant above-ground population is assumed to consist of N individuals, N being fixed, see e.g. Kaj et al. (2001) and Blath et al. (2013). A noisy substitute of this assumption is a logistic (fluctuating) population model (Anderson and King, 1970; King and Anderson, 1971; Charlesworth and Giesel, 1972a, b). For seeds, we find in the literature either the assumption of a finite, fixed number (Kaj et al., 2001) of seeds, or, a deterministic infinite seed density (Koopmann et al., 2017). In the latter, the Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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Fig. 1. Schematic sketch of the four models. (a), (b) Above ground population (bullets type A, circles type B), (a) fixed population size, (b) logistic population dynamics with empty spots. (c), (d) Below ground population (type A black, type B white), (c) deterministic seed bank with continuous state governed by an ODE, (d) discrete state governed by a stochastic birth–death process. The table shows the four possible combinations of above- and below ground dynamics.

assumption is that the number of seeds per plant is large enough, such that intrinsic fluctuations in number and consistence of the seed bank are negligible. As a result, if we know the history of the above-ground population, we know the state of the seed bank. We suggest, here, an alternative model, in which each plant produces single seeds at time points that are distributed according to a Poisson process. The seeds also die (or loose their ability to germinate) after a random time. Throughout the paper, we only consider Markovian processes, that is, all waiting times are exponentially distributed. The ‘‘fluctuating seed bank’’ assumption assumes stochastically varying seed bank size. We therefore consider four models defined by all possible combinations of fixed or logistic above-ground population and deterministic or fluctuating seed bank (Fig. 1). Notation: X1,t denotes the number of allele-A plants. In a model with fixed above-ground population size N, the number of allele-B plants reads X2,t = N − X1,t ; in the logistic fluctuating version, we set up a stochastic process for X1,t and X2,t . In that case, N does denote the carrying capacity of the population (i.e., the maximal possible size). The average population size (conditioned on non-extinction) is κ N for some κ ∈ (0, 1). Yt (Zt ) always refers to the amount of allele A (allele B) seeds in the bank. For the deterministic seed bank, Yt (Zt ) are real numbers that follow (conditioned on X1,t , X2,t ) an ordinary differential equation (ODE). In the case of stochastically fluctuating seed banks, Yt (Zt ) are non-negative integers, that follow a stochastic birth–death process. We allow for weak natural selection. The rates for allele B individuals slightly differ from those for allele A individuals on a scale of 1/N, as it is usual for weak effects. If σi > 0, allele B has a disadvantage in comparison with allele A in the respective process (where σ1 addresses the death of a plant, σ2 the death of seeds, σ3 the production of seeds, and σ4 the germination of seeds). Of course, the signs of σi can be chosen in an arbitrary way to consider a genotype B that has an advantage above genotype A (σi < 0), or for example a situation where B has a disadvantage above ground (σ1 , σ3 > 0) and an advantage below ground (σ2 , σ4 < 0). The parameters of our models are summarized in Table 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

2.1. Fixed population size and deterministic seed bank

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This model has been developed and discussed before (Koopmann et al., 2017). We augment this model with weak selection in the plant component. The total above-ground population has fixed size N, and the transitions for the allele-A plant population X1,t ∈ {0, . . . , N } given the seed densities Yt and Zt ∈ R+ are presented in Table 2. E.g., an A-plant dies at rate ζ X1,t . In the standard Moran model,

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L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx Table 1 Parameters of the models. Meaning

Symbol (Allele A)

Symbol (Allele B)

Death rate of plants Death rate of seeds Production rate of seeds Germination rate of seeds (log. pop. only)

ζ µ β γ

ζ (1 + σ1 /N) µ (1 + σ2 /N) β (1 − σ3 /N) γ (1 − σ4 /N)

Table 2 Possible transitions and their rates for the model with fixed population size and deterministic seed bank. Event

Offset

Rate

Death of A, birth of B Death of B, birth of A

X1,t ↦ → X1,t − 1 X1,t ↦ → X1,t + 1

ζ X1,t Zt /(Yt + Zt ) ζ (1 + σ1 /N)(N − X1,t ) Yt /(Yt + Zt )

Table 3 Possible transitions and their rates for the model with logistic population dynamics and deterministic seed bank. Event

Offset

Death of A Death of B Birth of A Birth of B

X 1 ,t X 2 ,t X 1 ,t X2,t

→ X1,t → X2,t → X1,t → X2,t

Rate

−1 −1 +1 +1

ζ X1,t (1 + σ1 /N)ζ X2,t γ (1 − (X1,t + X2,t )/N)Yt (1 − σ4 /N)γ (1 − (X1,t + X2,t )/N)Zt

Table 4 Possible transitions and their rates for the model with fixed population size and fluctuating seed bank.

1 2 3 4 5 6 7

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9 10 11 12 13

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Event

Offset

Rate

Death of A, birth of A Death of A, birth of B Death of B, birth of A Death of B, birth of B Birth of A-seed Death of A-seed Birth of B-seed Death of B-seed

(X1,t , Yt ) → (X1,t , Yt − 1) (X1,t , Zt ) → (X1,t − 1, Zt − 1) (X1,t , Yt ) → (X1,t + 1, Yt − 1) (X1,t , Zt ) → (X1,t , Zt − 1) Yt → Yt + 1 Yt → Yt − 1 Zt → Zt + 1 Zt → Zt − 1

ζ X1,t Yt /(Yt + Zt ) ζ X1,t Zt /(Yt + Zt ) (1 + σ1 /N)ζ (N − X1,t )Yt /(Yt + Zt ) (1 + σ1 /N)ζ (N − X1,t )Zt /(Yt + Zt ) β X1,t µYt (1 − σ3 /N)β (N − X1,t ) (1 + σ2 /N)µZt

it is instantaneously replaced by a B-individual with probability 1 − X1,t /N. In our case, the seeds determine the probability for a Bindividual, where this probability is given by Zt /(Yt + Zt ). In the same way we obtain the rate at which a B-individual dies (the rate is ζ (1 + σ1 /N) (N − X1,t )) and is replaced by an A-individual (the probability is Yt /(Yt + Zt )). In this model we assume that the number of seeds a plant produces is large (basically infinitely large), such that the seed density in the soil follows a deterministic process, given the history of the above ground population. We obtain a Davis’ piecewise deterministic process (Davis, 1984), where plants produce seeds at rate β (resp. β (1 − σ3 /N)) and seeds die at rate µ (resp. µ(1 + σ4 /N)) Y˙t = β X1,t − µYt ,

Z˙t = β (1 − σ3 /N)(N − X1,t ) − µ(1 + σ2 /N)Zt .

2.2. Logistic population dynamics and deterministic seed bank For the logistic model, we do not couple death and birth of a plant as it is usually done in Moran-type models to keep the total population size constant. We generalize the logistic dynamics as investigated, e.g., by Nasell (2011), or Parsons and Quince (2007) and Parsons et al. (2008) for the situation at hand and separate birth and death events. If the seed densities Yt , Zt ∈ R+ are given, the transitions for X1,t , X2,t ∈ {0, . . . , N }, X1,t + X2,t ≤ N, read as summarized in Table 3. Conditioned on X1,t and X2,t , the dynamics of seeds are again deterministic, Y˙ = β X1 − µY ,

(

Z˙ = 1 −

σ3 ) N

( σ2 ) β X2 − 1 + µZ . N

2.3. Fixed population size and fluctuating seed bank

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For the above-ground population we return to a fixed population size, s.t. it is sufficient to follow X1,t as X2,t = N − X1,t . In the present model we address the fluctuations in the number of seeds, Yt , Zt ∈ N0 . The seeds follow a stochastic birth–death process, where the death rate is kept constant, and the birth rate is proportional to the number of corresponding above-ground plants. We obtain the transitions summarized in Table 4.

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2.4. Logistic population dynamics and fluctuating seed bank

16 17 18

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The last model incorporates logistic growth and a stochastically fluctuating seed bank. This model is an obvious combination of the last two models. The transitions are stated in Table 5. Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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Table 5 Possible transitions and their rates for the model with logistic population dynamics and fluctuating seed bank. Event

Offset

Rate

Death of A Death of B Birth of A Birth of B Birth of A-seed Death of A-seed Birth of B-seed Death of B-seed

X1,t → X1,t − 1 X2,t → X2,t − 1 (X1,t , Yt ) → (X1,t + 1, Yt − 1) (X2,t , Zt ) → (X2,t + 1, Zt − 1) Yt → Yt + 1 Yt → Yt − 1 Zt → Zt + 1 Zt → Zt − 1

ζ X1,t (1 + σ1 /N)ζ X2,t γ (1 − (X1,t + X2,t )/N)Yt (1 − σ4 /N)γ (1 − (X1,t + X2,t )/N)Zt β X1,t µYt (1 − σ3 /N)β X2,t (1 + σ2 /N)µZt

Fig. 2. Simulated trajectory (X1,t number of A-plants, X2,t number of B-plants, Y number of A-seeds, Z number of B-seeds) for model 2.4 (logistic population dynamics and fluctuating seed bank) in the neutral case (according to the transition given in Table 5), (left) A-plants X1 vs. B-plants X2 , (center) A-plants X1 vs. A-seeds Y , (right) A-plants X1 vs. B-seeds Z . Population size N = 200, production rate β = 1, death rate of seeds µ = 1, death rate of plants ζ = 0.5, germination rate of seeds γ = 10, no selection σi = 0. The solid line indicates the coexistence line described in Proposition 2.1, scaled by the population size N.

2.5. Strategy for the analysis of the models

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The aim of the analysis is the reduction of the four models to a one-dimensional diffusion approximation of a Moran model representing the fraction of allele-A individuals within the population. As the details of the analysis are tedious, we present them in the Appendix and only outline informally the basic ideas in the present section. An approximation theorem derived by Kurtz (1973) (see also Ethier and Kurtz (1986, chapter 1, corollary 7.8)) and Majda et al. (2001)) and an algorithmic approach stated in Kogan et al. (2014) are given in Appendix A.1. This method is consequently used for the model with fixed population size and deterministic seed banks (Appendix A.2), logistic population dynamics and deterministic seed banks (Appendix A.4), fixed population size and fluctuating seed banks (Appendix A.5), and logistic population dynamics and fluctuating seed banks (Appendix A.6). We informally describe the approach used for the dimension reduction. The key insight here is that any realization settles fast on a one-dimensional manifold. If we consider, e.g., the deterministic version of the logistic population dynamics with stochastically fluctuating seed bank, we find, according to arguments by, e.g., Kurtz (1980), for the deterministic limit as N → ∞ (with xi (t) = Xt ,i /N, y(t) = Yt /N, z(t) = Zt /N)

2 3 4 5 6 7 8

x˙1 = γ (1 − x1 − x2 )y − ζ x1 x˙2 = γ (1 − x1 − x2 )z − ζ x2 y˙ = β x1 − µy z˙ = β x2 − µz It is straightforward to show that a line of stable equilibria, the so-called coexistence line, exists:

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Proposition 2.1. Let ϑ := (β − ζ )/µ > 0, κ := (γ ϑ − ζ )/(γ ϑ ) ∈ [0, 1]. Then, there is a line of stationary points in [0, κ]2 × R2+ given by (x1 , x2 , y, z) = (x, κ − x, ϑ x, ϑ (κ − x)),

x ∈ [0, κ].

The line of stationary points is transversally stable (locally and globally in the positive cone). It turns out that the stochastic process rapidly approaches this line of equilibria, and performs a random walk close to it (see Fig. 2). The analysis reveals that the distribution on a transversal cut is just a normal distribution with a variance of O(1/N). Along the line of stationary points, however, the realizations will move according to a one-dimensional diffusion approximation of a Moran process. This approximative process is a combination of one component of the full process parallel to this line, and a second component that results from an interaction between a component perpendicular to this line with the deterministic vector field directed towards this line. In order to reveal this structure, we first use a large population limit (Kramers–Moyal expansion) to obtain a Fokker–Planck/Kolmogorov forward equation for the full process. In the second step we apply singular perturbation methods to perform the dimension reduction to the one-dimensional Moran model. For the model with fixed population size and deterministic seed bank, we present a second approach that is interesting in itself. It is based on the small delay approximation (Guillouzic et al., 1999) (although, to our knowledge, no hard convergence theorem is available for this approximation, which is used in theoretical physics): if ecological time t is not considered but rather the evolutionary time τ = t /N Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx Table 6 Drift, diffusion, and selection coefficients for the different population/seed bank models. Model/scale term

a (selection)

b (genetic drift)

σ (selection coeff.)

fix. pop., det. s.b. fix. pop., fluct. s.b. log. pop., det. s.b. log. pop., fluct. s.b.

ζ ζ ζ ζ

2ζ 2ζ 2ζ 2ζ

σ1 + σ2 + σ3 σ1 + (1 − 1/K )σ2 + σ3 σ1 + σ2 + σ3 + σ4 σ1 + σ2 + σ3 + (1 − 1/K )σ4

G −1 G −1 G −1 G −1

G −2 G −2 G −2 κ −1 G −2 κ −1

4

(population size N large), the delay of a weak seed bank is small. In this case, the solution can be expanded w.r.t. the delay. As a result, the seed bank can be removed from the stochastic process and replaced by appropriately rescaled parameters (Appendix A.3). Since for the time being, the short delay approximation is only a heuristic approach, we formulate also a proof for the result based on time scale arguments in Appendix A.2. In this way, we find for this special case a proof for the validity of the small delay approximation.

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3. Results

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3.1. Timescales for different seed bank models

1 2 3

7 8

For all of our models, the resulting one-dimensional Fokker–Planck equation assumes the form of a diffusion approximation of a Moranmodel with weak selection, 1

∂τ u = −σ a∂x˜ {˜x(1 − x˜ )u} + b∂x˜2 {˜x(1 − x˜ )u}

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(1) 2 where a and b describe the speed of selection and genetic drift, respectively. The term σ represents selective coefficients and is a generic parameter for notation including σi , i = 1, . . . , 4. In order to formulate the results for a and b, let us introduce three composite parameters: G = ζ /µ is the number of plant generations a seed survives on average; K = β/ζ is the average number of seeds produced by a plant; κ already defined in Proposition 2.1 is the average fraction of the above-ground population size in the logistic model, in comparison with the maximal possible population size N. Furthermore, for deterministic seed banks, we define G := (1 + G)−1 , while in case of a fluctuating seed bank we define G := (1 + (1 − 1/K )G)−1 . G can be seen as the number of plant generations that seeds survive on average corrected by the size of the seed bank. Using these abbreviations, the parameters a, b and σ for our four models are summarized in Table 6. The basic seed bank model (fixed population size, deterministic seed bank) and the standard Moran model without seed bank can be used as reference models. A seed bank slows down the time scale of selection as well as that of genetic drift (Koopmann et al., 2017), where selection is less affected (by a factor of G −1 ) than genetic drift (by a factor of G −2 ). We find that fluctuations in the above-ground population and in the seeds have different effects. Fluctuations in the seed number reduce the storage effect of seed banks. The additional fluctuations driven by a stochastic birth–death process of the seeds yield a reduction of the effective time a seed spends within the seed bank, and thus increases the rate of genetic drift. For K → ∞ (fluctuations in seed bank are only due to the coupling to the stochastic above ground population), we obtain the result for the deterministic seed bank, for K → 1 (intrinsic fluctuations due to the birth–death process in the seeds is maximized), the model converges towards the standard Moran model without seed bank. Note that K is not the average number of seeds per plant directly measured but the effective number of seeds per plants. For example, a certain fraction of seeds might be getting lost due to other environmental reasons (abiotic or biotic factors). These seeds do not contribute to the bank. The fluctuations in the above-ground population driven by the birth–death process of plants only affect genetic drift and does not appear in the selection term. This result reflects that the actual competition between alleles A and B only happens above ground. Nonlinear terms in the transition rates only appear in the birth term of the plants. By increasing solely genetic drift, the above-ground fluctuations can counteract the amplification of selection by seed banks. The scaling of selection by G −1 and that of genetic drift by G −2 is somehow expected. All mutations are affected in the same way by the above-ground fluctuations (birth–death process of plants). Our result concerning the lumped selection coefficient σ , however, is unexpected: Mutations for some fitness components (mortality of seeds, σ2 , and germination ability, σ4 ) show reduced selection while this is not the case for selective coefficients of other fitness components (σ1 and σ3 ). This means that if the number of seeds per plant is not too large, beneficial mutations in the mortality of seeds (σ2 ) have a reduced chance to reach fixation compared with a beneficial mutation for the production of seeds (σ3 ).

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3.2. Site-frequency spectrum (SFS)

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

39 40 41 42 43 44 45

46

47

48

The SFS is a commonly used statistic for the analysis of genomewide distributed SNPs. It is defined as the distribution of the number of times a mutation is observed in a population or a sample of n sequences conditional on segregation. Herein, this distribution is taken over numerous unlinked sites and mutations occur only on previously monomorphic ones (Kimura, 1969) at rate θ/2 per N generations. Each mutant allele that arises from the wildtype at such an independent site is assumed to marginally follow the diffusion model specified in (1) so that in particular all mutants have equivalent selective effects among sites. Mutations are allowed to occur in plants and seeds, but the following results can be easily adapted to the scenario, where mutations may only arise in plants. The proportion of sites at equilibrium, where the mutant frequency is in (y, y + dy), is routinely obtained as (e.g., Griffiths 2003; Koopmann et al. 2017) fˆ (y) =

θ

1 − exp{−2a/b σ (1 − y)}

b y(1 − y)

1 − exp{−2a/b σ }

,

(2)

where a, b and σ ̸ = 0 are given in Table 6. The sample SFS at equilibrium can be immediately obtained from (2) via binomial sampling as fˆn,k = θ

n

1−1 F1 (k; n; 2a/b σ )e−2a/b σ

b k(n − k)

1 − e−2a/b σ

,

(3)

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Fig. 3. Continuous SFS fˆ (y) over mutant frequency y according to Eq. (2); neutral (solid) vs. selection (dashed, σ1 = 50, σ2 = σ3 = σ4 = 0, selection only in the death rate of plants) resp. (dotted, σ1 = σ2 = σ3 = 0, σ4 = 50, selection only in the germination rate of seeds). Further parameters: death rates of seeds µ = 0.25, death rates of plants ζ = 1, germination rate of seeds γ = 1.5, scaled mutation rate θ = 2; (left) production rate of seeds β = 2.4, s.t. average number of plant generations a seed survives G = 4 and average number of seeds per plant K = 2.4, (right) production rate of seeds β = 1.2, s.t. average number of plant generations a seed survives G = 4 and average number of seeds per plant K = 1.2. Note that in the left panel the dashed and the dotted curve are almost on top of each other.

where 1 F1 denotes the confluent hypergeometric function of the first kind (Abramowitz and Stegun, 1964). The neutral versions of (2) and (3) are respectively given by fˆ (y) = θ/(by) and fˆn,k = θ/(bk). Fig. 3 indicates the striking effects of the fluctuations in the seed bank size for the model with logistic population dynamics and fluctuating seed banks. When the fluctuations in the size are small (K high) the number of segregating sites is expected to be high, and mutations involved in the selection coefficients σ1 and σ4 lead to similar SFS. The SFS shows the typical U-shape expected under positive pervasive selection (left panel). If, however, K becomes small, the number of segregating sites decreases, and the selection coefficient σ1 shows a U-shaped SFS while mutations under σ4 do not (right panel). 4. Discussion We considered four models to investigate the effect of combined fluctuations of the population size in the above-ground population and below-ground seed bank. In all four cases time scale arguments allow us a reduction to a diffusion approximation of a Moran model. Our results extend the findings that a seed bank without intrinsic fluctuations (seed bank couples deterministically to the plant population) yields a change of time scale in selection and genetic drift that amplifies the effect of weak selection (Koopmann et al., 2017). The first main result of this present work is that there is no direct interaction between the fluctuations above ground and below ground (fluctuations in the number of seeds/plants, types of seeds/plants). The above-ground fluctuations in the number of plants increase the effect of genetic drift compared to a fixed above-ground population size, but does not affect selection. One can propose a new definition of the effective population size Ne = κ G 2 N /2 describing the change in genetic drift due to these fluctuations (according to Etheridge (2011, Definition 2.9)) . This allows us to redefine the evolutionary time scale T = ζ t /Ne . As a result of this procedure the parameter κ , which represents the reduction of the average total population size and indicates the increase of the above-ground fluctuations by the logistic population dynamics, appears as a factor in front of the selection term in the Moran model. Note that in this notation we find that the terms for below-ground (G 2 ) and above-ground (κ ) fluctuations are multiplied indicating their independence. The second result is that the below-ground fluctuations in the number of seeds affect the scaling of time for both the selection and the genetic drift term. We introduce the concept of the mean effective number of seeds per plant in a similar definition to the effective population size (Wright, 1931). If this average seed number tends to infinity, we recover the effect of a deterministic seed bank (seed bank is deterministic given history of the above ground population), and if this number tends to one, the fluctuations in the size of the seed bank accelerate the time scale such that the seed bank has no effect at all. The magnitude of these fluctuations is thus tuned between the two extreme cases of ‘‘no seed bank’’ (minimal effect) and ‘‘deterministic seed bank’’ (maximal effect). The third result of the present study is the insight that below-ground fluctuations in seed bank size may affect the four plants’ fitness components of viability and fecundity differentially. If the below-ground fluctuation is large, the selective effect of mutations involved in seed death and germination may even be cancelled out. Other mutations involved in fitness traits of the above-ground population, such as plant death or seed production are not affected. In other words, while finite size and fluctuations in the plant number above ground do not affect selection, fluctuations in the seed bank size and finite size of the seed bank do change the selection coefficients. In biological terms we interpret this result as follows. Above ground, the fate of an allele under positive selection is classically determined by the strength of genetic drift, which depends on the population size (the diffusion term in the Fokker–Planck equation) compared to the strength of selection which depends on the selection coefficient (the drift term in the forward model). Under a deterministic seed bank (as in the present models and in Koopmann et al. (2017)), selection is efficient because it occurs on plants, when they are above ground, with a probability G −1 and genetic drift occurs on a coalescent scale of G −2 . Any change in the allele frequencies above ground translates directly into the deterministic seed bank, just with a small time delay. However, when the seed bank has a fluctuating finite size, the strength of selection on seed fitness (the coefficients σ2 and σ4 ) is decreased by the fluctuation in the size of the seed compartment. This occurs because selection and genetic drift in the bank occur at every generation and not only when seeds germinate. This observation implies that, in species with seed banks, competition experiments, which measure and compare the fitness effect of mutations and/or allow to determine selection coefficients on the ecological time scale should be extrapolated with caution to the evolutionary time scale. Two difficulties indeed arise in experimental work. First, the plant fitness is usually measured as seed production (our coefficient σ3 in Tables 1, 6). While this coefficient σ3 is not affected by the population fluctuation below-ground (Table 6), it represents only a part of the plant fitness (also defined by the other coefficients σ1 , σ2 and σ4 ). The effects of the seed death and germination rate Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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on fitness are, however, modified by the seed bank size (Table 6), and unfortunately, this latter parameter is often unknown in field experiments. Second, measuring the other selection coefficients such as seed death rate and seed germination requires more complex experimental set-ups for example by surveying populations over time and genotyping plants and seeds over several years (see e.g. FalahatiAnbaran et al. (2014)). Therefore in species with seed banks, in order to measure accurately plant fitness at the evolutionary time scale, it is necessary to assess its four components (see review by Long et al. (2014)), and estimate the size of the seed bank reservoir. We finally discuss two ways of using genome polymorphism data to estimate seed bank parameters as well as the selection coefficients. We can attempt to infer the seed bank parameters based on neutral genetic diversity under the idealized conditions that the sample SFS reached an equilibrium as fˆn,k (i.e., there is no recent demographic impact) and that we can measure or estimate the population mutation rate θ and the death rate of plants ζ . It turns out that G (the number of plant generations a seed survives on average) and κ (the average above-ground population size) can then be identified, so that it is possible to disentangle the effects of the seed bank from those of fluctuations in the plant population on neutral evolution by utilizing fˆn,k . However, we cannot identify the effect of fluctuation in the size of the below-ground population, as G and K only appear in the composite parameter G . Note that if we only have information about ∑ the relative sample SFS fˆn,k / i fˆn,i , the multiplicative constant in fn,· cancels out, and therefore only κ G and the combined effect of aboveand below-ground fluctuations (with respect to size) can be estimated. Extending the work by Tellier et al. (2011), we suggest that the number of plant generations a seed survives on average can be estimated from the absolute SFS using for example a Bayesian inference method with priors on the census size of the above-ground population and the death rate of plants. We may also aim to infer the selection coefficients underpinning the various plants’ fitness components, namely the fecundity and viability of plants or seeds. With the abundance of gene expression, molecular and Gene Ontology data, it becomes feasible to group genes by categories of function or pathways, for example to know the genes involved in seed germination or seed integrity, viability and seed dormancy (e.g., Righetti et al., 2015). These functional groups of genes actually underlie the different plants’ fitness components investigated in this study. Using genome-wide polymorphism data of several individuals, the SFS for each functional groups of genes can be computed and used to draw inference of selection, for example as the distribution of selective effects (e.g., the method by Eyre-Walker and Keightley, 2009). Our prediction is thus that the SFS would reflect the differential selection on these fitness components and can be observed over the genes involved in the different functional groups. The limitation in current data lies so far on the functional side, as more gene expression studies are needed to assign genes to functional network and to different plant fitness components. As an extension, our expected SFS shows that the behaviour of the seed fitness coefficients can be affected by the fluctuations in the size of the seed bank. So, we predict that populations with a small sized seed bank should exhibit less selection signatures on genes related to seed fitness compared to populations with a larger seed bank compartment. The same analysis as above can be conducted, but now comparing the SFS and inferred selection for the different functional groups of genes across several populations with known ecological set-ups which define the seed bank size. We further suggest that such procedure can be applied to disentangle selection for fecundity from that for viability in age-structured populations, if functional groups of genes can be assigned to these traits (for example using gene expression at different life stages and ages). Our results differ thus from those of classic age-structured populations by the overlap of generations, as the seed bank can present its own rate of genetic drift. Moreover, selection acts differently above ground and below ground on the different plants’ fitness components, which may allow us to disentangle their effect on the overall selection coefficient.

36

Acknowledgements

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

38

This research is supported in part by Deutsche Forschungsgemeinschaft grants TE 809/1 (AT), STE 325/14 from the Priority Program 1590 (DZ), and MU2339/2-2 from the Priority Program 1617 (JM).

39

Appendix. Analysis of the models

37

44

Before we analyse the four models, one after the other, we first present the overall idea of the approximation approach, as one gets easily lost in the necessary but rather technical computations. Parts of the computations for the approximations are straightforward (based on Taylor expansion) but lengthy; therefore we only discuss the first model more in detail (Appendix A.2). For the other models we skip the trivial parts. The computations can be readily checked using computer algebra, e.g. by the package MAXIMA (Maxima, 2014) (see the available supplementary files).

45

A.1. Perturbation method for the Fokker–Planck equation

40 41 42 43

46 47 48 49 50 51 52 53 54 55 56 57

We consider Fokker–Planck equations incorporating a small parameter ε 2 = 1/N. Kurtz (1973) and others developed approximation theorems that allow to find a simplified, reduced Fokker–Planck equation that approximates the original one (see e.g. Ethier and Kurtz (1986, chapter 1, corollary 7.8) or the review article Givon et al. (2004, Section 6)). To be self-contained, we sketch the approximation method. We first describe the approximation theorem that is based on the backward equation following the nice presentation by Majda et al. (2001, chapter 4.4). The application of the approximation method is involving. Kogan et al. (2014) propose an appealing way to handle the equivalent forward equation in a way that bypasses some technical difficulties. We describe the principle of that approach in the second part of this section. We start off with a forward Fokker–Planck equation that includes a small parameter, ε 2 ∂t ρ ε (x, y, t) = Lε ρ ε (x, y, t). We concentrate on the long term behaviour of the equation, expressed by the scaled time ε −2 t, and have therefore ε 2 ∂t at the l.h.s. To be more specific, we assume that Lε can be expanded w.r.t ε , Lε ρ ε = L(0) ρ ε + ε L(1) ρ ε + ε 2 L(2) ρ ε + · · · . Since Lε depends on ε , so does the solution ρ ε of ε 2 ∂t ρ ε (x, y, t) = Lε ρ ε (x, y, t). Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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Approximation theorem: In order to formulate the approximation theorem, we consider the corresponding backward equation, ε

− ε ∂s w = L 2

(0)+

ε

w + εL

(1)+

ε

w +ε L

2 (2)+

1 2

ε

w + ···

3

where L(i)+ are the formal adjoint operators for L(i) . w ε denotes the solution of the backward equation, which again depends on ε (as the equation depends on ε ). Now we also expand w ε in powers of ε , ε

w =w

+ εw

(0)

(1)

+ε w 2

(2)

+ ··· .

7

w = 0 L w (1) = −L(1)+ w (0) (0)+ (2) L w = −∂s w(0) − L(2)+ w(0) − L(1)+ w(1) ... L

(0)

8

(0)+

9 10 11

These equations are solvable only if the r.h.s. belongs to the range of L semigroup that converges to a projection operator for t → ∞, lim e

L(0)+ t

t →∞

(0)+

. The next assumption is that the operator L

(0)+

generates a

w

·=P·

= −(L

14

(0)+ −1 (1)+

)

L

w . (0)

− ∂s w(0) = P L(2)+ w(0) − P(L(1)+ )(L(0)+ )−1 L(1)+ w(0) .

Theorem A.1. Let w ε (s, x, y|t) satisfy

− ε ∂s w = L where L

(0)+

,L

lim e

(1)+

,L

L(0)+ t

t →∞

(0)+

(2)+

ε

w + εL

(1)+

w +ε L

2 (2)+

ε

w ,

ε

w (t , x, y|t) = f (x) (0)+

− ∂s w ˆ+

are backward Fokker–Planck operators, L

with L = P (L

generates a stationary process such that

(0)

(1)+

(L

(0)+ −1 (1)+

24 25

)

L

(0)

L ρ

(0)

(x, y, t) = 0

L ρ

(1)

(x, y, t) = −L ρ

30 31 32

P.

33

Algorithmic approach: After we understand that Taylor expansion of operator and solution w.r.t. the small parameter ε can be used to obtain a simplified, approximate backward-Fokker–Planck equation, we turn to the problem how to obtain this equation in practice. At this point, we follow Kogan et al. (2014), who use the forward equation to demonstrate in particular how to avoid to explicitly compute the inverse of the operator L(1) . Basically, that paper shows an algorithmic way how to obtain the approximative equation. We start with the forward equation, where the operator Lε is expanded in terms of ε , ε 2 ∂t ρ ε = L(0) ρ ε + ε L(1) ρ ε + ε 2 L(2) ρ ε + · · · . As before, we also expand ρ ε (x, y, t) = ρ (0) + ερ (1) (x, y, t) + ε 2 ρ (2) (x, y, t) + · · · , and find (equating the coefficients of equal power in ε ) (0)

21 22

28

w (t , x|t) = f (x)

(0)

)P − PL

20

29

=L w ,

(2)+

19

27

· = P ·,

ˆ+

17

26

ε

and PL(2)+ P = 0. Assume Pf = f . Then, in the limit as ε → 0, w ε (s, x, y|t) tends to w (0) (s, x|t) for −T < s ≤ t, T < ∞, uniformly in x and y on compact sets, where w (0) satisfies Pw (0) = 0 and solves the backward equation (0)

16

23

Though we only used formal arguments, it is possible to prove an approximation theorem (which we cite according to Majda et al. (2001, chapter 4.4)).

ε

15

18

In the application of the approximation method, this step yields technical difficulties. We discuss below an algorithmic way how to bypass the (pseudo) inverse. Last, we use the solvability condition for the third equation (which reads 0 = P (−∂s w (0) − L(2)+ w (0) − L(1)+ w (1) )) together with our result for w (1) , and the invariance of w (0) under P. We obtain

2

12 13

As L(0)+ w (0) = 0, we have Pw (0) = w (0) . Next we turn to L(0)+ w (1) = −L(1)+ w (0) . The solvability condition reads P L(1)+ w (0) = 0. If this condition is true, there is a (pseudo)inverse of L(0)+ , and we can write (1)

5 6

We insert this expansion into the backward equation, and equate the coefficients of equal power in ε , (0)+

4

34 35 36 37 38 39 40 41

(1)

(0)

(x, y, t)

42

L(0) ρ (2) (x, y, t) = ∂t ρ (0) (x, y, t) − L(2) ρ (0) (x, y, t) − L(1) ρ (1) (x, y, t)

43

...

44

In the cases we will consider below, L(0) does not incorporate derivatives w.r.t. x, and the solution of L(0) ρ (0) (x, y, t) = 0 reads

45

ρ (0) (x, y, t) = f (x, t) ν (y; x)

46

where ν (y; x) is a probability measure (it turns out that ν is a normal distribution where the variance can depend on x). That is, ρ factorizes into a distribution ∫ ν (y; x) that is independent of time, multiplied with a time-dependent distribution f (x, t) which does not depend on y. Particularly, ρ (0) (x, y, t) dy = f (x, t). If we consider Fig. 2, we interpret ν (y; x) as the distribution perpendicular to the line (0)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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of stationary states, while f (x, t) characterizes the time evolution of the process along the line of stationary states. We aim at the time evolution of f (x, t). Furthermore, it turns out that the left eigenfunction for the eigenvalue 0 of operator L(0) is the constant function 1. If we integrate the three equations above w.r.t. y, the left hand side vanishes, and hence the integral of the r.h.s. also needs to vanish. If we look at the third equation, we obtain

∂t f (x, t) = ∂t

∫

11

13

∫

16

17 18 19

∫

L(2) ρ (0) (x, y, t) dy +

∫

L(1) ρ (1) (x, y, t) dy.

L(1) ρ (1) (x, y, t) dy = ∂x

∫

F (x, y)L(0) ρ (1) (x, y, t) dy.

This assumption seems to be rather special and difficult to satisfy, but it turns out that we are able to find such a function in the cases we consider. In this way, we avoid to explicitly determine the inverse of L(1) : Once we have this function F (x, y), we proceed using the second equation L(0) ρ (1) (x, y, t) = −L(1) ρ (0) (x, y, t) and write

14

15

ρ (0) (x, y, t) dy =

In order to find a proper equation for f (x, t), we directly evaluate the first integral, using ρ (0) (x, y, t) = f (x, t) ν (y; x). The second integral ∫ at the r.h.s. is more delicate, as we aim to express L(1) ρ (1) (x, y, t) dy in terms of ρ (0) (x, y, t) = f (x, t) ν (y; x). Therefore we assume that there is a function F (x, y) such that

10

12

∫

L(1) ρ (1) (x, y, t) dy = −∂x

∫

F (x, y)L(1) ρ (0) (x, y, t) dy.

Therewith, we have the approximate equation

∂t f (x, t) =

∫

L(2) f (x, t) ν (y; x) dy − ∂x

∫

F (x, y)L(1) f (x, t) ν (y; x) dy.

Note that L(2) only appears in the integral w.r.t. y. All full derivatives w.r.t. y vanish upon integration. Hence, when we determine L(2) , it is not necessary to work out all terms that are full derivatives w.r.t. y. We end up with four steps:

21

Step 1: Choose the coordinate system (x, y) in an appropriate way. In our case (see Fig. 2), that does mean that x parametrizes the line of stationary points, while y is perpendicular to this line.

22

Step 2: Expand the operator Lε w.r.t. ε , and determine L(i) , i = 0, 1, 2.

20

23 24

Step 3: It is not necessary to determine the stationary distribution ν (y; x) explicitly. Nevertheless, we start with u(0) = f (x, t) ν (y; x), and determine

∫ 25

26

Step 4: This is the most delicate step. We determine F (x, y) that satisfies for all suited u(x, y, t)

∫ 27

28

L(2) f (x, t) ν (y) dy.

L(1) u(x, y, t) dy = ∂x

∫

F (x, y)L(0) u(x, y, t) dy,

and find

∫ 29

L(1) ρ (1) (x, y, t) dy = −∂x

∫

F (x, y)L(1) ρ (0) (x, y, t) dy.

30

The approximate Fokker–Planck operator is given by adding the results of step 3 and step 4.

31

A.2. Fixed population size and deterministic seed bank — perturbation method

32 33

We demonstrate the method described above in the case of a deterministic seed bank and a fixed population size. Note that in this special case, the total number of seeds for the neutral model (σ1 = σ2 = σ3 = 0) satisfies an ODE, d

34

35 36 37

dt

(Yt + Zt ) = β N − µ(Yt + Zt ).

No stochastic effects perturbed the total number of seeds, that asymptotically for t → ∞ tends to β N /µ. We will find back this fact later in the analysis of the model. We start with the master equation. Let pi (k, l, t) = P(Xt = i, Yt ∈ (k, k + dk), Zt ∈ (l, l + dl)). The master equations are given by

) ] β i − µl pi (k, l, t) β (1 − σ3 /N)(N − i) − µ(1 + σ2 /N)k [ ] l k + ζ (1 + σ1 /N)(N − i) pi (k, l, t) = − ζi k+l k+l [ ] [ ] (i + 1)l (N − i + 1)k pi+1 (k, l, t) + ζ (1 + σ1 /N) pi−1 (k, l, t) + ζ k+l k+l

p˙ i (k, l, t) + ∇

38

[(

(A.1)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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The usual expansion yields the corresponding Fokker–Planck equation (x = i/N, y = k/N, z = l/N)

{[ ] } xz (1 − x)y ∂t u = ∂x ζ − ζ (1 + σ1 /N) u y+z y+z {[ ] } β x − µy − ∇y,z u β (1 − σ3 /N)(1 − x) − µ(1 + σ2 /N)z {[ ] } 1 2 xz (1 − x)y + ζ ∂x + ζ (1 + σ1 /N) u 2N y+z y+z

1

(A.2)

Note that we introduce here an error of order O(1/N 2 ). Below, we define ε 2 = 1/N, s.t. the error is of order O(ε 4 ). The dimension reduction method we will apply in the next steps comes with an approximation error of O(ε ), s.t. the error introduced by the diffusion limit does not play a role. A.2.1. Deterministic model The drift term of the Fokker–Planck equation defines the ODE model xz

x˙ = −ζ

y+z y˙ = β x − µy

+ζ

4 5

7

(1 − x)y 8

y+z

9 10

with the line of stationary points

11

(x, y, z) = (x, ϑ x, ϑ (1 − x)),

where ϑ = β/µ,

x ∈ [0, 1].

12

A.2.2. Dimension reduction by time scale analysis

13

Step 1: We introduce new coordinates, y = ϑ x˜ +

1 2

ε (y˜ + z˜ ),

14

z = ϑ (1 − x˜ ) +

1 2

ε (y˜ − z˜ ),

ρ (t , x˜ , y˜ , z˜ ) = u(t , x, y, z)

15

where ϑ = β/µ and ε 2 = 1/N. With x˜ = x,

3

6

z˙ = β (1 − x) − µz

x = x˜ ,

2

y˜ = ε −1 (y + z − ϑ ),

16

z˜ = ε −1 (y − z + ϑ (1 − 2x)),

(A.3)

we obtain

17 18

∂x = ∂x˜ − 2ε ϑ∂z˜ , ∂y = ε (∂y˜ + ∂z˜ ), ∂z = ε (∂y˜ − ∂z˜ ). −1

−1

−1

19

Step 2: We transform the Fokker–Planck equation, neglecting terms of O(ε 3 ). For ρ (x˜ , y˜ , z˜ , t ; ε ) we obtain

20

] } ) {[ x˜ (ϑ (1 − x˜ ) + 21 ε (y˜ − z˜ )) (1 − x˜ )(ϑ x˜ + 21 ε (ϑ + ε˜y)) 2 −1 ∂t ρ = ∂x˜ − 2ε ϑ∂z˜ ζ − ζ (1 + ε σ1 ) ρ ϑ + ε˜y ϑ + ε˜y ( ) {[ ] } 1 − ε −1 (∂y˜ + ∂z˜ ), β x˜ − µ(ϑ x˜ + ε (y˜ + z˜ )) ρ (

2

21

] } ( ) {[ 1 2 2 −1 − ε (∂y˜ − ∂z˜ ) β (1 − ε σ3 )(1 − x˜ ) − µ(1 + ε σ2 )(ϑ (1 − x˜ ) + ε (y˜ − z˜ )) u 2

] } )2 {[ x˜ (ϑ (1 − x˜ ) + 21 ε (y˜ − z˜ )) (1 − x˜ )(ϑ x˜ + 21 ε (ϑ + ε˜y)) ε −1 2 + ∂x˜ − 2ε ϑ∂z˜ ζ + ζ (1 + ε σ1 ) ρ . 2 ϑ + ε˜y ϑ + ε˜y ( 2

Taylor expansion in ε yields ∂t ρ = L(0) ρ + ε L(1) ρ + ε 2 L(2) ρ + · · · with (we use here that β − µϑ = 0)

[( ) ] [( ) ] [( ) ] µ y˜ ρ + ∂z˜ µ˜z + ζ (z˜ + y˜ (1 − 2x˜ )) ρ + ∂z˜2 4ζ ϑ 2 x˜ (1 − x˜ ) ρ [( ) ] [( ) ] (1) L ρ = − ∂x˜ ζ (z˜ + y˜ (1 − 2x˜ ))/(2ϑ ) ρ − ∂y˜ −σ3 β (1 − x˜ ) − σ2 µϑ (1 − x˜ ) ρ [( ) ] − ∂z˜ ζ (1/ϑ )y˜ (z˜ + y˜ (1 − 2x˜ )) − 2ζ ϑσ1 x˜ (1 − x˜ ) + β (σ2 + σ3 )(1 − x˜ ) ρ [( ) ] [( ) ] 2 + ∂z˜ ϑζ (y˜ + (1 − 2x˜ )z˜ ) − 4ζ ϑ x˜ (1 − x˜ )y˜ ρ − ∂x˜ ∂z˜ 4ϑζ x˜ (1 − x˜ ) ρ [( ) ] [( ) ] L(2) ρ =∂x˜ ζ y˜ (z˜ + y˜ (1 − 2x˜ ))/(2ϑ 2 ) − σ1 ζ x˜ (1 − x˜ ) ρ + ∂x˜2 ζ x˜ (1 − x˜ ) ρ [( ) ] [( ) ] + ∂y˜ · · · ρ + ∂z˜ · · · ρ L(0) ρ =∂y˜

(A.4)

(A.5)

(A.6)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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12 1 2 3 4 5 6 7

8 9 10 11 12

L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx

Please note: (a) The operator L(0) is only of first order w.r.t. y˜ . The solution of L(0) ν = 0 can be written as ν (y˜ , z˜ ; x˜ ) = δ0 (y˜ ) ν˜ (z˜ ; x˜ ), where ν˜ is a one-dimensional normal distribution with x˜ -dependent variance, characterized by ∂z˜ [(µ + ζ )z˜ ν˜ (z˜ ; x˜ )] + [4ζ ϑ 2 x˜ (1 − x˜ )]∂z˜2 ν (z˜ ; x˜ ) = 0. ν (y˜ , z˜ ; x˜ ) satisfies L(0) ν (y˜ , z˜ ; x˜ ) = 0 in the sense of generalized functions. This observation reflects the fact that the total number of seeds in a neutral population with a fixed population size is no random variable, but follows a deterministic time evolution, and tends asymptotically for t → ∞ to a given value. (b) As L(0) does not incorporate derivatives w.r.t. x˜ or t, the general solution reads (2) 2 ρ (0) (x˜ , y˜ , z˜ , t) =[( f (x˜ , t) ). (c) Below )ν (]y˜ , z˜ ; x˜[( ) ] we only use the integral of L w.r.t. (y˜ , z˜ ) over R , s.t. we do not need to know the exact shape of the terms ∂y˜

· · · ρ + ∂z˜

We introduce τ = ε 2 t = t /N, and get the appropriate scaling for ρ (x˜ , y˜ , z˜ , t) = ρ˜ (x˜ , y˜ , z˜ , ε 2 t),

ε 2 ∂τ ρ˜ (x˜ , y˜ , z˜ , τ ) = L(0) ρ˜ (x˜ , y˜ , z˜ , τ ) + ε L(1) ρ˜ (x˜ , y˜ , z˜ , τ ) + ε 2 L(2) ρ˜ (x˜ , y˜ , z˜ , τ ) + · · · From now on, we also use f (x, τ ) instead of f (x, t). Next we also expand ρ˜ w.r.t. ε ,

ρ˜ (x˜ , y˜ , z˜ , τ ) = ρ (0) (x˜ , y˜ , z˜ , τ ) + ερ (1) (x˜ , y˜ , z˜ , τ ) + ε 2 ρ (2) (x˜ , y˜ , z˜ , τ ) + O(ε 3 ) and obtain

13

L(0) ρ (0) = 0,

14

As seen above, from L ρ

15

16

· · · ρ ; they become zero upon integration.

L(0) ρ (1) = −L(1) ρ (0) ,

∂τ f =

(0)

(0)

In the following computations we use that

(

17

L(0)+ [(y˜ z˜ + (1 − 2x˜ )y˜ 2 )] = −(2µ + ζ )y˜ z˜ + y˜ (1 − 2x˜ )

20

L

∫

(0)+

(A.10)

[(z˜ + (1 − 2x˜ )y˜ ) ] = −2(µ + ζ )(z˜ + (1 − 2x˜ )y˜ ) + 8ζ ϑ x˜ (1 − x˜ ) 2

L(2) ρ (0) d(y˜ , zt) =

22

24

(A.9)

2

2

[( )∫ ] ∫ ζ (0) (0) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ x (1 − x ) ρ ( x , y , z , τ )d( y , z ) y ( z + y (1 − 2 x )) ρ d( y , z ) − σ ζ ∂ ∂ ˜ ˜ 1 x x 2ϑ 2 ] [( )∫ ρ (0) (x˜ , y˜ , z˜ , τ )d(y˜ , z˜ ) + ∂x˜2 ζ x˜ (1 − x˜ )

For the second and third integrals, we use ρ (0) (x˜ , y˜ , z˜ , τ )d(y˜ , z˜ ) = f (x˜ , τ ). With y˜ z˜ + y˜ (1 − 2x˜ ) = −L(0)+ [(y˜ z˜ + (1 − 2x˜ )y˜ 2 )]/(2µ + ζ ) and L(0) ρ (0) = 0, we conclude that the first integral at the r.h.s. is zero:

∫

∫ 25

[( ) ) ] ∫ ( −ζ (0)+ 2 (0) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ L [ ( y z + (1 − 2 x ) y ) ] ρ d( y , z ) − σ ζ ∂ x (1 − x ) f ( x , τ ) ∂ ˜ ˜ 1 x x 2ϑ 2 (2µ + ζ ) [( ) ] 2 ˜ ˜ ˜ + ∂x˜ ζ x(1 − x) f (x, τ ) [( ) ] ∫ −ζ 2 (0) (0) ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ = ∂ [ ( y z + (1 − 2 x ) y ) ] L ρ d( y , z ) − σ ζ ∂ x (1 − x ) f ( x , τ ) ˜ ˜ 1 x x 2ϑ 2 (2µ + ζ ) [( ) ] 2 ˜ ˜ ˜ + ∂x˜ ζ x(1 − x) f (x, τ ) [( ) ] [( ) ] = −σ1 ζ ∂x˜ x˜ (1 − x˜ ) f (x˜ , τ ) + ∂x˜2 ζ x˜ (1 − x˜ ) f (x˜ , τ ) .

L(2) ρ (0) d(y˜ , zt) =

26

27

28

29

30

We will use this trick with the adjoint operator of L(0) often throughout the analysis of the four models.

31

Step 4: Direct computation yields (recall that the integration of all full derivatives w.r.t. y˜ and z˜ vanishes)

∫ 32

33

35

36

37

L(1) ρ (1) d(y˜ , z˜ ) =

Let F (x˜ , y˜ , z˜ ) =

∫ 34

(A.11)

Step 3: Therewith we find

21

23

)

L(0)+ (z˜ + (1 − 2x˜ )y˜ ) = −(µ + ζ ) (z˜ + (1 − 2x˜ )y˜ )

18 19

(A.7)

= 0 we conclude that ρ (x˜ , y˜ , z˜ , τ ) = f (x˜ , τ ) ν (y˜ , z˜ ), and the reduced Fokker–Planck equation is given by ∫ L(1) ρ (1) d(y˜ , z˜ ) + L(2) ρ (0) d(y˜ , z˜ ). (A.8) (0)

∫

L(0) ρ (2) = ∂τ ρ (0) − L(1) ρ (1) − L(2) ρ (0) .

−1 µ+ζ

−ζ ∂x˜ 2ϑ

∫

(A.12)

(z˜ + y˜ (1 − 2x˜ )) ρ (1) d(y˜ , z˜ ).

(z˜ + y˜ (1 − 2x˜ )), then L(0)+ F (x˜ , y˜ , z˜ ) = (z˜ + y˜ (1 − 2x˜ )). Thus,

∫ ∫ −ζ −ζ ∂x˜ (z˜ + y˜ (1 − 2x˜ )) ρ (1) d(y˜ , z˜ ) = ∂x˜ L(0)+ F (x˜ , y˜ , z˜ ) ρ (1) d(y˜ , z˜ ) 2ϑ 2ϑ ∫ ∫ −ζ ζ = ∂x˜ ∂x˜ F (x˜ , y˜ , z˜ ) L(0) ρ (1) d(y˜ , z˜ ) = F (x˜ , y˜ , z˜ ) L(1) ρ (0) d(y˜ , z˜ ) 2ϑ 2ϑ ∫ −ζ = ∂x˜ (z˜ + y˜ (1 − 2x˜ )) L(1) ρ (0) d(y˜ , z˜ ) 2ϑ (µ + ζ ) ( ) −ζ ∂x˜ T1 + T2 + T3 + T4 + T5 = 2ϑ (µ + ζ )

L(1) ρ (1) d(y˜ , z˜ ) =

(A.13)

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13

where we define and evaluate the terms T1 ,. . . ,T5 below:

1

[( ) ] T1 = (z˜ + y˜ (1 − 2x˜ )) (−∂x˜ ) ζ (z˜ + y˜ (1 − 2x˜ ))/(2ϑ ) ρ (0) d(y˜ , z˜ ) [ ∫ ] ∫ ζ ∂x (z˜ + y˜ (1 − 2x˜ ))2 ρ (0) d(y˜ , z˜ ) + 2 y˜ (z˜ + y˜ (1 − 2x˜ ))ρ (0) d(y˜ , z˜ ) . =− 2ϑ ∫

2

3

For the second integral, we use y˜ (z˜ + y˜ (1 − 2x˜ )) = −L(0)+ [(y˜ z˜ + (1 − 2x˜ )y˜ 2 )]/(2µ + ζ ) and conclude from L(0) ρ (0) = 0 that this integral is zero. For the first integral, we note 2(µ + ζ )(z˜ + (1 − 2x˜ )y˜ )2 = −L(0)+ [(z˜ + (1 − 2x˜ )y˜ )2 ] + 8ζ ϑ 2 x˜ (1 − x˜ ), such that we obtain (again, using L(0) ρ (0) = 0) T1 = −

ζ ∂x 2ϑ

8ζ ϑ 2 x˜ (1 − x˜ )

∫

2(µ + ζ )

−2ϑζ 2 ∂x˜ x˜ (1 − x˜ )f ζ +µ

(

ρ (0) d(y˜ , z˜ ) =

4 5 6

) (A.14)

For the next three terms, we use partial integration w.r.t. y˜ (term T2 ), w.r.t. z˜ (term T4 ) resp. partial integration w.r.t. y˜ together with L(0)+ [(y˜ z˜ + (1 − 2x˜ )y˜ 2 )] = −(2µ + ζ )y˜ (z˜ + y˜ (1 − 2x˜ )) (term T3 )

( ) ˜ ˜ ˜ ˜ ˜ T2 = (z + y(1 − 2x)) (−∂y˜ ) −σ3 β (1 − x) − σ2 µϑ (1 − x) ρ (0) d(y˜ , z˜ ) ( ) = (1 − 2x˜ ) −(σ2 + σ3 )β (1 − x˜ ) f ( ) ∫ T3 = (z˜ + y˜ (1 − 2x˜ )) (−∂z˜ ) ζ (1/ϑ )y˜ (z˜ + y˜ (1 − 2x˜ )) − 2ζ ϑσ1 x˜ (1 − x˜ ) + β (σ2 + σ3 )(1 − x˜ ) ρ (0) d(y˜ , z˜ )

7

8 9

∫

= −2ζ ϑσ1 x˜ (1 − x˜ ) f + β (σ2 + σ3 )(1 − x˜ )f ( ) ∫ T4 = (z˜ + y˜ (1 − 2x˜ )) (∂z˜2 ) ϑζ (y˜ + (1 − 2x˜ )z˜ ) − 4ζ ϑ x˜ (1 − x˜ )y˜ ρ (0) d(y˜ , z˜ ) = 0 ( ) ( ) ∫ T5 = ∂x˜ 4ϑζ x˜ (1 − x˜ ) ρ (0) d(y˜ , z˜ ) = ∂x˜ 4ϑζ x˜ (1 − x˜ )f

10

(A.15)

12

(A.16)

13

(A.17)

14

(A.18)

15

Adding up the corresponding terms, Eq. (A.8) yields

∂τ f = −σ1 ζ ∂x˜

[(

)

]

x˜ (1 − x˜ ) f (x˜ , τ ) +

16

−ζ 2ϑ (µ + ζ )

{

∂x˜

−2ϑζ ∂x˜ x˜ (1 − x˜ )f ζ +µ 2

(

)

− 2ζ ϑσ1 x˜ (1 − x˜ ) f } + 2β (σ2 + σ3 )x˜ (1 − x˜ )f ∂x˜ (4ϑζ x˜ (1 − x˜ )f ) .

17

18

When we collect terms and use θ = β/µ, we have the reduced Fokker–Planck equation (G = ζ /µ) fτ =

ζ −ζ (σ1 + σ2 + σ3 ) ∂x˜ [˜x(1 − x˜ )f ] + ∂ 2 [˜x(1 − x˜ )f ]. 1+G (1 + G)2 x˜

19

(A.19)

A.3. Fixed population size and deterministic seed bank — small delay approximation

Yt = β

∞

e−µs Xt −s ds,

Zt = β (1 − σ3 /N)

∞

∫

0

20

21

We present a second approach for the model with fixed population size and deterministic seed bank, a nice but heuristic argument using the idea of a short delay approximation (Guillouzic et al., 1999). To our knowledge no rigorous approximation theorem is available, therefore we also did go through the more lengthy computations in the section above (Appendix A.2). Both approaches yield identical results. Using the variation-of constant formula, we find

∫

11

e−µ(1+σ2 /N)s (N − Xt −s ) ds.

22 23 24 25 26

27

0

Let xt = X1,t /N and ε 2 = 1/N. Then,

28

ε (1 + σ1 ε ) ζ (1 − xt ) ∫∞ −2

X1,t → X1,t + 1 at rate ∫ ∞

2

e−µs xt −s ds + (1 − σ3 ε 2 )

0

0

∫∞ 0

−µs

e

xt −s ds

e−µ(1+σ2 ε

2 )s

(1 − xt −s ) ds

,

29 30

X1,t → X1,t − 1 at rate ∫ ∞

ε

0

−2

ζ xt (1 − σ3 ε ) 2

−µ(1+σ2 ε 2 )(t −s)

∫∞

e

0

e−µs xt −s ds + (1 − σ3 ε 2 )

∫∞ 0

(1 − xt −s ) ds

e−µ(1+σ2 ε

2 )s

(1 − xt −s ) ds

.

31

With standard arguments, we obtain a stochastic delay differential equation (SDDE) at the evolutionary time scale τ = t ε 2 as

ε (1 + σ1 ε ) ζ (1 − xτ ) −4

( dxτ =

ε −2 −

∫∞ 0

2 e−µs/ε x

τ −s

2

ds +

ε −4 ζ xτ ε −2

∫∞ 0

ε −2 (1

∫∞ 0

− σ3 ε

∫∞

2)

e−µ(1+σ2 ε

0

2 )s/ε 2

e−µs/ε xτ −s ds + ε −2 (1 − σ3 ε 2 ) 2

e

∫0 ∞

xτ −s ds

2 2 e−µ(1+σ2 ε )s/ε (1

e−µ(1+σ2 ε

2 )s/ε 2

32

33

− xτ −s ) ds

)

(1 − xτ −s ) ds

∫∞ 0

−µs/ε 2

(1 − xτ −s ) ds

dτ

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34

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14

L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx

ε −2 (1 + σ1 ε 2 ) ζ (1 − xτ )

( +

1

ε −2 +

2

3 4

5

∫∞ 0

e−µs/ε xτ −s ds + ε −2 (1 − σ3 ε 2 ) 2

ε ε −2

∫∞ 0

−2

ζ xτ

∫∞ 0

9 10 11

12

13

14

0

e

e−µs/ε xτ −s ds + ε −2 (1 − σ3 ε 2 ) 2

e−µ(1+σ2 ε

2 )s/ε 2

(1 − xτ −s ) ds

)1/2

(1 − xτ −s ) ds

∫∞ 0

e−µ(1+σ2 ε

2 )s/ε 2

(1 − xτ −s ) ds

dWτ .

We aim a small delay approximation. Therefore, we note that for a function Φ (t), which is sufficiently smooth and bounded, we have (for µ > 0)

ε −2 µ

∞

∫

e−µ(τ −s)/ε Φ (τ − s) ds = ε −2 2

∫

∞

µ e−µs/ε (Φ (τ ) − sΦ ′ (τ ) + O(s2 )) ds 2

0

= Φ (τ ) − ε2 µ−1 Φ ′ (τ ) + O(ε 4 ).

6

8

2

∫0 ∞

−µ(1+σ2 ε 2 )s/ε2

0

7

e−µs/ε xτ −s ds

∫∞

Thus, at a formal level, lim ζ

(

ε→0

ε

−4

µ

∞

∫

e

−µs/ε2

xτ −s ds − ε

−2

) xτ

dτ = −

0

ζ dxτ = −G dxτ , µ

where G = ζ /µ. Note that this equation has to be interpreted in terms of the Euler–Maruyama approximation of an SDDE, where differential quotients are replaced by difference quotients. We again emphasize that this approach is only mend to be formal, as to our knowledge no rigorous approximation theorems for small delay approximations in the context of SDDE are available. If we add

( ) ∫∞ 2 − ζ ε −4 µ 0 e−µs/ε xτ −s ds − ε −2 xτ dτ ∫∞ ∫∞ ε −2 µ 0 e−µs/ε2 xτ −s ds + (1 − σ3 ε 2 ) ε −2 µ 0 e−µ(1+σ2 ε2 )s/ε2 (1 − xτ −s ) ds to both sides of the SDDE and let ε → 0, we obtain (1 + G)dxτ = ζ (σ1 + σ2 + σ3 ) xτ (1 − xτ ) dτ + (2ζ xτ (1 − xτ ))1/2 dWτ .

16

This equation yields the desired result with σ = σ1 + σ2 + σ3 and agrees with the approximate Fokker–Planck equation (A.19). We conjecture that the short delay approximation yields a valid approximation under suited (rather general) conditions.

17

A.4. Logistic population dynamics and deterministic seed bank

15

18 19

Let pi,j (k, l, t) = P(X1,t = i, X2,t = j, Yt ∈ (k, k + dk), Zt ∈ (l, l + dl)) be the joint probability density of the resulting stochastic process (with discrete i, j and continuous k, l). The corresponding master equation reads:

) ] β i − µk p˙ i,j (k, l, t) + ∇ p (k, l, t) β (1 − σ3 /N)j − µ (1 + σ2 /N)l i,j ] [ N −i−j σ4 σ1 = − ζ (i + (1 + )j) + γ (k + (1 − )l) pi,j (k, l, t) [(

N

20

N

+ ζ (i + 1)pi+1,j (k, l, t) + (1 + +γ 21 22

23

(N − i − j + 1)k

25

N

(A.20)

)ζ (j + 1)pi,j+1 (k, l, t)

pi−1,j (k, l, t) + (1 −

σ4 N

)γ

(N − i − j + 1)l N

pi,j−1 (k, l, t),

where the operator ∇ acts with respect to continuous state space variables k, l. Standard arguments yield the Fokker–Planck-approximation for large populations, where x1 = i/N, x2 = j/N, y = k/N and z = l/N, as

∂t u(x1 , x2 , y, z , t) {[ ] } = ∂x1 ζ x1 − γ (1 − x1 − x2 )y u(x1 , x2 , y, z , t) {[ ] } σ4 σ1 + ∂x2 (1 + )ζ x2 − (1 − )γ (1 − x1 − x2 )z u(x1 , x2 , y, z , t) N N {[ ] } + ∂y (µy − β x1 ) u(x1 , x2 , y, z , t) {[ ] } σ2 σ3 + ∂z (1 + )µz − (1 − )β x2 u(x1 , x2 , y, z , t) N N {[ ] } 1 2 + ∂x1 ζ x1 + γ (1 − x1 − x2 )y u(x1 , x2 , y, z , t) 2N {[ ] } 1 2 σ1 σ4 + ∂x2 (1 + )ζ x2 + (1 − )γ (1 − x1 − x2 )z u(x1 , x2 , y, z , t) . 2N

24

N

N

σ1

N

(A.21)

N

Note that the second order terms are solely due to x1 and x2 ; no component in the second order terms is added by the seed bank variables y and z. Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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15

A.4.1. Deterministic model The corresponding deterministic model (drift terms only) yields the ODEs

1

x˙1 = γ (1 − x1 − x2 )y − ζ x1 , x˙2 = γ (1 − x1 − x2 )z − ζ x2 , y˙ = β x1 − µy, z˙ = β x2 − µz . The lifetime reproductive success (or basic reproduction number) of a plant reads R0 = β γ /(µζ ). If R0 > 1, the plant population can persist. Under this condition, there is a line of stationary solutions:

2 3

Proposition A.2. Assume R0 > 0. Let ϑ := β/µ, κ := (γ ϑ − ζ )/γ ϑ . Then, κ ∈ [0, 1], and there is a line of stationary points in [0, κ]2 × R2+ given by (x1 , x2 , y, z) = (κ x, κ (1 − x), ϑ κ x, ϑ κ (1 − x)),

x ∈ [0, 1].

The line of stationary points is transversally stable (locally and globally).

4

Proof. It is straightforward to check that the line indicated above consists indeed of stationary points with non-negative values. The local stability is a consequence of Hartman–Grobman and the analysis of the Jacobian (due to the block structure of this matrix the eigenvalues can be stated explicitly). Note that one eigenvalue necessarily is zero with an eigenvector pointing in the direction of the line of stationary points. Now, to show global stability, we prove that the system will approach the equilibrium line from any starting point. We first respectively denote P = x1 + x2 and S = y + z as the total plant and seed populations and consider the resulting reduced system P˙ = γ (1 − P)S − ζ P ,

S˙ = β P − µS .

5 6 7 8 9 10 11

Here we use that we only consider weak selection: all selection effects tend to zero for N → ∞, and hence the ODE describes the neutral case. The divergence of this system is negative, and thus the combination of the theorems of Bendixon–Dulac and Pointcaré – Bendixon implies that trajectories (P , S) tend to stationary points. This observation yields the desired global stability. □ Note that in equilibrium, x1 + x2 = κ . That is, κ N represents the average above-ground population size of the model conditioned on non-extinction.

12 13 14 15 16

A.4.2. Dimension reduction by time scale analysis

17

Steps 1 and 2: First, new local variables for the boundary layer around the equilibrium line are defined as

18

ε

ε

v˜ , x2 = κ (1 − x˜ ) + v˜ , 2 ε ε y = κ ϑ x˜ + (y˜ + z˜ + ϑ v˜ ), z = κ ϑ (1 − x˜ ) + (y˜ − z˜ + ϑ v˜ ),

x1 = κ x˜ +

(A.22)

2

2

20

2

where x˜ =

19

21

x1 − x2 + κ 2κ

,

v˜ = ε (x1 + x2 − κ ), −1

y˜ = ε

−1

(y + z − ϑ (x1 + x2 )),

z˜ = ε

−1

(y − z − ϑ (x1 − x2 )),

22

and ε 2 = 1/N. For the transformed density ρ (x˜ , v˜ , y˜ , z˜ , t ; ε ) = u(x1 , x2 , y, z , t), we find

∂t ρ (x˜ , v˜ , y˜ , z˜ , t ; ε ) = L(0) + ε L(1) + ε 2 L

(

) (2)

ρ (x˜ , v˜ , y˜ , z˜ , t) + O(ε 3 ),

23

(A.23)

with linear differential operators (as seen above in Appendix A.2, we only apply Taylor expansion in a straightforward way and equate coefficients of equal order in ε . Thus we skip the details; the result can be readily checked via computer algebra). As discussed before, we can drop some terms of L(2) . It turns out that for similar reasons some terms of L(1) are not required.

] } {[ ] } −ζ y˜ + γ ϑκ v˜ ρ + ∂y˜ (µ + ζ )y˜ − γ ϑ 2 κ v˜ ρ ϑ {[ ] } + ∂z˜ (µ + ζ )z˜ + γ ϑ 2 κ (1 − 2x˜ )v˜ ρ { } { } { } + ∂v˜2 ζ κρ + ∂y˜2 ζ ϑ 2 κρ + ∂z˜2 ζ ϑ 2 κρ { } { } { } + ∂v˜ ∂y˜ −2ζ ϑκρ + ∂v˜ ∂z˜ 2ζ ϑκ (1 − 2x˜ )ρ + ∂y˜ ∂z˜ −2ζ ϑ 2 κ (1 − 2x˜ )ρ , ] } {[ ]} {[ γϑ −ζ 2 (1) z˜ − (1 − 2x˜ )v˜ ρ + ∂v˜ γ v˜ y˜ + γ ϑ v˜ + (σ4 + σ1 )ζ κ (1 − x˜ ) L ρ = ∂x˜ 2ϑκ 2 {[ ] } + ∂y˜ −γ ϑ v˜ y˜ − γ ϑ 2 v˜ 2 − (σ1 + σ4 )ζ ϑκ (1 − x˜ ) + (σ3 + σ2 )βκ (1 − x˜ ) ρ {[ ] } + ∂z˜ −γ ϑ v˜ z˜ + (σ4 + σ1 )ζ ϑκ (1 − x˜ ) − (σ2 + σ3 )βκ (1 − x˜ ) ρ L(0) ρ = ∂v˜

{[

(A.24)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

24

YTPBI: 2648

16

L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx

{

}

{

}

{ } + ∂x˜ ∂v˜ −ζ (1 − 2x˜ )ρ + ∂x˜ ∂y˜ ζ ϑ (1 − 2x˜ )ρ + ∂x˜ ∂z˜ −ζ ϑρ { } { } { } { } { } { } + ∂v˜2 . . . + ∂y˜2 . . . + ∂z˜2 . . . + ∂v˜ ∂y˜ . . . + ∂v˜ ∂z˜ . . . + ∂y˜ ∂z˜ . . . , ] } } { } { } { { } {[ (σ 1 + σ 4 ) ζ ζ γ 2 (2) v˜ z˜ − (1 − x˜ ) ρ + ∂x˜ ρ + ∂v˜ . . . + ∂y˜ . . . + ∂z˜ . . . . L ρ = ∂x˜ 2κ 2 4κ 1 2 3

4 5 6 7 8 9 10 11 12 13

14

15

ρ (x˜ , v˜ , y˜ , z˜ , t) = ρ (0) (x˜ , v˜ , y˜ , z˜ , ε2 t) + ερ (1) (x˜ , v˜ , y˜ , z˜ , ε2 t) + ε 2 ρ (2) (x˜ , v˜ , y˜ , z˜ , ε2 t) + O(ε 3 ). Plugging this into (A.23) and comparing same order terms, we have L(0) ρ (0) = 0,

(A.27)

with a time independent multivariate normal distribution ν (v˜ , y˜ , z˜ ; x˜ ) in v˜ , y˜ , z˜ that satisfies L(0) ρˆ = 0 (Kogan et al., 2014, appendix C), where in the distribution x˜ is a parameter that affects the correlation matrix. In contrast to the model with fixed population size and deterministic seed bank we obtain a proper normal distribution: In the present model the total population size is a random variable, such that not only the decomposition but also the size of the seed bank is a random variable. The function f (x˜ , τ ) modifies ν (v˜ , y˜ , z˜ ; x˜ ) and represents the time evolution of ρ (0) along the line of stationary points. Integrating the last equation of (A.27) from −∞ to ∞ with respect to v˜ , y˜ , z˜ – the left hand side is a total derivative w.r.t. variables of integration and becomes zero – yields the evolution equation

∂τ f =

∫

L(1) ρ (1) d(v˜ , y˜ , z˜ ) +

∫

L(2) ρ (0) d(v˜ , y˜ , z˜ ).

(A.28)

Step 3: We start by computing the second integral. Since all terms that are full derivatives w.r.t. v˜ , y˜ or z˜ vanish upon integration, we have L(2) ρ (0) d(v˜ , y˜ , z˜ )

{∫ ] } } γ ζ (0) (−σ1 − σ4 )ζ v˜ z˜ + (1 − x˜ ) ρ (0) d(v˜ , y˜ , z˜ ) + ∂x˜2 ρ d(v˜ , y˜ , z˜ ) 2κ 2 4κ {∫ [ } { } { } γ ] (0) (−σ1 − σ4 )ζ ζ = ∂x˜ v˜ z˜ ρ d(v˜ , y˜ , z˜ ) + ∂x˜ (1 − x˜ )f + ∂x˜2 f . 2κ 2 4κ

= ∂x˜

{∫ [

(A.29)

We take a closer look at the first integral. We aim to write v˜ z˜ = L(0)+ h+ − g(x˜ ) for suitable h+ , g, where L(0)+ is the adjoint of L(0) and g is independent of y˜ , z˜ , v˜ . If we can do so, then

19

21

L(0) ρ (2) = ∂τ ρ (0) − L(1) ρ (1) − L(2) ρ (0) ,

ρ (0) (x˜ , v˜ , y˜ , z˜ , τ ) = f (x˜ , τ )ρˆ (v˜ , y˜ , z˜ ; x˜ ),

∫

20

L(0) ρ (1) = −L(1) ρ (0) ,

which indicates that ρ (0) can be written as

16

18

(A.26)

We employ time scale separation and focus on a solution evolving on the slow time τ = ε 2 t = t /N using the Ansatz

∫

17

(A.25)

v˜ z˜ ρ

(0)

d(v˜ , y˜ , z˜ ) =

∫

h+ L(0) ρ (0) d(v˜ , y˜ , z˜ ) − g(x˜ )f (x˜ , τ ) = −g(x˜ )f (x˜ , τ ).

To identify h+ and g, we reduce the problem to linear algebra. Define the finite-dimensional vector space (for k ∈ N) Hk := {P : R4 → R, P polynomial homogeneous of degree k (w.r.t. variables v˜ , y˜ , z˜ )}, so that, e.g., γ /2κ v˜ z˜ ∈ H2 , while γ /2κ v˜ z˜ + c ̸ ∈ H2 for c ∈ R \ {0}. Examining L(0)+ , we find L(0)+ h+ = 0,

for h+ ∈ H0 .

L(0)+ h+ = h ∈ H1 ,

for h+ ∈ H1 .

L

h = h + g ∈ H2 ⊕ H0 ,

for h+ ∈ H2 .

(0)+ +

23

In particular the last observation L(0)+ H2 → H2 ⊕ H0 , h+ ↦ → (h, g) allows to define an operator M : H2 → H2 , h+ ↦ → h. To simplify notation, we identify vectors w.r.t. a given basis in Hk and the corresponding polynomials.

24

Proposition A.3. W.r.t. the canonical basis (v˜ 2 , y˜ 2 , z˜ 2 , v˜ y˜ , v˜ z˜ , y˜ z˜ ), the operator M has the representation

22

⎛

25

−2γ ϑκ

⎜ 0 ⎜ ⎜ 0 M=⎜ ⎜ 2ζ /ϑ ⎝ 0 0

26 27

28

0

−2(µ + ζ ) 0 2γ ϑ 2 κ 0 0

0 0

−2(µ + ζ ) 0

−2γ ϑ 2 κ (1 − 2x˜ ) 0

γ ϑ 2κ ζ /ϑ

−γ ϑ 2 κ (1 − 2x˜ ) 0 0 0

0

−(µ + ζ + γ ϑκ ) 0 0

−(µ + ζ + γ ϑκ ) ζ /ϑ

⎞

0 ⎟ 0 ⎟ 0 ⎟ ⎟, −γ ϑ 2 κ (1 − 2x˜ )⎟

γ ϑ 2κ −2(µ + ζ )

⎠

and is invertible, if all parameters are positive. Let h = (a1 (x˜ ), . . . , a6 (x˜ ))T ∈ H2 . Then, L(0)+ (h) = M h + g(x˜ ), where g(x˜ ) ∈ H0 is uniquely ˆ x˜ ) h and G( ˆ x˜ ) denoting the row-vector defined by g(x˜ ) = G(

ˆ x˜ ) = (2ζ κ, 2ζ ϑ 2 κ, 2ζ ϑ 2 κ, −2ζ ϑκ, 2ζ ϑκ (1 − 2x˜ ), −2ζ ϑ 2 κ (1 − 2x˜ )). G(

(A.30)

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L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx

17

˜ = h˜ + g(x˜ ) with g(x˜ ) = G( ˆ x˜ ) M −1 h, ˜ so that For h˜ ∈ H2 , we find L(0)+ (M −1 h)

∫

1

h˜ ρ (x˜ , v˜ , y˜ , z˜ , τ ) d(v˜ , y˜ , z˜ ) = −g(x˜ ) f (x˜ , τ ).

2

Proof. It is straightforward to obtain the representation of M and g by applying L(0) to the elements of the basis given above. Using, ˆ x˜ ), consider e.g., Gauß-elimination, we find that M is invertible if the parameters are positive. In order to obtain the i′ ’th component of G( the i’th entry of the canonical basis bi , compute L(0)+ bi , and identify the component that is in H0 . E.g., for i = 1 we find b1 = v˜ 2 , and ˆ x˜ ))1 = 2ζ κ . The equation L(0)+ v˜ 2 = −2v˜ [(−ζ /ϑ )y˜ + γ ϑκ v˜ ] + 2ζ κ , where −2v˜ [(−ζ /ϑ )y˜ + γ ϑκ v˜ ] ∈ H2 and 2ζ κ ∈ H0 . Hence, (G( (0)+ −1 − 1 ˆ (M h) = h + g(x˜ ) with g(x˜ ) = G(x˜ ) M h implies L

∫

∫

h ρ (x˜ , v˜ , y˜ , z˜ , τ ) d(v˜ , y˜ , z˜ ) =

[L(0)+ (M −1 h) − g(x˜ )] ρ (x˜ , v˜ , y˜ , z˜ , τ ) d(v˜ , y˜ , z˜ ) = 0 − g(x˜ ) f (x˜ , τ ).

□

(A.31)

by using the computer algebra package MAXIMA (Maxima, 2014). In summary, we have L(2) ρ (0) d(v˜ , y˜ , z˜ ) = ∂x˜

}

{

ζ 2ϑκ (µ + ζ )

z˜ + (1 − 2x˜ )y˜ +

(β + ζ ϑ )

ζ

(A.32)

13

(1 − 2x˜ )v˜ ,

(A.33)

] ∫ [ (0)+ + ] (1) −ζ γϑ L h0 ρ d(v˜ , y˜ , z˜ ) z˜ − (1 − 2x˜ )v˜ ρ (1) d(v˜ , y˜ , z˜ ) = ∂x˜ 2ϑκ 2 ∫ ∫ [ (1) (0) ] [ (0) (1) ] ˜ ˜ − h+ d(v˜ , y˜ , z˜ ). L ρ d( v ˜ , y , z ) = ∂ = ∂x˜ h+ ˜ x 0 L ρ 0

15

∫ [

(A.34)

The operator L(1) consists of 13 terms see Eq. (A.25). There are four types of terms: (a) ∂x˜ (..), (b) ∂v˜ (..), ∂y˜ (..), ∂z˜ (..), (c) ∂x˜ ∂v˜ (..), ∂x˜ ∂y˜ (..), ∂x˜ ∂z˜ (..), (d) remain terms, e.g. ∂v˜2 (..), ∂y˜2 (..) and so on. We indicate how to handle these terms in an exemplary way. For (a), we use the product rule, (β + ζ ϑ )

−ζ γϑ z˜ − (1 − 2x˜ )v˜ ρ (0) d(v˜ , y˜ , z˜ ) ζ 2ϑκ 2 ] {[ ] } ∫ [ (β + ζ ϑ ) −ζ γϑ = ∂x z˜ + (1 − 2x˜ )y˜ + (1 − 2x˜ )v˜ z˜ − (1 − 2x˜ )v˜ ρ (0) d(v˜ , y˜ , z˜ ) ζ 2ϑκ 2 [ ] {[ ] } ∫ (β + ζ ϑ ) −ζ γϑ − ∂x˜ z˜ + (1 − 2x˜ )y˜ + (1 − 2x˜ )v˜ z˜ − (1 − 2x˜ )v˜ ρ (0) d(v˜ , y˜ , z˜ ). ζ 2ϑκ 2

∫ [

z˜ + (1 − 2x˜ )y˜ +

(1 − 2x˜ )v˜

]

∂x˜

{[

]

∫ [ z˜ + (1 − 2x˜ )y˜ +

∫ [ =−

(β + ζ ϑ )

ζ

(β + ζ ϑ )

ζ

] (1 − 2x˜ )

z˜ + (1 − 2x˜ )y˜ +

[

∂v˜ y˜ z˜ + (1 − 2x˜ )y˜ +

]

∂v˜

19

{[ ] } γ v˜ y˜ + γ ϑ v˜ 2 + (σ4 + σ1 )ζ κ (1 − x˜ ) ρ (0) d(v˜ , y˜ , z˜ ) 22

γ v˜ y˜ + γ ϑ v˜ + (σ4 + σ1 )ζ κ (1 − x˜ ) ρ 2

]

(0)

}

d(v˜ , y˜ , z˜ )

(1 − 2x˜ )v˜

(β + ζ ϑ )

ζ

]

{

23 24

}

∂v˜ y˜ . . . d(v˜ , y˜ , z˜ )

(1 − 2x˜ )v˜

]{

25

}

. . . d(v˜ , y˜ , z˜ ) = 0.

In this way, we obtain

∫

18

21

(1 − 2x˜ )v˜

{[

(β + ζ ϑ )

ζ

17

20

For the terms in (c), we combine the methods for (a) and (b). Last, partial integration of the terms in group (d) shows that these terms vanish, e.g.

∫ [

16

}

For the terms (b) we integrate by parts, e.g.

=

14

L(1) ρ (1) d(v˜ , y˜ , z˜ )

= ∂x˜

∫

12

]

˜ ˜ ˜ . Remember that full derivatives w.r.t. v˜ , y˜ or z˜ vanish upon integration. So, we obtain L(0)+ h+ 0 = −ζ /(2ϑκ )z − γ ϑ/2(1 − 2x)v

∫

10

}

Step 4: Now, we turn to the computation of the first integral in (A.28). With h+ 0 :=

7

11

ζ γ ζ ϑ (1 − 2x˜ ) (−σ1 − σ4 )ζ (1 − x˜ ) + f (x˜ , τ ) + ∂x˜2 f (x˜ , τ ) . 2(µ + γ ϑ ) 2 4κ

)

[

6

9

γζϑ g(x˜ ) = − (1 − 2x˜ ) 2(µ + γ ϑ )

{(

4 5

8

Solving the system for h(x˜ , v˜ , y˜ , z˜ ) = γ /2κ v˜ z˜ , we obtain

∫

3

L(1) ρ (1) d(v˜ , y˜ , z˜ ) = T1 + T2 + T3 + T4 + T5

26

(A.35)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

27

YTPBI: 2648

18 1

2

L. Heinrich et al. / Theoretical Population Biology xx (xxxx) xxx–xxx

with T1 =

3

4

T2 =

5

T3 =

6

T4 =

7

T5 =

8

ζ ∂2 2ϑκ (µ + ζ ) x˜

[ γ ϑ 2 (µ + ζ ) ζ 2 γϑ (1 − 2x˜ )2 v˜ 2 + z˜ + (1 − 2x˜ )2 v˜ y˜ 2ζ 2ϑκ 2 ] ζ µ + γϑ (1 − 2x˜ )v˜ z˜ + (1 − 2x˜ )y˜ z˜ ρ (0) d(v˜ , y˜ , z˜ ) + 2κ 2ϑκ ] ∫ [ ζ µ+ζ µ + ζ − γ ϑκ ζ 2 2 2µ + ζ ∂x˜ (1 − 2x˜ )v˜ + γ ϑ (1 − 2x˜ )v˜ y˜ + v˜ z˜ + y˜ z˜ ρ (0) d(v˜ , y˜ , z˜ ) γϑ 2ϑκ (µ + ζ ) ζ ζ κ ϑκ ] ∫ [ ζ ζµ ∂x˜ (σ4 + σ1 ) (1 − x˜ ) − (σ4 + σ1 + σ3 + σ2 ) x˜ (1 − x˜ ) ρ (0) d(v˜ , y˜ , z˜ ) 2 µ+ζ ∫ −ζ ∂x˜ 2 β (1 − 2x˜ )ρ (0) d(v˜ , y˜ , z˜ ) 2ϑκ (µ + ζ ) ∫ −ζ ∂x˜2 [β (1 − 2x˜ )2 + ζ ϑ]ρ (0) d(v˜ , y˜ , z˜ ) 2ϑκ (µ + ζ )

We will treat all three terms separately starting with the last three terms, T3 + T4 + T5 = ∂x˜

9

10

11

∫

ζ

{[

ζµ

] }

x˜ (1 − x˜ ) f (σ4 + σ1 ) (1 − x˜ ) − (σ4 + σ1 + σ3 + σ2 ) 2 µ+ζ

{[ ] } { } ζ ζ − ∂x˜2 β (1 − 2x˜ )2 + ζ ϑ f − ∂x˜ 2β (1 − 2x˜ )f . 2ϑκ (µ + ζ ) 2ϑκ (µ + ζ ) ∫ (2) (0) To compute T2 , we proceed as in the computations for L ρ d(v˜ , y˜ , z˜ ), that is, we solve ⎛ ⎞ 2µ + ζ γ ϑ2 (1 − 2x˜ ) ζ ⎟ ⎞ ⎜ ⎛ ⎜ ⎟ a1 (x˜ ) 0 ⎜ ⎟ ⎜a2 (x˜ )⎟ ⎜ ⎟ 0 ⎟ ⎜ ⎜ ⎟ ⎜a3 (x˜ )⎟ ⎜ ⎟ µ+ζ = ⎜ γϑ M⎜ ⎟, ⎟ ˜ (1 − 2 x ) ⎜a4 (x˜ )⎟ ⎜ ⎟ ζ ⎟ ⎜ ⎝a5 (x˜ )⎠ ⎜ µ + ζ − γ ϑκ ⎟ ⎜ ⎟ a6 (x˜ ) κ ⎝ ⎠ ζ ϑκ   

(A.36)

= ˆh

12

13

14

and find

ζ ζ T2 = ∂x˜ {−g(x˜ )f (x˜ , τ )} = ∂x˜ 2ϑκ (µ + ζ ) 2ϑκ (µ + ζ )

] } γ ϑ 2 κ (µ + ζ ) 2β − (1 − 2x˜ )f (x˜ , τ ) . µ + γϑ

{[

(A.37)

For T1 , this can be done similarly, where the system to be solved is

⎞ γ ϑ 2 (µ + ζ ) 2 ˜ (1 − 2 x ) ⎟ ⎜ 2ζ ⎟ ⎛ ⎞ ⎜ ⎟ ⎜ 0 a1 (x˜ ) ⎟ ⎜ ζ ⎜a2 (x˜ )⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2ϑκ ⎜a3 (x˜ )⎟ ⎜ ⎟ M⎜ =⎜ ⎟, ⎟ γ ϑ 2 ⎜a4 (x˜ )⎟ ⎜ ⎟ (1 − 2x˜ ) ⎟ ⎝a5 (x˜ )⎠ ⎜ 2 ⎟ ⎜ µ + γϑ ⎜ a6 (x˜ ) ˜ (1 − 2x) ⎟ ⎟ ⎜ 2κ ⎠ ⎝ ζ (1 − 2x˜ ) 2ϑκ    ⎛

15

= ˆh

16

and as a result, we have

ζ ∂ 2 {−g(x˜ )f (x˜ , τ )} 2ϑκ (µ + ζ ) x˜ ] } {[ ] ζ 2ϑ [ ζ β + ζϑ = ∂x˜2 (1 − 2x˜ )2 − (1 − 2x˜ )2 − 1 f (x˜ , τ ) . 2ϑκ (µ + ζ ) 2 2(µ + ζ ) ∫ (1) (1) ∫ ∫ With L ρ d(v˜ , y˜ , z˜ ) = T1 + T2 + T3 + T4 + T5 and ∂τ f = L(1) ρ (1) d(v˜ , y˜ , z˜ ) + L(2) ρ (0) d(v˜ , y˜ , z˜ ), we obtain { } { } ζ ζ (−σ1 − σ4 − σ2 − σ3 ) 2 ˜ ˜ ˜ ˜ ∂τ f = ∂ x (1 − x )f + ∂ x (1 − x )f . x (1 + G)2 x 1+G T1 =

17

18

19

(A.38)

(A.39)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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19

A.5. Fixed population size and fluctuating seed bank

1

We only present the main steps for this and the following model, as the computations are lengthy and resemble that of the last model. Let pi,j,k (t) = P(Xt = i, Yt = j, Zt = k) be the probability distribution of the resulting stochastic process. The corresponding master equation reads:

2 3 4

[ ] σ1 σ3 σ2 p˙ i,j,k (t) = − ζ N + ζ (N − i) + β N − β (N − i) + µ(j + k) + µk pi,j,k (t) N

N

[

N

[ ] i(j + 1) (i + 1)(k + 1) ζ pi,j+1,k (t) + ζ pi+1,j,k+1 (t) j+k+1 j+k+1 ] [ σ1 (N − i + 1)(j + 1) )ζ pi−1,j+1,k (t) (1 + N j+k+1 [ ] [ ] [ ] σ1 (N − i)(k + 1) (1 + )ζ pi,j,k+1 (t) + β i pi,j−1,k (t) + µ(j + 1) pi,j+1,k (t) N j+k+1 [ ] [ ] σ3 σ2 (1 − )β (N − i) pi,j,k−1 (t) + (1 + )µ(k + 1) pi,j,k+1 (t).

+ + + +

]

N

(A.40)

N

We now obtain the Fokker–Planck equation for the approximating diffusion process as described before. The first step is to transform the system to a quasi-continuous state space by scaling with N −1 , i.e., defining x = i/N, y = j/N, z = k/N, h = 1/N and the quasi-continuous density u(x, y, z , t) = pi,j,k (t). The resulting PDE reads

∂t u(x, y, z , t) [( ) ] y σ1 (1 − x)y = ∂x ζ x − ζ − ζ u(x, y, z , t) y+z N y+z [( ) ] y σ1 (1 − x)y + ∂y ζ + (µy − β x) + ζ u(x, y, z , t) y+z N y+z [( ) ] z σ1 (1 − x)z σ3 σ2 ζ + ∂z + (µz − β (1 − x)) + ζ + β (1 − x) + µz u(x, y, z , t) y+z N y+z N N [( ) ] 1 2 xz σ1 (1 − x)y + ∂ ζ + (1 + )ζ u(x, y, z , t) 2N x y+z N y+z [( ) ] xy σ1 (1 − x)y 1 2 ∂y ζ + (1 + )ζ + β x + µy u(x, y, z , t) + 2N y+z N y+z [( ) ] 1 2 xz σ1 (1 − x)z σ3 σ2 + ∂ ζ + (1 + )ζ + (1 − )β (1 − x) + (1 + )µz u(x, y, z , t) 2N z y+z N y+z N N [ [ ] ] σ1 (1 − x)y xz 1 1 u(x, y, z , t) + ∂x ∂z ζ u(x, y, z , t) . + ∂x ∂y −(1 + )ζ N N y+z N y+z

(A.41)

A.5.1. Deterministic model For N → ∞, we obtain a deterministic model, governed by the ODEs x˙ = −ζ x + ζ

y y+z

y˙ = β x − µy − ζ

= −ζ x y

y+z

z˙ = β (1 − x) − µz − ζ

5

z y+z

+ ζ (1 − x)

y y+z

8

9

10 11

(A.42)

y+z

7

,

, z

6

12

,

which is a neutral competition model since the selection terms vanish in the limit.

13

Proposition A.4. Let ϑ = (β − ζ )/µ > 0. Then, there is a line of stationary points for (A.42) in [0, 1] × R2+ given by

14

(x, y, z) = (x, ϑ x, ϑ (1 − x)),

x ∈ [0, 1].

15

The line of stationary points is transversally stable. The eigenvectors perpendicular to the line of stationary points (together with the eigenvalues) read

( X1 =

0 x

)

1−x

) ζ X2 = −β , β

16 17

(

,

λ1 = −µ,

λ2 = −ζ − β/ϑ.

The proof of this proposition is straightforward, along the lines of the proof of Proposition A.2. Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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1

A.5.2. Dimension reduction by time scale analysis

2

Step 1: To formulate the inner solution, we introduce local coordinates

3

4

5

6 7 8

9

10 11

12

x˜ = x,

y˜ = ε −1 (y + z − ϑ ),

z˜ = ε −1 (y − z − ϑ (2x − 1)),

ρ (x˜ , y˜ , z˜ , t) = u(x, y, z , t),

(A.43)

where ϑ = (β − ζ )/µ again and ε 2 = 1/N. Alternatively formulated, we have x = x˜ ,

y=

ε 2

(y˜ + z˜ ) + ϑ x,

z=

ε 2

(y˜ − z˜ ) + ϑ (1 − x).

y˜ can be thought of as measuring the deviation of the total amount of seeds from its deterministic value ϑ and z˜ as measuring the deviation of the allele ratio in seeds from the allele ratio in plants. Both are scaled by ε −1 so we expect them to be of order O(1). By transforming derivatives, we have

∂x =

∂ y˜ ∂ z˜ ∂ x˜ ∂x˜ + ∂y˜ + ∂z˜ = ∂x˜ − 2ε −1 ϑ∂z˜ , ∂x ∂x ∂x

∂y = ε −1 (∂y˜ + ∂z˜ ),

∂z = ε −1 (∂y˜ − ∂z˜ ).

Step 2: We now approximate ∂t ρ (x˜ , y˜ , z˜ , t) by transforming all terms on the r.h.s. of (A.41), using Taylor expansion, and ignoring terms of O(ε 3 ). We find

) ( ∂t ρ (x˜ , y˜ , z˜ , t) = L(0) + ε L(1) + ε 2 L(2) ρ (x˜ , y˜ , z˜ , t) + O(ε 3 ), (0)

with linear differential operators L , L

(1)

and L

(2)

(A.44)

that take the form

{ } {[ ( ) ] } 1 (z˜ + y˜ (1 − 2x˜ )) ρ L(0) ρ = ∂y˜ µ˜yρ + ∂z˜ µ˜z + ζ 1 + ϑ {[ ] } { } { } 2 2 β + 4ζ ϑ (1 + ϑ )x˜ (1 − x˜ ) ρ , + ∂y˜ βρ − ∂y˜ ∂z˜ 2β (1 − 2x˜ )ρ + ∂z˜ { {[ ] } } ζ L(1) ρ = − ∂x˜ σ1 ζ (1 − x˜ ) + σ3 β (1 − x˜ ) + σ2 µϑ (1 − x˜ ) ρ (z˜ + y˜ (1 − 2x˜ ))ρ + ∂y˜ 2ϑ {[ ( ) ] } ζ y˜ 1 − ∂z˜ 1+ (z˜ + y˜ (1 − 2x˜ )) − σ1 ζ (2(ϑ + 1)x˜ − 1) (1 − x˜ ) + σ3 β (1 − x˜ ) + σ2 µϑ (1 − x˜ ) ρ ϑ ϑ {[ { } ] } µ˜y µ ζ (1 + ϑ )(z˜ + y˜ (1 − 2x˜ )) − 4ζ (ϑ + 1)y˜ x˜ (1 − x˜ ) + y˜ ρ + ∂z˜2 − 2x˜ ζ (z˜ − y˜ )(1 + ϑ ) ρ + ∂y˜2 2 2 { } { } ( ) − ∂x˜ ∂z˜ 2(2ϑ + 1)ζ x˜ (1 − x˜ )ρ + ∂y˜ ∂z˜ µ˜z + ζ (1 + 1/ϑ )(z˜ + y˜ (1 − 2x˜ )) ρ , { } { } {[ { } ] } ζ 2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ... . . . . + ∂ ζ x (1 − x ) ρ + ∂ L(2) ρ = ∂x˜ y ( z + y (1 − 2 x )) − σ ζ x (1 − x ) ρ + ∂ ˜ ˜ 1 z y x˜ 2ϑ 2

(A.45)

(A.46) (A.47)

Before we proceed, we define the functions h0 (y˜ , z˜ ; x˜ ) h1 (y˜ , z˜ ; x˜ ) h2 (y˜ , z˜ ; x˜ )

g(x˜ )

−[˜z + (1 − 2x˜ )y˜ ] , µ + ζ (1 + 1/ϑ ) −[(z˜ + (1 − 2x˜ )y˜ )y˜ ] = , 2µ + ζ (1 + 1/ϑ ) ((1 − 2x˜ )y˜ )2 ζ (1 + 1/ϑ ) = 2(2µ + ζ (1 + 1/ϑ ))(µ + ζ (1 + 1/ϑ )) (1 − 2x˜ )y˜ z˜ µ z˜ 2 − − , (2µ + ζ (1 + 1/ϑ ))(µ + ζ (1 + 1/ϑ )) 2(µ + ζ (1 + 1/ϑ )) β + ζ ϑ (1 + ϑ ) = 4(x˜ (1 − x˜ )) . µ + ζ (1 + 1/ϑ ) =

Below we will use the fact that

13 14 15

16

L(0)+ h0 (y˜ , z˜ ; x˜ ) = (z˜ + (1 − 2x˜ )y˜ ),

(A.48)

L(0)+ h1 (y˜ , z˜ ; x˜ ) = (z˜ + (1 − 2x˜ )y˜ )y˜ ,

(A.49)

L(0)+ h2 (y˜ , z˜ ; x˜ ) + g(x˜ ) = (z˜ + (1 − 2x˜ )y˜ )z˜ .

(A.50)

That is, h0 and h1 are eigenfunctions of L(0)+ . Observing the system on slow time τ = ε 2 t = t /N, we see rapid dynamics for τ close to zero, τ ∈ O(1/N) to be precise. This is the new boundary layer. For τ ∈ O(1), only slow drift effects along the equilibrium line should remain. Being interested in the outer solution that develops on the slow time scale, we make the Ansatz

ρ (x˜ , y˜ , z˜ , t ; ε ) = ρ (0) (x˜ , y˜ , z˜ , ε2 t) + ερ (1) (x˜ , y˜ , z˜ , ε2 t) + ε2 ρ (2) (x˜ , y˜ , z˜ , ε2 t) + O(ε 3 ), Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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21

assuming ∂t ρ = O(ε 2 ) to focus on the outer solution and neglect the boundary layer. We plug this Ansatz into (A.44), compare same order terms on both sides and obtain L(0) ρ (0) = 0, Step 3: We have

∫

∫

L(0) ρ (1) = −L(1) ρ (0) ,

L(0) ρ (2) = ∂τ ρ (0) − L(1) ρ (1) − L(2) ρ (0) .

y˜ (z˜ + y˜ (1 − 2x˜ ))ρ (0) d(y˜ , z˜ ) =

L(2) ρ (0) d(y˜ , z˜ ) = ∂x˜

ζ

∫ [

∫

(0) L(0)+ h+ d(y˜ , z˜ ) = 0, and hence 1ρ

]

2ϑ

(A.51)

y˜ (z˜ + y˜ (1 − 2x˜ )) − σ1 ζ x˜ (1 − x˜ ) ρ (0) d(y˜ , z˜ ) + ∂x˜2

)

(

(

ζ x˜ (1 − x˜ )ρ (0) d(y˜ , z˜ )

5

∫

(A.52)

6

L1 ρ 1 d(v˜ , y˜ , z˜ ), and by applying (A.48)–(A.50), we

7 8

L1 ρ 1 d(v˜ , y˜ , z˜ ) = Ta + Tb + Tc ,

9

where (for Ta and Tc we use Eq. (A.49), for the Tc additionally (A.50))

−ζ

10

∫

(z˜ + (1 − 2x˜ )y˜ )y˜ ρ (0) d(v˜ , y˜ , z˜ ) = 0, 2ϑ 2 ((ϑ + 1)ζ + µϑ ) ( ) ∫ −ζ x˜ (1 − x˜ ) (−ζ σ1 ϑ − ζ σ2 + β (σ2 + σ3 )) Tb = ∂x˜ ρ (0) d(v˜ , y˜ , z˜ ) , (ϑ + 1)ζ + µϑ Ta = ∂x˜

ζ2 (z˜ + (1 − 2x˜ )y˜ )z˜ + (z˜ + (1 − 2x˜ )y˜ ) y˜ (1 − 2x˜ ) ρ (0) d(v˜ , y˜ , z˜ ) 4ϑ 2 ((1 + 1/ϑ ) ζ + µ) ( ) ∫ ζ2 − ∂x˜2 (2ϑ + 1)x˜ (1 − x˜ ) ρ (0) d(v˜ , y˜ , z˜ ) ϑ ((1 + 1/ϑ ) ζ + µ) ( 2 ) ∫ −ζ [(ϑ + 1)2 ζ + µϑ (1 + 2ϑ ) + β]˜x(1 − x˜ ) (0) ˜ ˜ ρ d( v ˜ , y , z ) . = ∂x˜2 (ϑ ((1 + 1/ϑ )ζ + µ))2

Tc = ∂x˜2

3

)

Step 4: Using the same procedure as in Appendix A.4, consider (A.34)–(A.35), to handle find

2

2

4

∫

= −σ1 ζ ∂x˜ x˜ (1 − x˜ ) f (τ , x˜ ) + ζ ∂x˜2 ζ x˜ (1 − x˜ )f (τ , x˜ ) .

∫

1

(

(A.53)

11

(A.54)

12

)

∫

13

14

(A.55)

With

15

16

∂τ f =

∫

L(1) ρ (1) d(v˜ , y˜ , z˜ ) +

∫

L(2) ρ (0) d(v˜ , y˜ , z˜ ),

17

we find (G = ζ /µ, K = β/ζ )

∂τ f (x˜ , τ ) =

18

− ζ [σ1 + (1 − 1/K )σ2 + σ3 ] ∂x˜ x˜ (1 − x˜ )f (x˜ , τ ) (1 + (1 − 1/K )G) { } ζ 2 ˜ ˜ ˜ + ∂ x (1 − x )f ( x , τ ) . (1 + (1 − 1/K )G)2 x˜

{

} 19

(A.56)

A.6. Logistic population dynamics and fluctuating seed bank

20

21

Let pi,j,k,l (t) = P(X1,t = i, X2,t = j, Yt = k, Zt = l) be the probability distribution of the resulting stochastic process. The corresponding master equation reads:

22 23

σ1 N −i−j σ4 (k + (1 − )l) [ ζ (i + (1 + )j) + γ N N N ] p˙ i,j,k,l (t) = − σ2 σ3 + µ(k + (1 + )l) + β (i + (1 − )j) pi,j (k, l, t) N

N

+ ζ (i − 1)pi−1,j,k,l (t) + (1 −

−σ1 N

)ζ (j − 1)pi,j−1,k,l (t)

+ γ (1 − (i + 1)/N − j/N) (k + 1)pi−1,j,k+1,l (t) σ4 + (1 − )γ (1 − i/N − (j + 1)/N) (l + 1)pi,j−1,k,l+1 (t) N

+ β i pi,j,k−1,l (t) + (1 −

σ3 N

(A.57)

24

)β j pi,j,k,l−1 (t)

+ µ(k + 1)pi,j,k+1,l (t) + (1 +

σ2 N

)µ(l + 1)pi,j,k,l+1 (t).

We transform the system to the quasi-continuous state space with rescaled parameters x1 = i/N, x2 = j/N, y = k/N, z = l/N, h = 1/N and quasi-continuous density u(x1 , x2 , y, z , t) = pi,j,k,l (t). After expanding about (x1 , x2 , y, z) in terms up to order O(ε 2 ), we have Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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a 4-dimensional Fokker–Planck equation again:

{[ ] } ζ x1 − γ (1 − x1 − x2 )y u {[ ] } σ1 σ4 + ∂x2 (1 + )ζ x2 − (1 − )γ (1 − x1 − x2 )z u N N {[ ] } + ∂y µy − β x1 + γ (1 − x1 − x2 )y u {[ ] } σ2 σ3 σ4 + ∂z (1 + )µz − (1 − )β x2 + (1 − )γ (1 − x1 − x2 )z u N N N {[ ] } 1 2 ζ x1 + γ (1 − x1 − x2 )y u + ∂ 2N x1 {[ ] } 1 2 σ1 σ4 + ∂x2 (1 + )ζ x2 + (1 − )γ (1 − x1 − x2 )z u 2N N N {[ ] } 1 2 µy + β x1 + γ (1 − x1 − x2 )y u + ∂y 2N {[ ] } 1 2 σ2 σ3 σ4 + (1 + ∂z )µz + (1 − )β x2 + (1 − )γ (1 − x1 − x2 )z u 2N N N N {[ ] } {[ ] } 1 σ4 1 − ∂x1 ∂y γ (1 − x1 − x2 )y u − ∂x2 ∂z (1 − )γ (1 − x1 − x2 )z u .

∂t u = ∂x1

2

N

3

N

(A.58)

N

The limiting deterministic system for N → ∞ has dynamics x˙1 = γ (1 − x1 − x2 )y − ζ x1 , x˙2 = γ (1 − x1 − x2 )z − ζ x2 ,

4

(A.59)

y˙ = β x1 − µy − γ (1 − x1 − x2 )y, z˙ = β x2 − µz − γ (1 − x1 − x2 )z .

5

As before, we can show that a line of stable equilibria exists: Proposition A.5. Let ϑ := (β − ζ )/µ > 0, κ := (γ ϑ − ζ )/γ ϑ ∈ [0, 1]. Then, there is a line of stationary points in [0, κ]2 × R2+ given by (x1 , x2 , y, z) = (x, κ − x, ϑ x, ϑ (κ − x)),

x ∈ [0, κ].

6

The line of stationary points is transversally stable (locally and globally).

7

Since the proof is similar to the proof of Proposition A.2, it is omitted here.

8

A.6.1. Perturbation approximation

9

Step 1: As before, new local variables for the boundary layer around the equilibrium line are defined: x1 = κ x˜ +

10

13

14 15 16 17 18

19

2

y = κ ϑ x˜ +

11

12

1

ε v˜ , 1 2

x2 = κ (1 − x˜ ) +

ε (y˜ + z˜ + ϑ v˜ ),

1 2

ε v˜ ,

z = κ ϑ (1 − x˜ ) +

(A.60) 1 2

ε (y˜ − z˜ + ϑ v˜ ),

where x1 − x2 + κ

, v˜ = ε −1 (x1 + x2 − κ ), 2κ y˜ = ε −1 (y + z − ϑ (x1 + x2 )), z˜ = ε −1 (y − z − ϑ (x1 − x2 )), x˜ =

and ε 2 = 1/N. The considerations for the global behaviour are the same as for model 2. We choose the same local variables for the boundary layer around the equilibrium line. Step 2: Next we transform (A.58) up to terms of O(ε 3 ) and higher. The resulting reparametrized Fokker–Planck equation in the local variables is

( ) ∂t ρ (x˜ , v˜ , y˜ , z˜ , t) = L(0) + ε L(1) + ε 2 L(0) ρ (x˜ , v˜ , y˜ , z˜ , t) + O(ε2 ),

(A.61)

with linear differential operators

] } {[ ] } {[ ζ L ρ = ∂v˜ − y˜ + γ ϑκ v˜ ρ + ∂y˜ (β + ζ ϑ )y˜ /ϑ − γ ϑ (1 − ϑ )κ v˜ ρ ϑ {[ ] } + ∂z˜ (β + ζ ϑ )z˜ /ϑ + γ ϑ (1 + ϑ )κ (1 − 2x˜ )v˜ ρ { } {[ ] } {[ ] } 2 2 2 + ∂v ζ κ ρ + ∂y˜ κ (β + ζ ϑ (1 + ϑ )) ρ + ∂z˜ κ (β + ζ ϑ (1 + ϑ )) ρ (0)

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

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{ } { } − ∂v˜ ∂y˜ ζ (1 + 2ϑ )κρ + ∂v˜ ∂z˜ ζ (1 + 2ϑ )κ (1 − 2x˜ )ρ { } − ∂y˜ ∂z˜ [2 (1 + ϑ )ϑζ + 2β] κ (1 − 2x˜ ) ρ ,

(A.62)

] } {[ ] } γϑ ζ z˜ + (1 − 2x˜ )v˜ ρ + ∂v˜ v˜ γ (ϑ v˜ + y˜ ) + (σ4 + σ1 )ζ κ (1 − x˜ ) ρ 2ϑκ 2 {[ ] } + ∂y˜ −˜v γ (1 + ϑ )(ϑ v˜ + y˜ ) + (σ2 + σ3 )βκ (1 − x˜ ) − (σ4 (1 + 1/ϑ ) + σ1 )(1 − x˜ )ζ ϑκ ρ {[ ] } + ∂z˜ −˜v γ (1 + ϑ )z˜ − (σ2 + σ3 )βκ (1 − x˜ ) + (σ4 (1 + 1/ϑ ) + σ1 )(1 − x˜ )ζ ϑκ ρ { } { } { } − ∂x˜ ∂v˜ ζ (1 − 2x˜ )ρ + ∂x˜ ∂y˜ ζ (1 − 2x˜ )(ϑ + 1/2)ρ − ∂x˜ ∂z˜ ζ (ϑ + 1/2)ρ { } { } { } { } { } { } + ∂v˜2 . . . + ∂y˜2 . . . + ∂z˜2 . . . + ∂v˜ ∂y˜ . . . + ∂v˜ ∂z˜ . . . + ∂y˜ ∂z˜ . . . , ] } } { } { { } { } {[ (σ 1 + σ 4 ) ζ ζ γ 2 (2) v˜ z˜ − (1 − x˜ ) ρ + ∂x˜ ρ + ∂v˜ . . . + ∂y˜ . . . + ∂z˜ . . . . L ρ = ∂x˜ 2κ 2 4κ

L(1) ρ =

− ∂x˜

{[

(A.63) (A.64)

ˆ : H2 → H0 in a similar way as above (Proposition A.3). Step 3: To handle the integrals below, we define an operator M : H2 → H0 resp. G Recall that we use (v˜ 2 , y˜ 2 , z˜ 2 , v˜ y˜ , v˜ z˜ , y˜ z˜ ) as the basis in H2 . The operator M has the representation ⎛ −2γ ϑκ ⎜ 0 ⎜ ⎜ 0 M=⎜ ⎜ 2ζ /ϑ ⎝ 0 0

0 0 2γ ϑ (1 + ϑ )κ 0 0

γ ϑ (1 + ϑ )κ ζ /ϑ

0 0

−2(β + ζ ϑ )/ϑ

−2(β + ζ ϑ )/ϑ 0

−2γ ϑ (1 + ϑ )κ (1 − 2x˜ ) 0

−γ ϑ (1 + ϑ )κ (1 − 2x˜ )

0 −γ ϑκ − (β + ζ ϑ )/ϑ 0 0

0 0 0 −γ ϑκ − (β + ζ ϑ )/ϑ

ζ /ϑ

0 0 0

ˆ x˜ ) = κ G(

2

⎞

⎟ ⎟ ⎟ ⎟. −γ ϑ (1 + ϑ )κ (1 − 2x˜ )⎟ ⎠ γ ϑ (1 + ϑ )κ

3

−2(β + ζ ϑ )/ϑ

ˆ x˜ ) h, we find If we define g via g(x˜ ) = G(

(

1

4

)

2ζ , 2(ϑ (ϑ + 1)ζ + β ), 2(ϑ (ϑ + 1)ζ + β ), −(1 + 2ϑ )ζ , (1 + 2ϑ )ζ (1 − 2x˜ ), −2(ϑ (1 + ϑ )ζ + β )(1 − 2x˜ ) .

5

As before, we employ time scale separation and focus on a solution evolving on the slow time τ = ε 2 t = t /N, using the Ansatz

6

ρ (x˜ , v˜ , y˜ , z˜ , t) = ρ (0) (x˜ , v˜ , y˜ , z˜ , ε2 t) + ερ (1) (x˜ , v˜ , y˜ , z˜ , ε2 t) + ε 2 ρ (2) (x˜ , v˜ , y˜ , z˜ , ε2 t) + O(ε 2 ).

7

+

If we define h0 by

8

] ζ β + ζϑ ˜ ˜ ˜ ˜ h+ = (1 − 2 x ) v ˜ , z + (1 − 2 x ) y + 0 2κ (β + ζ ϑ ) ζ [

9

we find

10

γϑ ζ L(0)+ h+ z˜ + (1 − 2x˜ )v˜ . 0 = 2ϑκ 2

11

By now, we have all ingredients together to go along the same route as in Appendix A.4. We start with

∫

L(2) ρ (0) d(v˜ , y˜ , z˜ ). If we use

( γ ) γ ζ (β + ζ ϑ − βϑ ) γ v˜ z˜ = L(0)+ M −1 v˜ z˜ + (1 − 2x˜ ), 2κ 2κ 2µ(β + γ ϑ 2 )

13

we obtain

14

γ ζ (β + ζ ϑ − βϑ ) (σ1 + σ4 )ζ (1 − x˜ ) (1 − 2x˜ ) − f 2µ(β + γ ϑ 2 ) 2 ∫ ∫ (1) (0) Step 4: Next, we turn to L(1) ρ (1) d(v˜ , y˜ , z˜ ) = − h+ d(v˜ , y˜ , z˜ ) with 0 L ρ

∫

L(2) ρ (0) d(v˜ , y˜ , z˜ ) = ∂x˜

h+ 0 =

{[

] }

+ ∂x˜2

{

ζ 4κ

} f

.

(A.65)

ζ (z˜ + (1 − 2x˜ )y˜ + (β + ζ ϑ )(1 − 2x˜ )v˜ /ζ ) . (2κ (β + ζ ϑ ))

We integrate by parts w.r.t. all derivatives ∂v˜ , ∂y˜ , ∂z˜ , and respectively move the derivatives ∂x˜ in front of the integral by means of the chain rule, so that we obtain

∫ −

12

(1) (0) h+ d(v˜ , y˜ , z˜ ) = ∂x˜ (Ta + Tb + Tc + Td,1 ) + ∂x˜2 Td,2 , 0 L ρ

Please cite this article in press as: Heinrich L., et al., Effects of population- and seed bank size fluctuations on neutral evolution and efficacy of natural selection. Theoretical Population Biology (2018), https://doi.org/10.1016/j.tpb.2018.05.003.

15

16

17

18 19

20

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with

( γ ϑ (β + ζ ϑ ) ρ (0) ζ ζ z˜ 2 γϑ (1 − 2x˜ )2 v˜ 2 + + (1 − 2x˜ )2 v˜ y˜ 2κ (β + ζ ϑ ) 2ζ 2ϑκ 2 ) ζ β + γ ϑ2 (1 − 2x˜ )v˜ z˜ + (1 − 2x˜ )y˜ z˜ d(v˜ , y˜ , z˜ ), + 2ϑκ 2ϑκ ( ∫ γ ϑ (2β + ζ ϑ − ζ ) γ ϑ (µ + ζ ) ζ ρ (0) (1 − 2x˜ )v˜ 2 + (1 − 2x˜ )v˜ y˜ 2κ (β + ζ ϑ ) ζ ζ ) β + ζ ϑ − γ ϑ (1 + ϑ )κ ζ + v˜ z˜ + y˜ z˜ d(v˜ , y˜ , z˜ ), ϑκ ϑκ ) ∫ ( β (σ1 + σ4 (1 − ζ /β ) − σ2 − σ3 ) 1 ζ (σ1 + σ4 ) (1 − x˜ ) − x˜ (1 − x˜ ) ρ (0) d(v˜ , y˜ , z˜ ), 2 β + ζϑ ) ∫ ( ζ (ζ − 2β ) (1 − 2x˜ )ρ (0) d(v˜ , y˜ , z˜ ), 2κ (β + ζ ϑ ) ) ∫ ( 2 −ζ (2x˜ (x˜ − 1) − ϑ ) − ζ β (4x˜ (1 − x˜ ) − 1) ρ (0) d(v˜ , y˜ , z˜ ). − 2κ (β + ζ ϑ ) ∫

2

Ta =

3

4

Tb =

5

6

Tc =

7

Td,1 =

8

Td,2 =

9

10

11

12

13

14

15

16

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

(A.66)

(A.67) (A.68) (A.69) (A.70)

ˆ In particular, the integrals in Ta and Tb can be transformed using the operators M and G,

) ∫ ( ζ β + ζϑ 2 ζ (β + ζ ϑ (1 + ϑ )) 2 Ta = (1 − 2x˜ ) − x˜ (1 − x˜ ) ρ (0) d(v˜ , y˜ , z˜ ), 2 κ (β + ζ ϑ ) 2 2(β + ζ ϑ ) ) ∫ ( ζ γ κ (β + ζ ϑ − βϑ ) (β + ζ ϑ ) Tb = 2β − ζ + (1 − 2x˜ ) ρ (0) d(v˜ , y˜ , z˜ ). 2 κ (β + ζ ϑ ) µ (β + γ ϑ 2 )

(A.71) (A.72)

Recall that in the present variant of the model ϑ = (β − ζ )/µ, G = ζ /µ and K = β/ζ . With

∂τ f =

∫

L ρ (1)

(1)

d(v˜ , y˜ , z˜ ) +

∫

L(2) ρ (0) d(v˜ , y˜ , z˜ ),

we find (G = ζ /µ, K = β/ζ )

∂τ f =

{ } { } ζ ζ (−σ1 − (1 − 1/K )σ4 − σ2 − σ3 ) 2 ˜ ˜ ˜ ˜ x (1 − x )f + x (1 − x )f . ∂ ∂ x (1 + (1 − 1/Y )G)2 x 1 + (1 − 1/K )G

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