In this work, a partial differential equation model for evolutionary dynamics is presented that describes changes in densities of phenotypes in a population. We consider that the traits of individuals of a population are distributed at an interval of real numbers where a mortality rate is assigned for each value of this interval. We present some conditions for stability of stationary solutions and apply the model in theoretical scenarios of natural selection. Particularly we approach cases of stabilising, disruptive and directional selection, including the scenario of the survival of the flattest. Some computational simulations are performed to illustrate the results obtained.

#### Keywords

- evolution equation
- natural selection
- dynamical systems

#### MSC 2010

- 92D25
- 92D40

The Theory of Evolution through Natural Selection [3, 12, 15] is central to the understanding of patterns observed in biological systems. The body of theory had gone through many revisions [17], which are natural, if we recall that even the mechanisms of inheritance were still unknown when it was first conceived.

Given the immense scope of Evolution Theory, there are variations in its formulations, depending on the biological phenomenon approached. Nevertheless, it is possible to identify a central invariant core that is present in almost any evolutionary explanation of biological observations. Authors in [20,21] indicate three fundamental components:

There is variation in morphological, physiological, and behavioural traits among members of a species (the principle of variation).

The variation is in part inheritable, so that individuals resemble their relatives more than they resemble unrelated individuals and, in particular, offspring resemble their parents (the principle of heredity).

Different variants produce different numbers of offspring either in immediate or remote generations (the principle of differential fitness).

Thus, the concept of

In all of these formulations, the

There are many examples of traits that influence mortality rates of organisms. Mutations affecting senescence mechanisms [5], resistance to toxin produced by prey [7, 27] and the degree of dominance of a gene that provides insecticide resistance [6] are examples of such traits. In fact, any adaptation for defence against predators or parasites (speed, thick shells, and quick reflexes) can be related to the increase of fitness through the reduction of mortality rates.

Many methods can be used to model evolution: Game theory [28], population genetics [16, 19] and dynamical systems related to Game Theory [25] are examples. To model selection through differential mortality rates, we are going to use an

Here we consider scenarios in which all phenotypes in the aspect space have the same reproduction rates but possess distinct mortality rates. We deduce some conditions, depending on the distribution of mortality rates, for the permanence of the population. We also deduce some properties of equilibrium solutions, investigating the particular cases of directional, stabilising and disruptive selection. Finally, we show some differences and similarities of the process of selection through differential mortality rates when compared with the process of differential reproduction rates [2].

We begin by stating the basic assumptions of the model:

Individuals are distributed in a phenotype space Ω = [−

Individuals reproduce at constant rate

There is a non-negative function that for each trait on the domain assigns a mortality rate value.

Traits are inherited and undergo mutations with no bias.

Individuals are in competition for limited resources.

Hypothesis 1 simply means that we are describing the evolution of a quantitative trait. Given possible discontinuities in the morphogenesis [23], it is possible that similar quantitative traits lead to dissimilar phenotypes, leading to discontinuities in the mortality function. Hypothesis 2 and 3 are related to the discussion in consideration here, i.e., the focus of selection is on mortality rates and not on differential reproduction rates.

Hypothesis 4 simply states that there is imperfect heredity of traits. Variation on traits might come from recombination of genes, mutations or even changes in environmental factors. The change may be considered to be “small” in the timescale of reproduction and has no bias to either increasing or decreasing the trait described. Additive genetic models [29] can display this kind of behaviour as long as a quantitative trait is influenced by a sufficient large number of alleles. In [13] the authors estimate that a particular quantitative trait in

Finally, hypothesis 5 includes intra-specific competition. Although models that include unlimited population growth can display a certain type of selection represented by distinct fractions of types [30], a more realistic selection may be obtained with the inclusion of competition and limited population growth.

Let _{i}_{i±1}. This makes a substantial difference also with standard reaction-diffusion equations in population theory.

The above assumptions lead us to the following partial differential equation

In this section, we analyse what happens with the solution as time evolves when describing the equilibrium solution.

We start by rescaling the variables as follows

In this way, we end up with the following dimensionless equation

We could easily check that

By integrating the previous equation on the domain

The last equality implies the following:

As we are looking for positive equilibrium solutions, if we consider homogeneous Dirichlet boundary conditions, since ^{′}^{′}^{″}

Now, replacing

Since the right hand side of the previous equation is non-negative it follows that both sides are equal only when

As we are going to see further on, in case of

For non-zero solutions we have the following result:

_{0} ^{′}_{0}) = 0, _{0} _{0}) > 1 − _{0}) < 1 −

By means of Eq. (3), since

Now, let us consider a special case in which the mortality rate function _{0} = 0, in which we can apply the last criteria. Yet, in case of considering Neumann boundary conditions then both −

Applying the fundamental theorem of calculus on Eq. (3) it turns out that
^{′}

We can also state the following result about the shape of a non-zero equilibrium solution

We prove the statements by contradiction. Let us assume ^{″}^{″}^{′}^{′}

Let ^{″}^{′}^{′}_{0} ^{′}_{0}) = 0. As we have _{0} ^{′}_{0}) = ^{′}^{′}_{1} _{0},^{″}_{1}) = 0. But this is a contradiction because

Using a similar argument we can show the dual result

Previous results show us that

Applying last inequality to Eq. (4) it turns out that

It is not difficult to check that the same inequality is true when

Eq. (6) implies that the existence of a non-zero equilibrium solution relies on how much the value of the integral changes according to changes in the values of ^{p}^{1+s},

Before proceeding further, we consider the nonlinear functional

It is not difficult to verify that

Now, let

Assuming that

Now, let us assume that the solution of Eq. (2) takes the form

In order to analyse the sensitivity of equilibrium solutions, let us assume that ^{″}

The eigenvectors/eigenvalues of the second order differential operator are given by

for homogeneous Dirichlet boundary conditions:

for homogeneous Neumann boundary conditions:

In both cases, _{1} is the leading eigenvalue. Thus, since ∫_{D}_{1} (

Therefore, by integrating both sides over

In this way, the criteria of stability of an equilibrium solution

In summary we have the following result:

For Neumann boundary conditions, if we consider a constant mortality rate function ^{−1} that is asymptotically stable.

From (14) and (15), the stability of a non-zero solution relies on the value of the unknown quantity

Looking at Eq (14) we see that the condition for stability of non-zero equilibrium solutions is less restrictive than the one of the zero solution, namely

This means that Eq. (2) could have a stable zero solution and, if it exists, a stable non-zero equilibrium solution as well.

Now, let us revert to the original variables in order to infer the influence of the original parameters in the instability conditions. In this case, Eq. (16) is rewritten as

Let us assume that the natality rate is greater than the mortality rate,

Thus, as we can clearly see, the mutation rate

In the next subsections, we study the circumstances under which the selection by differential mortality rate could lead to the three most basic types of evolution: directional, disruptive and stabilising selection. We remark that condition (16) is always satisfied when we consider Neumann boundary conditions.

Directional selection occurs when an extreme phenotype is selected over the others on the phenotype space. In our context, it means that the mortality rate function is monotone over the domain

Thus, the criterion (16) for establishing a population becomes

In other words, when the adaptive value is increasing linearly with respect to the measurement of the trait then a population is experiencing directional selection if the maximum of the mortality is less than the double of the natality rate (recall that for the dimensionless equation

Now, let us consider the shape of the non-zero equilibrium solution. First of all, let us observe that, in the context of directional selection, equilibrium solutions on bounded domains are a kind of transient solution. As we have previously discussed, according to Proposition 5 if

In case of Dirichlet boundary conditions, there is a critical point in the interior of the domain. If

In Figure 2 we have the non-zero equilibrium solution of Eq. (2) with mortality rate given by

Now, let us consider the case in which the mortality rate is defined as

This function describes a situation in which the extreme traits have greater adaptive value over the mean trait. That is, the mortality rate decreases with respect to distance to the mean trait. The zero phenotype has the greatest mortality rate

For this function, it turns out that

In the last expression, the value of

According to Eq. (4), we have 0 <

The same statement is true when we consider Dirichlet boundary conditions. That is, for a non-constant equilibrium solution,

In Figure 4, we see numerical solutions illustrating this behaviour.

Let us consider now the situation in which the mean phenotype has the greatest adaptive value over all the others. This situation can be described by the following mortality rate function:

For this function, it is not difficult to check that

Now, assuming that

Since

We finally address another important case of selection: the survival of the flattest. That is, we consider the case in which the fitness landscape has two peaks: one low and flat and another one higher but narrower.

In terms of mortality rates, we are interested in discussing the shape of possible equilibrium solutions when the mortality function

As shown in previous sections, according to Proposition 2, the existence of local maximum points for an equilibrium solution relies on whether

If we look at Eq. (5) we can see that the difference between

However, more interesting is the role played by the mutation rate

We also set

To measure the influence of

In words,

In Figure 8 we have the graphs of two equilibrium solutions when

In Figure 9 we have the graphs of two equilibrium solutions when

In this paper we presented some results relative to evolutionary selection through differential mortality rates. The question of population permanence has been discussed, given fixed mortality distributions and boundary conditions. Two main effects may push the population to extinction: high mutation rates leading to population dissipation or high mortality rates resulting in insufficient reproduction.

The first effect is closely related to the classical problem of critical patch size for population permanence under diffusion [4,22]. In this case, the coefficient of phenotype change,

The model has shown coherence in the results of all three particular cases analysed: directional, stabilising and disruptive selection, with results mimicking qualitatively the expected and observed biological results, see Chapter 12 of [15]. Selection of quasispecies was another characteristic displayed by the model, which is also observed in other theoretical approaches [24], experiments [8] and in an aspect space model with differential reproduction rates [2].

However, an important difference in the quasispecies selection based on differential mortality rates was found. The transition from the selection of one quasispecies to another one is smooth. While replicator and differential reproduction models lead to abrupt transitions from one type to another, selection through differential reproduction rates can display coexistence of types and smooth transitions from one type to another (as shown in Figure 7). Also, in this type of selection, increased mutation rates favoured the ‘selection of the flattest’, being coherent with the results obtained in other models and experiments [24, 25].

There was also an important difference observed between the dynamics of selection through mortality rates and selection based on differential reproduction rates. In the scenario with just one adaptive peak, regardless of mutation rates, selection of the fittest is observed in the model with differential mortality rates. The same cannot be said to occur in classical approaches [14] or even in the aspect space model with focus on reproduction rates [2]. In such models, the mutation rate must be below a certain threshold for the fittest to be selected, a result that is coherent with biological observations.

Since mortality is just one of the many components of fitness in real systems, the results obtained by the model should not be considered as a contradiction, since a mutation rate below a certain threshold is still needed for the selection of the fittest when its other components are included. Finally, we point out that the analytical results concerning the aspect space model are original and represent the important development in understanding the dynamics of such models.

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