Modelling the evolution of naturally bounded traits in a population

Abstract

Eco-evolutionary models commonly assume that traits are normally distributed in a population, and that the trait bounds do not influence the adaptation of traits. However, recent empirical evidence suggests that at least some traits are not normally distributed, and there is theoretical support for the view that trait bounds can be fundamental to trait adaptation. These attributes suggest that a beta distribution, which can accommodate unbounded (i.e. normal), singly bounded (i.e. gamma) or doubly bounded (beta) trait distributions, may be an appropriate alternative assumption for eco-evolutionary models. We develop an evolutionary model that represents how the mean values of a population’s traits change. Implementation of the model requires assumptions to be made regarding the relative fitness of the individuals in the population, and how their traits are distributed within natural bounds. We compare the numerical results of “population” models that evolve a plant population and the means of its two traits using our eco-evolutionary equations with those of “phenotype” models that evolve 10,000 phenotypes, each defined by a pair of trait values, of a plant population. The phenotype models do not assume any particular trait distribution or fitness, and allow phenotypes to wax and wane according to their ability to compete with other phenotypes in the population for a finite resource. Comparison of the trait distributions obtained by solving 3 coupled population odes with those obtained by solving 10,000 coupled phenotype odes reveals very good agreement between the approaches for each of four mortality functions. Further, it supports the ubiquity of the beta distribution in describing evolutionary processes in populations. An advantage of the simple population model is that it provides insights into why particular results are obtained, which augments the predictive power of the modelling, suggesting that in fact a simplified, abstracted modelling approach is sometimes preferable to the detailed, complicated alternative.

Appendices

Appendix 1: Resource-limited ecological modelling

Here we outline the framework that imposes ecological bounds on the trait values in the eco-evolutionary model—a detailed exposition of the framework is provided in Cropp and Norbury (2015). We consider n living populations in an ecosystem that has a limiting resource (often a food web limited by nutrient, but a limiting resource might also be light, for example) represented by the variables x_i, $i=1,2,\dots ,n$. The rate of change of each population may be written in the general form:

$$ \frac{dx_{i}}{dt} = x_{i} f_{i}(x_{1},x_{2},\dots,x_{n}; {\Upsilon}) \quad \text{for} \quad i=1,2,\dots,n, $$

(7)

where n is the number of populations modelled and Υ is a set of parameters that quantify how the populations interact with each other and their environment. The system conserves the total quantity of the limiting resource, so implicit in the model is an additional equation, not of the form of Eq. (7), which keeps track of the size of the resource pool R, often an inorganic nutrient pool N. The functions $f_{i}(x_{1},x_{2},\dots ,x_{n};{\Upsilon })$, which for convenience we shall refer to as f_i, are key to the ecological and evolutionary dynamics of the system.

The resource-limited ecosystem modelling approach is implemented by the following process:

• We identify a limiting resource, most commonly a limiting nutrient, which we use as a common currency to measure all populations. We sum the limiting nutrient cycling in both the living populations and the nonliving nutrient pool, and scale all the (living and nonliving) variables by this total;

• Each population grows (dies) according to the constraints of the other populations and the environment, represented here in its simplest form by the nonliving nutrient pool, independently of how we measure the populations and the scale of the ecosystem;

• The mass of limiting nutrient is conserved as it flows through the food web: we assume that immigration equals emigration and nutrient influx equals nutrient efflux (most delineated ecosystems recycle at least 90% of their limiting nutrient Loreau 2010; Vitousek and Matson 2012);

• The growth of every population is explicitly limited by the resources available to it: when its resources are maximal the population increases, when its resources are zero, the population decreases;

• Negative nutrient concentrations are not permitted, that is, when any living population or the nutrient in the abiotic resource pool are zero, these variables can only increase.

The first three principles ensure the setup of model ecosystem equations that are internally consistent. The fourth provides checks to ensure that the populations have sensible properties and that they are viable but not immortal.

The final principle ensures that there is no capacity for living populations to “borrow” resources from the environment, and reflects that the total amount of limiting nutrient is finite. We note that this condition does not apply to other nutrients or resources, solely to the limiting resource. Also note that the fifth condition, which precludes negative values, does not imply that the system may take negative values and that we over-ride these, as is apparently done in some complex ecological simulation model examples. The condition in our example arises from the structure of the equations and is a property of the system. Finally, note that we usually scale all limiting nutrient amounts by the total amount in the ecosystem, or food web. Then our populations P (and N) are measured as fractions or percentages. Time t is scaled to be days which, in the case of microscopic plants that we use as a model system, equates approximately to generations.

Appendix 2: Resource-limited ecological modelling

Consider a population x_i(t) divided into phenotypes classified by the trait parameter u which continuously extends from 0 to 1 (after scaling suitably), for each value of time t > 0. We define X(t, u) to be the mass measure of limiting nutrient across the phenotype range, at each t, measured so that:

$$ X(t,0) = 0; \quad X(t,1) = x_{i}(t). $$

(8)

Then:

$$ \frac{1}{x_{i}(t)} {{\int}_{0}^{1}}\frac{\partial X}{\partial u} du = \frac{X(t,1)-X(t,0)}{x_{i}(t)} = 1, \quad \textrm{for } t\!>\!0, $$

(9)

so that X(t, u) defines a cumulative distribution and ∂X/∂u a probability density function on u in 0 < u < 1.

We can now define a trait mean u_i and a trait variance ${\sigma _{i}^{2}}$ (at any time t > 0) by using this probability distribution on u. So we have:

$$ \begin{array}{@{}rcl@{}} u_{i}(t) & =& {{\int}_{0}^{1}} u\left [ \frac{1}{x_{i}(t)} \frac{\partial X}{\partial u} \right ] du, \end{array} $$

(10a)

$$ \begin{array}{@{}rcl@{}} {\sigma_{i}^{2}}(t) & =& {{\int}_{0}^{1}} (u-u_{i})^{2} \left [ \frac{1}{x_{i}(t)} \frac{\partial X}{\partial u} \right ] du, \end{array} $$

(10b)

$$ \begin{array}{@{}rcl@{}} & =& {{\int}_{0}^{1}} u(u-u_{i}) \frac{\partial X}{\partial u} du \frac{1}{x_{i}(t)}, \end{array} $$

because ${{\int \limits }_{0}^{1}} u_{i}(u-u_{i}) \frac {\partial X}{\partial u} du=0$. We now want to find an expression for the change of the trait mean u_i as time evolves, when the variance ${\sigma _{i}^{2}}(t)$ of the trait distribution in the population remains small.

Phenotype dynamics are given by:

$$ \frac{d}{dt} \left (\frac{\partial X}{\partial u} \right ) = \frac{\partial X}{\partial u} f_{i} (u; \underline{x},\underline{\Upsilon}), \quad \textrm{for } t>0, $$

(11)

where the life function f_i for the population x_i is here written in terms of all other populations $\underline {x}$ and parameters $\underline {\Upsilon }$, but only the trait u is adapting together with X(t, u), which measures the phenotype share of the x_i population at each t > 0. To recover the population dynamics equation we sum all the phenotype behaviours by summing Eq. (11), using Eq. (9),

$$ \begin{array}{@{}rcl@{}} \frac{d}{dt} x_{i}(t) & =& \frac{d}{dt} {{\int}_{0}^{1}} \frac{\partial X}{\partial u} du = {{\int}_{0}^{1}} \frac{\partial X}{\partial u} f_{i}(u) du, \\ & =& {{\int}_{0}^{1}} \frac{\partial X}{\partial u} f_{i}(u_{i}) du + {{\int}_{0}^{1}} \frac{\partial X}{\partial u} (f_{i}(u)-f_{i}(u_{i})) du, \\ & =& x_{i}(t) f_{i}(u_{i}) + \mathfrak{R}_{1}. \end{array} $$

(12)

The remainder $\mathfrak {R}_{1}$ in Eq. (12) is small when the trait variance ${\sigma _{i}^{2}}$ is small, as discussed later. Next we differentiate Eq. (10a) with respect to t:

$$ \frac{d}{dt} u_{i}(t) = {{\int}_{0}^{1}} u \left [ \!- \frac{\frac{d}{dt} x_{i}(t)}{{x_{i}^{2}}(t)} \frac{\partial X}{\partial u} + \frac{1}{x_{i}(t)} \frac{d}{dt} \left (\!\frac{\partial X}{\partial u}\! \right )\! \right ] du. $$

(13)

Substitute Eqs. (12) and (11):

$$ \begin{array}{@{}rcl@{}} \frac{d}{dt} u_{i}(t) & =& {{\int}_{0}^{1}} u \left [ - \frac{x_{i}(t)f_{i}(u_{i})}{{x_{i}^{2}}(t)} \frac{\partial X}{\partial u} + \frac{1}{x_{i}(t)} \frac{\partial X}{\partial u} f_{i}(u) \right ] du\\&& +~ \mathfrak{R}_{2}, \\ & =& {{\int}_{0}^{1}} u \left [ -f_{i}(u_{i}) + f_{i}(u) \right ] \frac{1}{x_{i}(t)} \frac{\partial X}{\partial u} du + \mathfrak{R}_{2}, \\ & =& {{\int}_{0}^{1}} u \left. \frac{\partial f_{i}}{\partial u} \right |_{u=u_{i}} (u-u_{i}) \frac{1}{x_{i}(t)} \frac{\partial X}{\partial u} du + \mathfrak{R}_{3}, \\ & =& {{\int}_{0}^{1}} u (u-u_{i}) \frac{1}{x_{i}(t)} \frac{\partial X}{\partial u} du \left. \frac{\partial f_{i}}{\partial u} \right |_{u=u_{i}} + \mathfrak{R}_{3}, \\ & =& {\sigma_{i}^{2}} \left. \frac{\partial f_{i}}{\partial u} \right |_{u=u_{i}} + \mathfrak{R}_{3}, \end{array} $$

(14)

where we have used Eq. (10b) for the variance. The remainder term $\mathfrak {R}_{3}$ has a similar form to $\mathfrak {R}_{2}$, and also is small when the variance is small. It arises from:

$$ \begin{array}{@{}rcl@{}} f_{i}(u) - f_{i}(u_{i}) &=& (u-u_{i}) \left. \frac{\partial f_{i}}{\partial u} \right |_{u=u^{*}} = (u-u_{i}) \left. \frac{\partial f_{i}}{\partial u} \right |_{u=u_{i}} \\&&+ O(u-u_{i})^{2}, \end{array} $$

(15)

where u^∗ lies between u_i and u, and where the second derivative of f_i is bounded. The terms $\mathfrak {R}_{1}$, $\mathfrak {R}_{2}$, and $\mathfrak {R}_{3}$ are all small because of the form of the integral of the probability measure in Eq. (10b) when the variance is small and the function f_i is smoothly twice differentiable in all its variables.

Appendix 3: Resource-limited ecological modelling

We implement the equations describing how the aggregate properties of trait distributions change (see derivations in the Appendix B), assuming that beta distributions provide the greatest utility in describing many trait distributions. We obtain different evolutionary equations for the mean value of a trait u depending on whether or not the ecosystem constraints dictate that the trait has bounds:

$$ \begin{array}{@{}rcl@{}} &&\textrm{Unbounded: }\\ & \dot{u_{i}} & = \sigma^{2} \left. \frac{\partial f}{\partial u} \right |_{u=u_{i}} = \left (\frac{L^{2}}{\alpha + \beta + 1} \right ) \left. \frac{\partial f}{\partial u}\right |_{u=u_{i}}, \\ &&\textrm{Lower bounded: }\\ & \dot{u_{i}} & = \left (\frac{(u_{i}-u^{L})L}{\alpha + \beta + 1} \right ) \left. \frac{\partial f}{\partial u}\right |_{u=u_{i}}, \\ &&\textrm{Upper bounded: }\\ & \dot{u_{i}} & = \left (\frac{(u^{U}-u_{i})L}{\alpha + \beta + 1} \right ) \left. \frac{\partial f}{\partial u} \right |_{u=u_{i}}, \\ &&\textrm{Lower and upper bounded: }\\ & \dot{u_{i}} & = \left (\frac{(u^{U}-u_{i})(u_{i}-u^{L})}{\alpha + \beta + 1} \right ) \left. \frac{\partial f}{\partial u} \right |_{u=u_{i}}. \end{array} $$

(16)

Here, f is the function that describes the per-capita growth of the population that contains the trait with mean value u_i. The term L = u^U − u^L provides the appropriate trait variable scaling for the equations that are singly bounded or unbounded, where u^L and u^U are the lower and upper bound of the trait distribution respectively. In the case of the normal distribution, u^U and u^L are notional symmetric values, for example the mean u_i plus or minus a standard deviation σ.

The variances of the trait distributions are given by σ² = L²/(α + β + 1) (notionally a normal distribution), (u_i − u^L)L/(α + β + 1) and (u^U − u_i)L/(α + β + 1) (notionally gamma or skewed beta distributions), and (u^U − u_i)(u_i − u^L)/(α + β + 1) for the beta distribution. Note that (u^U − u_i)(u_i − u^L)/(α + β + 1) is an exact expression for the variance of a beta distribution, but that (u_i − u^L)L/(α + β + 1) and (u^U − u_i)L/(α + β + 1) are approximations as α << β and α >> β respectively, for α + β >> 1. Note that in some cases the trait scalings and dimensions may be included in the ecosystem equations, and then the trait equations just involve standard beta distributions. We use this approach in this paper.

Appendix 4: Conservation of α + β?

The denominators of the variances in Eq. (16) contain the sum α + β + 1, where α and β are commonly referred to as shape parameters of the beta distribution. The sum α + β > 0 has several interpretations, for example as a notional sample size in Bayesian statistics. Here, we consider its application in genetics where it may be interpreted as a statistical estimate of the allele frequencies in the components of a sub-divided population. We somewhat abstract that interpretation to phenotypes:

$$ \alpha + \beta = \frac{1-F}{F}, \quad 0<F<1, $$

(17)

where F is Wright’s genetic distance between two populations (Wright 1950). Small values of F indicate little differentiation between allele frequencies (phenotypes or trait values) within a population, implying a high level of mixing within the population. Small values of F imply large values of α + β, so that the related trait distribution has a small variance and is tightly focussed about the mean.

There likely exist instances where the mixing properties of a population are independent of the dynamics of a particular trait, in which case the sum α + β would be constant over simulations of trait evolution. Populations with large, constant values of α + β may remain symmetrically distributed away from their bounds under stabilising selection, or move homogeneously to one side or the other of their trait distribution under directional selection, but not split into two distinct phenotypes as occurs under disruptive selection. We observe directional selection in the simulations of both models. We chose to maintain α + β constant in the population model, but the phenotype model does not. If we assume that the phenotype model is the “reality” that we test the population model against, then an interesting question is how does α + β, or Wright’s F, actually change?