λ = w’i + w’j
Each couple produced a number of offspring randomly drawn from a Poisson
distribution with expectancy λ .
2.7 Inheritance
After reproduction, each offspring inherited one strain copy or
haplotype from each parent. Only the genetic component (or breeding
value) a was inherited. Allele values of the inherited haplotype
were picked randomly from the corresponding parental locus. Each
haplotype mutated with probability μ of mutation per locus. In
case that a mutation occurred, its effect was randomly drawn from a
normal distribution with zero mean and variance equal to the effect size
of mutations MV , which was an input parameter (Vincenzi 2014).
The assumption of a Gaussian distribution is consistent with analysis of
mutation effects (Lynch and Walsh 1998; Martin et al. 2006).
All adults died after reproduction (non-overlapping generations), and
the environmental state was updated before repeating the loop.
2.8 Simulation experiments
With the model we compared different implementations of plasticity
(objective 1) and studied the effect of these types of plasticity
(non-adaptive and adaptive phenotypic plasticity) on the persistence of
populations with different life histories under scenarios of directional
stochastic environmental change (objective 2). Probability of
persistence was computed as the proportion of times a persistence event
(or success) occurs from the total number of observations (the 100
replicates). Error bars were estimated from the binomial distribution. A
total of 100 replicates lasting 250 generations each were performed for
different combinations of rate of directional climate change, degree of
temporal autocorrelation (noise color), type of phenotypic plasticity
(including genetic determinism), and density dependence effect (life
history strategy) (see table I). To illustrate selected parameter
values: under no stochasticity, η = 0.02 imposed a directional
trend on the optimum phenotype of 0.02 phenotypic units per generation
(or per year, assuming one generation a year). This means that after 250
generations of simulation time, the optimum θtwould have changed in five phenotypic units, and a population with no
density compensation (i.e., ψ = 1 ) could avoid extinction
via evolutionary rescue or some sort of phenotypic plasticity. To our
experimental setup, the contribution of mutations was important.
Otherwise, populations failed to cope with the changing environmental
conditions (Fig. S1, Appendix B: Supplementary material). It is
important to highlight that, in our model, limits of plasticity were
reached well before the simulation ends. Following the same example, an
initially well adapted individual (zi =
θ0 ) reached its limits of plasticity (i.e., saturation,
by logistic plasticity; decay in ability of plastic response, by
sinusoidal plasticity) at θt = 1.58 . This
corresponds to generation t = 80 .
To complement the findings on life history strategies of intermediate
and strong density dependence effects, additional scenarios of rate of
environmental change and of stochastic fluctuations in carrying capacity
were implemented to unravel whether, in the model, life history
strategies differently tolerate fluctuations in K. Specifically, each
time unit of model iteration (or generation), the value of carrying
capacity was drawn from a normal distribution centered in K and
variance sd 2, where sd was a percentage
of K . For instance, if K = 1000 , the scenario of 10% ofK means sd = 100 . This is, around 70% of the cases, the
value of carrying capacity is expected to occur within the interval
(K – sd , K + sd ).
Simulation experiments were also evaluated under scenarios of lower
mutation rate, and mutational effects according to the model of slightly
deleterious mutations (Ohta 1973; Eyre-Walker et al. 2002;
Romero-Mujalli et al. 2019a), which imposed higher genetic constraints.
The analysis of data and plotting was performed in r v3.5.2.
Table I Parameter values and description