Natural selection in compartmentalized environment with reshuffling

Abstract

The emerging field of high-throughput compartmentalized in vitro evolution is a promising new approach to protein engineering. In these experiments, libraries of mutant genotypes are randomly distributed and expressed in microscopic compartments—droplets of an emulsion. The selection of desirable variants is performed according to the phenotype of each compartment. The random partitioning leads to a fraction of compartments receiving more than one genotype making the whole process a lab implementation of the group selection. From a practical point of view (where efficient selection is typically sought), it is important to know the impact of the increase in the mean occupancy of compartments on the selection efficiency. We carried out a theoretical investigation of this problem in the context of selection dynamics for an infinite non-mutating subdivided population that randomly colonizes an infinite number of patches (compartments) at each reproduction cycle. We derive here an update equation for any distribution of phenotypes and any value of the mean occupancy. Using this result, we demonstrate that, for the linear additive fitness, the best genotype is still selected regardless of the mean occupancy. Furthermore, the selection process is remarkably resilient to the presence of multiple genotypes per compartments, and slows down approximately inversely proportional to the mean occupancy at high values. We extend out results to more general expressions that cover nonadditive and non-linear fitnesses, as well non-Poissonian distribution among compartments. Our conclusions may also apply to natural genetic compartmentalized replicators, such as viruses or early trans-acting RNA replicators.

Appendices

Appendices

1.1 Random variables and generalized functions

This appendix does not contain new results. Its purpose is to be an introduction to the framework used in the article.

A random variables X on ${\mathbb {R}}$ is associated with a positive Radon measure $\mu _X$, which can be understood as a non-negative generalized function (linear continuous functional) $\rho _X$ defined on the space of continuous functions with compact supports $C_c$ endowed with the appropriate topology [see, for example, the book by Schwartz (1997)]. This topology is conventionally induced by the following convergence rule: we say $\varphi _n \rightarrow \varphi$ in $C_c$, if there is a compact K such that $\forall n$${\text {supp}}\varphi _n \subset K$ and $\varphi _n \rightarrow \varphi$ homogeneously. We will denote the topological linear space of generalized functions on $C_c$ endowed with the weak topology as $C_c'$, so the topology on $C_c'$ is induced by the convergence $\rho _n \rightarrow \rho \Leftrightarrow \forall \varphi \in C_c$$\langle \rho _n, \varphi \rangle \rightarrow \langle \rho , \varphi \rangle$. Such convergence is equivalent to the weak convergence of the corresponding measures in the measure-theoretical language. As this is the only convergence of generalized functions that we consider in the article, the use of the sign $\rightarrow$ to denote it does not bring any confusion.

A generalized function $\rho$ is called non-negative, if for any non-negative $\varphi \in C_c$ (that is $\forall x \in {\mathbb {R}}$$\varphi (x) \geqslant 0$) we have $\langle \rho , \varphi \rangle \geqslant 0$. We will denote the subset of non-negative generalized functions with the subset topology as $C'_{c+}$.

It is possible to extend the action of $\rho$ from functions in $C_c$ to any indicator function of a Borel set $\chi _B$, $B \in {\mathfrak {B}}$, where ${\mathfrak {B}}$ is the Borel $\sigma$-algebra. This is done using the so called upper and lower value of $\rho$ on an indicator of a set [in the book by Schwartz (1997) this corresponds to the upper and the lower measure of a set]. The upper value of $\chi _A$ is defined by

(69)

where U are open sets in the standard topology on ${\mathbb {R}}$, and $\varphi \in C_c$. The lower value is defined as

$$\begin{aligned} \langle \check{\rho }, \chi _A \rangle = \sup _{K \subset A} \inf _{{\begin{matrix}\varphi \geqslant 0 \\ \varphi |_K = 1\end{matrix}}} \langle \rho , \varphi \rangle , \end{aligned}$$

(70)

where K are compact. If, for compactly supported non-negative$\rho$, $\langle {\hat{\rho }}, \chi _A \rangle = \langle \check{\rho }, \chi _A \rangle$, the set A is called $\rho$-measurable and the value $\langle \rho , \chi _A \rangle$ is defined as this common value and is called the measure of A. It can be proven that for any compactly supported non-negative generalized function $\rho$ any Borel set is $\rho$-measurable [it is also $\sigma$-regular, see the book by Schwartz (1997)]. In particular, the whole ${\mathbb {R}}$ is measurable, and in fact, for $\rho$ to be a (generalized) probability density functions, we require $\langle \rho , \chi _{{\mathbb {R}}} \rangle \overset{{\text {def}}}{=}\langle \rho , 1 \rangle = 1$. For non-negative$\rho$that have$\langle \rho , 1 \rangle < \infty$, any set A such that $\langle {\hat{\rho }}, \chi _A \rangle = \langle \check{\rho }, \chi _A \rangle$ is called $\rho$-measurable, whether $\rho$ is compactly supported or not. Note that for a non-negative $\rho$ and a continuous f, $\langle \rho , f\rangle$ is well defined as $\langle f\rho ,1\rangle$ and one does not need any further development of the theory.

Any point-set is measurable, too, and we can compute a measure of a point x as

$$\begin{aligned} \langle \rho , \chi _{\{x\}} \rangle = \inf _{ {\begin{matrix}\varphi \geqslant 0 \\ \varphi (x) = 1\end{matrix}}} \langle \rho , \varphi \rangle . \end{aligned}$$

(71)

From now on we will call non-negative generalized functions with $\langle \rho , 1 \rangle = 1$ (probability) densities (of random variables). The meaning of a density $\rho _\xi$ of a random variable $\xi$ is that the probability to find the value of $\xi$ in a set A is equal to $\langle \rho _\xi , \chi _A \rangle$. A (cumulative) distribution function of $\xi$ is the function $F_\xi :x \mapsto \langle \rho _\xi , \chi _{(-\infty ,x]} \rangle$. The density is the (generalized) derivative of its own distribution function. The mathematical expectation (the mean) of the random variable is then computed as ${\bar{x}} = \langle \rho , x \rangle$.

Any density $\rho$ can be uniquely decomposed into a sum

$$\begin{aligned} \rho = \rho _a + \rho _p + \rho _r, \end{aligned}$$

(72)

where $\rho _a$ is the regular part, $\rho _p$ is the point-mass part, and $\rho _r$ is the residual singular part.

The regular part $\rho _a$ is a regular generalized function, i.e. its action on any $\varphi \in C_c$ can be represented by $\langle \rho _a, \varphi \rangle = \int f\varphi \,dx$ for a unique $f \in L^1_{\mathrm {loc}}$ (the integration is in the sens of Lebesgue). The regular part is also called the absolutely continuous part (hence the notation $\rho _a$). A density that has only this part is also called absolutely continuous. Any point has a zero measure in respect to an absolutely continuous density. It is convenient to identify (a representative of) f with $\rho _a$ and to write $\rho _a$ instead of f.

The point-mass part $\rho _p$ is an at most countable sum of $\delta$-functions

$$\begin{aligned} \rho _p = \sum _{n \in {\mathbb {N}}} a_n \delta _{x_n}, \quad a_n \geqslant 0, \quad n \ne m \Rightarrow x_n \ne x_m,\quad \sum _{n \in {\mathbb {N}}} a_n \leqslant 1. \end{aligned}$$

(73)

It follows that $\forall x \ne x_n$$\langle \rho _p, \chi _{\{x\}} \rangle = 0$, and $\forall x_n$$\langle \rho _p, \chi _{\{x_n\}} \rangle = a_n$. Physically speaking, $\rho _p$ represents all fitness values that are present in macroscopic quantities in the population.

Finally, the residual singular part $\rho _r$ is characterized by the zero (Lebesgue) measure of its support and, at the same time, $\forall x$$\langle \rho _r, \chi _{\{x\}} \rangle = 0$. Its support is Cantor set-like and its distribution function is Cantor function-like.

The regular and the residual singular parts form together the continuous part $\rho _c = \rho _a + \rho _r$. The sum of the point-mass and the residual singular parts is the singular part $\rho _s = \rho _p + \rho _r$. It is tempting to disregard $\rho _r$ as unphysical. However, it may turn to be a good tool to model libraries obtained by a random mutagenesis from a single mutant in a very rugged fitness landscape. It might be possibly a good approximation to a library generated on a smooth landscape but by an error-prone PCR with large number of cycles.

1.2 Continuity of operator A and of the operators that generate $\sigma _x$ and $\sigma$ for the linear fitness case

Let ${\mathbb {P}}$ be the space of probability densities with, so ${\mathbb {P}} = \{\rho \in C_{c+}'\,|\, \langle \rho ,1 \rangle \ = 1\}$. Let ${\mathbb {P}}_p$ be the space of finite point-mass probability densities, so ${\mathbb {P}}_p = \{\rho \in {\mathbb {P}} \,|\, \exists \, n \in {\mathbb {N}} \,, \rho = \sum \nolimits _{k=0}^n a_k \delta _{x_k}\}$. Let us fix some very large positive number ${\mathscr {L}}$. Let us denote ${\mathscr {I}} \overset{{\text {def}}}{=}[0,{\mathscr {L}}]$. Let ${\mathbb {P}}^{\mathscr {I}}$ be the space of probability densities concentrated in ${\mathscr {I}}$, so ${\mathbb {P}}^{\mathscr {I}} = \{\rho \in {\mathbb {P}} \,|\, {\text {supp}}\rho \subset {\mathscr {I}}\}$. Let ${\mathbb {P}}^{\mathscr {I}}_p$ be the space of finite point-mass probability densities from ${\mathbb {P}}^{\mathscr {I}}$, so ${\mathbb {P}}^{\mathscr {I}}_p = {\mathbb {P}}^{\mathscr {I}} \cap {\mathbb {P}}_p$. Let the topologies on all these spaces be inherited from $C_c'$, subspaces of which all of them are.

In Sect. 2, we derived the update operator $\rho _{t+1} = A(\rho _t)$ given by the formula (17) for densities from ${\mathbb {P}}_p$. This formula is generalizable on any generalized function $\rho$ from ${\mathbb {P}}$ with nonzero finite mean $\langle \rho , x \rangle$. Let us denote $N \overset{{\text {def}}}{=}\{\rho \in {\mathbb {P}}\,|\, \langle \rho ,x\rangle = 0\}$. Let us also denote $N_p \overset{{\text {def}}}{=}N \cap {\mathbb {P}}_p$. If $N_p$ was nowhere dense in ${\mathbb {P}}_p$, if, in addition, ${\mathbb {P}}_p$ was dense in ${\mathbb {P}}$ and (17) happened to be continuous both in ${\mathbb {P}}_p{\setminus } N_p$ and ${\mathbb {P}}{\setminus } N$, we could take this expression as an extension by continuity of A from ${\mathbb {P}}_p$ to ${\mathbb {P}}$. Unfortunately, this assertion is not true.

Proposition 1

For any $\lambda > 0$, the operator $A:{\mathbb {P}}_p {\setminus } N_p \rightarrow {\mathbb {P}}_p$ defined by (17) is nowhere continuous.

Proof

Take any $\rho \in {\mathbb {P}}_p {\setminus } N_p$. By the definition of ${\mathbb {P}}_p$, we have $0< |\langle \rho ,x\rangle | < \infty$. Consider the sequence

$$\begin{aligned} \rho _n = \dfrac{n}{n+1}\rho + \dfrac{1}{n+1}\delta _{n^2}. \end{aligned}$$

(74)

For any large enough n, $\rho _n \in {\mathbb {P}}_p {\setminus } N_p$. Furthermore, $\rho _n \rightarrow \rho$ in the topology of ${\mathbb {P}}_p {\setminus } N_p$. Indeed, for any test function $\varphi \in C_c$ there is a number $n_0$ such that for any $n > n_0$ the point $n^2$ does not belong to the support of $\varphi$. Therefore, for $n > n_0$ we have

$$\begin{aligned} \langle \rho _n, \varphi \rangle = \dfrac{n}{n+1}\langle \rho ,\varphi \rangle \rightarrow \langle \rho ,\varphi \rangle . \end{aligned}$$

(75)

From the other hand, $\rho _n$ does not converge to $\rho$ in mean. Indeed,

$$\begin{aligned} \langle \rho _n,x\rangle = \dfrac{n}{n+1}\langle \rho ,x\rangle + \dfrac{n^2}{n+1} \rightarrow +\infty . \end{aligned}$$

(76)

Therefore,

$$\begin{aligned} A(\rho _n) \rightarrow \Big (1 - g(\lambda )\Big ) \rho \ne A(\rho ) = \left( 1 - g(\lambda ) + g(\lambda )\frac{x}{\langle \rho ,x\rangle }\right) \rho . \end{aligned}$$

(77)

$\square$

This deplorable fact, however, can be remedied by the restriction of the space of the considered probability densities to ${\mathbb {P}}^{\mathscr {I}}$ for some ${\mathscr {I}}$. This restriction has purely technical meaning and does not reflect any physical constraints. Nevertheless, some justification comes from the fact that there is a universal upper bound on the activity of any enzyme of the considered class of enzymes. This upper bound is reflected by the number ${\mathscr {L}}$ that defines ${\mathscr {I}}$. Note that ${\mathbb {P}}^{\mathscr {I}} \cap N = {\mathbb {P}}^{\mathscr {I}}_p \cap N = \{\delta _0\}$.

Theorem 1

For any ${\mathscr {I}}$, the operator $A:{\mathbb {P}}^{\mathscr {I}} {\setminus } \{\delta _0\} \rightarrow {\mathbb {P}}^{\mathscr {I}}$ defined by the formula (17) is continuous.

Proof

Let $\rho _n$ be some sequence from ${\mathbb {P}}^{\mathscr {I}}{\setminus } \{\delta _0\}$ that converges to some $\rho \in {\mathbb {P}}^{\mathscr {I}}{\setminus } \{\delta _0\}$. Then

$$\begin{aligned} |\langle \rho _n,x\rangle - \langle \rho ,x\rangle | = |\langle \rho _n - \rho , x \rangle | = \mathscr {|}\langle \rho _n - \rho , \eta \rangle | \rightarrow 0, \end{aligned}$$

(78)

where $\eta$ is some function from $C_c$ such that $\forall x \in {\mathscr {I}}$$\eta (x) = x$.

It follows that

$$\begin{aligned} \left( 1-g(\lambda ) + g(\lambda )\frac{x}{\langle \rho _n, x\rangle }\right) \rightarrow \left( 1-g(\lambda ) + g(\lambda )\frac{x}{\langle \rho , x\rangle }\right) \end{aligned}$$

(79)

point-wise on ${\mathscr {I}}$. As all the involved functions are also continuous on the compact set ${\mathscr {I}}$, the convergence is uniform.

Let us denote the function on the right-hand side of (79) as ${\tilde{\varphi }}$ and the functions on the left-hand side as ${\tilde{\varphi }}_n$ (for each n, ${\tilde{\varphi }}_n$ corresponds to the function generated by $\rho _n$). Then it is always possible to find functions $\varphi ,\varphi _n \in C_c$ such that $\varphi _n \rightarrow \varphi$ in $C_c$ and $\left. \varphi \right| _{\mathscr {I}} = \left. {\tilde{\varphi }}\right| _{\mathscr {I}}$, $\left. \varphi _n\right| _{\mathscr {I}} = \left. {\tilde{\varphi }}_n\right| _{\mathscr {I}}$ and, therefore, ${\tilde{\varphi }}_n\rho = \varphi _n\rho$, ${\tilde{\varphi }}\rho = \varphi \rho$.

As $A(\rho _n) = \varphi _n\rho _n$ and $A(\rho ) = \varphi \rho$, what is left to be proven is that given $\rho _n \rightarrow \rho$ and $\varphi _n \rightarrow \varphi$ we have $\varphi _n \rho _n \rightarrow \varphi \rho$. First notice that $\sup |\varphi _n\psi - \varphi \psi | \rightarrow 0$ for any $\psi \in C_c$ (from which it follows that $\varphi _n \psi \rightarrow \varphi \psi$ in $C_c$). Therefore, for any $\epsilon > 0$ there is $n_0$ such that for any $n > n_0$ we have $\sup |\varphi _n\psi - \varphi \psi | < \epsilon$. We have, therefore, for any $n > n_0$

$$\begin{aligned}&|\langle \varphi _n\rho _n - \varphi \rho ,\psi \rangle | \leqslant |\langle \rho _n - \rho , \varphi \psi \rangle | + |\langle \rho _n,\varphi _n\psi - \varphi \psi \rangle | \nonumber \\&\quad \leqslant |\langle \rho _n - \rho ,\varphi \psi \rangle | + \sup |\varphi _n\psi - \varphi \psi | < |\langle \rho _n - \rho ,\varphi \psi \rangle | + \epsilon . \end{aligned}$$

(80)

But $\rho _n \rightarrow \rho$ means that there is $n_1 > n_0$ such that for any $n > n_1$ we have $|\langle \rho _n - \rho ,\varphi \psi \rangle | < \epsilon$. Therefore, for any $\epsilon$ and for any $\psi$ we have $|\langle A(\rho _n) - A(\rho ),\psi \rangle | < 2\epsilon$ starting from $n_1$. $\square$

As it is seen from the proof, the statement of the theorem stays correct, if we replace ${\mathscr {I}}$ with any compact set K with nonempty interior and if we replace ${\mathbb {P}}^{\mathscr {I}}{\setminus } \{\delta _0\}$ with ${\mathbb {P}}^K {\setminus } ({\mathbb {P}}^K \cap N)$, where ${\mathbb {P}}^K = \{\rho \in {\mathbb {P}}\,|\,{\text {supp}}\rho \subset K\}$. Furthermore, the set ${\mathbb {P}}^K_p \cap N$ is nowhere dense in ${\mathbb {P}}^K_p$ and the set ${\mathbb {P}}^K \cap N$ is nowhere dense in ${\mathbb {P}}^K$.

The only thing that is left to be proven is that the space of finite discrete densities is dense in the space of general densities.

Theorem 2

For any ${\mathscr {I}}$, the space ${\mathbb {P}}^{\mathscr {I}}_p$ is dense in ${\mathbb {P}}^{\mathscr {I}}$ as its subset.

Proof

First let us prove that the space ${\mathbb {P}}^{\mathscr {I}} \cap C_c$ is dense in ${\mathbb {P}}^{\mathscr {I}}$, where $C_c$ is understood as being naturally embedded into $C_c'$. That is, any density from ${\mathbb {P}}^{\mathscr {I}}$ can be approximated by a sequence of densities from $C_c$ with the supports in ${\mathscr {I}}$.

Consider some $\omega \in C_{c+}$ such that $\forall x$$\omega (-x) = \omega (x)$ and $\int \omega (x)\,dx = 1$. Let us denote $r = {\text {diam}}{\text {supp}}\omega$ and $\omega _n(x) = \dfrac{1}{n}\omega \left( \dfrac{x}{n}\right)$. Let us also consider the sequence of mappings $F_n: {\mathbb {R}} \rightarrow {\mathbb {R}}$, $x \mapsto \dfrac{{\mathscr {L}} }{{\mathscr {L}} + 2\frac{r}{n}}\left( x + \dfrac{r}{n}\right)$. For each n, $F_n$ bijectively maps the interval $\left[ -\dfrac{r}{n}, {\mathscr {L}} + \dfrac{r}{n}\right]$ to the interval $[0,{\mathscr {L}}] = {\mathscr {I}}$.

For any generalized function $\rho \in {\mathbb {P}}^{\mathscr {I}}$ take the sequence of $\psi _n = (F_n)_\star (\rho * \omega _n) \in {\mathbb {P}}^{\mathscr {I}} \cap C_c$. Since $\omega _n \rightarrow \delta _0$ in ${\mathbb {P}}$, as $n \rightarrow \infty$, we have $\rho * \omega _n \rightarrow \rho$ in ${\mathbb {P}}$. Let us prove that $\psi _n \rightarrow \rho$.

For any $\varphi \in C_c$ we have $\langle (F_n)_\star (\rho * \omega _n), \varphi \rangle = \langle \rho * \omega _n, \varphi \circ F_n\rangle$. It is not difficult to show that $\varphi \circ F_n \rightarrow \varphi$ in $C_c$. The point $x_0 = {\mathscr {L}}/2$ is the stationary point for all $F_n$. All $F_n$ are affine and contracting with the contraction coefficient ${\mathscr {L}}/({\mathscr {L}} + 2r/n)$. Their inverses $F_n^{-1}$ are expanding with the expansion coefficient $\kappa _n \overset{{\text {def}}}{=}1 + 2r/(n{\mathscr {L}})$. Note that $\forall n$, $\kappa \overset{{\text {def}}}{=}\kappa _1 \geqslant \kappa _n$. Let $\varDelta \overset{{\text {def}}}{=}\max \Big (|x_0 - \inf {\text {supp}}\varphi |, |x_0 - \sup {\text {supp}}\varphi |\Big )$ and $K \overset{{\text {def}}}{=}[x_0 - \kappa \varDelta , x_0 + \kappa \varDelta ]$. Then $\forall n > 0$, ${\text {supp}}\varphi \circ F_n \subset K$ and ${\text {supp}}\varphi \subset K$. As $F_n \rightarrow {\text {Id}}_{{\mathbb {R}}}$ pointwise on ${\mathbb {R}}$ and $F_n$ are continuous, this convergence is uniform on K. That is $\sup \nolimits _{x \in K} |F_n(x) - x| \rightarrow 0$. As $\varphi$ is continuous and compactly supported, it is also uniformly continuous. Therefore,

$$\begin{aligned} \sup _{x \in {\mathbb {R}}}\Big |\varphi \Big (F_n(x)\Big ) - \varphi (x)\Big | \rightarrow 0. \end{aligned}$$

(81)

But together with ${\text {supp}}\varphi \circ F_n \subset K$ this proves that $\varphi \circ F_n \rightarrow \varphi$ in $C_c$.

We have $\rho * \omega _n \rightarrow \rho$ and $\varphi \circ F_n \rightarrow \varphi$. Using the same reasoning as in the proof of Theorem 1, we conclude that $\langle \psi _n, \varphi \rangle = \langle \rho * \omega _n, \varphi \circ F_n\rangle \rightarrow \langle \rho ,\varphi \rangle$, and thus, $\psi _n \rightarrow \rho$.

From the other hand, any $\psi \in {\mathbb {P}}^{\mathscr {I}} \cap C_c$ can be approximated by a sequence from ${\mathbb {P}}^{\mathscr {I}}_p$. Indeed, we can select some sequence of conventionally ordered Darboux partitions $\{\varDelta _n\}$ of some interval [a, b] that contains ${\text {supp}}\psi$ (one can take tho whole ${\mathscr {I}}$) with the graininess of the partitions going to 0 with $n \rightarrow \infty$, where $\varDelta _n = \{x^{(n)}_k\}$, $k \in \{0, 1, \ldots , K_n\}$, $x^{(n)}_k < x^{(n)}_{k+1}$, $x^{(n)}_0 = a$, $x^{(n)}_{K_n} = b$, $\varDelta x^{(n)}_k = x^{(n)}_{k+1}-x^{(n)}_k$, and $\max \nolimits _{k < K_n} \varDelta x^{(n)}_k \rightarrow 0$, when $n \rightarrow \infty$. Then we can take the sequence of $\rho _n \in {\mathbb {P}}^{\mathscr {I}}_p$ of the following form

$$\begin{aligned} \rho _n = \sum _{k = 0}^{K_n - 1}\varDelta x^{(n)}_k\psi (\xi ^{(n)}_k)\delta _{\xi ^{(n)}_k}, \end{aligned}$$

(82)

where $\xi ^{(n)}_k \in [x^{(n)}_k, x^{(n)}_{k+1}]$ such that

$$\begin{aligned} \psi (\xi ^{(n)}_k) \varDelta x^{(n)}_k = \int _{x^{(n)}_k}^{x^{(n)}_{k+1}} \psi (x)\,dx. \end{aligned}$$

(83)

Such $\xi ^{(n)}_k$ always exist by the mean value theorem. Their role is to enforce $\langle \rho _n,1\rangle = 1$.

Then for any $\varphi \in C_c$ we have

$$\begin{aligned} \langle \rho _n, \varphi \rangle= & {} \left\langle \sum _{k = 0}^{K_n - 1} \varDelta x^{(n)}_k\psi (\xi ^{(n)}_k)\delta _{\xi ^{(n)}_k}, \varphi \right\rangle \nonumber \\= & {} \sum _{k = 0}^{K_n - 1}\varDelta x^{(n)}_k\psi (\xi ^{(n)}_k)\varphi (\xi ^{(n)}_k) \rightarrow \int _a^b\psi (x)\varphi (x)\,dx = \langle \psi ,\varphi \rangle , \end{aligned}$$

(84)

as both $\psi$ and $\varphi$ are continuous, and thus, $\psi \varphi$ is Riemann integrable. As ${\mathbb {P}}^{\mathscr {I}} \cap C_c$ is dense in ${\mathbb {P}}^{\mathscr {I}}$, it follows that ${\mathbb {P}}^{\mathscr {I}}_p$ is dense in ${\mathbb {P}}^{\mathscr {I}}$. $\square$

Note that the proof of the theorem is easily extended to probability densities on ${\mathbb {R}}^n$, the case important for a multitrait selection considered in Sect. 5.3. The only difference is that in this case Riemann sums are built on the base of Jordan partitions of a Jordan-measurable set (a simple rectangular parallelepiped is enough) that contains ${\text {supp}}\psi$.

We now will prove that the operators that generate $\sigma _x$ and $\sigma$ from $\rho$ defined by formulas (11) and (12) are continuous, too, and that they can be thus extended to any probability density. It means that they can be used independently of A, if the situation demands it. Let us denote these operators from ${\mathbb {P}}$ to itself as $\varSigma _x$ and $\varSigma$. Let us also denote $\sigma ^\rho _x \overset{{\text {def}}}{=}\varSigma _x(\rho )$ and $\sigma ^\rho \overset{{\text {def}}}{=}\varSigma (\rho )$ to be able to distinguish fitness distributions generated by different phenotypic distributions.

Theorem 3

For any $x \in {\mathbb {R}}$, the operators $\varSigma _x, \varSigma :{\mathbb {P}} \rightarrow {\mathbb {P}}$ are continuous.

Proof

We will prove the theorem only for $\varSigma _x$. The proof for $\varSigma$ is analogous. The proof is essentially based on the absolute convergence of all the involved numerical series.

Let us choose any sequence of $\rho _n \in {\mathbb {P}}$ that converges to some $\rho \in {\mathbb {P}}$ in ${\mathbb {P}}$. We need to prove that $\varSigma _x(\rho _n) \rightarrow \varSigma _x(\rho )$. Let us choose some $\varphi \in C_c$. The value of $\langle \varSigma _x(\rho ), \varphi \rangle$ is equal to

$$\begin{aligned} \langle \varSigma _x(\rho ), \varphi \rangle = \sum _{k=0}^\infty P_k \langle \delta _x * \rho ^{*k}, \varphi \circ h_{k+1}\rangle , \end{aligned}$$

(85)

where $P_k = e^{-\lambda }\lambda ^k/k!$ and $h_k:x \mapsto x/k$.

First note that $\sup \nolimits _x |\varphi \circ h_k (x)| \leqslant \sup \nolimits _x |\varphi (x)|$. Let us denote $\varPhi \overset{{\text {def}}}{=}\sup \nolimits _{x} |\varphi (x)|$. Then the following estimate is correct

$$\begin{aligned} |\langle \delta _x * \rho _n^{*k} - \delta _x * \rho ^{*k}, \varphi \circ h_{k+1} \rangle |\leqslant & {} |\langle \delta _x * \rho _n^{*k},\varphi \circ h_{k+1}\rangle | + |\langle \delta _x * \rho ^{*k}, \varphi \circ h_{k+1} \rangle | \nonumber \\\leqslant & {} \varPhi \Big (\langle \delta _x * \rho _n^{*k},1\rangle + \langle \delta _x * \rho ^{*k}, 1\rangle \Big ) = 2\varPhi . \end{aligned}$$

(86)

This, in turn, implies

$$\begin{aligned} |\langle \varSigma _x(\rho _n) - \varSigma _x(\rho ),\varphi \rangle |\leqslant & {} \sum _{k=0}^\infty P_k|\langle \delta _x*\rho _n^{*k}-\delta _x*\rho ^{*k},\varphi \circ h_{k+1}\rangle |\nonumber \\\leqslant & {} 2\varPhi \sum _{k=0}^\infty P_k = 2\varPhi < \infty . \end{aligned}$$

(87)

Therefore, for any $\epsilon > 0$ there is $k_0$ such that $\sum \nolimits _{k=k_0}^\infty 2\varPhi P_k < \epsilon$, and, thus, by (86), for any n

$$\begin{aligned} \sum _{k=k_0}^\infty P_k|\langle \delta _x*\rho _n^{*k}-\delta _x*\rho ^{*k},\varphi \circ h_{k+1}\rangle | < \epsilon . \end{aligned}$$

(88)

From the other hand, for any k, $\rho _n \rightarrow \rho$ implies $\delta _x*\rho _n^{*k} \rightarrow \delta _x*\rho ^{*k}$. Therefore, for any $\epsilon > 0$ and any $k_0$ there is $n_0$ such that for any $n > n_0$

$$\begin{aligned} \sum _{k=0}^{k_0-1} P_k|\langle \delta _x*\rho _n^{*k}-\delta _x*\rho ^{*k},\varphi \circ h_{k+1}\rangle | < \epsilon . \end{aligned}$$

(89)

Using the following intuitive logical formula

$$\begin{aligned} \Big (\forall x\exists y\forall z\, A(x,y,z)\Big ) \wedge \Big (\forall x\forall y \exists z\, B(x,y,z) \Big ) \Rightarrow \forall x\exists y\exists z\ \Big (A(x,y,z) \wedge B(x,y,z)\Big ), \nonumber \\ \end{aligned}$$

(90)

where A and B are some propositional functions in three variables, we conclude that for any $\epsilon > 0$ there exists $n_0$ such that for any $n > n_0$

$$\begin{aligned} \sum _{k=0}^\infty P_k|\langle \delta _x*\rho _n^{*k}-\delta _x*\rho ^{*k},\varphi \circ h_{k+1}\rangle | < 2\epsilon , \end{aligned}$$

(91)

and thus, $|\varSigma _x(\rho _n) - \varSigma _x(\rho )| < 2\epsilon$. $\square$

The only assertion that is left to be proven to justify the extension of $\varSigma _x$ and $\varSigma$ from ${\mathbb {P}}_p$ to ${\mathbb {P}}$ is that ${\mathbb {P}}_p$ is dense in ${\mathbb {P}}$. This theorem can be proven in the same way as Theorem 2 but simpler. One can simply take $\psi _n = \rho *\omega _n$.

1.3 Continuity of A and the operators that generate $\sigma$ and $\sigma _x$ for a nonlinear selection function f majorated by an exponential function

The assumption of the Poisson distribution of the individuals in the compartments is essential here. We also assume that the phenotype-fitness relation is defined by a continuous selection function f. By its meaning, f is expected to be nonnegative on the positive semiaxis. We, however, will treat a more general case, which will be useful for the question of an approximation of f. The phenotype is considered to be additive. The notations are the same as in “Appendix A.2”, except that by $\sigma _x^{f,\rho }$ and $\sigma ^{f,\rho }$ we will denote the expressions (11) and (12), respectively, where $h_n :x \mapsto f(x)/n$, and we explicitly indicate the dependence on f and $\rho$. By $\varSigma _x^f$ and $\varSigma ^f$ we will denote the operators $\varSigma _x^f:\rho \mapsto \sigma _x^{f,\rho }$ and $\varSigma ^f :\rho \mapsto \sigma ^{f,\rho }$. We will also denote $N^f \overset{{\text {def}}}{=}\{\rho \in {\mathbb {P}} \,|\, \langle \sigma ^{f,\rho },x\rangle = 0\}$, $N^{f,{\mathscr {I}}} \overset{{\text {def}}}{=}N^f \cap {\mathbb {P}}^{\mathscr {I}}$, and $N^{f,{\mathscr {I}}}_p \overset{{\text {def}}}{=}N^f \cap {\mathbb {P}}^{\mathscr {I}}_p$.

With a nonlinear selection function f, the situation becomes more complicated. In general, both $\sigma ^{f,\rho }$ and $\sigma _x^{f,\rho }$ are not compactly supported densities anymore. Thus, the update operator $A^f :\rho \mapsto \dfrac{\langle \sigma _x^{f,\rho },y\rangle }{\langle \sigma ^{f,\rho },y\rangle }\rho$ may not even be defined on all densities from ${\mathbb {P}}{\setminus } N^f$ or even from ${\mathbb {P}}_p^{\mathscr {I}}{\setminus } N^{f,{\mathscr {I}}}_p$.

We will prove first that the operator in question is, indeed, well defined for some class of functions f.

Theorem 4

Let f be a continuous function. Let a and b be positive real numbers such that $|f(x)| \leqslant a{\text {ch}}bx$ for all $x \in {\mathbb {R}}$. Then for any compactly supported probability density $\rho$ and for any $\lambda > 0$ the expectations of $\sigma _x^{f,\rho }$ and $\sigma ^{f,\rho }$ are finite.

Proof

To prove the theorem we will show that the series involved in $\langle \sigma _x^{f,\rho }, y \rangle$ and in $\langle \sigma ^{f,\rho }, y \rangle$ converge absolutely, namely that

$$\begin{aligned} \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}|\langle \delta _x * \rho ^{*n}, f(y)\rangle |< \infty ,\quad \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}|\langle \rho ^{*n+1}, f(y)\rangle | < \infty . \end{aligned}$$

(92)

First, for any compactly supported probability density $\nu$ we have $|\langle \nu , f(x) \rangle | \leqslant \langle \nu , |f(x)|\rangle$. Then, using the estimate $|f(x)| \leqslant a{\text {ch}}bx$, the expression for the moment generating function $\psi _\rho (s) \overset{{\text {def}}}{=}\langle \rho , e^{sx}\rangle$, and the fact that $\psi _{\delta _a}(s) = e^{as}$, we obtain the estimates

$$\begin{aligned}&\sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}|\langle \delta _x * \rho ^{*n}, f(y)\rangle | \leqslant \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\frac{a}{2}(e^{bx}\psi _\rho (b)^n+e^{-bx}\psi _\rho (-b)^n) \nonumber \\&\quad \leqslant \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}ae^{|bx|}{\tilde{\psi }}_\rho (b)^n = ae^{|bx|-\lambda }\frac{e^{\lambda {\tilde{\psi }}_\rho (b)}-1}{\lambda {\tilde{\psi }}_\rho (b)} < \infty , \end{aligned}$$

(93)

where ${\tilde{\psi }}_\rho (x) = \max \Big (\psi _\rho (x),\psi _\rho (-x)\Big )$, and

$$\begin{aligned}&\sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}|\langle \rho ^{*n+1}, f(y)\rangle | \leqslant \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\frac{a}{2}(\psi _\rho (b)^{(n+1)}+\psi _\rho (-b)^{(n+1)}) \nonumber \\&\quad \leqslant \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}a{\tilde{\psi }}_\rho (b)^{(n+1)} = a\frac{e^{\lambda {\tilde{\psi }}_\rho (b)} - 1}{\lambda e^\lambda } < \infty . \end{aligned}$$

(94)

The last relations in the chains follow from the fact that for any compactly supported probability density $\rho$ its moment generating function $\psi _\rho$ is positive and finite for any value of the argument.$\square$

A counterexample to the theorem’s statement with the dropped condition $|f(x)| \leqslant a{\text {ch}}bx$ is given by the function $e^{x^2}$ and the density $\delta _1$. Indeed, in this case the expressions for $\sigma _x$ and $\sigma$ coincide and we have

$$\begin{aligned} \langle \sigma ^{e^{x^2},\delta _1}_1,y\rangle = \langle \sigma ^{e^{x^2},\delta _1},y\rangle = \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}e^{(n+1)^2}. \end{aligned}$$

(95)

This series is divergent as its positive terms increase with the increase of n.

The statement of theorem can be extended to a wider class of functions f than merely continuous (keeping the majorating condition). This requires a construction of the fully developed theory of integration for Radon measures, which is possible but is not of an interest in this work.

The next two theorem establish the continuity of $A^f$, $\varSigma _x^f$, and $\varSigma ^f$.

Theorem 5

For any interval ${\mathscr {I}}$, under conditions of Theorem 4, the operator $A^f$ is continuous on the subset ${\mathbb {P}}^{\mathscr {I}}{\setminus } N^{f,{\mathscr {I}}}$.

Proof

The proof essentially repeats the proof of Theorem 3.

Let $\rho _n \in {\mathbb {P}}^{\mathscr {I}}$ be a sequence that approaches some $\rho \in {\mathbb {P}}^{\mathscr {I}}$ in ${\mathbb {P}}^{\mathscr {I}}$. We will prove that $\langle \sigma _x^{f,\rho _n},y\rangle$, as functions of x, converge to $\langle \sigma _x^{f,\rho },y\rangle$ uniformly on ${\mathscr {I}}$. The convergence $\langle \sigma ^{f,\rho _n},y\rangle \rightarrow \langle \sigma ^{f,\rho },y\rangle$ is proven analogously. The both facts will imply that, if $\rho \in {\mathbb {P}}^{\mathscr {I}}{\setminus } N^{f,{\mathscr {I}}}$, then

$$\begin{aligned} \frac{\langle \sigma _x^{f,\rho _n},y\rangle }{\langle \sigma ^{f,\rho _n},y\rangle } \rightarrow \frac{\langle \sigma _x^{f,\rho },y\rangle }{\langle \sigma ^{f,\rho },y\rangle } \end{aligned}$$

(96)

uniformly on ${\mathscr {I}}$, and thus, $A^f(\rho _n) \rightarrow A^f(\rho )$ in ${\mathbb {P}}^{\mathscr {I}}{\setminus } N^{f,{\mathscr {I}}}$.

As ${\text {supp}}\rho _n \subset {\mathscr {I}}$ and ${\text {supp}}\rho \subset {\mathscr {I}}$, we have the pointwise convergence $\psi _{\rho _n} \rightarrow \psi _\rho$. Indeed, for any $t \in {\mathbb {R}}$ we have

$$\begin{aligned} \psi _{\rho _n}(t) - \psi _\rho (t) = \langle \rho _n - \rho ,e^{xt}\rangle = \langle \rho _n - \rho ,\eta _t(x)\rangle \rightarrow 0, \end{aligned}$$

(97)

where $\eta _t \in C_{c+}$ such that $\forall x \in {\mathscr {I}}$$\eta _t(x) = e^{xt}$.

Therefore, for any n, we have the following estimate

$$\begin{aligned} |\langle \delta _x*(\rho _n^{*k} - \rho ^{*k}),f(y)\rangle |\leqslant & {} ae^{b{\mathscr {L}}}\left( \left( \sup _m {\tilde{\psi }}_{\rho _m}(b)\right) ^k + {\tilde{\psi }}_\rho (b)^k\right) \nonumber \\\leqslant & {} ae^{b{\mathscr {L}}}\left( \sup _m {\tilde{\psi }}_{\rho _m}(b) + {\tilde{\psi }}_\rho (b)\right) ^k, \end{aligned}$$

(98)

where the notation is the same is in the proof of Theorem 4.

Let us denote $c = \sup \limits _m {\tilde{\psi }}_{\rho _m}(b) + {\tilde{\psi }}_\rho (b)$. It follows that for any n

$$\begin{aligned}&\sum _{k=0}^\infty \frac{e^{-\lambda }\lambda ^k}{(k+1)!}|\langle \delta _x * (\rho _n^{*k} - \rho ^{*k}), f(y)\rangle | \nonumber \\&\quad \leqslant \sum _{k=0}^\infty \frac{ae^{b{\mathscr {L}}-\lambda }(c\lambda )^k}{(k+1)!} = \frac{ae^{b{\mathscr {L}}-\lambda }\left( e^{c\lambda } - 1\right) }{c\lambda } < \infty , \end{aligned}$$

(99)

and therefore, for any $\epsilon > 0$ there is $k_0$ such that $\sum \nolimits _{k = k_0}^\infty \frac{ae^{b{\mathscr {L}}-\lambda }(c\lambda )^k}{(k+1)!} < \epsilon$, and, thus, for any n

$$\begin{aligned} \sum _{k=k_0}^\infty \frac{e^{-\lambda }\lambda ^k}{(k+1)!}|\langle \delta _x * (\rho _n^{*k} - \rho ^{*k}), f(y)\rangle | < \epsilon . \end{aligned}$$

(100)

From the other hand, as $\rho _n \rightarrow \rho$, for any $\epsilon > 0$ and any $k_0$ there is $n_0$ such that for any $n > n_0$

$$\begin{aligned} \sum _{k=0}^{k_0 - 1}\frac{e^{-\lambda }\lambda ^k}{(k+1)!}|\langle \delta _x * (\rho _n^{*k} - \rho ^{*k}), f(y)\rangle | < \epsilon . \end{aligned}$$

(101)

Therefore, by (90), it follows that for any x, $\langle \sigma _x^{f,\rho _n}, y\rangle \rightarrow \langle \sigma _x^{f,\rho }, y\rangle$.

For any $\rho \in {\mathbb {P}}$ with bounded support the function $\langle \delta _x*\rho ,f(y)\rangle$ is continuous in x. Indeed, we have $\langle (\delta _{x'}-\delta _x)*\rho ,f(y)\rangle = \langle \rho , f(y+x') - f(y+x)\rangle$ and $f(y + x') \rightarrow f(y+x)$ uniformly on ${\text {supp}}\rho$ as $x' \rightarrow x$. Therefore, $\langle \sigma _x^{f,\rho _n},y\rangle$ and $\langle \sigma _x^{f,\rho },y\rangle$ are continuous in x as absolutely convergent series of continuous functions. This, in turn, implies that $\langle \sigma _x^{f,\rho _n}, y\rangle \rightarrow \langle \sigma _x^{f,\rho }, y\rangle$ uniformly on ${\mathscr {I}}$ as functions of x. $\square$

Note that $N^{f,{\mathscr {I}}}$ is nowhere dense in ${\mathbb {P}}^{\mathscr {I}}$ and $N^{f,{\mathscr {I}}}_p$ is nowhere dense in ${\mathbb {P}}^{\mathscr {I}}_p$. Indeed, this follows from the implication $\lnot (\langle \sigma ^{f,\rho _n},x\rangle \rightarrow \langle \sigma ^{f,\rho },x\rangle )$$\Rightarrow$$\lnot (\rho _n \rightarrow \rho )$, which, in turn, is equivalent to the proven implication $\rho _n \rightarrow \rho$$\Rightarrow$$\langle \sigma ^{f,\rho _n},x\rangle \rightarrow \langle \sigma ^{f,\rho },x\rangle$. Note also that the statement of the theorem can be extended to ${\mathbb {P}}^K$, where K is any compact set with nonempty interior. In the case when $\forall x > 0$, $f(x) > 0$, we have either $N^{f,{\mathscr {I}}} = \{\delta _0\}$ or $N^{f,{\mathscr {I}}} = \varnothing$, so all these subtleties become irrelevant for the extensions by continuity of $A^f$ from ${\mathbb {P}}^{\mathscr {I}}_p$ to ${\mathbb {P}}^{\mathscr {I}}$.

Theorem 6

Under conditions of Theorem 4, the operators $\varSigma _x^f$ and $\varSigma ^f$ are continuous on ${\mathbb {P}}$.

Proof

Note that for any $\varphi \in C_c$ we have $\sup \nolimits _x \left| \varphi \left( \dfrac{f(x)}{n}\right) \right| \leqslant \sup \nolimits _x |\varphi (x)|$. After that, the proof literally repeats the proof of Theorem 3. $\square$

1.4 Polynomial selection function and sums of exponential with polynomial coefficients

Let the total fitness in a compartment with n individuals characterized by phenotypes $x_1$, ..., $x_n$ be given by $f(x_1 + \cdots + x_n)$, where

$$\begin{aligned} f(x) = a_0 + a_1 x + \cdots + a_m x^m. \end{aligned}$$

(102)

As it is shown in Sect. 4.1, to find $\sigma _x$ and $\sigma$ it is enough to consider $(s^k_n)_x = \langle \delta _x*\rho ^{*n}, y^k \rangle$ and $s^k_n = \langle \rho ^{*n+1},y^k\rangle$ for any k, $0 \leqslant k \leqslant m$, and then to find the sums $\sum \nolimits _n\frac{P_n}{n+1}(s^k_n)_x$ and $\sum \nolimits _n\frac{P_n}{n+1}s^k_n$, $P_n = e_{}^{-\lambda }\lambda ^n/n!$.

Note that the operation of the multiplication of a generalized function $\rho$ by the parameter ($x\rho$) is a derivation on a convolution algebra, that is it is linear and obeys the Leibniz rule:

$$\begin{aligned} x(\rho _1 * \rho _2) = (x\rho _1)*\rho _2 + \rho _1*(x\rho _2). \end{aligned}$$

(103)

Indeed, for any $\varphi \in C_c$ we have

$$\begin{aligned}&\langle x(\rho _1*\rho _2), \varphi \rangle = \langle \rho _1*\rho _2, x\varphi \rangle = \langle \rho _1\otimes \rho _2,(x_1+x_2)\varphi (x_1+x_2)\rangle \nonumber \\&\quad =\langle (x_1\rho _1)\otimes \rho _2 + \rho _1\otimes (x_2\rho _2),\varphi (x_1+x_2)\rangle = \langle (x\rho _1)*\rho _2 + \rho _1*(x\rho _2),\varphi \rangle .\nonumber \\ \end{aligned}$$

(104)

As $\langle \rho ,y^m\rangle = \overline{x^m}$ and $\langle \delta _x,y^m\rangle = x^m$, $(s^k_n)_x = \langle y^k(\delta _x*\rho ^{*n}),1\rangle$ is a linear combination of all expressions of the form $x^\alpha \prod \nolimits _i\overline{x^{\beta _i}}^{\gamma _i}$, where all $\beta _i$ are different and $\alpha + \sum \nolimits _i(\beta _i + \gamma _i) = k$. For example, for $k=2$ we have $x^2$, $x {\bar{x}}$, ${\bar{x}}^2$, $\overline{x^2}$, for $k = 3$ we have $x^3$, $x^2 {\bar{x}}$, $x {\bar{x}}^2$, $x \overline{x^2}$, ${\bar{x}}^3$, ${\bar{x}} \overline{x^2}$, $\overline{x^3}$, for $k = 4$ we have $x^4$, $x^3{\bar{x}}$, $x^2{\bar{x}}^2$, $x^2 \overline{x^2}$, $x {\bar{x}}^3$, $x {\bar{x}} \overline{x^2}$, $x \overline{x^3}$, ${\bar{x}}^4$, ${\bar{x}}^2\overline{x^2}$, ${\bar{x}} \overline{x^3}$, $\overline{x^2}^2$, $\overline{x^4}$, etc. These expressions enter $(s^k_n)_x$ with coefficients that are (nonnegative) polynomials in n of the form $n(n-1)\ldots (n-l+1)a_l(k)$ or just $a_0(k)$ ($a_i(k) \in {\mathbb {N}}$). The polynomial (in k) functions $a_i(k)$ can be in principle found using some combinatorics. At the very least, they are algorithmically computable.

The sums $\sum \nolimits _n P_n (s^k_n)_x/(n+1)$ can be evaluated using the identity

$$\begin{aligned} \sum _{n=0}^\infty \frac{\lambda ^n}{n!}\frac{n(n-1)(n-2)\ldots (n-p+1)}{n+1} = \lambda ^p\sum _{n=0}^\infty \frac{\lambda ^n}{n!}\frac{1}{n+p+1} = \lambda ^p \frac{d}{d\lambda ^p}\frac{e_{}^\lambda - 1}{\lambda }.\nonumber \\ \end{aligned}$$

(105)

Likewise, $s^k_n$, being $\langle x^k \rho ^{*n+1},1 \rangle$, is a linear combination of all expressions of the form $\prod \nolimits _i\overline{x^{\beta _i}}^{\gamma _i}$, where all $\beta _i$ are different and $\sum \nolimits _i(\beta _i + \gamma _i) = k$. The coefficients in front these expressions in $s^k_n$ are of the form $(n+1)n(n-1)\ldots (n-l+1)a_l(k)$, $l \in {\mathbb {N}}$. The summation of $\sum \nolimits _n P_n s^k_n/(n+1)$ is trivial.

Let us now consider that the total fitness in a compartment with n individuals with phenotypes $x_1,\ldots ,x_n$ is given by $f(x_1+\cdots +x_n)$, where f is a linear combination of exponential functions with polynomial coefficients:

$$\begin{aligned} f(x) = \sum _{j=1}^m p_j(x) e_{}^{a_j x}, \end{aligned}$$

(106)

where $p_j$ are polynomials. It is again sufficient to find $\langle \delta _x*\rho ^{*n}, y^k e_{}^{ay} \rangle$ and $\langle \rho ^{*n+1}, y^k e_{}^{a y} \rangle$ for any k and a. Note that $\langle x\rho , e_{}^{ax}\rangle = \left. \dfrac{d}{ds}\psi (s)\right| _{s=a}$, where $\psi (a) = \langle \rho , e_{}^{ax} \rangle$ [this is directly related to (103)]. Therefore, $\langle \delta _x*\rho ^{*n},y^k e_{}^{ay}\rangle = \left. \dfrac{d^k}{ds^k}(e_{}^{sx}\psi (s)^n)\right| _{s=a}$ and $\langle \rho ^{*n+1},y^k e_{}^{ay}\rangle = \left. \dfrac{d^k}{ds^k}\psi (s)^{n+1}\right| _{s=a}$. These expressions can be summed with $e_{}^{-\lambda }\lambda ^n/(n+1)!$ in the same way as in the pure polynomial case. Thus, every function f of the form (106) allows to write the update equation in closed form.

1.5 Approximation of f using truncated Fourier series

We assume the Poisson distribution of the individuals in the compartments. We also assume that the phenotype-fitness relation is defined by a continuous selection function f. The activity is considered to be additive. By $\sigma _x^{f,\rho }$ and $\sigma ^{f,\rho }$ we will denote the expressions (11) and (12), respectively, where $h_n :x \mapsto f(x)/n$, and we explicitly indicate the dependence on f and $\rho$.

We will call a trigonometric polynomial of order n and period T any function from ${\mathbb {R}}$ to ${\mathbb {C}}$ of the form

$$\begin{aligned} p(x) = \sum _{k=-\,n}^{n}c_k e^{\frac{2\pi ikx}{T}},\quad c_k \in {\mathbb {C}},\quad c_{-n}c_n \ne 0. \end{aligned}$$

(107)

The trigonometric polynomial p is called real if $p({\mathbb {R}}) \subset {\mathbb {R}}$, where the second ${\mathbb {R}}$ is understood as the natural embedding in ${\mathbb {C}}$. p is real if and only if $\forall k$$c_{-k} = {\bar{c}}_k$.

Let L(I) mean the length of the interval I. We call the truncated to order n Fourier series of the function f on the interval I the trigonometric polynomial

$$\begin{aligned} f_n(x) = \sum _{k=-\,n}^n a_k e^{\frac{2\pi ikx}{L(I)}},\quad a_k = \frac{1}{L(I)}\int _I f(x)e^{-\frac{2\pi ikx}{L(I)}}dx. \end{aligned}$$

(108)

We need the following known fact.

Theorem 7

For any $\epsilon > 0$ and any periodic continuous function f with period T there exists a trigonometric polynomial p with period T such that $\sup \nolimits _x |f(x) - p(x)| < \epsilon$. Furthermore, p can be constructed from the Fourier series of f, namely if $f_n$ is the truncated to order n Fourier series of f, then, for large enough n, one can take $p = (f_0 + f_1 + \cdots + f_n)/(n+1)$.

The proof can be found, for example, in the book by Rudin (1976). The last statement is known as Fejér’s theorem. Note that all p constructed in this way for a real function f are real.

We start with an observation that for any probability density $\rho$ the following holds. If ${\text {supp}}\rho \subset [-d,d]$, then ${\text {supp}}\rho ^{*n} \subset [-nd, nd]$ and ${\text {supp}}\delta _x * \rho ^{*n-1} \subset [-nd, nd]$ for any $x \in {\text {supp}}\rho$. For any $\rho$ we will denote $I^\rho$ some interval $I^\rho = [-d,d]$ such that ${\text {supp}}\rho \subset I^\rho$. For example, one can use the smallest interval with these properties. We will also introduce intervals $I^\rho _n = [-nd,nd]$ (not to be confused with $I_k$ used later) for the same d that defines $I^\rho$.

We will prove the following main theorem.

Theorem 8

Let f be a continuous function $f:{\mathbb {R}} \rightarrow {\mathbb {R}}$ exponentially bounded in the following sense: There are positive numbers a and b such that $|f(x)| \leqslant a{\text {ch}}bx$ for all x. Then for any compactly supported probability density $\rho$ there exists a sequence $k \mapsto (I_k,p_k)$ of pairs of closed intervals $I_k$ and of real trigonometric polynomials $p_k$, where $p_k$ approximates f on $I_k$, such that $\langle \sigma _x^{p_k,\rho },y\rangle \rightarrow \langle \sigma _x^{f,\rho },y\rangle$ homogeneously on ${\text {supp}}\rho$ as a function of x, $\langle \sigma ^{p_k,\rho },y\rangle \rightarrow \langle \sigma ^{f,\rho },y\rangle$ as a sequence of numbers, while $\sigma _x^{p_k,\rho } \rightarrow \sigma _x^{f,\rho }$ for any $x \in {\text {supp}}\rho$ and $\sigma ^{p_k,\rho } \rightarrow \sigma ^{f,\rho }$ in the sense of generalized functions. $p_k$ can be constructed using Fourier series approximations of f on the appropriate intervals.

As the logic of the proof is slightly convoluted, we will first formulate and prove two auxiliary lemmas.

Lemma 2

Under conditions of Theorem 8, for any $\epsilon > 0$ there exists $n_0 > 0$ such that for any function p that is bound by $\sup \limits _{x \in {\mathbb {R}}} |p(x)| < \sup \limits _{x \in I^\rho _{n_0}} |f(x)| + \epsilon$ the following holds:

$$\begin{aligned}&\forall x \in {\text {supp}}\rho \quad \displaystyle \left| \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\langle \delta _x * \rho ^{*n}, f - p \rangle \right| < \epsilon \quad \text {and} \end{aligned}$$

(109)

$$\begin{aligned}&\qquad \qquad \qquad \qquad \displaystyle \left| \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\langle \ \rho ^{*n+1}, f - p \rangle \right| < \epsilon . \end{aligned}$$

(110)

Proof

By the virtue of the estimate on f from the conditions of Theorem 8, the relations of the statement of the lemma follow from the relations

$$\begin{aligned}&\forall x \in {\text {supp}}\rho \quad \displaystyle \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\langle \delta _x * \rho ^{*n}, |f| + \epsilon + a {\text {ch}}\big (b L(I^\rho _{n_0})\big ) \rangle < \epsilon \quad \text {and} \end{aligned}$$

(111)

$$\begin{aligned}&\qquad \qquad \qquad \qquad \displaystyle \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\langle \ \rho ^{*n+1}, |f| + \epsilon + a {\text {ch}}\big (b L(I^\rho _{n_0})\big ) \rangle < \epsilon . \end{aligned}$$

(112)

Let us prove (111). Indeed, using the same reasoning as in the proof of Theorem 4, we have $\forall x \in {\text {supp}}\rho$

$$\begin{aligned}&\sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\langle \delta _x * \rho ^{*n}, |f(y)| + a {\text {ch}}\big (b L(I^\rho _{n_0})\big ) + \epsilon \rangle \nonumber \\&\quad \leqslant \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}\langle \delta _x * \rho ^{*n}, a {\text {ch}}\big (by) + a {\text {ch}}\big (b L(I^\rho _{n_0})\big ) + \epsilon \rangle \nonumber \\&\quad < \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \left( ae^{b|x|}{\tilde{\psi }}_\rho (b)^n + a\left( e^{L(I^\rho )}\right) ^{n_0} + \epsilon \right) \nonumber \\&\quad \leqslant a\sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!} e^{bx_0}{\tilde{\psi }}_\rho (b)^n + a\sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \left( e^{L(I^\rho )}\right) ^n + \epsilon \sum _{n=n_0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!}, \nonumber \\ \end{aligned}$$

(113)

where $\psi _\rho (t) = \langle \rho ,e^{tx}\rangle$, ${\tilde{\psi }}_\rho (t) = \max \Big (\psi _\rho (t), \psi _\rho (-t)\Big )$, and $x_0 = \max (|\inf {\text {supp}}\rho |, |\sup {\text {supp}}\rho |)$.

The statement (111) follows from the fact that the series in the last expression have positive terms and converge to finite numbers, if summed started from $n = 0$. Indeed, this means that for any $\epsilon > 0$ there is $n_0$ such that the last expression is smaller than $\epsilon$. It can be proven analogously that the same $n_0$ fulfills the statement (112). $\square$

Lemma 3

Under conditions of Theorem 8, for any $\epsilon > 0$ and any $n_0 > 0$ there exists a real trigonometric polynomial p such that $\sup \nolimits _{x \in I^\rho _{n_0}} |f(x) - p(x)| < \epsilon$, $\sup \nolimits _{x \in {\mathbb {R}}}|p(x)| < \sup \nolimits _{x \in I^\rho _{n_0}}|f(x)| + \epsilon$, as well as

$$\begin{aligned}&\forall x \in {\text {supp}}\rho \quad \displaystyle \left| \sum _{n=0}^{n_0-1} \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \langle \delta _x * \rho ^{*n}, f - p \rangle \right| < \epsilon \quad \text {and} \end{aligned}$$

(114)

$$\begin{aligned}&\qquad \qquad \qquad \qquad \displaystyle \left| \sum _{n=0}^{n_0-1} \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \langle \rho ^{*n+1}, f - p \rangle \right| < \epsilon . \end{aligned}$$

(115)

Proof

Let d be the number such that $I^\rho = [-d,d]$. For any $n \leqslant n_0-1$, we have ${\text {supp}}\delta _x * \rho ^{*n} \subset I^\rho _{n_0}$ and ${\text {supp}}\rho ^{*n+1} \subset I^\rho _{n_0}$. Let us extend $f|_{I^\rho _{n_0}}$ to $I^\rho _{n_0+1}$ by a function ${\hat{f}}$ such that ${\hat{f}}(x) = \alpha x + \beta _1$ for any $x \in [-(n_0+1)d, -n_0 df]$ and ${\hat{f}}(x) = \alpha x + \beta _2$ for any $x \in [n_0 d, (n_0+1)d]$, where $\alpha$, $\beta _1$, and $\beta _2$ are selected to fulfill

$$\begin{aligned} {\hat{f}}\big (-(n_0+1)d\big )= & {} {\hat{f}}\big ((n_0+1) d\big ) = \frac{f(-n_0 d) + f(n_0 d)}{2},\quad \nonumber \\ {\hat{f}}(-n_0 d)= & {} f(-n_0 d),\quad {\hat{f}}(n_0 d) = f(n_0 d). \end{aligned}$$

(116)

Function ${\hat{f}}$ is continuous on $I^\rho _{n_0+1}$ and can be extended to the whole ${\mathbb {R}}$ as a periodic continuous function $\check{f}$ with period $L(I^\rho _{n_0+1})$. By Theorem 7, using the truncations of the Fourier series for $\check{f}$ up to some order, we can construct a real trigonometric polynomial p such that for any $\epsilon > 0$  $\sup \nolimits _{x \in {\mathbb {R}}} |\check{f}(x) - p(x)| < \epsilon$. It follows that for any $\epsilon > 0$ we can find p such that $\sup \nolimits _{x \in I^\rho _{n_0}} |f(x) - p(x)| < \epsilon$. Furthermore, by construction, $\sup \nolimits _{x \in {\mathbb {R}}}|p(x)| < \sup \nolimits _{x \in I^\rho _{n_0}}|f(x)| + \epsilon$.

Let us consider the right-hand side of the statement (114). For any $x \in {\text {supp}}\rho$ we have

$$\begin{aligned} \sum _{n=0}^{n_0-1} \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \langle \delta _x * \rho ^{*n}, |f - p| \rangle< \epsilon \sum _{n=0}^{n_0-1} \frac{e^{-\lambda }\lambda ^n}{(n+1)!} < \epsilon g(\lambda ) \leqslant \epsilon . \end{aligned}$$

(117)

Here we essentially used the inclusion ${\text {supp}}\delta _x * \rho ^{*n} \subset I^\rho _{n_0}$ for any $x \in {\text {supp}}\rho$. The relation (117) implies the relation (114).

The statement (115) can be proven analogously, using the fact that ${\text {supp}}\rho ^{*n+1} \subset I^\rho _{n_0}$ for any $n < n_0$. $\square$

Proof of Theorem 8

Let us symbolically rewrite the statement of Lemma 2 as $\forall \epsilon> 0\,\exists n_0 > 0\,\forall p:A(\epsilon , n_0, p)$. In the same manner, let us symbolically rewrite the statement of Lemma 3 as $\forall \epsilon> 0\,\forall n_0 > 0\,\exists p:B(\epsilon , n_0, p)$. Then (90) implies $\forall \epsilon>0\, \exists n_0 > 0\, \exists p:A(\epsilon , n_0, p) \wedge B(\epsilon , n_0, p)$. As the statement A estimates the sums from 0 to $n_0 - 1$, and the statement B estimates the sums from $n_0$ to $\infty$, the statement $A \wedge B$ gives estimates on the sums from 0 to $\infty$. Therefore, the consequence of the above is explicitly read as following. For any positive $\epsilon$ there exists a number $n_0$ and a trigonometric polynomial p (which can be constructed based on an approximation of f by truncated Fourier series, as in the proof of Lemma 3) such that $\sup \nolimits _{x \in I^\rho _{n_0}} |f(x) - p(x)| < \epsilon$ and

$$\begin{aligned}&\forall x \in {\text {supp}}\rho \quad \displaystyle \left| \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \langle \delta _x * \rho ^{*n}, f - p \rangle \right| < \epsilon \quad \text {and} \end{aligned}$$

(118)

$$\begin{aligned}&\qquad \qquad \qquad \quad \,\,\displaystyle \left| \sum _{n=0}^\infty \frac{e^{-\lambda }\lambda ^n}{(n+1)!} \langle \rho ^{*n+1}, f - p \rangle \right| < \epsilon . \end{aligned}$$

(119)

The part of the theorem with $\langle \sigma _x^{p_k,\rho },y\rangle \rightarrow \langle \sigma _x^{f,\rho },y\rangle$ and $\langle \sigma ^{p_k,\rho },y\rangle \rightarrow \langle \sigma ^{f,\rho },y\rangle$ is proven, if we take, for example, $\epsilon _k = 1/(1 + k)$, $k \in {\mathbb {N}}$, and if we take as $I_k$ the interval $I^\rho _{n_0}$ and as $p_k$ the trigonometric polynomial p, where $n_0$ and p are provided by the above statement for $\epsilon = \epsilon _k$.

Let us choose any $\varphi \in C_c$. Let us denote $\varPhi \overset{{\text {def}}}{=}\sup \nolimits _x |\varphi (x)|$ and $s_k \overset{{\text {def}}}{=}\sup \nolimits _{x \in I_k} |\varphi \left( p_k(x)\right) - \varphi \left( f(x)\right) |$. As $\varphi$ is continuous and compactly supported it is also uniformly continuous on ${\mathbb {R}}$. Therefore, we have $s_k \rightarrow 0$, as $k \rightarrow \infty$. Furthermore, for any n we have

$$\begin{aligned} \sup _{x \in I_k} \left| \varphi \left( \frac{p_k(x)}{n+1}\right) - \varphi \left( \frac{f(x)}{n+1}\right) \right| \leqslant s_k \end{aligned}$$

(120)

and

$$\begin{aligned} \sup _x \left| \varphi \left( \frac{p_k(x)}{n+1}\right) - \varphi \left( \frac{f(x)}{n+1}\right) \right| \leqslant 2\varPhi . \end{aligned}$$

(121)

Using the same technique of splitting the series into two parts by $n_0(k) = n_0(\epsilon _k)$ and using the former estimate for the initial part of the sum of the series and the latter estimate for the rest of the series by the same logic and taking into account that $n_0(\epsilon )$ provided by the Lemma 2 ever grows with the decay of $\epsilon$, we conclude that

$$\begin{aligned} |\langle \sigma _x^{p_k,\rho } - \sigma _x^{f,\rho },\varphi \rangle | \leqslant s_k + 2\varPhi e^{-\lambda }e_{n_0(k)}^\infty (\lambda ) \rightarrow 0,\quad k \rightarrow \infty , \end{aligned}$$

(122)

where $\displaystyle e_m^\infty (x) \overset{{\text {def}}}{=}\sum \nolimits _{j = m}^\infty \frac{x^j}{j!}$.

In the same way we prove $\sigma ^{p_k,\rho } \rightarrow \sigma ^{f,\rho }$. $\square$

This theorem implies that for any compactly supported $\rho$ with $\langle \sigma ^{f,\rho },x\rangle \ne 0$ we have the convergence $A^{p_k}(\rho ) \rightarrow A^f(\rho )$, as $k \rightarrow \infty$. Note that the sequence $p_k$ depends on $\rho$. This is due to the involvement of $\psi _\rho$ in (113). However, if f is a bounded function, then this dependence can be dropped from (113), and the choice of $n_0$, and thus of $p_k$, becomes independent of the current distribution. In this case we can state the pointwise convergence $A^{p_k} \rightarrow A^f$ (on ${\mathbb {P}}^K$ for some compact K). The proof of the theorem is constructive. However, it is not optimized for applications.