3 Sensitivity to sexual
reproductive variance: Ne_sensitivity_Vk()
3.1 What is Vk/k̄ and why does it matter?
The variance-to-mean ratio of sexual reproductive output Vk/k̄ measures how equally seeds are distributed among reproductive adults. Under Poisson distribution (Vk/k̄ = 1), all adults contribute equally to the seed pool. Values above 1 mean some individuals dominate — a few adults produce most of the seeds, so the effective number of parents is smaller than the census count.
In Lepanthes, this is biologically plausible because:
- Pollination by fungus gnats is patchy — some plants are visited far more frequently than others
- Microhabitat variation means some individuals are far more fecund than others
- In small populations, one or two dominant individuals can produce most seeds in a given year
The Poisson default (Vk/k̄ = 1) is almost certainly an underestimate of the true variance for most natural populations.
3.2 Running
Ne_sensitivity_Vk() for the reproductive stage
For L. rupestris (6 stages), stage 5 (single-shoot reproductive adults) is the primary sexual reproductive stage. We sweep Vk/k̄ from 0.5 to 8.
sens_Vk_Lrup1 <- Ne_sensitivity_Vk(
model_fn = Ne_sexual_Y2000,
T_mat = MatU_Lrup1,
F_vec = F_Lrup1,
D = D_Lrup1,
stage_index = 5, # stage 5: single-shoot reproductive adults
Vk_range = seq(0.5, 8, by = 0.5),
Ne_target = 50,
population = "L. rupestris pop 1"
)
print(sens_Vk_Lrup1)
#>
#> === NeStage Sensitivity Analysis ===
#> Parameter swept : Vk/k_bar (sexual reproductive variance)
#> Focal stage : s5
#> Model : Ne_sexual_Y2000
#> Population : L. rupestris pop 1
#>
#> --- Range swept ---
#> Vk_over_k from 0.5 to 8 (16 values)
#>
#> --- Ne/N summary ---
#> At reference (Vk_over_k = 1) : Ne/N = 0.572 | Min N = 88
#> Maximum Ne/N (Vk_over_k = 0.5) : Ne/N = 0.604 | Min N = 83
#> Minimum Ne/N (Vk_over_k = 8) : Ne/N = 0.331 | Min N = 152
#> Ne/N range across sweep : 0.273
#>
#> --- Elasticity (Caswell 2001; Morris & Doak 2002) ---
#> Local elasticity at reference point : E = -0.104
#> Global elasticity (mean over range) : E = 0.285
#>
#> Interpretation (local elasticity):
#> A 10% increase in Vk_over_k at the reference point
#> produces a 1.0% change in Ne/N.
#> Direction: NEGATIVE -- higher variance reduces Ne/N (typical)
#>
#> --- Conservation implications ---
#> Min N at reference point : 88
#> Min N at worst case (Vk_over_k = 8) : 152
#> Ratio worst/reference : 1.7x more individuals
#> Ne target used : 50
#>
#> --- Comparison to empirical Ne/N (Frankham 1995) ---
#> Ne/N at reference (0.572) is ABOVE the empirical median of ~0.10
#> across 102 wildlife species (Frankham 1995). This population
#> has relatively high effective size.
#> Frankham (1995) empirical range: 0.01 -- 0.99
#>
#> --- References ---
#> Caswell H. (2001). Matrix Population Models, 2nd ed.
#> Sinauer Associates. [elasticity methodology]
#> Morris W.F. & Doak D.F. (2002). Quantitative Conservation
#> Biology. Sinauer Associates. Ch. 9. [sensitivity analysis]
#> Frankham R. (1995). Effective population size/adult population
#> size ratios in wildlife: a review.
#> Genetical Research 66: 95-107. [Ne/N empirical benchmarks]
#> Yonezawa et al. (2000). Evolution 54(6): 2007-2013. [model]
3.3 Interpretation
The elasticity printed above tells us how strongly Ne/N responds to Vk/k̄. A negative local elasticity is expected — higher reproductive variance always reduces Ne/N. The conservation ratio (worst-case Ne/N relative to Poisson baseline) tells a manager: if reproductive inequality is at the high end of the plausible range, how many more individuals are needed?
# Also sweep stage 6 (two-or-more-shoot adults) for comparison
sens_Vk_Lrup1_s6 <- Ne_sensitivity_Vk(
model_fn = Ne_sexual_Y2000,
T_mat = MatU_Lrup1,
F_vec = F_Lrup1,
D = D_Lrup1,
stage_index = 6,
Vk_range = seq(0.5, 8, by = 0.5),
Ne_target = 50,
population = "L. rupestris pop 1 — stage 6"
)
# Combined data for comparison plot
df_vk_compare <- bind_rows(
sens_Vk_Lrup1$data %>% mutate(Stage = "Stage 5 (1-shoot repro)"),
sens_Vk_Lrup1_s6$data %>% mutate(Stage = "Stage 6 (2+-shoot repro)")
)
ggplot(df_vk_compare, aes(x = Vk_over_k, y = NeN, color = Stage)) +
geom_line(linewidth = 1) +
geom_point(size = 2) +
geom_vline(xintercept = 1, linetype = "dashed", color = "grey50") +
scale_color_manual(values = c("#2166ac", "#d6604d")) +
labs(
title = expression(paste("Ne/N vs. sexual reproductive variance ", V[k]/bar(k))),
subtitle = "L. rupestris pop 1 — comparing focal stage",
x = expression(V[k]/bar(k)),
y = expression(N[e]/N),
color = NULL,
caption = "Dashed line = Poisson default (Vk/k̄ = 1). Monthly matrices."
) +
theme_classic(base_size = 12) +
theme(
plot.title = element_text(face = "bold"),
legend.position = "bottom",
panel.grid.major.y = element_line(color = "grey92", linewidth = 0.4)
)
Stage 6 (multi-shoot plants) has a larger effect because its fecundity (F = 0.414 per month) is more than three times that of stage 5 (F = 0.120). When high-fecundity adults have unequal reproductive output, the genetic consequences are proportionally larger.
Management implication: Strategies that equalise seed production — for example, hand-pollination programmes that ensure all reproductive adults contribute pollen — would have the greatest effect in stage 6 individuals.
