Adaptation, fitness landscape learning and fast evolution

2.1 Genome

We assume that the genotype can be described by Boolean strings of length N _g , where N _g is the number of genes. Then

where s _i = 1 means that gene i is activated (switched on) and s _i = 0 means that it is repressed (switched off). Correspondence between Boolean hypercube and genotypes is considered for example in 12.

2.2 Phenotypic traits

Although phenotype is controlled by genes, it is also influenced by environmental conditions and various epigenetic processes. In this paper, we suppose that phenotypic traits are controlled by genotype only. We consider levels f _j of expressions of those traits as real variables in the interval (0, 1). Then the vector f = ( f ₁, . . . , f _{N b} ) can be considered to represent the organismal phenotype. We suppose that

where f _j ∈ (0, 1) is a real valued function of the Boolean string s, the genotype.

Only a part of s _i is involved in f _j . Namely, for each j we have a set of indices K _j = { i ₁, i ₂, . . ., i _{n j} } such that f _j depends on s _i with i ∈ K _j , so that

where i _l ∈ K _j and n _j is the number of genes involved in the control of the trait expression. The representation of phenotype by the quantities f _j is suggestive of quantitative traits because the f _j are real valued. The limiting values of 0 or 1 suggest another interpretation, however, in terms of cell type. Multicellular organisms consist of cells of different types. One can suppose that the organismal phenotype is defined completely by the corresponding cell pattern. The cell type j is determined by morphogenes, which can be identified as gene products or signaling molecules that can change cell type (or genes that code for signaling molecules that can determine cell types or cell-cell interactions and then finally the cell pattern). The morphogene activity is defined by ( 2.2).

We further suppose:

Assumption M. Assume activities f _j have the following properties.

The sets K _j are independent uniformly random subsets of S _g = {1, . . . N _g }

We denote the total number of genes involved in regulation of all f _j by N _r , where $N_r=\sum n_j\le K{N}_b\mathrm{.}$

Assumption M implies that the genetic control of the phenotype is organized, in a sense, randomly, and that only a portion of the full set of genes controls phenotypic traits. That modularity of gene control is well known from experimental data (see 18, 19) and for evolution computation problems it was studied, for example, in 16.

Consider an example, where the assumption M holds, where we have a saturated expression, inspired by earlier work ^{20, 21} . Let

where j = 1, . . . , N _b . Here σ( z) is a sigmoidal function of real z such that

and w _ij , h _j are some coefficients (their meaning will be explained below). As an example, we can take σ( S) = (1 + exp( −bS)) ⁻¹, where b > 0 is a sharpness parameter. Note that for large b this sigmoidal function tends to the step function and for b = + ∞ our model becomes a Boolean one. The parameters h _j defines thresholds for trait expression ²⁰ . The relation ( 2.4) can be interpreted as a simple mathematical model for quantitative trait locus (QTL) action.

To understand the role of h _j consider a trait f _j and suppose that for a well adapted organism f _j ≈ 1. Let, for simplicity, w _ij take the values 1, 0, or −1. Then the parameter h _j defines how many genes involved in the control of the f _j expression should be activators and how many should be repressors. Let the numbers of activator and repressor genes be $n_j^{\pm }$ respectively.

Then f _j ≈ 1 if $n_j^{+}-n_j^{-}\gg h_j\mathrm{.}$

One can suppose that h _j describes a direct influence of environment on phenotype, such as stress, that can exert epigenetic effects. In Section 2.5 using data from 3 we will show that the model defined by ( 2.4) are capable to describe main topological characteristics of really observed fitness functions in the case of mimicry, camouflage and thermoregulation for insects.

Let us introduce the matrix W of size N _b × N _g with the entries w _ij . The coefficients w _ji determine the effects of terminal differentiation genes (see 23), and hence encodes the genotype-phenotype map. We assume that the coefficients w _ji are random, with the probability that w _ji > 0 or that w _ji < 0 is β/2 N, where β > 0 is a parameter. This quantity β << N defines a genetic redundancy, i.e., averaged numbers of genes that control a trait. Note that then large β ≫ 1 one has n _j < 4 β with the probability Pr _β , which is exponentially close to 1: Pr _β > 1 – exp(−0.1 β), thus, the number n _j are bounded.

2.3 Fitness

We know little about the details of how fitness relates to the phenotype of multicellular organisms, and for that reason classic neo-Darwinian theory takes fitness to be a function of genotype. Some models which take account of epistasis have been proposed ²⁴ . The random field models assign fitness values to genotypes independently from a fixed probability distribution. They are close to mutation selection models introduced by 25, and can be named House of Cards (HoC) model. The best known model of this kind is the NK model introduced by Kauffman and Weinberger ²⁶ , where each locus interacts with K other loci. Rough Mount Fuji (RMF) models are obtained by combining a random HoC landscape with an additive landscape models ²⁷ . In evolution computations (EC) some artificial fitness models were used, for example OneMax and Leading Ones to test evolution algorithms, see for example ¹⁵ .

In this work, we use the classical approach of R. Fisher by introducing an explicit representation of phenotype, f, and allow it to determine fitness through interaction with an environment b. That is, we assume that the phenotype is completely determined by the phenotype trait expression, and thus the fitness depends on the genotype s via f _j .

We express the relative fitness F and its dependence on environment b via an auxiliary function W via the relation

where K _F is a positive constant and b = ( b ₁,..., b _{N b} ) is a vector consisting of coefficients b _j , respectively. Below we refer to W as a fitness potential, and we assume that

Sometimes, if the parameter b is fixed, we shall omit the corresponding argument in notation for W and F.

We consider fitness as a numerical measure of interactions between the phenotype and an environment. For a fixed environment, this idea gives us the fitness of classical population genetics. A part of the fitness, however, depends on the organism developing properly and for now we represent it as independent of the environment, although we are aware that this is not always the case. Note that some coefficients b _j may be negative and others may be positive, and that the model ( 2.7) can describe gene epistatic effects via dependence of f _j on s if f _j are nonlinear in s.

The expression ( 2.7) can serve as a rough approximation of the fitness function in the case of insects such as grasshoppers or fruit flies. In fact, important factors, which determine insect survival, are thermoregulation, mimicry and camouflage levels ^{18, 22, 28} . All those factors depend on colour pigmentation pattern. Blackwhite pigmentation patterns can be roughly described by vectors f = ( f ₁, f ₂,..., f _{N b} ), where f _j ≈ 1 and f _j ≈ 0 mean that the cell j is black, or white, respectively, Then thermoregulation depends on ${\displaystyle {\sum }_j{f}_j}.$ The mimicry level can be approximately defined by expression ${\displaystyle {\sum }_j\left|f_j-f_j^{*}\right|}$ , where f _* is a target pattern corresponding to an insect to mimic. Colour patterns can be also described by classical RGB formalism.

The representation of the fitness as a sum of terms ( 2.7) is of course a rough approximation; however if assumption M holds that representation is consistent with important observed facts. First, mutations have been identified that alter one part of the pigment pattern without affecting any other. This independence of different pattern parts can be explained by the modular organization of the genetic regulation that controls pigmentation. In the course of evolution, different aspects of the pigment pattern have clearly evolved independently of each other ¹⁸ . Second, the topology of the fitness landscapes was studied in 22 by field experiments in the case of insect mimicry. Main conclusions are as follows. A number of studies of fitness landscapes in natural populations have demonstrated low fitness of intermediate phenotypes, i.e., existence of valley in the fitness landscape. It is found ²² that natural selection promotes genetic architecture preventing the expression of intermediate phenotypes. Close fitness peaks are separated by ridges, favouring colour pattern switches and allowing drift from local peaks.

In Section 2.5 we will show that the fitness model defined by ( 2.4) and ( 2.7) have those topological properties.

2.4 Population dynamics model

For simplicity, we consider populations with asexual reproduction. (Although a part of the results remain valid for sexual reproduction, as we discuss at the end of this subsection). We choose initial genotypes randomly from a gene pool and assign them to organisms. This choice is invariant with respect to the population member, i.e,. the probability to assign a given genotype s to a member of the population does not depend on that member.

In each generation, there are N _pop( t) individuals, the genome of each of which is denoted by s( t), where t = 0, 1, 2,... stands for the evolution step number). Following the classical Wright-Fisher ideas, we suppose that generations do not overlap. In each generation (i.e., for each t), the following three steps are performed:

Conditions 1 and 2 imply that mutations in the genotypes create a new genetic pool and then a new round of selection starts. Condition 3 expresses the fundamental ecological limitation that all environments can only support populations of a limited size. If N _popmax ≫ 1 then by ( 2.8) and the Central Limit Theorem one can show, under some additional conditions, (see Section 4) that fluctuations of the population size are small, and thus the population is ecologically stable and N _pop( t) ≈ N _popmax.

In the limit case of infinitely large populations we will write the discrete dynamical equation for the time evolution of the frequency X( s, t) of the genotype s in the population as

where $\overline{F}$ ( t) is the average fitness of the population at the moment t defined by

where S( t) is the set of genotypes existing in the population at time t (the genetic pool) and X( s, t) = N( s, t) /N _pop( t) is the frequency of the genotype s. Here N( s, t) denotes the number of the population members with the genotype s at the step t.

The equations ( 2.9) do not take mutations into account. They describe only changes in the genotype frequencies because of selection at the t-th time step. The same equations govern evolution in the case of sexual reproduction in the limit of weak selection ^{6, 10} . Note that for an evolution defined by ( 2.9), the average fitness $\overline{F}$ ( t) defined by ( 2.10) satisfies Fisher’s theorem, so that this function increases at each time step t: $\overline{F}$ ( t + 1) ≥ $\overline{F}$ ( t).

2.5 Adaptation as a hard combinatorial problem

Adaptation (i.e., maximization of fitness in a changing environment) is a very hard problem since over evolutionary history we observe the coevolution of many traits accompanied by changes in many genes. In its general context, this is a problem in the theory of macroevolution, which in general requires the integration of population genetics and developmental biology for its full understanding. There are two key components of this problem. First, development is itself a dynamical process operating over time. Second, there is a combinatorial component of development wherein different combinations of gene must be expressed in different cell types. This combinatorial aspect of the problem means that straightforward theoretical methods of considering the relationship between gene expression and a changing environment that have been very successful in single celled organisms ²⁹ cannot be applied to metazoa. In this work, for the sake of tractability, we focus on the combinatorial aspect of the problem and neglect developmental dynamics. Even at the highly simplified level of our model, adaptation is a hard computational problem, as we now demonstrate.

Consider the case, where f _j are defined by relations ( 2.4) and assume that

As a consequence of the second assumption, F attains its maximum for f ₁ = 1, f ₂ = 1,..., f _{N b} = 1. Let us show that, even in this particular case, the problem of the fitness maximization with respect to s is very complex. In fact, for a choice of h _j it reduces to the famous NP-complete problem, so-called K-SAT, which has received a great deal of attention in the last few decades (see 30– 35). The K-SAT can be formulated as follows.

K -SAT problem. Let us consider the set V _n = { s ₁,..., s _n } of Boolean variables s _i ∈ {0, 1} and a set _m of m clauses. The clauses C _j are disjunctions (logical ORs) involving K literals z _{i 1} , z _{i 2} ,..., z _{i k} , where each z _i is either s _i or the negation $\overline{s}$ _i of s _i. The problem is to test whether one can satisfy all of the clauses by an assignment of Boolean variables.

Cook and Levin ^{30, 31} have shown that the K-SAT problem is NP-complete and therefore in general it is not feasible in a reasonable running time. In subsequent studies—for instance, by 32 —it was shown that K-SAT of a random structure is feasible under the condition that N _b < α _c ( K) N _g , where α _c ( K) ≈ 2 ^K log 2 for large K.

The set _K of solutions of random K-SAT has a nontrivial structure depending on parameter α = N _b/N _g ^{33, 35} . For sufficiently small α < α _g ( K), where α _g ( K) ≈ 2 ^K log( K) /K is some critical parameter, the set _K forms a giant cluster, where nearest solutions are connected by a single flip and one can go from a solution to another by a sequence of single flips (pointed mutations) ³⁵ . For α ∈ ( α _g , α _d ), where α _d ( k) < α _c is another critical value, solutions form a set of disconnected clusters. The local search algorithms do not work in the domain α > α _g .

Probably, for evolution context K-SAT was applied first in 36, where it was used for an investigation of speciation problem.

To see the connection of our model with K-SAT, consider equation ( 2.4) supposing that w _ij ∈ {1, 0, −1} and h _j = −C _j + 0.5, where C _j is the number of negative w _ji in the sum $S_j={\displaystyle {\sum }_{i=1}^{N_g}{w}_{ji}{s}_i\mathrm{.}}$ We set m = N _b and n = N _g . Under this choice of h _j , the terms σ( S _j ) can be represented as disjunctions of literals z _j . Each literal z _j equals either s _j or $\overline{s}$ _j , where $\overline{s}$ _j denotes negation of s _j . To maximize the fitness, we must assign s _j such that all disjunctions will be satisfied. If we fix the number n _j of the literals participating in each disjunction (clause) and set n _j = K, this assignment problem is precisely the K-SAT problem formulated above.

Reduction to the K-SAT problem is a transparent way of representing the idea that multiple constraints need to be satisfied. The number K defines the gene redundancy and the probability of gene pleiotropy. Remind that pleiotropy occurs when one gene influences two or more seemingly unrelated phenotypic traits. The threshold h _j and K define the number of genes which need be flipped in order to attain a high expression of the trait f _j . Note that gene pleiotropy is a fundamental characteristics ³⁷ , which is studied for real organisms only recently (see 19). We can compare experimental observations and consequence of model ( 2.4), which is a generalisation of K-SAT (compare plot Figure 1 and plots on Figure 1 in 19). So, we can fit our model parameters using real data. Moreover, we can check validity of our model by the following arguments.

Figure 1. Frequency distributions of degree of gene pleiotropy for model ( 2.4) with the parameters N _g = 4000, β = 4, h = 0, N _b = 3000.

We note that, in the case of giant cluster formation, the topological properties of the solution set _K , mentioned above, outline the properties of really observed fitness landscapes ²² : existence of many peaks, valleys and ridges connecting peaks. Namely, existence of many solutions of K-SAT, when a giant cluster exists, means that the landscape has a number of peaks separated by valleys. On the other hand, connectance of solutions within the giant cluster can be interpreted that there exist ridges that connect peaks.

Note that there are important differences between K-SAT in Theoretical Computer Science and fitness maximization problems. First, the signs of b _j are unknown for real biological situations since the fitness landscape is unknown. Second, our adaptation problem involves the threshold parameters h _j (see ( 2.4)). In contrast to K-SAT, in our case the Boolean circuit is plastic, because the h _j are not fixed.

If the b _j are unknown, the adaptation (fitness maximization) problem becomes even harder because we do not know the function to optimize. Therefore, many algorithms for K-SAT are useless for biological adaptation problems. Below we will nonetheless obtain some analytical results based on the assumption that b _j are random.

3.1 Fitness landscape learning theorems

For simplicity, we consider asexual reproduction. To obtain similar results for sexual reproduction, one can consider a weak selection regime and use the results of 10, where eq. ( 2.9) are derived.

Let us introduce two sets of indices I ₊ and I ₋ , such that I ₊ ∪ I ₋ = {1,..., N _b }. We refer to these sets in the sequel as positive and negative sets, respectively. We have

The biological interpretation of that definition is transparent: the expression of the traits f _j with j ∊ I ₊ increases the fitness and for j ∊ I _– expression of the trait decreases the fitness.

Let s and $\overline{s}$ be two genotypes. Then we denote by Diff( s, $\overline{s}$ ) the set of positions i such that s _i ≠ $\overline{s}$ _i :

The set Diff( s, $\overline{s}$ ) indicates which genes in s should be flipped in order to obtain $\overline{s}$ .

We formulate two theorems on fitness landscape learning. First we consider the case of infinitely large populations.

Theorem 3.1. Suppose that the evolution of the genotype frequencies X( s, t) is determined by equations ( 2.9) and ( 2.10). Moreover, assume that

I for all t ∊ [ T ₁, T ₁ + T _c ] , where T ₁, T _c > 0 are integers, the population contains two genotypes s and $\overline{s}$ such that the frequencies X( s, t) and X( $\overline{s}$ , t) satisfy

II we have

for some j. In other words, the genes s _i such that s _i ≠ $\overline{s}$ _i are involved in a single regulation set K _j ; and finally,

III Let

and

Then, if

we have j ∊ I ₊. If f _j ( s) > f _j ( $\overline{s}$ ), then j ∊ I _– .

Before proving this, let us make some comments. The biological meaning of the theorem is simple: for simple fitness models, where unknown parameters b _j are involved in a linear way, in the limit of infinitely large populations fitness landscape learning is possible.

Moreover, note that we do not make any specific assumptions about the nature of mutation, but only that all genetic variation between s and $\overline{s}$ are contained in a single regulatory set K _j .

The assertion of Theorem 3.1 is not valid if the set Diff( s, $\overline{s}$ ) belongs to a union of different regulation sets K _j , j = j ₁, . . . , j _p with p > 1. This effect of belonging to different sets K _j is pleiotropy in gene regulation. Note that if N _b ≪ N _g then the pleiotropy probability is small for large genome lengths N _g . On the contrary, if N _b ≫ N _g then assumption II is invalid.

Assumption II looks natural if when we deal with point mutations. In fact, if $\overline{s}$ is obtained from s by a single point mutation then condition ( 3.4) always holds for some j. For small mutation rates the probability of two point mutations is essentially below than the probability of a single mutation.

To conclude let us note that Theorem gives a rough estimate for the learning time T _c :

Proof. The main idea is simple. Negative mutations lead to elimination of mutant genotypes from the population, and the corresponding frequencies become, for large times, exponentially small.

Assume that ( 3.7) holds. Let j ∊ I ₋ , and thus b _j < 0. Consider the quantity

According to assumption II

Assumption III entails that

Relations ( 2.6) and ( 3.10) imply

By ( 2.9) and the last inequality we find that for T > T ₁

Consider inequality ( 3.11) for T = T ₁ + T _c . Let us note that in the relation Q( T ₁) = X( s, T ₁) /X( $\overline{s}$ , T ₁) the numerator is p ₀ whereas the denominator ≤ 1. Thus, Q( T ₁) ≥ p ₀. The same arguments show that Q( T ₁ + T _c ) ≤ 1 /p ₁. Therefore, by ( 3.11) one obtains that

This inequality leads to a contradiction for T _c satisfying ( 3.6), thus completing the proof.

3.2 The case of finite populations

Theorem 3.1 can be extended to the case of finite populations and non-zero mutation rates. is small. To formulate this generalization, we need an additional assumption about the fitness function. Suppose that

where c _F , C _F > 0 are constants independent of t. For example, if

then c _F = K _F exp(– γ) and C _F = K _F exp( γ) and ( 3.13) holds if K _F > exp( γ).

Condition ( 3.13) means that each individual gives birth to at least c _F and at most C _F descendants, where those bounds do not depend on the population size and evolution step.

Let

Note that for simplicity in the next Theorem 3.2 we consider point mutations (bit flipping) only. The model used here cannot represent mutations of arbitrarily small effect, but it can include insertions or deletions. In contrast to Theorem 3.2, Theorem 3.1 is valid for all kinds of mutations.

Then we have

Theorem 3.2. Consider the population dynamics defined by model 1 - 3 in Subsection 2.4. Assume conditions ( 3.14) and M hold, and assumptions ( 3.3), ( 3.4), ( 3.5), ( 3.7) of Theorem 3.1 are satisfied. Suppose

Then if j ∊ I _– the inequality

is fulfilled with the probability Pr _v such that

where for large N _popmax and p _mut → 0

Interpretation of Theorem 3.2

It is interesting to compare Theorem3.1 and Theorem3.2. The previous one asserts that for infinite populations the probability of the event j ∊ I ₋ is zero whereas the second one claims that this probability becomes exponentially small as the population size increases.

This theorem also shows that evolution can make a statistical test checking the hypothesis H _– that j ∊ I _– against the hypothesis H ₊ that j ∊ I ₊. Suppose that H _– is true. Let V be the event that the frequency X( $\overline{s}$ , t) of the genotype $\overline{s}$ in the population is larger than p ₁ within a sufficiently large time T _c . According to estimate ( 3.18), the probability of the event V is so small that it is almost unbelievable. Therefore, the hypothesis H _– should be rejected. We will refer T _c as the checking time.

Ideas for the proof. The main idea is the same as that for the previous theorem: we compare the frequencies of the organisms with the genotype $\overline{s}$ and the organisms with the genotype s. However, the proof includes a number of technical details connected with estimates of mutation effects and fluctuations. The formal proof can be found in Section 4. It is based on estimates of the accuracy of the Nagylaki equations (in the case of asexual reproduction).

4.1 Main tools and auxiliary Lemmas

Let us introduce notation and make some preliminary remarks. Remind that we denote by N ( s, t) the number of the population members with the genotype s at the moment t. Let X( t) be the set of all population members at the moment t. For each x ∊ X( t) let us denote by $N^{\prime }$ ( x, t) the number of progeny born by the individual x at the moment t before the massacre (see point 3 of model from Subsection 2.4). Let s _g ( x) be the genotype of x. Then, according to ( 2.8), the mean of $N^{\prime }$ ( x, t) is

where EX denotes the expected value of X. By $\stackrel{-}{N}$ ( s, t) we denote the number of all progeny born by individuals with the genotype s at the moment t before the massacre. Since all progeny are produced independently and randomly, the previous relation gives

Our main analytical tools are the Chernoff bounds and the Hoeffding inequalities. We also use the Markov inequality: for a positive random quantity X and a > 0 one has

Moreover, we use two elementary estimates. Let be an event in stochastic population dynamics. We denote by Not the negation (complement) of and by Pr( |ℬ) the conditional probability of under the condition ℬ. For events , ℬ ₁, . . . , ℬ _n we have

For two events , ℬ one has

Lemma 4.1. Let X _i be independent random quantities, where i = 1, . . . , n. Let each X _i be distributed according to the Poisson law with the average EX _i = μ _i. Let us denote

Then for all δ > 0

where

Similarly,

Proof. Note that for any λ > 0

Since X _j are independent quantities, we have

The straight forward computation shows that

Therefore, due to the Markov inequality ( 4.3) and estimate ( 4.8) one has

where

We minimize f with respect to λ and obtain ( 4.6). To derive ( 4.7), we use

and repeat the same arguments. The Lemma is proved.

Lemma 4.2. Let X _i be independent random quantities, where i = 1, . . . , n such that X _i ∊ {0, 1} and EX _i = p. Then

where

Proof. Note that for any λ > 0

Since X _j are independent quantities, we have

Note that E exp( −λX _j ) = p exp( −λ) + 1 − p. Let

We take λ = ln 2 and find that G(ln 2, p) = −g( p). Now by using the Markov inequality ( 4.3) and estimate ( 4.12) one obtains ( 4.10). The Lemma is proved.

We also use the following Chernoff-Hoeffding theorem. Let X _i be i.i.d. quantities such that X _i ∊ {0, 1} and EX _i = p, where i = 1, . . . , n. Then for $X={\displaystyle {\sum }_{j=1}^n{X}_j}$ one has

where D( x||y) is the Kullback-Leibler divergence

Moreover, we will use the Hoeffding Theorem: if i.i.d. quantities X _i ∊ [0, 1] with the probability 1 then

4.2 Main lemmas

First we estimate the population size fluctuations.

Lemma 4.3. Let $\overline{N}$ ( t) be the number of all progeny, born in the population at the moment t before the massacre, and ε ₁ > 0 be a small number. Then

with probability

where

Proof. Let $n^{\prime }$ ( x, t) denote the number of progeny produced by the individual x before the massacre at the t-th evolution step. The number $\overline{N}$ ( t) is the sum

of the mutually independent random quantities. According to ( 4.2), the average E $N^{\prime }$ ( x, t) is F( s _g ( x)). Therefore,

We set

and use the Lemma 4.1 that gives us ( 4.17).

Lemma 4.4. Let ε ₂ ∊ (0, 1) be fixed and condition ( 3.13) be fulfilled. Assume, moreover, that

where

and c _F > 1 is defined by ( 3.13). Let us define the event _{ε 2} ( t) by

Then one has

where

and

Proof. Let ξ( x) be random quantities defined as follows: ξ( x) = 1 if the individual x is survived as a result of massacre (see point 3 of our model from Subsection 2.4), and ξ( x) = 0 otherwise. Let $X^{\prime }$ ( t) be the set of progeny produced by all individuals from the population. Then the number N _sur( t) = N _pop( t + 1) of finally survived progeny can be computed as follows:

Note that | $X^{\prime }$ ( t) | = $\overline{N}$ ( t). Moreover, Eξ( x ) = N _popmax/ $\overline{N}$ ( t) for $\overline{N}$ ( t) ≥ N _popmax. Therefore, if $\overline{N}$ ( t) ≥ N _popmax then

Let us define the event

where the interval J _ε ( t) is defined by ( 4.16) and $\tilde{\epsilon }$ is defined by ( 4.27). By ( 4.4) we have

Now we apply the Hoeffding inequality ( 4.15). For each ε ₂ > 0 we obtain

If ℬ ( t) takes place, then $\overline{N}$ ( t) ≥ N _popmax and consequently

Therefore,

Moreover, by Lemma 4.3

Inequalities ( 4.30), ( 4.32) and ( 4.33) prove ( 4.25).

The following lemma, in particular, allows us to obtain equations ( 2.9) and ( 2.10) in the limit of infinite populations and for small mutation probabilities.

Recall that $\overline{N}(s,t)$ denotes the number of non-mutated progeny generated by the individuals with the genotype s before the massacre. Let N _sur( s, t) be the number of those progeny that survived after that massacre.

Lemma 4.5. Let ε ₀ be a positive number satisfying ( 4.75) and

Then one has

with the probability Pr _{s, t,+} such that

where

Similarly,

with the probability Pr _{s, t,−} such that

Proof. Step 1, estimates of fluctuations. First let us estimate the fluctuations of the number $\overline{N}(s,t)$ . For each ε ₂ > 0 let us define the event

By Lemma 4.1 one has

Note that

As a result, by ( 4.46) we obtain

Step 2. Here we estimate the number of progeny that survived as a result of the massacre procedure (point 3 of the population dynamics model, see subsection 2.9). Let X' ( s, t) be the set of progeny produced by individuals with the genotype s. Then the number N _sur( s, t) of survived progeny x for individuals x belonging to the set Z' ( s, t) is

where ξ( x) are defined in the proof of the previous Lemma. For ε ₃ > 0 we consider the event

Let us estimate the probability Pr( _{sur, s} ( t)). According to the Hoeffding Theorem ( 4.15)

Note that ξ( x) and ξ( y) are independent quantities for different x and y, thus under the condition $\overline{N}(t)$ > N _popmax

therefore,

Let us define the events ℬ _s ( t) and ℬ( t) by

Then using ( 4.4) one has

We observe that under conditions ℬ _s ( t) and ℬ( t)

In that estimate let us set ε ₂ = 0.5 and ε ₁ = 1. Taking into account that E $\overline{N}(t)$ = $\overline{F}(t)$ N _pop( t) < 2C _FN _popmax, we have that

where

Moreover, according to ( 4.47)

and due to ( 4.17)

where R ₁, R ₂ are defined by ( 4.37) and ( 4.38). Finally,

Step 3, estimate of the number of mutants.

Let us estimate how many individuals with genotypes s can mutate. The probability of mutation is p _mut. Let N _mut( s, t) be the number of such mutants. Let us define the event _{mut, s} ( t) by

Since the random quantity N _mut( s, t) is subject to the Bernoulli law, we can apply the Chernoff-Hoeffding inequality ( 4.13). Then we obtain that

where, according to definition ( 4.14) of D( x|| y), one has

and g is defined by ( 4.11).

Using ( 4.4) one has

As a result, by Lemma 4.3 one finds

where R ₄, R ₅ are defined by ( 4.40) and ( 4.41).

To prove ( 4.35), we set ε ₃ = ε ₀/2. Taking into account condition ( 4.75) for ε ₀ we see that if the both events Not _{mut, s} ( t) and Not _{sur, s, ε 0/2}( t) take place, then inequality ( 4.35) is fulfilled. Thus

where R _i are defined by ( 4.37)–( 4.41).

Finally, taking into account the results of steps 1, 2 and 3 we see that estimate ( 4.35) holds with the probability Pr _t,+. It completes the proof of ( 4.35). The second inequality ( 4.42) can be obtained in the same way.

4.3 Remaining part of the proof of Theorem 3.2

We use the same idea that in the proof of Theorem 3.1 but first we establish uniform bounds for the population size and other quantities involved in the proof.

Step 1 Here we estimate the population size. Let us set

in Lemma 4.4. Let us consider the events _{ε 2} ( t) defined by ( 4.24) in Lemma 4.4. If the events _{ε 2} ( t) take place for all t ∈ [ T ₁, T ₁ + T _c ] and N _pop(0) = N _popmax, we have that

Then conditions ( 3.15), ( 3.16) of Theorem 3.2 imply

Those inequalities imply the following estimates for the quantities R _i defined by ( 4.37) –( 4.41):

where q _i are defined by