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