Niche construction in evolutionary theory: the construction of an academic niche?

Appendices

Appendix 1

Models of niche construction

Table 1 Matrix of two-locus genotypic fitnesses for the first two models (Laland et al. 1996, 1999).

Here we discuss, the three early and often cited models of niche construction that have been described as extending the understanding possible through SET, and as constituting an extensive body of formal theory (Laland et al. 2016). Two of these models (Laland et al. 1996, 1999) are based on standard diallelic two-locus population genetic models with multiplicative fitnesses (general discussion in Hartl and Clark 1989), where one locus specifies a niche constructing phenotype which, in turn, affects fitness through genotypes at the second locus as a result of the specific environmental perturbations it causes. The third model (Laland et al. 2001) is a gene-culture coevolution model where the niche constructing trait is culturally inherited. We start by describing the diallelic two-locus genetic models (Laland et al. 1996, 1999). This appendix is aimed at readers familiar with population genetics models to show clearly how these foundational NCT models do not add anything substantial to what is already known from standard two-locus viability selection models incorporating frequency-dependent and epistatic effects on fitness.

Genetic models

In these models, the niche constructing activity is controlled by a locus labelled E. At this locus, there are two alleles, E and e, and the niche constructing ability of the population, reflected entirely by resource levels, is directly proportional to the frequency of the allele E, given by p. A resource, R, is defined such that its amount is directly proportional to the niche constructing activities of the present and past generations. In the first model (Laland et al. 1996), R depends on n previous generations of niche construction and the n generations can either weigh in equally, or there can be recency or primacy effects. A recency effect entails generations closer to the present generation having a larger effect on R than the ones further in the past. A primacy effect entails generations further into the past having a larger effect on R than the ones which are more recent. In the second model (Laland et al. 1999), R is also affected by autonomous ecological mechanisms of resource depletion or recovery, besides the niche constructing activity of the population. In this case, R is given by the following recursion equation:

$$\begin{aligned} R_{t} = \lambda _{1}R_{t-1}(1 - \gamma {p}_{t}) + \lambda _{2}p_{t} + \lambda _{3}, \end{aligned}$$

where $R_{t}$ is the amount of resource in the present generation, $R_{t-1}$ is the amount of resource in the previous generation, $\lambda _{1}$ determines independent depletion, $\lambda _{2}$ determines effect of positive niche construction, $\lambda _{3}$ determines independent renewal, and $\gamma$ determines effect of negative niche construction. In both models, R is constrained to be between 0 and 1 $(0<R<1)$ and the value of R determines fitness through genotypes at a second locus, A, with alleles, A and a. A is favoured when R is high (above 0.5), whereas a is favoured when R is low (below 0.5). The two-locus genotypic fitnesses are given in table 1 in terms of R and marginal one locus genotypic fitnesses.

In table 1, $\varepsilon$ gives the strength and direction of niche construction $(-1<\varepsilon <1)$. The two locus gametic frequencies are given by $x_{1}$, $x_{2}$, $x_{3}$, and $x_{4}$ for EA, Ea, eA, and ea, respectively. The recombination rate is given by r. Then, the gametic recursions are given by:

$$\begin{aligned} { Wx}_{1}^{*}= & {} [x_{1}(x_{1}w_{11}+x_{2}w_{21}+x_{3}w_{12}\\&+\,x_{4}w_{22})] - { rw}_{22}D,\\ { Wx}_{2}^{*}= & {} [x_{2}(x_{1}w_{21}+x_{2}w_{31}+x_{3}w_{22}\\&+\,x_{4}w_{32})] + { rw}_{22}D,\\ { Wx}_{3}^{*}= & {} [x_{3}(x_{1}w_{12}+x_{2}w_{22}+x_{3}w_{13}\\&+\,x_{4}w_{23})] + { rw}_{22}D,\\ { Wx}_{4}^{*}= & {} [x_{4}(x_{1}w_{22}+x_{2}w_{32}+x_{3}w_{23}\\&+\,x_{4}w_{33})] - { rw}_{22}D, \end{aligned}$$

where $x_{1}^{*}$, $x_{2}^{*}$, $x_{3}^{*}$, and $x_{4}^{*}$ are two locus gametic frequencies in the next generation, and D is the linkage disequilibrium given by,

$$\begin{aligned} D=x_{1}x_{4} - x_{2}x_{3}, \end{aligned}$$

and W, the mean fitness of the population, is the sum of all the right-hand sides in the gametic recursions. The dynamics of this model were studied under four conditions, namely, no external selection at E or A loci, external selection at the A locus, external selection at the E locus, and overdominance at both loci.

Table 2 Matrix of pheno-genotypic fitnesses in terms of marginal trait/genotypic fitnesses given in brackets.

No external selection

No external selection means that frequency of the E allele remains constant $( \alpha _{1} = \beta _{1} = \alpha _{2} = \beta _{2} = 1)$. Thus, the amount of resource remains constant in the first model and attains an equilibrium value in the second model. If these values are above 0.5, then allele A gets fixed and otherwise allele a gets fixed due to the selection generated by niche construction. The line, $R = 0.5$, defines a neutrally stable equilibrium in both models. Fixation of allele A is unstable below $R = 0.5$ and fixation of allele a is unstable above $R = 0.5$.

External selection at the E locus

If selection favours allele $E\, (\alpha _{1}> 1 > \beta _{1}$, $\alpha _{2} = \beta _{2} = 1)$, alleles E and A get fixed $(x_{1} = 1)$ if $\varepsilon$ is positive and alleles E and a get fixed $(x_{2} = 1)$ if $\varepsilon$ is negative. If selection favours allele $e \,(\alpha _{1}< 1 < \beta _{1}$, $\alpha _{2} = \beta _{2} = 1)$, alleles e and a get fixed $(x_{4} = 1)$ if $\varepsilon$ is positive and alleles e and A get fixed $(x_{3} = 1)$ if $\varepsilon$ is negative. In the model with independent renewal and depletion of R, an additional caveat on the result is whether R is less than one half or more at the fixation value of E or e, respectively. Allele A gets fixed if $R> 1/2$ at fixation of either allele at the E locus, whereas allele a gets fixed if $R < 1/2$. A range of polymorphic equilibria are obtained if $R = 1/2$.

In the first model, if more than one previous generation of niche construction affects $R\, (n > 1)$, then time lags between the start of selection at locus E and response at locus A can occur as R builds up slower than the rate of fixation at locus E. This evolutionary inertia is largest when there is a primacy effect and smallest when there is a recency effect. A similar process can lead to an evolutionary momentum type of effect as well when the selection at locus E stops or reverses because there is a lag between the frequency of allele E and the amount of resource accumulated. Such effects are not seen in the second model as there are no primacy effects in it. But, they can be obtained by making $\lambda _{2} = 1/n$, i.e., a primacy effect.

In the second model, the rate of fixation of allele A when allele E is being favoured by external selection is dependent on magnitude of the impact of niche construction on the resource $( \lambda _{2})$. Increasing the value of $\lambda _{2}$ increases the value of R, and, if R is near 1, this reduces the difference in fitness between the genotypes AaEE and aaEE, thus, remarkably reducing the rate of fixation of allele A.

External selection at the A locus

In the first model, if external selection favours allele $A \,( \alpha _{1} = \beta _{1} = 1$, $\alpha _{2}> 1 > \beta _{2})$ and there is no niche construction $(\varepsilon = 0)$, or selection due to it is very weak $(1-\alpha _{2}< \varepsilon < 1- \beta _{2})$, allele A always gets fixed if it is present. Near $x_{4} = 1$, if $\varepsilon > 1-\beta _{2}$, fixation of allele a is neutrally stable. A set of polymorphic equilibria are possible near $x_{1} = 1$, with alleles e and a increasing, if $1-\alpha _{2}> \varepsilon$.

In the second model, if external selection favours allele $A\, ( \alpha _{1} = \beta _{1} = 1$, $\alpha _{2}> 1 > \beta _{2})$ and is weak, niche construction is positive $(\lambda _{2} > 0$, $\gamma = 0)$, and $\varepsilon$ is greater than zero, then for small values of R selection due to niche construction can take allele a to fixation. Also, there are a range of values of R, for which stable equilibria for fixation of alleles A and a overlap. If external selection is strong then fixation of allele a becomes less probable. If niche construction is negative $(\lambda _{2} = 0$, $\gamma > 0)$ fixation of allele a still happens at low values of R, but, now those correspond to higher values of p instead of lower in the case of positive niche construction. For negative values of $\varepsilon$ and positive niche construction, fixation of both a and A alleles becomes unstable and a set of stable polymorphisms are possible. If niche construction becomes negative, stable polymorphisms are possible near $x_{3} = 1$ and allele A gets fixed for rest of the parameter space.

Overdominance at both loci

Diallelic two locus viability models can have a maximum of four gamete fixation states, four allelic fixation states, and seven interior fixation states (Karlin 1975). The results from overdominance $(\alpha _{1}$, $\alpha _{2}$, $\beta _{3}$, $\beta _{4} < 1)$ are too complicated and varied to go into detail here, but generally, the effect of niche construction is to move the interior polymorphic equilibria and the edge equilibria (when they exist) towards higher values of q, when R is more than one half and towards lower values of q when R is less than one half. The magnitude of shift depends on the how far frequency of allele E is from one half. For high values of $\varepsilon$ the edge equilibria can even merge with the respective gamete fixation states. For tightly linked loci (small r), niche construction can either increase or decrease linkage disequilibrium at genetic equilibrium. Equilibrium frequencies of allele E greater than one half $(p > 1/2)$ result in increase in equilibrium frequencies of gametes AE and Ae and equilibrium frequencies of allele E less than one half $(p> 1/2)$ result in decrease in equilibrium frequencies of gametes aE and ae. In the second model, these effects of niche construction persist, for some sets of parameter values, even when there is external renewal or depletion of the resource.

It is important to note that these two models have different meanings of positive and negative niche construction (Laland et al. 2005). For the first model, positive niche construction $( \varepsilon > 0)$ means that increase in R increases the fitness of allele A. For the second model, positive niche construction implies that $\lambda _{2} > 0$, $\gamma = 0$; negative niche construction implies that $\lambda _{2} = 0$, $\gamma > 0$, meaning that increase in frequency of allele $E \,(\,p)$ results in an increase in R, even though the sign of $\varepsilon$ still mediates the effect of R on selection at the A locus.

Cultural model

We turn now to the third model in which the niche constructing trait is culturally inherited. A niche constructing trait E with variants E and e is postulated as a culturally inherited trait. A resource R depends on either n previous generations of niche construction, i.e., the frequency of trait variant E(x) (model 1), or on niche construction and independent renewal or depletion following the same equation for R as in the second model (model 2). A genetic locus A is postulated with alleles A and a, and its fitness is affected by amount of resource present with allele A being favoured when $R > 1/2$ and allele a being favoured when $R < 1/2$. The six pheno-genotypes, AAE, AAe, AaE, Aae, aaE and aae, have frequencies $z_{1}{-}z_{6}$ and their fitnesses are given in table 2. Rules for vertical cultural transmission are given in table 3.

Table 3 Probabilities of offspring having trait E or e for each combination of parental mating.

Three specific cultural transmission scenarios were analysed: unbiased transmission $(b_{3} = 1$, $b_{2}=b_{1} = 0.5$, $b_{0} = 0)$, biased transmission $(b_{3} = 1$, $b_{2}=b_{1}=b$, $b_{0} = 0$, $b\ne 0.5)$, and incomplete transmission $(b_{3} = 1 - \delta$, $b_{2}=b_{1}=b$, $b_{0} = \delta$, $\delta > 0)$. For ease of analysis, the recursions were written in terms of allelo-phenotypic frequencies, namely, AE, aE, Ae, and ae (for the equations see Laland et al. 2001). Similar results were obtained for both model 1 and model 2 unless otherwise stated.

No external selection

For unbiased transmission, the results for model 1 are analogous to Laland et al. (1996; see above) and the results for model 2 are analogous to Laland et al. (1999; see above).

For biased transmission, frequency of trait E increases if $b > 0.5$ and that of trait e increases if $b < 0.5$. For positive values of $\varepsilon$ and $b< 0.5$, ae is fixed at equilibrium values of $R < 0.5$ and Ae is fixed at equilibrium values of $R > 0.5$. If $b > 0.5$, trait E gets fixed instead of trait e. Symmetric results are obtained when $\varepsilon$ is negative.

For incomplete transmission, when $\delta > 0$ and $b = 0.5$ the cultural trait remains polymorphic and a line of neutrally stable equilibria is obtained for locus A. If $b \ne 0$ and $\varepsilon$ is positive allele A gets fixed for equilibrium values of $R > 0.5$ and allele a get fixed for equilibrium values of $R < 0.5$. Symmetric results are obtained when $\varepsilon$ is negative.

External selection at the A locus

Again, for unbiased transmission, the results for model 1 are analogous to Laland et al. (1996; see above) and the results for model 2 are analogous to Laland et al. (1999; see above).

For biased transmission, when cultural transmission favours trait $E\, (b > 0.5)$ and $\varepsilon$ is positive, whether external selection at the A locus is opposed or not depends on the value of R at fixation of trait E. The positive $\varepsilon$ and increasing frequency of trait E make it improbable that R will be lower than 0.5 at equilibrium. For cultural transmission favouring trait $e\, (b < 0.5)$, R ends up being low enough for fixation of allele a instead of A more often. When $\varepsilon$ is negative, three polymorphic equilibria are possible depending on value of R, namely, fixation of AE, fixation of aE, or an equilibrium polymorphic for alleles A and a. Symmetrically opposite results are obtained when niche construction is negative, i.e., trait E is responsible for depletion of the resource.

For incomplete transmission, a polymorphism for the cultural trait is obtained, and if $\varepsilon$ is positive, either of the alleles A or a get fixed, depending on value of R at the equilibrium frequency of the cultural trait. If $\varepsilon$ is negative, a fully polymorphic equilibrium is possible for very high values of R at equilibrium.

Selection at the cultural trait

Again, for unbiased transmission, the results for model 1 are analogous to Laland et al. (1996; see above) and the results for model 2 are analogous to Laland et al. (1999; see above).

For biased transmission, when natural selection and transmission bias reinforce each other by either favouring $E\, ( \alpha _{1}> 1 > \alpha _{2}$; $b > 0.5)$ or $e\, (\alpha _{1}< 1 < \alpha _{2}$; $b < 0.5)$, AE or ae get fixed for positive values of $\varepsilon$. When these processes work against each other than their relative strength determines the final equilibrium. In such a scenario, cultural transmission can fix the trait which is not favoured by selection, if transmission bias is strong enough.

For incomplete transmission, the frequency of the cultural trait is given by a cubic equation (see Laland et al. 2001). For model 1, if $n > 1$ then time lags are obtained as in the analogous genetic model (Laland et al. 1996). The length of the time lag depends on both the selection coefficients and transmission bias with cultural transmission usually shortening the lags as compared to completely genetic models.

Overdominance at the A locus

For unbiased transmission, polymorphisms at the locus A are possible if the selection due to niche construction does not completely overcome the external selection at the A locus, i.e., R is either too large or too small.

For biased transmission, polymorphisms at the E trait no longer exist and either trait E or e gets fixed. Frequency of alleles at the A locus depends on the interplay of external selection and selection due to niche construction.

For incomplete transmission, if there is no statistical association between the cultural trait and the genetic locus then a single-polymorphic equilibrium is obtained. Selection due to niche construction shifts this equilibrium from the point where it would have been had niche construction not been acting.