Evolutionary branching via replicator-mutator equations

Matthieu Alfaro and Mario Veruete IMAG, Université de Montpellier, CC051, 34095, Montpellier, France matthieu.alfaro@umontpellier.fr, mario.veruete@umontpellier.fr

Abstract.

We consider a class of non-local reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schrödinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur.

Key words and phrases:

Evolutionary genetics, dynamics of adaptation, branching phenomena, long time behaviour, Schrödinger eigenelements

2010 Mathematics Subject Classification:

92B05, 92D15, 35K15, 45K05

1. Introduction

In this paper we first study the existence, uniqueness and long time behaviour of solutions $u = u (t, x)$ , $t > 0$ , $x \in R$ , to the integro-differential Cauchy problem

(1)

which serves as a model for the dynamics of adaptation, and where $W$ is a confining fitness function (see below for details). Next, we enquire on the possibility, depending on the function $W$ and the parameter $σ > 0$ , for a solution to split from uni-modal to multi-modal shape, thus reproducing evolutionary branching.

The above equation is referred to as a replicator-mutator model. This type of model has found applications in different fields such as economics and biology [25], [4]. In the field of evolutionary genetics, a free spatial version of equation (1) was introduced by Tsimring, Levine and Kessler in [40], where they propose a mean-field theory for the evolution of RNA virus population. Without mutations, and under the constraint of constant mass $\int_{R} u (t, x) d x = 1$ , the dynamics is given by

\frac{\partial u}{\partial t} = u (W (x) - \int_{R} W (y) u (t, y) d y),

(2)

with $W (x) = x$ in [40]. In this context, $u (t, x)$ represents the density of a population (at time $t$ and per unit of phenotypic trait) on a one-dimensional phenotypic trait space. The function $W (x)$ represents the fitness of the phenotype $x$ and models the individual reproductive success; thus the non-local term

¯ ¯ ¯ u (t) := \int_{R} W (y) u (t, y) d y

stands for the mean fitness at time $t$ .

As a first step to take into account evolutionary phenomena, mutations are modelled by the local diffusion operator $σ^{2} \partial_{x}^{2}$ , where $σ^{2}$ is the mutation rate, so that equation (2) is transferred into (1). We refer to the recent paper [41] for a rigorous derivation of the replicator-mutator problem (1) from individual based models.

Equation (1) is supplemented with a non-negative and bounded, initial data $u_{0} (\cdot) \geq 0$ such that $\int_{R} u_{0} (x) d x = 1$ , so that, formally, $\int_{R} u (t, x) d x = 1$ for later times. Indeed, integrating formally (1) over $x \in R$ , the total mass

m (t) := \int_{R} u (t, x) d x

solves the initial value problem

\frac{d}{d t} m (t) = (1 - m (t)) ¯ ¯ ¯ u (t), m (0) = 1.

Hence, by Gronwall’s lemma, $m (t) = 1$ , as long as $¯ ¯ ¯ u (t)$ is meaningful.

The case of linear fitness function, $W (x) = x$ , was the first introduced in [40], but little was known concerning existence and behaviours of solutions. Let us here mention the main result of Biktashev [5]: for compactly supported initial data, solutions converge, as $t$ goes to infinity, to a Gaussian profile, where the convergence is understood in terms of the moments of $u (t, \cdot)$ . In a recent paper [2], Alfaro and Carles proved that, thanks to a tricky change of unknown based on the Avron-Herbst formula (coming from quantum mechanics), equation (1) can be reduced to the heat equation. This enables to compute the solution explicitly and describe contrasted behaviours depending on the tails of the initial datum: either the solution is global and tends, as $t$ tends to infinity, to a Gaussian profile which is centred around $x (t) \sim t^{2}$ (acceleration) and is flattening (extinction in infinite horizon), or the solution becomes extinct in finite time (or even immediately) thus contradicting the conservation of the mass, previously formally observed.

For quadratic fitness functions, $W (x) = \pm x^{2}$ , it turns out that the equation can again be reduced to the heat equation [3], up to an additional use of the generalized lens transform of the Schrödinger equation. In the case $W (x) = x^{2}$ , for any initial data, there is extinction at a finite time which is always bounded from above by $T^{*} = \frac{π}{4 σ}$ . Roughly speaking, both the right and left tails quickly enlarge, so that, in order to conserve the mass, the central part is quickly decreasing. Then the non-local mean fitness term $\int_{R} y^{2} u (t, y) d y$ becomes infinite very quickly and equation (1) becomes meaningless (extinction). On the other hand, when $W (x) = - x^{2}$ , for any initial data, the solution is global and tends, as $t$ tends to infinity, to an universal stationary Gaussian profile.

The aforementioned cases $W (x) = x$ and $W (x) = x^{2}$ share the property of being unbounded from above, meaning that some phenotypes are infinitely well-adapted. This unlimited growth rate of $u (t, x)$ in (1) yields rich mathematical behaviours (acceleration, extinction) but is not admissible for biological applications. To deal with such a problem, for the linear fitness case, some works consider a “cut-off version” of (1) at large $x$ [40], [35], [37], or provide a proper stochastic treatment for large phenotypic trait region [34].

On the other hand, $W (x) = - x^{2}$ is referred to as a confining fitness function, typically preventing extinction phenomena. However, it does not suffice to take into account more realistic cases for which fitness functions are defined by a linear combination of two components (e.g. birth and death rates), each maximized by different optimal values of the underlying trait, a typical case being $W (x) = x^{2} - x^{4}$ .

Our main goal is thus to provide a rigorous treatment of the Cauchy problem (1) when the fitness function $W$ is confining. For a relatively large class of such fitness functions, we prove well-posedness, and show that the solution of (1) converges to the principal eigenfunction (or ground state) of the underlying Schrödinger operator divided by its mass. This requires rather non-standard estimates on the eigenelements. Also, from a modelling perspective, this enables to reproduce evolutionary branching, consisting of the spontaneous splitting from uni-modal to multi-modal distribution of the trait.

Such splitting phenomena have long been discussed and analysed in different frameworks, see e.g. [29] via Hamilton-Jacobi technics, [42] within finite populations, or [27] for a Lotka-Volterra system in a bounded domain. In a replicator-mutator context, let us notice that, while branching in (1) is mainly induced by the fitness function, it was recently obtained in [18] through different means. Precisely, the authors study the case of linear fitness $W (x) = x$ but non-local diffusion $J * u - u$ (mutation kernel), namely

\partial_{t} u = J * u - u + u (x - \int_{R} y u (t, y) d y) .

Their approach [30], [18] is based on Cumulant Generating Functions (CGF): it turns out that the CGF satisfies a first order non-local partial differential equation that can be explicitly solved, thus giving access to many informations such as mean trait, variance, position of the leading edge. When a purely deleterious mutation kernel $J$ balances the infinite growth rate of $W (x) = x$ , they reveal some branching scenarios.

The paper is organized as follows. In Section 2 we present the underlying linear material. In Section 3 we prove the well-possessedness of the Cauchy problem associated to (1). We also provide an explicit expression of the solution and studies its long time behaviour. In Section 4 we discuss, through rigorous details or numerical explorations, the conditions on the shape of the fitness function $W$ and on the mutation parameter $σ > 0$ for branching phenomenon to occur. Finally, we briefly conclude in Section 5.

2. Some spectral properties

In this section, we present some linear material. We first quote some very classical results [39], [33], [1], [21], [22], [20], [15] for Schrödinger operators, and then prove less standard estimates on the eigenfunctions, which are crucial for later analysis.

2.1. Confining fitness functions and eigenvalues properties

Confining fitness functions tend to $- \infty$ at infinity. In quantum mechanics, this corresponds to potentials describing the evolution of quantum particles subject to an external field force that prevents them from escaping to infinity, that is, particles have a high probability of presence in a bounded spatial region.

Assumption 1 (Confining fitness function).

The fitness function $W$ is continuous and confining, that is

lim | x | \to \infty W (x) = - \infty .

Proposition 2.1 (Spectral basis).

Let $W$ satisfy Assumption 1. Then the operator

H := - σ^{2} \frac{d^{2}}{d x^{2}} - W (x)

(3)

is essentially self-adjoint on $C_{c}^{\infty} (R)$ , and has discrete spectrum: there exists an orthonormal basis ${ϕ_{k}}_{k \in N}$ of $L^{2} (R)$ consisting of eigenfunctions of $H$

H ϕ_{k} = λ_{k} ϕ_{k}, ∥ ϕ_{k} ∥_{L^{2} (R)} = 1,

with corresponding eigenvalues

λ_{0} < λ_{1} \leq λ_{2} \leq \dots \leq λ_{k} \to + \infty,

of finite multiplicity.

Remark 1.

Proposition 2.1 is a classical result [33], [38, Chapter 3, Theorem 1.4 and Theorem 1.6] for Schrödinger operators with confining potential $V_{c o n f} (x) = - W (x)$ . In this paper, the minus sign is due to the biological interpretation of the fitness function $W$ .

In the quantum mechanics terminology, $ϕ_{0}$ is known as the ground state, corresponding to the bound-state of minimal energy $λ_{0}$ . In this paper we refer to the couple $(ϕ_{0}, λ_{0})$ indistinctly as ground state/ground state energy or as principal eigenfunction/principal eigenvalue.

The principal eigenvalue $λ_{0}$ can be characterised by the variational formulation

λ_{0} = inf {E (u) : u \in C_{c}^{\infty} (R), {∥ u ∥}_{L^{2} (R)} = 1},

(4)

where $E$ is the energy functional given by

E (u) = σ^{2} \int_{R} {∣ ∣ ∣ \frac{\partial u}{\partial x} ∣ ∣ ∣}^{2} d x + \int_{R} - W (x) {| u (x) |}^{2} d x .

Using concentrated test functions, the above formula enables to understand the behaviour of the principal eigenvalue $λ_{0} = λ_{0} (σ)$ as the mutation rate $σ$ tends to 0. The following will be used in Section 4 to prove some branching phenomena.

Proposition 2.2 (Asymptotics for $λ_{0} (σ)$ as $σ \to 0$ ).

Let $W$ satisfy Assumption 1. Assume that $W$ reaches a global maximum $M$ at $x = α$ . Then ${lim}_{σ \to 0^{+}} λ_{0} (σ) = - M$ .

Proof.

For the convenience of the reader, we give the proof of this standard fact. Let $p$ be a smooth, non-negative, and compactly supported in $[- 1, 1]$ function with $∥ p ∥_{L^{2} (R)} = 1$ . We define the test function

p_{σ} (x) := \frac{1}{σ^{1 / 4}} p (\frac{x - α}{\sqrt{σ}}) .

From the variational formula (4), we have

- M \leq λ_{0} (σ) \leq σ^{2} \int_{R} {| \partial_{x} p_{σ} (x) |}^{2} d x + \int_{R} - W (x) {| p_{σ} (x) |}^{2} d x .

The first integral in the right hand side is given by

σ^{2} \int_{R} {| \partial_{x} p_{σ} (x) |}^{2} d x = σ {∥ ∥ p^{'} ∥ ∥}_{L^{2} (R)} \to 0,

as $σ \to 0^{+}$ . The second integral gives

\int_{R} - W (x) {| p_{σ} (x) |}^{2} d x = \int_{R} - W (α + \sqrt{σ} y) p (y)^{2} d y

which, by the $L^{1}$ -dominated convergence theorem tends to $- M {∥ p ∥}_{L^{2} (R)} = - M$ as $σ \to 0^{+}$ . ∎

In the subsequent sections, we will quote results on the spectral properties of Schrödinger operators, in particular an asymptotics for the eigenvalues $λ_{k}$ as $k \to + \infty$ . As far as we know, the available results require to assume that the fitness $W$ is polynomial.

Assumption 2 (Polynomial confining fitness function).

The fitness function $W$ is a real polynomial of degree $2 s$ :

W (x) = - x^{2 s} + 2 s - 1 \sum k = 0 w_{k} x^{k},

for some integer $s \geq 1$ and some real numbers $w_{k}$ , $0 \leq k \leq 2 s - 1$ .

Under Assumption 2, elliptic regularity theory insures that the eigenfunctions are infinitely differentiable. Furthermore, all the derivatives of each eigenfunction are square-integrable [17]. Notice that it is also known that all eigenfunctions actually belong to the Schwartz space $S (R)$ .

Proposition 2.3 (Asymptotics for eigenvalues).

Let $W$ satisfy Assumption 2. Then all eigenvalues of $H$ are simple and

λ_{k} \sim {C_{s, σ} k}^{\frac{2 s}{s + 1}} as k \to + \infty,

(5)

where $C_{s, σ} := {(\frac{σ \sqrt{π} Γ (\frac{3}{2} + \frac{1}{2 s})}{Γ (1 + \frac{1}{2 s})})}^{\frac{2 s}{s + 1}}$ , with $Γ (z) = \int R^{+} t^{z - 1} e^{- t} d t$ being the gamma function.

We refer to [39], [15] and the references therein for more details on the above asymptotic formula. Furthermore, in the case of a symmetric fitness $W (- x) = W (x)$ , the simplicity of eigenvalues enforce all eigenfunctions to be even or odd. In particular the principal eigenfunction $ϕ_{0}$ (ground state) is even since it is known to have constant sign.

Remark 2.

Assume that $W$ is such that $P - c \leq W \leq P + c$ for some polynomial $P$ as in Assumption 2 and some constant $c > 0$ . From Courant-Fisher’s theorem, that is the variational characterization of the eigenvalues, we deduce that $λ_{k}^{'} - c \leq λ_{k} \leq λ_{k}^{'} + c$ , where $λ_{k}^{'}$ are the eigenvalues of the Hamiltonian with potential $P$ . Hence, $λ_{k}$ share with $λ_{k}^{'}$ the asymptotics (5), which is the keystone for deriving the estimates on eigenfunctions in subsection 2.2, and thereafter our main results in Section 3. Hence, our results apply to such fitness functions, covering in particular the case of the so-called pseudo-polynomials (i.e. smooth functions which coincide, outside of a compact region, with a polynomial $P$ as in Assumption 2), which are relevant for numerical computations.

2.2. $L^{1}$ , $L^{\infty}$ and weighted $L^{1}$ norms of the eigenfunctions

In the study of spectral properties of Schrödinger operators, efforts tend to concentrate around asymptotic estimates of eigenvalues or on the regularity and decay of eigenfunctions [39], [1], [10], [16]. Much less attention has been given to estimate the $L^{1}$ and $L^{\infty}$ norms of eigenfunctions. One reason is that the natural framework for eigenfunctions of the Hamiltonian $H$ , defined in (3), is $L^{2} (R)$ . On the other hand, the biological nature of problem (1) suggests $L^{1} (R)$ and $L^{\infty} (R)$ as natural spaces for the solution $u (t, x)$ . We therefore provide in this subsection rather non-standard estimates on the eigenfunctions.

We define

m_{k} := \int_{R} ϕ_{k} (x) d x,

the mass of the $k$ -th eigenfunction $ϕ_{k}$ of the Hamiltonian $H$ . In the sequel, by

A_{k} ≲ B_{k}

we mean that there is $c > 0$ such that, for all $k \geq 1$ , $A_{k} \leq c B_{k}$ .

Proposition 2.4 ( $L^{1}$ norm of eigenfunctions).

Let $W$ satisfy Assumption 2. Then we have

| m_{k} | \leq {∥ ϕ_{k} ∥}_{L^{1} (R)} ≲ k^{\frac{1}{2 (s + 1)}} .

(6)

Before proving the above proposition, we need the following lemma which is of independent interest.

Lemma 2.5.

Let $d \in N^{*}$ and $N \in (d, + \infty)$ be given. Then there is a constant $C = C (d, N) > 0$ such that, for all $f : R^{d} \to R$ ,

{∥ f ∥}_{L^{1} (R^{d})} \leq C {∥ f ∥}_{L^{2} (R^{d})}^{1 - δ} {∥ ∥ x^{N / 2} f ∥ ∥}_{L^{2} (R^{d})}^{δ}, δ := d / N .

(7)

Proof.

Let $B_{R}$ denote the open $d$ -dimensional ball of radius $R > 0$ and center $0_{R^{d}}$ . We write

{∥ f ∥}_{L^{1} (R^{d})} = \int B_{R} | f (x) | d x + \int R^{d} ∖ B_{R} \frac{1}{{| x |}^{N / 2}} {| x |}^{N / 2} | f (x) | d x =: I_{1} + I_{2} .

By the Cauchy-Schwarz inequality we have

and

$I_{2}$	$\leq$

	$\leq$	$C (\int_{R}^{+ \infty} \frac{1}{r^{N}} r^{d - 1} d r)^{1 / 2} {∥ ∥ x^{N / 2} f ∥ ∥}_{L^{2} (R^{d})}$
	$\leq$	$C_{2} R^{(d - N) / 2} {∥ ∥ x^{N / 2} f ∥ ∥}_{L^{2} (R^{d})},$

for some $C = C (d) > 0$ and $C_{2} = C_{2} (d, N) > 0$ . Summarizing,

(8)

Now, we select

R = {⎛ ⎜ ⎝ \frac{C_{2} (N - d)}{C_{1} d} \frac{{∥ ∥ x^{N / 2} f ∥ ∥}_{L^{2} (R^{d})}}{{∥ f ∥}_{L^{2} (R^{d})}} ⎞ ⎟ ⎠}^{2 / N}

which minimizes the right hand side of (8) and yields (7). ∎

Remark 3.

The correct power $δ$ in (7) can be retrieved by a standard homogeneity argument. Indeed, defining $f_{λ} (x) := f (λ x)$ for $λ > 0$ , we get

{∥ f_{λ} ∥}_{L^{1} (R^{d})} = \frac{1}{λ^{d}} {∥ f ∥}_{L^{1} (R^{d})},

{∥ f_{λ} ∥}_{L^{2} (R^{d})} = \frac{1}{λ^{d / 2}} {∥ f ∥}_{L^{2} (R^{d})},

{∥ ∥ x^{N / 2} f_{λ} ∥ ∥}_{L^{2} (R^{d})} = \frac{1}{λ^{(N + d) / 2}} {∥ ∥ x^{N / 2} f ∥ ∥}_{L^{2} (R^{d})},

so that

\frac{1}{λ^{d}} ≲ {(\frac{1}{λ^{d / 2}})}^{1 - δ} {(\frac{1}{λ^{(N + d) / 2}})}^{δ} .

Powers of $λ$ in both sides must coincide, which enforces $δ = d / N$ .

We can now estimate the mass of the eigenfunctions.

Proof of Proposition 2.4.

Up to subtracting a constant to $W$ , we can assume without loss of generality that $W < 0$ . Multiplying by $ϕ_{k}$ the eigenvalue equation

- σ^{2} ϕ_{k}^{''} - W ϕ_{k} = λ_{k} ϕ_{k}

and integrating over $x \in R$ , we get

\int_{R} - σ^{2} ϕ_{k}^{''} (x) ϕ_{k} (x) d x + \int_{R} - W (x) ϕ_{k} (x)^{2} d x = λ_{k} \int_{R} ϕ_{k}^{2} (x) d x .

Integrating by parts and recalling that eigenfunctions $ϕ_{k}$ are normalized in $L^{2} (R)$ , we obtain

σ^{2} \int_{R} ϕ_{k}^{'} (x)^{2} d x + \int_{R} - W (x) ϕ_{k} (x)^{2} d x = λ_{k},

so that

\int_{R} - W (x) ϕ_{k} (x)^{2} d x \leq λ_{k} .

Next, it follows from Assumption 2 (and $W < 0$ ) that there is $γ > 0$ such that $- W (x) \geq γ x^{2 s}$ for all $x \in R$ , and thus

{∥ x^{s} ϕ_{k} ∥}_{L^{2} (R)}^{2} \leq \frac{λ_{k}}{γ} .

Now, by Lemma 2.5, we have

{∥ ϕ_{k} ∥}_{L^{1} (R)} ≲ {∥ x^{s} ϕ_{k} ∥}_{L^{2} (R)}^{1 / (2 s)} ≲ λ_{k}^{1 / 4 s},

which, combined with (5), implies (6). The proposition is proved. ∎

Proposition 2.6 ( $L^{\infty}$ norm of eigenfunctions).

Let $W$ satisfy Assumption 2. Then we have

{∥ ϕ_{k} ∥}_{L^{\infty} (R)} ≲ k^{\frac{}{s} 2 (s + 1)} .

(9)

Proof.

Since $- \frac{1}{2} ϕ_{k}^{2} (x) = \int_{x}^{+ \infty} ϕ_{k} (s) ϕ_{k}^{'} (s) d s$ , we have

{∥ ϕ_{k} ∥}_{L^{\infty} (R)}^{2} ≲ {∥ ϕ_{k} ∥}_{L^{2} (R)} {∥ ∥ ϕ_{k}^{'} ∥ ∥}_{L^{2} (R)} ≲ λ_{k}^{1 / 2},

and the conclusion follows from (5). ∎

Proposition 2.7 (Weighted $L^{1}$ norm of eigenfunctions).

Let $W$ satisfy Assumption 2. Then we have

{∥ W ϕ_{k} ∥}_{L^{1} (R)} ≲ k^{\frac{5 s + 2}{2 s + 2}} .

Proof.

From Assumption 2 and $λ_{k} \to + \infty$ , we can find $k_{0} \geq 0$ large enough so that the following facts hold for all $k \geq k_{0}$ : there are $- y_{k} < 0$ and $x_{k} > 0$ such that

	$W (x) + λ_{k} \geq 0$		$\forall x \in (- y_{k}, x_{k})$
	$W (x) + λ_{k} = 0$		$\forall x \in {- y_{k}, x_{k}}$
	$W (x) + λ_{k} \leq 0$		$\forall x \in R ∖ (- y_{k}, x_{k}),$

and $W$ is decreasing on $(- \infty, - y_{k}) \cup (x_{k}, + \infty)$ . Assumption 2 implies that $x_{k}, y_{k} \sim λ_{k}^{\frac{1}{2 s}}$ and thus, from Proposition 2.3, $x_{k}, y_{k} ≲ k^{\frac{1}{s + 1}}$ . Next, up to enlarging $k_{0}$ if necessary, it follows from Assumption 2 that $W (x) + λ_{k} \leq - 1$ for all $x \in (2 x_{k}, + \infty)$ . As a result, functions

ϕ^{\pm} (x) := \pm {∥ ϕ_{k} ∥}_{\infty} e^{- (x - 2 x_{k})}

are respectively super and sub-solutions of the eigenvalue equation

- ϕ_{k}^{''} - (W (x) + λ_{k}) ϕ_{k} = 0,

so that

| ϕ_{k} (x) | \leq {∥ ϕ_{k} ∥}_{\infty} e^{- (x - 2 x_{k})} \forall x \in (2 x_{k}, + \infty),

(10)

by the comparison principle. An analogous estimate holds on $(- \infty, 2 y_{k})$ .

In order to estimate ${∥ W ϕ_{k} ∥}_{L^{1} (R)}$ , we split the domain of integration into three parts: $Ω_{1} := R ∖ (- 2 y_{k}, 2 x_{k})$ , $Ω_{2} := (- 2 y_{k}, - y_{k}) \cup (x_{k}, 2 x_{k})$ and $Ω_{3} := (- y_{k}, x_{k})$ . Setting

I_{i} := \int_{Ω_{i}} | W (x) ϕ_{k} (x) | d x = \int_{Ω_{i}} - W (x) | ϕ_{k} (x) | d x,

we decompose ${∥ W ϕ_{k} ∥}_{L^{1} (R)} = I_{1} + I_{2} + I_{3}$ . Notice that,

\int_{2 x_{k}}^{+ \infty} - W (x) e^{- (x - 2 x_{k})} d x = {[- P (x) e^{- (x - 2 x_{k})}]}_{2 x_{k}}^{- (x - 2 x_{k})} = P (2 x_{k}),

where $P$ is a polynomial of same degree as $W$ . Hence, from (10) and Proposition 2.6, we get

	$I_{1}$	$≲$	${∥ ϕ_{k} ∥}_{\infty} \int_{2 x_{k}}^{+ \infty} \| W (x) \| e^{- (x - 2 x_{k})} d x ≲ k^{\frac{s}{2 (s + 1)}} P (2 x_{k})$
		$≲$	$k^{\frac{s}{2 (s + 1)}} x_{k}^{2 s} ≲ k^{\frac{s}{2 (s + 1)}} k^{\frac{2 s}{s + 1}} = k^{\frac{5 s}{2 s + 2}} .$

By monotonicity of $W$ on $Ω_{2}$ ,

I_{2} ≲ x_{k} | W (2 x_{k}) | {∥ ϕ_{k} ∥}_{\infty} ≲ x_{k}^{1 + 2 s} k^{\frac{s}{2 (s + 1)}} ≲ k^{\frac{5 s + 2}{2 s + 2}} .

Remember that $- W (x) \leq λ_{k}$ in $Ω_{3}$ so that

I_{3} ≲ x_{k} λ_{k} {∥ ϕ_{k} ∥}_{\infty} ≲ x_{k} k^{\frac{2 s}{s + 1}} k^{\frac{s}{2 (s + 1)}} ≲ k^{\frac{5 s + 2}{2 s + 2}} .

Finally, ${∥ W ϕ_{k} ∥}_{L^{1} (R)} ≲ k^{\frac{5 s + 2}{2 s + 2}}$ . ∎

3. Well-posedness and long time behaviour

In this section we show that the Cauchy problem (1) has a unique smooth solution which is global in time. Keystones are the change of variable (13) that links the non-local equation (1) to a linear parabolic problem, and our previous estimates on the underlying eigenelements. Equipped with the representation (12) of the solution, we then prove convergence in any $L^{p}$ , $1 \leq p \leq + \infty$ , to the principal eigenfunction normalized by its mass.

Up to subtracting a constant to the confining fitness function $W$ , we can assume without loss of generality (recall the mass conservation property) that $W \leq - 1$ .

3.1. Functional framework

For $W$ a negative confining fitness function (see Assumption 1), we set

L_{- W}^{2} (R) := {v : R \to R, ∥ v ∥_{L_{- W}^{2} (R)}^{2} := \int_{R} - W (x) v^{2} (x) d x < + \infty} .

Recall that the Sobolev space $W^{1, 2} (R)$ is defined as

W^{1, 2} (R) = {f : R \to R, f \in L^{2} (R), f^{'} \in L^{2} (R)},

where the derivative $f^{'}$ is understood in the distributional sense. We denote by $V := W^{1, 2} (R) \cap L_{- W}^{2} (R)$ the Hilbert space with inner product defined by

{(v, v)}_{V} := \int_{R} \frac{d v}{d x} (x) \frac{d v}{d x} (x) d x + \int_{R} - W (x) v (x) v (x) d x,

and $H := L^{2} (R)$ with usual inner product

{(v, v)}_{H} = \int_{R} v (x) v (x) d x .

By Assumption 1, it is straightforward that $L_{- W}^{2} (R) \subset L^{2} (R)$ , so that $V \subset H$ . Moreover, the following holds.

Lemma 3.1.

The embedding $V ↪ H$ is dense, continuous and compact.

Proof.

This is very classical but, for the convenience of the reader, we present the details. Since $C_{c}^{\infty} (R) \subset V$ and $C_{c}^{\infty} (R)$ is dense in $L^{2} (R) = H$ , it follows that $V$ is dense in $L^{2} (R)$ . Next, since for $v \in V$ ,

the embedding $V ↪ H$ is continuous.

The proof of compactness follows by the Riesz-Fréchet-Kolmogorov theorem, see e.g. [8, Theorem 4.26]. Let ${(v_{n})}_{n \geq 0}$ a bounded sequence of functions of $V$ : there is $M > 0$ such that, for all $n \geq 0$ ,

{∥ v_{n} ∥}_{V}^{2} = \int_{R} v_{n}^{'} (x)^{2} d x + \int_{R} - W (x) v_{n} (x)^{2} d x < M .

We first need to show the uniform smallness of the tails of $v_{n}^{2}$ . Let $ε > 0$ . Select $a > 0$ large enough so that $\frac{1}{- W (x)} \leq ε$ for all $| x | \geq a$ . Then

	${∥ v_{n} ∥}_{L^{2} (R ∖ [- a, a])}^{2}$	$= \int_{- \infty}^{- a} v_{n} (x)^{2} d x + \int_{a}^{+ \infty} v_{n} (x)^{2} d x$
		$\leq ε \int_{- \infty}^{- a} - W (x) v_{n} (x)^{2} d x + ε \int_{a}^{+ \infty} - W (x) v_{n} (x)^{2} d x$
		$\leq M ε .$

Next, for a compact set $K$ , we need to show the uniform smallness of the $L^{2} (K)$ norm of $v_{n} (\cdot + h) - v_{n}$ . Let $ε > 0$ . By Morrey’s theorem, there is $C > 0$ such that, for all $h \in R$ and $n \geq 0$ ,

| v_{n} (x + h) - v_{n} (x) | \leq C {| h |}^{1 / 2} {∥ v_{n}^{'} ∥}_{L^{2} (R)} \leq C {| h |}^{1 / 2} M^{1 / 2},

so that

{∥ v_{n} (\cdot + h) - v_{n} ∥}_{L^{2} (K)}^{2} \leq C^{2} M | K | | h | \leq ε,

for all $n \geq 0$ , if $| h |$ is sufficiently small. The lemma is proved. ∎

3.2. Main results

We first define the notion of solution to the Cauchy problem (1).

Definition 3.2 (Admissible initial data).

We say that a function $u_{0}$ is an admissible initial data if $u_{0} \in L^{1} (R) \cap L^{\infty} (R)$ , $u_{0} (\cdot) \geq 0$ and $\int_{R} u_{0} (x) d x = 1$ .

Definition 3.3 (Solution of the Cauchy problem (1)).

Let $u_{0}$ be an admissible initial data. We say that $u = u (t, x)$ is a (global) solution of the Cauchy problem (1) if, for any $T > 0$ , $u \in C^{0} (0, T; H) \cap L^{2} (0, T; V)$ , $u (0, \cdot) = u_{0}$ , and

[label=()]
For all $v \in V$ , all $t \in (0, T]$ ,

$\begin{matrix} \frac{d}{d t} \int_{R} u (t, x) v (x) d x + σ^{2} \int_{R} \frac{\partial u}{\partial x} (t, x) \frac{d v}{d x} (x) d x + \int_{R} - W (x) u (t, x) v (x) d x + ¯ ¯ ¯ u (t) \int_{R} u (t, x) v (x) d x = 0, \end{matrix}$

where the time derivative is understood in the distributional sense. Equivalently, for all $v \in V$ , all $φ \in C_{c}^{1} (0, T)$ ,

(11)
$¯ ¯ ¯ u : t \mapsto \int_{R} W (y) u (t, y) d y$ is a continuous function on $(0, T]$ .
$C_{T} := \int_{0}^{T} ∣ ∣ ¯ ¯ ¯ u (t) ∣ ∣ d t < + \infty$ .

Here is our main mathematical result.

Theorem 3.4 (Solving replicator-mutator problem).

Let $W$ satisfy Assumption 2. For any admissible initial condition $u_{0}$ , there is a unique solution $u = u (t, x)$ to the Cauchy problem (1), in the sense of Definition 3.3. Moreover the solution is smooth on $(0, + \infty) \times R$ and is given by

u (t, x) = \frac{+ \infty \sum k = 0 {(u_{0}, ϕ_{k})}_{L^{2} (R)} ϕ_{k} (x) e^{- λ_{k} t}}{+ \infty \sum k = 0 {(u_{0}, ϕ_{k})}_{L^{2} (R)} m_{k} e^{- λ_{k} t}}, t > 0, x \in R,

(12)

where $(λ_{k}, ϕ_{k})$ are the eigenelements defined in Proposition 2.1, and

m_{k} = \int_{R} ϕ_{k} (x) d x .

Proof.

We proceed by necessary and sufficient condition. Let $u$ be a solution, in the sense of Definition 3.3. We define the function $v$ as

v (t, x) := u (t, x) exp (\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ), 0 \leq t \leq T, x \in R .

(13)

This function is well defined since by Definition 3.3 $(i i i)$ , the integral in the exponential is finite for all $t \in [0, T]$ . Since $C_{T} < \infty$ and $u (t, \cdot) \in H \cap V$ , it is straightforward to see that, for all $t \in (0, T]$ , $v (t, \cdot) \in H \cap V \equiv V$ . Additionally, from $C_{T} < \infty$ and $u \in C^{0} (0, T; H)$ , one can see that $v \in C^{0} (0, T; H)$ . Last, $v \in L^{2} (0, T; V)$ due to

\int_{0}^{T} {∥ v (t, \cdot) ∥}_{V}^{2} d t

since $u \in L^{2} (0, T; H)$ .

We now show that $v$ solves the linear Cauchy problem

{\begin{matrix} \partial_{t} v = σ^{2} \partial_{x}^{2} v + W (x) v v (0, x) = u_{0} (x) . \end{matrix}

(14)

Indeed, formally for the moment,

\begin{matrix} \partial_{t} v = (\partial_{t} u) e^{\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ} + u ¯ ¯ ¯ u \partial_{x}^{2} v = (\partial_{x}^{2} u) e^{\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ}, \end{matrix}

so that

\partial_{t} v - σ^{2} \partial_{x}^{2} v - W (x) v = (\partial_{t} u + u ¯ ¯ ¯ u - σ^{2} \partial_{x}^{2} u - W (x) u) e^{\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ} = 0

since $u$ solves (1). Those computations can be made rigorous in the distributional sense. Indeed, for a test function $ψ \in C_{c}^{1} (0, T)$ , set

φ (t) := ψ (t) e^{\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ},

and by Definition 3.3 $(i i)$ , $φ$ belongs to $C_{c}^{1} (0, T)$ . Writing (11) with $φ$ as test function yields the weak formulation of (14) with $ψ$ as test function, that is

for all $v \in V$ .

The well-posedness of the linear Cauchy problem (14) is postponed to the next subsection: from Proposition 3.6, we know that, for all $t \in (0, T]$ ,

v (t) = + \infty \sum k = 0 {(u_{0}, ϕ_{k})}_{L^{2} (R)} ϕ_{k} e^{- λ_{k} t} in L^{2} (R) .

Now, the estimates on the eigenvalues and the $L^{\infty}$ norm of eigenfunctions, namely Proposition 2.3 and Proposition 2.6, allow to write

v (t, x) = + \infty \sum k = 0 {(u_{0}, ϕ_{k})}_{L^{2} (R)} ϕ_{k} (x) e^{- λ_{k} t}, 0 < t \leq T, x \in R .

Also, we know from the parabolic regularity theory and the comparison principle, that $v \in C^{\infty} ((0, T) \times R)$ and that $v (t, x) > 0$ for all $t > 0$ , $x \in R$ .

Now, we show that the change of variable (13) can be inverted. For $t > 0$ , multiplying (13) by $W (x)$ and integrating over $x \in R$ , we get

	$¯ ¯ ¯ v (t) := \int_{R} W (x) v (t, x) d x$	$= ¯ ¯ ¯ u (t) exp (\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ)$
		$= \frac{d}{d t} (exp (\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ)) .$		(15)

On the other hand, we claim that, for all $t > 0$ ,

\frac{d}{d t} m_{v} (t) = ¯ ¯ ¯ v (t),

(16)

which follows formally by integrating (14) over $x \in R$ . To prove (16) rigorously, notice first that by Proposition 2.3 and 2.4, the series

+ \infty \sum k = 0 ∣ ∣ {(u_{0}, ϕ_{k})}_{L^{2} (R)} ∣ ∣ e^{- λ_{k} t} \int_{R} | ϕ_{k} (x) | d x

converges for all $t > 0$ . Hence $m_{v} (t)$ , the total mass of $v$ , is given by

m_{v} (t) = + \infty \sum k = 0 {(u_{0}, ϕ_{k})}_{L^{2} (R)} m_{k} e^{- λ_{k} t} .

Next, for any $t_{0} > 0$ , $\sum_{k = 0}^{+ \infty} ∣ ∣ {(u_{0}, ϕ_{k})}_{L^{2} (R)} ∣ ∣ | m_{k} | λ_{k} e^{- λ_{k} t_{0}} < + \infty$ thanks to Proposition 2.3 and 2.4, so that $m_{v}$ is differentiable on $(0, T]$ and

\frac{d}{d t} m_{v} (t) = + \infty \sum k = 0 {(u_{0}, ϕ_{k})}_{L^{2} (R)} m_{k} (- λ_{k}) e^{- λ_{k} t} = ¯ ¯ ¯ v (t),

the last equality following by similar arguments based on Proposition 2.7. Hence (16) is proved. From (15), (16) and $m_{v} (0) = 1$ , we deduce that

exp (\int_{0}^{t} ¯ ¯ ¯ u (τ) d τ) = m_{v} (t),

for all $t \geq 0$ . As a conclusion, (13) is inverted into

u (t, x) = \frac{v (t, x)}{m_{v} (t)}, m_{v} (t) := \int_{R} v (t, x) d x > 0,

(17)

for all $0 \leq t \leq T$ , $x \in R$ .

Conversely, we need to show that the function $u$ given by (17) is the solution of (1) in the sense of Definition 3.3. Let $T > 0$ .

Since $¯ ¯ ¯ u (t) = \frac{¯ v (t)}{m_{v} (t)} = \frac{\frac{d}{d t} m_{v} (t)}{} m_{v} (t)$ , the function $¯ ¯ ¯ u$ is continuous on $(0, T]$ , which shows item $(i i)$ of Definition 3.3.

Next, since $m_{v} > 0$ and $¯ ¯ ¯ v < 0$ ,

\int_{0}^{T} ∣ ∣ ¯ ¯ ¯ u (t) ∣ ∣ d t = - \int_{0}^{T} \frac{\frac{d}{d t} m_{v} (t)}{m_{v} (t)} d t = - ln

Evolutionary branching via replicator-mutator equations

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Some spectral properties

2.1. Confining fitness functions and eigenvalues properties

Assumption 1 (Confining fitness function).

Proposition 2.1 (Spectral basis).

Remark 1.

Proposition 2.2 (Asymptotics for λ0(σ) as σ→0).

Proof.

Assumption 2 (Polynomial confining fitness function).

Proposition 2.3 (Asymptotics for eigenvalues).

Remark 2.

2.2. L1, L∞ and weighted L1 norms of the eigenfunctions

Proposition 2.4 (L1 norm of eigenfunctions).

Lemma 2.5.

Proof.

Remark 3.

Proof of Proposition 2.4.

Proposition 2.6 (L∞ norm of eigenfunctions).

Proof.

Proposition 2.7 (Weighted L1 norm of eigenfunctions).

Proof.

3. Well-posedness and long time behaviour

3.1. Functional framework

Lemma 3.1.

Proof.

3.2. Main results

Definition 3.2 (Admissible initial data).

Definition 3.3 (Solution of the Cauchy problem (1)).

Theorem 3.4 (Solving replicator-mutator problem).

Proof.

Proposition 2.2 (Asymptotics for $λ_{0} (σ)$ as $σ \to 0$ ).

2.2. $L^{1}$ , $L^{\infty}$ and weighted $L^{1}$ norms of the eigenfunctions

Proposition 2.4 ( $L^{1}$ norm of eigenfunctions).

Proposition 2.6 ( $L^{\infty}$ norm of eigenfunctions).

Proposition 2.7 (Weighted $L^{1}$ norm of eigenfunctions).