Ancestral Lineages in Mutation Selection Equilibria with Moving Optimum

Abstract

Many populations can somehow adapt to rapid environmental changes. To understand this fast evolution, we investigate the genealogy of individuals inside those populations. More precisely, we use a deterministic model to describe the phenotypic density of a population under selection when the fitness optimum moves at constant speed. We study the inside dynamics of this population using the neutral fractions approach. We then define a Markov process characterizing the distribution of ancestral phenotypic lineages inside the equilibrium. This construction yields qualitative as well as quantitative properties on the phenotype of typical ancestors. In particular, we show that in asexual populations typical ancestors of present individuals carried traits much closer to the fitness optimum than most individuals alive at the same time. We also investigate more deeply the asymptotic regime of small mutation effects. In this regime, we obtain an explicit formula for the typical ancestral lineage using the description of the solutions of the Hamilton Jacobi equation as a minimizer of an optimization problem. In addition, we compare our deterministic results on lineages with the lineages of stochastic models.

Appendices

Appendices

A. The Regime of Small Variance: Further Developments

1.1 A.1. Derivation of the Hamilton-Jacobi Equation

In this section, we explain formally how to derive a Hamilton–Jacobi equation from the pulse characterization (7), in the regime of small mutations.

From the work of Bouin et al. (2020), we know that the regime of small mutation effects can be described by the following scaling factor:

$$\begin{aligned} \varepsilon := \sqrt{\sigma ^2 \frac{\alpha }{\beta }}, \text {with } \alpha := \mu '' (0)>0. \end{aligned}$$

(56)

When $\varepsilon \rightarrow 0$ our Eq. (7) converges to a Hamilton–Jacobi equation which allows us to use the rigorous results of Lorz et al. (2011). This limiting regime captures the weak selection regime, in the sense that either the variance vanishes, or the selection is weak compared to birth:

$$\begin{aligned} \sigma \ll 1 \quad \text { or } \quad \frac{\alpha }{\beta }\ll 1. \end{aligned}$$

We show here how the pulse F concentrates around a mean trait value, $z^*$, when $\sigma \rightarrow 0$.

Recall that F is a solution of the integro-differential equation (7), which we rewrite here:

$$\begin{aligned} \lambda _\sigma F_\sigma (z) - c F'_\sigma (z) + \mu (z) F_\sigma (z) = \dfrac{\beta }{\sigma } \int _{\mathbb {R}}K\left( \dfrac{z-z'}{\sigma }\right) F_\sigma (z')\, \text {d}z' . \end{aligned}$$

(57)

When the variance of the mutation kernel vanishes, that is $\sigma \rightarrow 0$, we may expect the function F to concentrate around a specific trait, guided by selection, see Barles et al. (2009) and Lorz et al. (2011) for instance. To capture this phenomenon, we use the following Hopf–Cole transform, identical to (17):

$$\begin{aligned} F_\sigma (z) = \exp \left( - \frac{U_\sigma (z)}{\sigma }\right) . \end{aligned}$$

(58)

The scaled quantity $U_\sigma$ then satisfies the following equation:

$$\begin{aligned} \lambda _\sigma + \frac{c}{\sigma } U'_\sigma (z) + \mu (z) = \dfrac{\beta }{\sigma } \int _{\mathbb {R}}K\left( \dfrac{z-z'}{\sigma }\right) \exp \left( \frac{U_\sigma (z)-U_\sigma (z')}{\sigma }\right) \, \text {d}z' . \end{aligned}$$

(59)

To avoid degeneration of terms in the equations when $\sigma \rightarrow 0$,

we rescale the speed of environmental change by $\sigma$, as in (16):

$$\begin{aligned} c := \sigma c'. \end{aligned}$$

(60)

In order to find the limit equation, we use the following Taylor expansion of the exponential term inside the integral:

$$\begin{aligned} \forall z'\in {\mathbb {R}}, \quad \exp \left( \frac{U_\sigma (z)-U_\sigma (z')}{\sigma }\right) \approx \exp \left( -\frac{(z-z')}{\sigma } \, U'_\sigma (z)\right) . \end{aligned}$$

Plugging this approximation into the integral term of (59), with an affine change of variable, we obtain formally when $\sigma \rightarrow 0$:

$$\begin{aligned} \lambda _0 + c' U'_0(z) + \mu (z) = \beta \int _{\mathbb {R}}K(y) \exp \Big ( y\, U'_0(z) \Big ) \, \text {d}y . \end{aligned}$$

In the following, we omit the index 0, as in (17). We then recover Eq. (43). Let us observe that the limit equation is well defined thanks to assumption (2) on the exponential decay of K. It also guarantees the finiteness of the Hamiltonian H in (17) for all $p \in {\mathbb {R}}$.

1.2 A.2. Lagrangian Point of View: Qualitative Formulas

The Hamilton–Jacobi Eq. (43) provides analytical formula as well as qualitative behavior on U and $\lambda$, which are expected to be a good approximation when $\sigma$ is small. Let $\mu _0:=\mu (0)$ be the minimum mortality rate.

We first start with the following formula on $\lambda$:

$$\begin{aligned} \lambda = \beta -\mu _0- \beta L \left( \frac{c'}{ \beta }\right) . \end{aligned}$$

(61)

where L is the Lagrangian associated with the Hamiltonian H and related to the mutation kernel K, see (44). The quantity $\beta -\mu _0$ corresponds to the intrinsic growth rate (fitness) of the population, while the additional quantity $\beta L( c'/ \beta )$ measures the lag load induced by the changing environment.

A short argument for (61) consists in assuming that the asymptotic behavior of $\Gamma$ stated in (48) holds true. Then, plugging this into (47), it prescribes the value of $\lambda$ such that U does not take infinite values:

$$\begin{aligned} \beta L\left( \frac{c'}{\beta }\right) - \beta + \mu (0) + \lambda = 0. \end{aligned}$$

Since $\mu (0)=\mu _0$, this formula coincides with (61).

The proof of this analytical formula relies on convex analysis methods (see Bouin et al. 2020). First, the function $p\mapsto c'p-\beta H(p)$ admits a maximum value: $\beta L(c'/\beta )$. Indeed, the functions H and L have reciprocal derivatives functions: $\partial _p H \circ \partial _v L = Id$. Adding this maximum value on each side of the Hamilton–Jacobi Eq. (43), we obtain

$$\begin{aligned} \mu (z)-\mu _0 + \left[ \lambda -\beta + \mu _0 + \beta L \left( \frac{c'}{ \beta }\right) \right] = \beta H( U'(z)) -c' U'(z) + \beta L \left( \frac{c'}{ \beta }\right) . \end{aligned}$$

(62)

We claim that on the left-hand side of this relationship, the term between brackets must vanish. Otherwise, it would lead to a contradiction as follows.

Indeed, the function ${\mathcal {C}}: p\mapsto \beta H(p) - c'p + \beta L(c'/\beta )$ is convex, nonnegative and reaches zero from the properties of the Hamiltonian H and the Lagrangian L. If the term between brackets does not vanish in (62), the function $z \mapsto {\mathcal {C}} \circ U'$ takes only (strictly) positive values. As a consequence, $U'$ only takes values in one of the two branches of the convex function ${\mathcal {C}}$, and on each of these branches, ${\mathcal {C}}$ is invertible. Therefore, for each $z \in {\bar{{\mathbb {R}}}}$, we can invert the relationship (62) to deduce the value taken by $U'(z)$. From (3), $\mu$ has the same infinite value at $\pm \infty$. Inverting (62) for $z = \pm \infty$ yields

$$\begin{aligned} \lim _{z \rightarrow + \infty } U'(z)= \lim _{z \rightarrow - \infty } U'(z). \end{aligned}$$

This is in contradiction with the assumption $U(\pm \infty )= + \infty$, or equivalently, $F(\pm \infty )=0$, i.e., the population density vanishes at infinity. Therefore, the term between brackets in (62) vanishes, exactly as in the desired formula (61).

As a side, $z=z^*$ leads to a formula that dictates the position of the dominant trait $z^*$:

$$\begin{aligned} \lambda + \mu (z^*) =\beta . \end{aligned}$$

Combined with (61), we find

$$\begin{aligned} \mu (z^*) =\mu _0 + \beta L \left( \frac{c'}{ \beta }\right) . \end{aligned}$$

(63)

As a matter of fact, those formulas are consistent with those in Bouin et al. (2020),

$$\begin{aligned} \lambda \approx \beta - \mu _0- \beta L\left( \frac{c'}{ \beta } \right) + O(\varepsilon ).\end{aligned}$$

They further show how to obtain more accurate expansions, up to an arbitrary order (in $\varepsilon$ defined in (56)), and compute explicitly the following corrector terms.

In addition, we know from (8) that $\lambda$ is a measure of the size of the population. Thus, this size should remain positive which gives us a critical threshold for the speed $c'$ so that the population does not go extinct:

$$\begin{aligned} c_{max} = \sigma \beta \, L^{-1} \left( \frac{\beta -\mu _0}{\beta }\right) . \end{aligned}$$

Moreover, we can check by integrating (7) that

$$\begin{aligned} \lambda = \int _{\mathbb {R}}\dfrac{\mu (x)F(x)}{\int _{\mathbb {R}}F(x')\text {d}x'}\text {d}x \end{aligned}$$

which corresponds to the mean fitness of the population, or its mean intrinsic rate of increase.

1.3 A.3. Long Time Behavior of $\Gamma _z$

In this part, we show a somehow stronger version of Proposition 1.2: the typical ancestral lineages $\Gamma _z$ converges toward 0, the optimal trait, when s goes to infinity. In the regime of small variance, this result concerns the ODE (20).

Let $z_0$ be a steady state of (20). Then,

$$\begin{aligned} 0 = -c'+\beta \, \partial _p H(U'(z_0)). \end{aligned}$$

(64)

Let $p_0:= U'(z_0)$. Then, from (64), $p_0$ is a critical point of the function $p\mapsto - c' \, p + \beta \, H(p)$. Since this is a convex function, we deduce that

$$\begin{aligned} p_0 \in {{\,\mathrm{argmin}\,}}_{p} \Big (-c' p + \beta \, H(p)\Big ). \end{aligned}$$

Therefore, by definition of the Lagrangian function L in (44)

$$\begin{aligned} - \beta L\left( \frac{c'}{\beta }\right) = -c'p_0 +\beta \, H(p_0). \end{aligned}$$

(65)

On the other hand, by evaluating the Hamilton Jacobi Eq. (43) in $z = z_0$, one finds

$$\begin{aligned} \mu (z_0) + \lambda = \beta +\beta \, H(p_0) -c'p_0. \end{aligned}$$

Plugging in (65)

$$\begin{aligned} \mu (z_0) = \beta - \lambda +\beta L\left( \frac{c'}{\beta }\right) . \end{aligned}$$

(66)

Finally, thanks to the formula (61) for $\lambda$, we get

$$\begin{aligned} \mu (z_0) = \mu _0. \end{aligned}$$

(67)

According to our assumptions on $\mu$ stated in (3), 0 is the only global point of minimum of the convex function $\mu$, therefore $z_0=0$. We conclude that 0 is the unique steady state of the ODE (20). Moreover, it is established in Lorz et al. (2011) that under the assumptions (2), U is a convex function. Therefore, the flow in (20) is an increasing function, and it is straightforward that $\Gamma _z$ converges to this unique steady state of (20):

$$\begin{aligned} \Gamma _z(s) \xrightarrow [s \rightarrow \infty ]{} 0. \end{aligned}$$

1.4 A.4. Approximation of the Mean of the Ancestral Lineages

Working further on the Hamilton Jacobi Eq. (43), we can make an analytic approximation of $\Gamma _z$, the typical lineage. Let us first differentiate (43) with respect to z, and then divide by $U''$ on each side. We obtain for all $z\in {\mathbb {R}}$,

$$\begin{aligned} -c'+ \beta \, \partial _p H \big ( U' (z)\big ) = \dfrac{\mu '(z)}{U'' (z)}. \end{aligned}$$

(68)

Now, our idea is to link $U''(z)$ with the variance of F. Using results from Bouin et al. (2020), we obtain the following approximation of the variance of F at the leading order in $\sigma$:

$$\begin{aligned} \mathrm {Var}(F) = \frac{\sigma }{U''(z^*)} + o(\sigma ), \end{aligned}$$

(69)

where $z^*$ is the dominant trait in our population. This approximation comes from a Taylor expansion (with Laplace’s method) of the integrals defining the variance:

$$\begin{aligned} \mathrm {Var}(F) = \frac{\displaystyle \int _{{\mathbb {R}}} (z-z^*)^2 \exp \left( - \frac{U(z)}{\sigma } \right) \text {d}z}{\displaystyle \int _{{\mathbb {R}}} \exp \left( - \frac{U(z)}{\sigma }\right) \text {d}z }. \end{aligned}$$

In addition, we make the following rough approximation, valid if z is close to $z^*$: $U''(z)\approx U''(z^*)$. Plugging these approximations into (68), we find that

$$\begin{aligned} -c'+ \beta \, \partial _p H \big ( U'(z)\big ) \approx \dfrac{\mu '(z)\mathrm {Var}(F)}{\sigma } \ \hbox { for } \ z \ \hbox { close to } \ z^*. \end{aligned}$$

The ODE (20) satisfied by $\Gamma _z$ then becomes:

$$\begin{aligned} {{\dot{\Gamma }}}_z(s)&= - \dfrac{\mathrm {Var}(F)}{\sigma } \mu '\big (\Gamma _z(s)\big ) ,\nonumber \\ \Gamma _z(0)&=z . \end{aligned}$$

(70)

In particular, if the selection function is quadratic, $\mu (z)= \mu _0 + z^2/2$, the solution of (70) is precisely

$$\begin{aligned} \Gamma _z(s) = z \exp \left( -\dfrac{\mathrm {Var}(F)}{\sigma }\,s\right) . \end{aligned}$$

(71)

Moreover, on this case, we can derive an explicit formula for the variance from the HJ equation.

More precisely, we compute $U''(z^*)$ from (43) which provides the following formula from (69):

$$\begin{aligned} \mathrm {Var}(F)= -\frac{c' \sigma }{\mu '(z^*)}. \end{aligned}$$

Since $\mu$ is quadratic and $z^*$ satisfies (63), we obtain

$$\begin{aligned} \mathrm {Var}(F) = \dfrac{c'\sigma }{\sqrt{2\beta }\sqrt{L\left( \dfrac{c'}{\beta }\right) }}. \end{aligned}$$

(72)

Finally, we get the following approximation for the mean trajectories of the ancestral process:

$$\begin{aligned} \Gamma _z(s) = z \exp \left( -\frac{c'\,s}{\sqrt{2\beta }\sqrt{L\left( c'/\beta \right) }} \right) . \end{aligned}$$

(73)

This formula is explicit, since it only depends on the mutation kernel through the Lagrangian L. However, this approximation only applies to the case of a quadratic selection and for traits close to the dominant trait. But we can check from numerical simulations, that this approximation is quite robust (see Fig. 12).

B. Numerical Methods

1.1 B.1. Ancestral Lineages in the Stochastic Model

We detail in this section the numerical simulations how we deal with the simulations of the individual based model and the lineages. The population at each time t is made of a number N(t) of alive individuals. Let us consider an individual, denoted i, that is alive at time t, with $1\leqslant i \leqslant N(t)$. It has the trait $z_i\in {\mathbb {R}}$. The first event for this individual is one of the following:

$\triangleright$:

Birth of a descendant: it happens at an uniform rate among individuals $r_B^i= \beta$.

$\triangleright$:

Death of the individual i: it is decomposed in two separate events:

(a):

Death by selection The individual may die because its phenotype is ill-adapted in the phenotype landscape. This happens at the rate:

$$\begin{aligned} r_{Ds}^i = \mu (z_i-ct). \end{aligned}$$

(b):

Death by competition Alternatively, an individual may die because of the density dependence in the population, at a rate that that depends on the total size of population at time t, and on the carrying capacity N

$$\begin{aligned} r_{Ddd} = \displaystyle \sum _{j=1}^{N(t)}\frac{1}{N}=\frac{N(t)}{N}. \end{aligned}$$

Next event: incrementation of the time step The time step is the smallest time for all individuals to go through one of the previous steps. Thanks to the Markov property, each event occurs following an exponential law of parameter:

$$\begin{aligned} dt \sim \min _{1 \leqslant i \leqslant N(t) }{\mathcal {E}}(r_B^i+r_{Ds}^i + r_{Ddd}). \end{aligned}$$

(B.1)

By the memory loss property of the exponential distribution, dt also can be drawn from an exponential law which rate is the sum of the rates of all the independent events:

$$\begin{aligned} dt \sim {\mathcal {E}}\left( \displaystyle \sum _i (r_B^i+r_{Ds}^i + r_{Ddd})\right) . \end{aligned}$$

Update of the population: Once the next event is decided, according to the law (B.1), the population at time $t+dt$ is deduced by either adding the individual that was born ($N(t+dt)=N(t)+1$) or subtracting the one that died ($N(t+dt)=N(t)-1$). In the case of a birth event, the trait of the offspring is drawn according to the operator ${\mathcal {B}}$ in (1):

$$\begin{aligned} z_\text {offspring} = z_\text {parent} + \sigma dK. \end{aligned}$$

We repeat all the steps until reaching the final time of simulation. Numerically, this model has a very high computational cost, because it needs a relatively high number of individuals to approach the deterministic model given by (4). As a consequence, we performed the simulations using an approximating model, by first fixing dt to a small but deterministic value. Then, for each individual, we draw a time of birth following the law ${\mathcal {E}}(\beta )$ and a time of death following the law ${\mathcal {E}}(\mu (z_i)+N(t)/N)$. Then, we simply count which individuals led to a reproduction event and which died on the time-window $\left[ t,t+dt\right]$. This amounts to the supposition that on this time interval, individuals cannot reproduce more than once.

Finally, to follow the lineage of individuals, we create a huge matrix at the start of the simulation. We will stock the lineage of every individual in this matrix, filling it progressively. Every time an individual appears, its lineage is similar to the one of its parent, translated by one generation. The numerical procedure works as described in Fig. 5, where different columns correspond to different individuals, and each line corresponds to a new generation.

This algorithm led to Figs. 6 and 4 with the following parameters:

$$\begin{aligned} \alpha = 2, \quad \beta = 2, \quad \sigma = 0.1, \quad N=20000, \quad c=0.2\, . \end{aligned}$$

Fig. 5

Keeping track of the lineages: a schematic representation of the ancestral lineages algorithm described in Appendix B.1 (Color figure online)

Fig. 6

Population density profile F in the moving frame obtained from deterministic model (4) (plain green curve) and from the IBM model with a number of individuals of order $2\cdot 10^4$ (blue histogram). The selection function $\mu$ is quadratic with minimum at $z=0$ (red dashed curve). The value $z^*$ corresponds to the mean phenotypic trait of the deterministic density (Color figure online)

1.2 B.2. Mean Traits Along the Lineages

We extend the simulations of Fig. 4 to different scenarios corresponding to various mutational variances $\sigma$ and speeds c. More precisely, for two different couples of mutational variance and speed, we compare the mean and the variance of the stochastic lineages with the mean and the variance of the ancestral process defined by our PDE model. We show that our deterministic model provides agreed with the individual based model. In addition, we see that the variance of the ancestral process increases with the mutational variance $\sigma$, while it slightly increases with the speed c (Fig. 7).

Fig. 7

Mean (left) and variance (right) of the ancestral process $Y_s$ for different set of parameters: red curves correspond to the model (12) and the black dashed curves correspond to the IBM model averaged over 50 replicates and the gray regions correspond to the $5\%$ and $95\%$ confident interval. On the left panel, the horizontal blue lines are the optimal trait in the moving frame, 0. The green lines are the dominant trait of the equilibrium F, denoted $z^*$. On the right panel, the horizontal cyan line corresponds to the asymptotic variance given by the deterministic model see Proposition 1.2 (Color figure online)

1.3 B.3. Coalescent Time

We first investigate the time $T_2$ before two individuals lineages meet in the past. More precisely, for each individual in the population, we compute the minimal time such that its lineage coalesces with an other lineage in the population. From Fig. 8, we can observe that after a time delay $T_0=1/(\sigma \sqrt{\alpha \beta })$, the time $T_2$ is exponentially distributed. The exponential distribution of $T_2$ is apparent from the inset in Fig. 8 which shows the complementary of the cumulative distribution function ${\mathbf {P}}(T_2>t)$.

Fig. 8

Distribution of the time $T_2$ to the most recent common ancestor of two individuals for different speed c a $c=0.1$ and b $c=0.2$) and typical size N of the population. The inset shows the complementary of the cumulative distribution function ${\mathbf {P}}(T_2>t)$ in the semilog coordinates. An exponential $\exp (-t/T_0)$ is indicated as a black dashed line. Different line style and color corresponds to $N=10^4$ (plain blue), $N=10^5$ (dashed red) and $N=10^6$ (dot-dashed orange). The mutational variance is fixed to $\sigma ^2=0.1$ and $\alpha =\beta =2$ (Color figure online)

The time $T_2$ is different from the time ${\widetilde{T}}_2$ before the lineages of two individuals sampled randomly in the population coalesce (see Fig. 9).

Fig. 9

Distribution of the time ${\widetilde{T}}_2$ to the most recent common ancestor of two individuals sampled randomly in the population for different speed c and typical size N of the population. The inset shows the cumulative distribution function ${\mathbf {P}}(T_2\leqslant t)$. Different line style and color corresponds to $N=10^4$ (plain blue), $N=10^5$ (dashed red) and $N=10^6$ (dot-dashed orange). The mutational variance is fixed to $\sigma ^2=0.1$ and $\alpha =\beta =2$ (Color figure online)

1.4 B.4. The PDE Approach of Sect. 1.4

In this section, we compare the realizations of the IBM model with the ancestral process $Y_s$, obtained by the simulation of the fundamental solution defined by (21). As mentioned in Sect. 1.4, the distribution $w^z(s,y)$ of the ancestral process $Y_s$ starting from a trait z is given by

$$\begin{aligned} w^z(s,y) = \dfrac{\upsilon ^y(-s,y)}{F(z)} \end{aligned}$$

where $\upsilon ^z$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t \upsilon ^y(t,z) &{} = {\mathcal {L}}(\upsilon ^y(t,\cdot ))(z),\\ \upsilon ^y(0,z) &{} = \delta (z-y)F(z). \end{array}\right. \end{aligned}$$

(B.2)

In order to solve numerically this equation, we replace the Dirac $\delta (z-y)$ with the characteristic function of a small interval around y of the form $[y-\text {d}y,y+\text {d}y]$ as pictured in Fig. 1. More precisely, we solve Eq. (B.2) on a finite interval of the form $[z_{min},z_{max}]$ with $z_{min} = -z_{max} = - 10$, and we add Dirichlet boundary condition. For the initial conditions, we evenly decompose the interval $[z_{min},z_{max}]$ into $n=200$ intervals $I_k$ of size $\text {d}y = (z_{max}-z_{min})/n$ and centered around $y_k$. The numerical solution of $\upsilon ^k$ satisfying (B.2) starting from $\upsilon ^k_0(z) = {\mathbf {1}}_{I_k}(z)$ is obtain using a semi-explicit Euler scheme. The advantage of the decomposition is that $\sum _k \upsilon ^k(t,z) = F(z)$ for all $t>0$ and $z\in [z_{min,z_{max}}].$ Then, the numerical distribution $w^i(s)$ of $Y_s$ starting from the trait $z_i$ is given by $w^i(s,y_k) = \dfrac{\upsilon ^k(-s,z_i)}{F(z_i)}$ (see video and Fig. 10). In our simulation, we look at the particular point $z^*$ which corresponds to the dominant trait of the population, that is the mean of F.

As expected, we observe a good fit between the distribution of the ancestral lineage of the IBM model and the distribution of the ancestral process (see Fig. 10 and video Fig. 11). In particular, the means coincide and each ancestral lineage of the IBM model lies in the region where the ancestral process is the most likely to be (blue region in Fig. 1).

Fig. 10

Ancestral lineages over time of individuals with the dominant trait $z^*$: distribution of the ancestral process $Y_s$ (blue region) and the mean of the ancestral lineages of one realization of the IBM model (dashed black curve). The plain black curve and blue curve correspond to the confidence interval at $5\%$ around the mean for the IBM lineages (black) and the ancestral process (blue). The mean of the ancestral lineages corresponds to the dashed black curve, and the mean of the ancestral process $E_z(Y_s)$ is represented by the plain red curve. The gray and green dot-dashed curves represent, respectively, the optimal and the dominant trait (Color figure online)

Fig. 11

Ancestral lineages distribution over time of individuals with the dominant trait $z^*$. DynlineagePDEvsIBM.mp4 (Color figure online)

1.5 B.5. Deterministic Approximation of the Ancestral Lineage

In this section, we aim to compare the diffusive approximation stated in Sect. 4 and the Hamilton–Jacobi approximation stated in Sect. 5.1 with the ancestral lineage defined in Theorem 1.1. We compute numerically the mean of the ancestral lineage ${\mathbf {E}}_z(Y_s)$ using the fraction approach detailed in the above section, and we compare it to the formula (42) and to the solution of the ODE (46) (Fig. 12).

Fig. 12

Mean trait of the ancestral lineage along time (plain curve), diffusive approximation (circle marked curve) and the Hamilton Jacobi approximation (diamond marked curve) in the semilog representation. The color corresponds to different parameters: blue curve ($\sigma =0.05$, $c=0.025$), red curve ($\sigma =0.05$, $c=0.05$), orange curve ($\sigma =0.1$, $c=0.05$), purple curve ($\sigma =0.1$, $c=0.1$) (Color figure online)