Model of phenotypic evolution in hermaphroditic populations.

We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of phenotypic distribution. The main result of the paper is a theorem on asymptotic stability of trait distribution. This theorem is applied to models with the offspring trait distribution given by additive and multiplicative random perturbations of the parental mean trait.

Introduction

This paper studies the evolution of phenotypic traits in hermaphroditic populations, i.e. populations where every individual has both male and female reproductive system. A great part of these populations has formed various defense mechanisms against self-fertilization (autogamy) to guarantee genetic diversification (e.g. a proper shape of a flower can inhibit self-pollination in some species of plants). In that case, individuals can only mate with others to copulate and cross-fertilize. Nonetheless, cross-fertilisation occurs in some hermaphroditic species. Hermaphroditic populations are plentiful among both water and terrestrial animals as well as plants. Some of examples include Sponge (Porifera), Turbelleria, Cestoda (Cestoidea), Lumbricidae, some of mollusks such as sea slug Blue Dragon (Glaucus atlanticus) and various kinds of land snails or majority of flowering plants (angiosperms).

A considerable amount of literature has been published on modelling asexual populations by means of microscopic description of trait evolution. Macroscopic approximations of that models were derived in the forms of deterministic processes or superprocesses (see Champagnat 2006; Champagnat et al. 2008; Fournier and Méléard 2004; Ferrière and Tran 2009; Méléard and Tran 2009). In this paper we formulate an individual based model to describe the phenotypic evolution in hermaphroditic populations. We consider a large population of small individuals characterized by their traits. The traits are assumed to be unchanged during lifetime, and their examples include skin colour, the shape of a leaf and shell pattern. All the individuals are capable of mating or self-fertilizing to give birth to an offspring.

We consider a general model of mating, which includes both random and assortative mating. The first particular case is a semi-random mating model. This model is based on the assumption that each individual has an initial capability of mating depending on its trait. This mating model is similar to models describing aggregation processes in phytoplankton dynamics (see Arino and Rudnicki 2004; Rudnicki and Wieczorek 2006a, b). The second particular case is an assortative mating model. In this model, the individuals with similar traits mate more often than they would choose a partner randomly. We adapt a model based on a preference function Doebeli et al. (2007), Gavrilets and Boake (1998), Matessi et al. (2001), Polechová and Barton (2005), Schneider and Bürger (2006), Schneider and Peischl (2011), usually used in two-sex populations, to the hermaphrodites. The consequence of mating or self-fertilization is a birth of a new individual. The trait of this individual is given by a random variable that depends only on traits of the parents. Each individual can die naturally or when competing with others. We consider a continuous time model, and we assume that all above-mentioned events happen randomly.

The model presented in this paper is a hermaphroditic analogue of the asexual model introduced by Bolker and Pacala (1997), Law and Dieckmann (2002) and studied by Fournier and Méléard (2004). Despite the vast literature concerning individual based models and their macroscopic approximations, only a few models involving mating processes have appeared so far (Collet et al. 2013; Remenik 2009).

One of our aims is to study a macroscopic deterministic approximation of the model. We obtain it by increasing the number of individuals in the population to infinity, with simultaneous decrease in the mass of each individual. After suitable scaling of parameters, the limit passage leads to an integro-differential equation. Solutions of the equation describe the evolution of trait distribution. We also study the existence and uniqueness of the solutions. We investigate extinction and persistence of the population and convergence of its size to some stable level.

The main aim of our paper is to prove asymptotic stability of trait distribution. Asymptotic behavior of the solutions is characterized by conservation of mean phenotypic trait. We apply our main theorem to two specific models. In these models the offspring trait is the parental mean trait randomly perturbed by some external environmental effects or genetic mutations. In the first model, the noise is additive. The property of additivity allows us to derive a formula for the stationary phenotypic distribution. The second model contains multiplicative noise, and it includes, as a special case, the Tjon–Wu version of the classic Boltzmann equation (see Bobylev 1976; Krook and Wu 1977; Tjon and Wu 1979). The Tjon–Wu equation describes the distribution of energy of particles. As a by-product of our investigation we give a simple proof of the theorem of Lasota and Traple (see Lasota 2002; Lasota and Traple 1999) concerning asymptotic stability of this equation. Addtionally, an example of the trait reduction is given. In this case, in a long period of time, all traits reduce to a particular one, which is the mean trait of the initial population.

The scheme of the paper is following. In Sect. 2 we collect all assumptions on the dynamics of the population. In Sect. 3 we introduce a stochastic process corresponding to our individual based model. Section 4 is devoted to the macroscopic approximation, the limiting equation and its solutions. In particular, we give results about extinction and persistence of the population and stabilization of its size. In Sect. 5 we formulate the results concerning the asymptotic stability, and we give examples of their applications. Finally, in the last section we discuss problems for future investigation concerning assortative mating models.

Individual-based model

Let us fix a positive integer . We assume that every individual is described by a phenotypic trait , which belongs to some closed and convex subset of , whose interior is nonempty. The trait of an individual does not change during its lifetime.

Random mating

In sexually reproducing populations a mating process highly depends on a given species. We will consider both random and assortative mating. In classical genetics individuals mate randomly—the choice of partner is not influenced by the traits (panmixia). Random mating occurs often in plants, but it is also observed in some hermaphroditic animals (Baur 1992). We study a semi-random mating model in which the mating rate depends on the trait. An individual described by the trait is capable of mating/self-fertilizing with rate , where is a positive function of the trait.

Consider a population which consists of individuals with traits . Since two different individuals may have the same trait, it is useful to describe the state of the population as the multisetWe recall that a multiset (or bag) is a generalization of the notion of a set in which the members are allowed to appear more than once. We suppose that any individual can mate with an individual of trait with the following probabilityThus the mating rate of individuals with traits and is given byThe figure is a self-fertilization rate. In the case of populations without self-fertilization we assume thatif and . Let us observe that in both cases the mating rate is a symmetric function of and but only in the first case we have . If we pass with the number of individuals to infinity, and replace the discrete model by the infinitesimal model with trait distribution described by a continuous measure , then the mating rate in both cases is given by

Assortative mating

Now we consider models with assortative mating, i.e. when individuals of the similar traits mate more often than they choose a partner randomly. Assortative mating can be modelled in different ways. For example one can use matching theory, according to that, each participant ranks all the potential partners according to its preferences, and attempts to pair with the highest-ranking one (Almeida and de Abreu 2003; Puebla et al. 2012). Such models are very interesting but difficult to analyze. The most popular models of assortative mating are based on the assumption that a random encounter between two individuals with traits and depends on a preference function (Doebeli et al. 2007; Gavrilets and Boake 1998; Matessi et al. 2001; Polechová and Barton 2005; Schneider and Bürger 2006; Schneider and Peischl 2011). We consider only the case when all the individuals have the same initial capability of mating .

Usually, it is assumed that , where is a continuous and decreasing function. It means that if the population consists of members with traits , then individuals of traits , mate with rateNote that in general, the function is not symmetric in and , and usually it describes mating in two-sex populations. Then the first argument in refers to a female. Females are assumed to mate only once, whereas males may participate in multiple matings. We have for each , which means that all females mate with the same rate. The mating rate in the infinitesimal model is of the formWhile considering hermaphroditic populations, one can expect a model with a symmetric mating rate. We obtain such a model assuming that the mating rate is of the formwhere is a symmetric nonnegative preference function, e.g. (in the case of populations without self-fertilization we eliminate the terms with and from the denominators). The mating rate in the infinitesimal model is now of the formIn the rest of the paper we will assume that the mating rate is of the form (1) or (6).

Birth of a new individual

After mating/self-fertilization an offspring is born with probability 1. The trait of the offspring is drawn from a distribution , where and are parental traits. We suppose that for every the measure is a Borel probability measure with support contained in the set , and assume that there exist positive constants such thatandThe above condition has a simple biological interpretation, namely, the expected offspring’s trait is the parental mean trait. Moreover, we suppose that for every and for every Borel set and the functionis measurable.

Competition and death rates

An individual from the population can die naturally or when competing with others. Let us denote by the rate of interaction of the individual with trait . We assume that is a nonnegative function. For individuals with traits and we define a competition kernel which is assumed to be a nonnegative and symmetric function. Competition always leads to death of one of the competitors. We assume that the natural death rate of the individual with trait is expressed by a number , and suppose that is a nonnegative function.

Stochastic process corresponding to the model

The dynamics of the population

We present the dynamics of the ecological system that we are interested in. The process starts at time from an initial distribution. Individuals with traits and can mate at rate of the form (1) or (6). After mating an offspring is born with probability one. An individual with trait dies naturally at rate or by competition at rate where the sum extends over all living individuals at time , and are their traits. We assume that all the events (mating, natural death, competition) are independent.

The phase space

We denote by the set of all positive integers, stands for the Dirac measure concentrated at a point and denotes the indicator function of a set . We consider the space of all finite positive Borel measures on equipped with the topology of weak convergence of measures. We introduce the set of the formFor any measure and any measurable function we define . In particular, if . We write for the Skorokhod space of all cad-lag functions from the interval to the set (see for details, e.g., Ethier and Kurtz 1986, Skorokhod 1956).

Generator of the process

We consider a continuous time -valued stochastic process with the infinitesimal generator given for all bounded and measurable functions by the formulaThe first term in the right-hand side describes the mating and birth processes with the dispersal of traits. The second term stands for two kinds of death. The death part of the generator was previously studied in Fournier and Méléard (2004). Notice that, on the contrary to Fournier and Méléard (2004), the operator has the first term nonlinear with respect to .

We assume that there are positive constants such that for every Under the above assumptions, if the initial measure satisfies for some number , thenfor any . Consequently, the standard approach of Fournier and Méléard can be easily applied to prove the existence of the Markov process with the infinitesimal generator given by formula (13) (see Fournier and Méléard 2004, Remenik (2009) for detailed proofs).

Macroscopic model

Macroscopic approximation

This section contains an approximation of the process which was introduced and studied in the previous sections. The idea is to normalize the initial model and pass with the number of individuals to infinity, assuming that the “mass” of each individual becomes negligible. In this approximation mating and death rates remain unchanged. Only the intensity of interaction is rescaled, and tends to 0 with an unbounded growth of the population. This approach leads to a deterministic nonlinear integro-differential equation whose solutions describe an evolution of trait distribution.

We consider a sequence of populations indexed by numbers . If the th population consists of individuals , thenThe th population is described by a process which is defined in the same way as the process but with the corresponding coefficients. We define the -valued Markov process by the formula . The value space for the process is thusThe generator of the process is given byfor any measurable and bounded map .

individuals with traits and can mate with rate of the form (1) or (6),

a new offspring’s trait is drawn from a distribution , where are the traits of parents,

an individual with trait can die with rate ,

an individual with trait interacts with other individuals with intensity ,

a competition kernel of individuals with traits is a symmetric, nonnegative function .

Theorem 1

We assume that condition (14) holds, and the functions are continuous. We suppose that for some and all , and almost each sequence converges weakly to a deterministic finite measure , as . Then for all the sequence of processes converges in distribution in to a deterministic and continuous measure-valued function , satisfying the following equationfor every bounded and measurable function

The standard proof of the above theorem is based on Ethier and Kurtz (1986) Corollary 8.16, Chapter 4, and since mating is described by the Lipschitz continuous operator on the space of positive and finite Borel measures with total variation norm, it can be directly adapted for example, from Rudnicki and Wieczorek (2006a).

Strong solutions in the space of measures

According to Theorem 1 the solutions of (15) are continuous in the topology of weak convergence of measures. In this part we show a stronger result that they are also continuous in the total variation norm . We use the following formal notationof equation (15) in the space of positive, finite Borel measures on the set with the total variation norm.

Theorem 2

Assume that the functions and (11) are measurable, and condition (14) holds. Moreover, suppose that there exist positive constants such thatfor all . If , then there exists a unique solution , , of Eq. (16) with the initial condition . The function is bounded and continuous in the norm .

Proof

Let us fix , and consider the space with the norm . Define the operator by the formulawhere is some measure. Notice that from assumption (14) there are constants depending on in the case of semi-random mating, and on in the case of assortative mating, such that for any and measures andTake functions from the ball . ThenandTaking sufficiently small, from (20) and (21) it follows that transforms the ball into itself, and is Lipschitz continuous with some constant . By the Banach fixed point theorem the operator has a unique fixed point and, consequently, (16) has a unique local solution.

In order to extend a local solution to the whole interval it is sufficient to show that the solution is bounded. Notice thatand consequentlywhich completes the proof of the global existence and uniqueness.

Eventually, we show that the measure is positive for . Indeed, for any Borel set we can write , where Hence

A straight-forward conclusion is the following statement about solutions in space.

Corollary 1

Suppose that for each the measure is absolutely continuous with respect to the Lebesgue measure andUnder the assumptions of Theorem 2, if has a density with respect to the Lebesgue measure, then also has a density and is the unique solution of the following equationwith the initial condition .

Take a Borel set with zero Lebesgue measure. Since the measure is absolute continuous with respect to the Lebesgue measure, for every and consequently , and . Therefore for all , and the statement comes from the Radon-Nikodym theorem.

Boundedness, extinction and persistence

From the proof of Theorem 2 it follows that the function is upper-bounded. Now we analyze further properties of . Let us recall that a population becomes extinct if , and is persistent provided .

Proposition 1

If in the case of random mating and in the case of assortative mating, then the population becomes extinct. If in the case of random mating and in the case of assortative mating, then the population is persistent.

In the case of random mating these properties follow simply from the inequalitiesandIn the case of assortative mating we use similar inequalities with and replaced by 1.

Equation on a global attractor

In order to describe more precisely asymptotic behavior of , we need to assume that the functions do not depend on , and are positive. To avoid extinction of the population, we additionally assume that in the case of random mating and in the case of assortative mating (see Proposition 1). Then satisfies the following equationand is a stationary solution of this equation. Using basic facts from the theory of differential equations, it is easy to see thatThe number is an analogue of carrying capacity studied in Bolker and Pacala (1997) and Fournier and Méléard (2004). In our case is a number of individuals per unit of volume after long time.

From (25) it follows that all positive solutions converge to the setwhich is invariant with respect to Eq. (16), i.e., if an initial condition belongs to , then for . It means that is a global attractor for Eq. (16). If then the function satisfies the following equationLet be a positive solution of (16). If we substitute , then the function also satisfies (26). Therefore, the long-time behaviour of solutions of (26) is completely characterized by the dynamics on the attractor .

Now we consider only solutions on the set . If we replace by and by in (26), then becomes a probability measure for all . The function satisfiesin the case of random mating, andin the case of assortative mating, where

Asymptotic stability in the case of random mating

General remarks

In this section we study the convergence of solutions of Eq. (27) to some stationary solutions. Equation (27) can be treated as an evolution equationwhere the operator acting on the space of all probability Borel measures on is given by the formulaThe solution of (29) with an initial measure is the deterministic process given by Theorem 2. The set is called the orbit of .

Since the problem of the asymptotic stability of the solutions of Eq. (29) in an arbitrary -dimensional space seems to be quite difficult, we consider only the case when and is a closed interval with nonempty interior. Generally, Eq. (29) has a lot of different stationary measures and it is rather difficult to predict a limit of a given solution. Assumption (9) allows us to omit this difficulty. Indeed, if a measure has a finite first moment , then according to (8) and (9) we haveandTherefore, any solution of Eq. (29) has the same first moment for all . It means that we can restrict our consideration only to probability Borel measures with the same first moment.

The following example shows why we consider solutions of Eq. (29) with values in the space of probability Borel measures instead of the space of probability densities. In this example all the stationary solutions are the Dirac measures, and any solution converges in the weak sense to some stationary measure.

Example 1

Let be a random variable with values in the interval such that and Assume that if and are parental traits, then the trait of an offspring is given byi.e., the trait of an offspring is distributed between the traits of parents according to the law of . For a random variable we denote by and its first and second moments and by its variance, i.e., . Let , , be a solution of (29) with a finite second moment, and let , , be random variables with distribution measures . Then is a constant, andSince , we have . Consequently, converges weakly to .

The Wasserstein distance

In order to investigate asymptotic properties of the solutions, we recall some basic facts concerning the Wasserstein distance between measures. For we denote by the set of all probability Borel measures on such that and by the subset of which contains all the measures such that . For any two measures , we define the Wasserstein distance by the formulawhere is the set of all continuous functions such that for any The following lemma is of a great importance in the subsequent part of the paper.

Lemma 1

The Wasserstein distance between measures can be computed by the formulawhere is the cumulative distribution function of the signed measure .

Let and be the cumulative distribution functions of the measures and . Since these measures have finite first absolute moments we haveandThis givesand if is bounded from below or from above, then we have for . Since is a locally absolutely continuous function, integrating by parts leads to the formulaClearly the supremum is taken when .

Consider probability measures and , , on the set . We recall that the sequence converges weakly (or in a weak sens) to , if for any continuous and bounded function as . It is well-known that the convergence in the Wasserstein distance implies the weak convergence of measures. Moreover, the space of probability Borel measures on any complete metric space is also a complete metric space with the Wasserstein distance (see e.g. Bolley 1934; Rachev 1991). The convergence of the sequence to in the space is equivalent to the following condition (see Villani (2008), Definition 6.7 and Theorem 6.8)where Fix , , and . Consider a set such thatfor all Then the set is relatively compact in . Indeed, by Markov inequality, for all , what means that the set is tight, and thus is relatively compact in the topology of weak convergence (see e.g. Billingsley (1995)). Moreover, for we havewhich implies the second condition in (C). Consequently, the set is relatively compact in .

Theorems on asymptotic stability

We use the script letter for the cumulative distribution function of the measure , i.e.,The main result of this section is the following.

Theorem 3

Fix . Suppose that

for all the function is absolutely continuous with respect to and for each we have

there are constants , , and such that for every we have Then for every initial measure there exists a unique solution , , of Eq. (27) with values in . Moreover, there exists a unique measure such that and for every initial measure the solution , , of Eq. (27) converges to in the space .

We split the proof of Theorem 3 into a sequence of lemmas. Denote by the set of all cumulative distribution functions of the signed measures of the form , where .

Lemma 2

Suppose that for all and , , we haveThenfor , . In particular, for every initial measure there exists a unique solution , , of Eq. (27) with values in .

Since , we can writewhere . If is the cumulative distribution function of , then the signed measure has the cumulative distribution function of the formHence

Lemma 3

Suppose that condition (i) of Theorem 3 is fulfilled. Then condition (37) holds.

Take a and denote and . Since is the cumulative distribution function of , where , we haveand, consequently,Since and are nonnegative functions and have the same integral, condition (35) impliesIntegrating by parts we obtainFrom (40) and (41) it follows

Lemma 4

Assume that for all , . Let and denote by and , respectively, the solutions of Eq. (29) in the space of probability Borel measures on . Then for and for provided that .

Since every solution of (29) with an initial value from the set remains in this set for all . Any solution of (29) satisfies the following integral equationLet and be solutions of (29) with values in and such that . Then for and from (42) it follows thatfor . Let . Thenand from Gronwall’s lemma it follows that , which gives .

Lemma 5

Assume that condition (ii) of Theorem 3 is fulfilled. Then for every initial measure its orbit is a relatively compact subset of . Moreover, , where denotes the closure of in .

Take a and let be a solution of (29) with the initial condition . Let , , and . We check that for andTo see this, we define the setThen is a closed and convex subset of . Hence is a closed subset of . For a sufficiently small , the mapis a contraction on the space and the function , , is its fixed point. In order to prove that for and that (43) holds, it is sufficient to check that the set is invariant with respect to . Indeed, since we haveThus, there exists depending on and such that (43) holds. Consequently, the orbit is a relatively compact subset of and .

Let be a family of transformations of defined by , where is the solution of (29) with the initial condition . For we define the -limit set by

Proof of Theorem 3

Take a measure . According to Lemma 5 the orbit of is a relatively compact subset of . From this it follows that is a nonempty compact set and for we have . First we check that is a singleton. Indeed, if has more than one element, then since is a compact set, we can find two elements and in with maximum distance . But since , then for given there exist and in such that and . Now from condition (i) and Lemmas 2, 3, and 4 it follows thatwhich contradicts the definition of and . Let . Then for and, consequently, . Since the orbit is relatively compact, we have . According to Lemmas 2 and 3 the operator has only one fixed point what means that the limit does not depend on . Now, we consider a measure . The set is dense in the space . Thus, for every we can find such that . Moreover, since we find such that for . As the operators are contractions we havefor , which completes the proof.

We can strengthen the thesis of Theorem 3, if we additionally assume that for all the measure has a density and is a bounded and continuous function.

Theorem 4

Assume that is a bounded and continuous function, and satisfies assumptions of Theorem 3. Then the stationary measure is absolutely continuous with respect to the Lebesgue measure and has a continuous and bounded density . Moreover, for every the solution of Eq. (27) can be written in the form , where are absolutely continuous measures, have continuous and bounded densities , which converge uniformly to .

Since is a continuous and bounded function and is a probability measure,is a continuous bounded function and is a density of because is a fixed point of the operator . For any initial measure the solution of (27) satisfies the equationFor each the measure has a continuous and bounded density . Since the function given by is continuous and , the function given by is continuous and . Thus the measures have continuous and bounded densities and converges uniformly to as .

Examples

Now, we study two biologically reasonable forms of , which satisfy conditions (i) and (ii) of Theorem 3.

Example 2

We suppose that if and are parental traits, then the trait of the offspring is of the formwhere is a 0-mean random variable, and has a positive density . ThenandThe condition (i) is equivalent to the inequalityfor all , which is a simple consequence of the assumption that is a positive density. Now we check that condition (ii) holds with . We haveIf we additionally assume that the density is a continuous function, then according to Theorem 4 the limit measure has a continuous and bounded density , , and converges uniformly to .

Now we determine the limiting distribution . Densities of the measures and have the same first moment and satisfies the equationObserve that if a probability density satisfies (45) and , then also satisfies (45) and has the first moment . Since is a unique solution of (45) with the first moment we have for . Now we construct the density . Consider an infinite sequence of i.i.d. random variableswith density and define the random variableThen . If and are two independent copies of and is a random variable with density independent of and , thenIt means that the density of satisfies (45). Let for and . From the definition of the random variable it follows thatwhere denotes the convolution of and .

From (46) it follows immediately that . For instance, if has a normal distribution with zero mean and standard deviation , then has also a normal distribution with zero mean and standard deviation .

Example 3

As in Example 2 we suppose that if and are parental traits, then the trait of the offspring is of the formwhere is a random variable with values in the interval , and has a density such thatThen and the function is of the formfor and otherwise. Equation (29) with kernel given by (48) is known as the general version of Tjon–Wu equation. If , then this equation is the Tjon–Wu version of the Boltzmann equation (see Bobylev 1976; Krook and Wu 1977; Tjon and Wu 1979). The asymptotic stability of the classical Tjon–Wu equation in space was proven by Kiełek (see Kiełek 1990). Lasota and Traple (see Lasota 2002; Lasota and Traple 1999) proved stability in the general case but in the sense of the weak convergence of measures. If we assume additionally that the support of contains an interval , , then this result follows immediately from Theorem 3. Indeed, in that case one can easily computeNow, condition (35) is equivalent to the inequalityfor all . This inequality is a simple consequence of positivity of on the interval and of the following conditionNow we check that the condition (ii) holds with . We havewhere . Since , we have .

Remark 1

The kernel in Example 3 is not a continuous function even if the density is a continuous and we cannot apply directly Theorem 4 in this case. But it not difficult to check that if then and to prove that the invariant measure has a density , and is a continuous function on the interval . Moreover, repeating the proof of Theorem 4 one can check that , and converges uniformly to on the sets , . In particular, if we consider Eq. (29) on the space of probability densities, then every solution converges to in .

Conclusion

In the paper we presented some phenotype structured population models with a sexual reproduction. We consider both random and assortative mating. Our starting point is the individual-based model which clearly explains all interactions between individuals. The limit passage with the number of individuals to infinity leads to the macroscopic model which is a nonlinear evolution equation. We give some conditions which guarantee the global existence of solutions, persistence of the population and convergence of its size to some stable level. Next, we consider only a population with random mating and under suitable assumptions we prove that a phenotypic profile of the population converges to a stationary profile.

It would be interesting to study analytically long-time behavior of a phenotypic profile of population with assortative mating. Some numerical results presented in the paper Doebeli et al. (2007) suggest that also in this case one can expect convergence of a phenotypic profile to multimodal limit distributions. This result suggests that assortative mating can lead to a polymorphic population and adaptive speciation. We hope that our methods invented to asymptotic analysis of populations with random mating will be also useful in the case of assortative mating. In order to do it, we probably need to modify the model of assortative mating (7) presented in Sect. 2, because it has a disadvantage that the mating rate does not satisfy the condition for all . We can construct a new model which corresponds to the same preference function with a symmetric mating rate which has the above property. In order to do this we look for constants depending on the state of population such thatand for all . In this way we obtain a system of linear equations for :where for and . Since the matrix has positive entries and the dominated main diagonal, system (51) has a unique and positive solution. The passage with the number of individuals to infinity leads to the following mating ratewhere the function depends on a phenotypic distribution , and satisfies the following Fredholm equation of the second kindOne can introduce a general model which covers both semi-random and assortative mating. Let be the initial capability of mating of an individual with trait and be a symmetric nonnegative preference function. Now we can define a cumulative preference function by . The mating rate is a symmetric function given by (50) with replaced by and we assume that for each . The mating rate in an infinitesimal model is of the formwhere the function satisfies the following equationIn particular, in the semi-random case we have and and the mating rate is given by (3). Let us recall that in the general case is not only a function of but it is also depends on and therefore, the proofs of results from Sects. 3 and 4 cannot be automatically adopted to these models.