Abstract
This study evaluates the sustainability of harvesting practices for Pyromycis' native flora, specifically the Dark Wizard Mushroom, Witch’s Ringlet, Brimmed Bonnet, and Glowy Rose. Utilizing observed maximum daily harvested specimens as initial populations (P₀), we model population dynamics under a regime of complete daily harvesting followed by instantaneous seed propagation when planted. The analysis reveals that species with seed yield ratios below 1 face inevitable extinction, with unskilled harvesting (Skill Level 0) drastically accelerating this process compared to skilled harvesting (Skill Level 3). The Glowy Rose, with a ratio exceeding 1, exhibits exponential growth and was chosen as a control flora. We present precise extinction timelines, visual data representations, and enhanced recommendations to ensure species preservation, addressing both immediate regulatory needs and long-term conservation strategies. When other strategies are implemented such as reducing daily harvesting will further delay the inevitable, providing in some case over 2.0x days before extinction. It is not until we better understand the natural propagation rate that we will be able to determine if with proper harvesting controls, we are able to sustain different species on Pyromycis.
Introduction
Pyromycis hosts a unique fungal ecosystem critical to its biodiversity, yet current harvesting practices threaten its native species. This research quantifies the impact of harvester skill levels on seed yield and population sustainability, using empirical data on maximum daily harvested specimens (P₀) and seed yield ratios. By modeling extinction trajectories for the Dark Wizard Mushroom (at Skill Levels 0 and 3), Witch’s Ringlet, Brimmed Bonnet, and Glowy Rose (all at Skill Level 3), we identify critical thresholds and propose robust conservation measures.
Methodology
Data Collection
Field data were gathered on Pyromycis, detailing:
- Harvested Specimens (Max Daily): Maximum specimens harvested in a single day, set as P₀.
- Seeds Collected: Total seeds obtained per harvest.
- Seed Yield Ratio: Seeds per specimen (ratio = (Seeds Collected)⁄(P₀)).
Table 0: Raw Collected Data
| Specimen | Skill | Harvested | Seeds | Ratio |
|---|---|---|---|---|
| Dark Wizard | 0 | 275 | 93 | 0.338181818 |
| Dark Wizard | 3 | 150 | 108 | 0.72 |
| Brimmed Bonnet | 0 | 109 | 36 | 0.330275229 |
| Brimmed Bonnet | 3 | 67 | 40 | 0.597014925 |
| Witch's Ringlet | 0 | 228 | 80 | 0.350877193 |
| Witch's Ringlet | 3 | 161 | 118 | 0.732919255 |
| Glowy Rose | 0 | 206 | 206 | 1 |
| Glowy Rose | 3 | 173 | 346 | 2 |
**Note - Table 0 corresponds to the raw collected data over multiple different days for testing purposes to gather the proper harvest to seed ratio. The maximum species found in a single day are used going forward for all modeling and calculations, this maximum is denoted as P₀
Table 1: Harvesting Data
| Species | Skill Level | P₀ | Seeds Collected | Seed Yield Ratio |
|---|---|---|---|---|
| Dark Wizard Mushroom | 0 | 112 | 37.876 | 0.338181818 |
| Dark Wizard Mushroom | 3 | 112 | 80.64 | 0.72 |
| Brimmed Bonnet | 0 | 109 | 36 | 0.330275229 |
| Brimmed Bonnet | 3 | 109 | 65.075 | 0.597014925 |
| Witch's Ringlet | 0 | 228 | 80 | 0.350877193 |
| Witch's Ringlet | 3 | 228 | 167.006 | 0.732919255 |
| Glowy Rose | 0 | 206 | 206 | 1.0 |
| Glowy Rose | 3 | 206 | 412 | 2.0 |
Assumptions and Limitations
- Instantaneous Propagation: Assumes instant seed germination, possibly overestimating recovery rates if planting processes are modified.
- 0 Natural Growth: Assumes 0 natural growth as the current implementation of natural growth is competitive in nature amongst all flora on the planet and thus the likelihood of natural propagation is unlikely. In Table 5, it discusses the minimum propagation rate that would be needed to prevent extinction when it is implemented in accordance to a true natural propagation.
- Single Harvester Model: Assumes one harvester collects P₀ daily; multiple harvesters could hasten depletion.
Mathematical Model
We assume daily harvesting of the entire population Pₙ, yielding Sₙ = Pₙ × ratio seeds, which instantly propagate to form the next day’s population Pₙ₊₁ = Sₙ. This forms a geometric sequence:
Pₙ₊₁ = Pₙ × ratioPₙ = P₀ × ratioⁿExtinction occurs when Pₙ < 1 (no whole specimens remain). For ratio < 1, the days to extinction (neₓtiₙct) is derived as follows:
P₀ × ratioⁿ < 1ratioⁿ < (1)⁄(P₀)n ln(ratio) < -ln(P₀)n > (-ln(P₀))⁄(ln(ratio)) = (ln(P₀))⁄(-ln(ratio)) (since ln(ratio) < 0, inequality flips)neₓtiₙct = ⌈ (ln(P₀))⁄(-ln(ratio)) ⌉For ratio ≥ 1, the population sustains or grows indefinitely.
Verification of Calculations
- Dark Wizard Mushroom (Skill 0):
- P₀ = 112, ratio = 0.338181818
- n > (ln(112))⁄(-ln(0.338181818)) ≈ (4.7185)⁄(1.0846) ≈ 4.351
- neₓtiₙct = lceil 4.351 rceil = 5
- Check: P₄ = 112 × 0.338181818⁴ ≈ 1.47 > 1, P₅ ≈ 0.50 < 1
- Dark Wizard Mushroom (Skill 3):
- P₀ = 112, ratio = 0.72
- n > (ln(112))⁄(-ln(0.72)) ≈ (4.7185)⁄(0.3285) ≈ 14.36
- neₓtiₙct = lceil 14.36 rceil = 15
- Check: P₁₄ = 112 × 0.72¹⁴ ≈ 1.62 > 1, P₁₅ ≈ 1.17 > 1, P₁₆ ≈ 0.84 < 1
- Brimmed Bonnet (Skill 0):
- P₀ = 109, ratio = 0.330275229
- n > (ln(109))⁄(-ln(0.330275229)) ≈ (4.6913)⁄(1.108) ≈ 4.23
- neₓtiₙct = lceil 4.23 rceil = 5
- Check: P₄ = 109 × 0.330275229⁴ ≈ 1.30 > 1, P₅ ≈ 0.43 < 1
- Brimmed Bonnet (Skill 3):
- P₀ = 109, ratio = 0.597014925
- n > (ln(109))⁄(-ln(0.597014925)) ≈ (4.6913)⁄(0.5158) ≈ 9.09
- neₓtiₙct = lceil 9.09 rceil = 10
- Check: P₉ = 109 × 0.597014925⁹ ≈ 1.14 > 1, P₁₀ ≈ 0.68 < 1
- Witch's Ringlet (Skill 0):
- P₀ = 228, ratio = 0.350877193
- n > (ln(228))⁄(-ln(0.350877193)) ≈ (5.4293)⁄(1.0477) ≈ 5.18
- neₓtiₙct = lceil 5.18 rceil = 6
- Check: P₅ = 228 × 0.350877193⁵ ≈ 1.21 > 1, P₆ ≈ 0.42 < 1
- Witch's Ringlet (Skill 3):
- P₀ = 228, ratio = 0.732919255
- n > (ln(228))⁄(-ln(0.732919255)) ≈ (5.4293)⁄(0.3106) ≈ 17.48
- neₓtiₙct = lceil 17.48 rceil = 18
- Check: P₁₇ = 228 × 0.732919255¹⁷ ≈ 1.12 > 1, P₁₈ ≈ 0.82 < 1
- Glowy Rose (Skill 0):
- P₀ = 206, ratio = 1.0
- Pₙ = 206 × 1ⁿ = 206, stable, no extinction.
- Glowy Rose (Skill 3):
- P₀ = 206, ratio = 2.0
- Pₙ = 206 × 2ⁿ, grows exponentially, no extinction.
Results
Extinction Timelines
Table 2: Days to Extinction
| Species | Skill Level | P₀ | Ratio | Days to Extinction |
|---|---|---|---|---|
| Dark Wizard Mushroom | 0 | 112 | 0.338181818 | 5 |
| Dark Wizard Mushroom | 3 | 112 | 0.72 | 15 |
| Brimmed Bonnet | 0 | 109 | 0.330275229 | 5 |
| Brimmed Bonnet | 3 | 109 | 0.597014925 | 10 |
| Witch's Ringlet | 0 | 228 | 0.350877193 | 6 |
| Witch's Ringlet | 3 | 228 | 0.732919255 | 18 |
| Glowy Rose | 0 | 206 | 1.0 | Never |
| Glowy Rose | 3 | 206 | 2.0 | Never |
Discussion
Comparative Analysis
- Extinction Rate Difference:
- Dark Wizard Mushroom: Skill 0 yields extinction in 5 days (ratio 0.338181818), 66.7% faster than Skill 3’s 15 days (ratio 0.72), a 3-fold increase in survival time with skill.
- Brimmed Bonnet: Skill 0 leads to extinction in 5 days (ratio 0.330275229), versus 10 days at Skill 3 (ratio 0.597014925), a 100% increase.
- Witch's Ringlet: Skill 0 results in 6 days (ratio 0.350877193), while Skill 3 extends to 18 days (ratio 0.732919255), a 3-fold increase.
- Across Skill 0: Dark Wizard Mushroom and Brimmed Bonnet are most vulnerable (5 days), Witch's Ringlet lasts slightly longer (6 days).
- Across Skill 3: Brimmed Bonnet extinguishes fastest (10 days), Witch's Ringlet lasts longest (18 days).
- Sustainability through Skill Enhancement:
- Training to Skill 3 markedly delays extinction for species with ratio < 1, emphasizing skill’s role in conservation. Glowy Rose’s stability (Skill 0) and growth (Skill 3) highlight that ratio ≥ 1 ensures sustainability.
Implications for Conservation
- Urgent Need for Regulation: Rapid extinction at Skill 0 demands bans on unskilled harvesting.
- Education and Training: Skill enhancement delays extinction, but ratios below 1 require reduced harvest intensity for long-term survival.
Mathematical Model
- Pₙ: Population on day n
- f: Fraction harvested daily (e.g., 0.3 for 30%)
- r: Seed yield ratio
Daily process:
- Harvest f Pₙ specimens, yielding f Pₙ × r seeds.
- Plant these seeds to produce f Pₙ × r new specimens.
- Unharvested specimens: (1 - f) Pₙ
- Next day’s population:
Pₙ₊₁ = (1 - f) Pₙ + f Pₙ × r- Simplified:
Pₙ₊₁ = Pₙ [(1 - f) + f r]This forms a geometric sequence with growth factor G = (1 - f) + f r. For species with r < 1, G < 1 when f > 0, leading to population decline until extinction (when Pₙ < 1). The number of days to extinction is:
neₓtiₙct = ⌈ (ln P₀)⁄(-ln G) ⌉For species with r ≥ 1 (e.g., Glowy Rose), G ≥ 1, so the population remains stable or grows, avoiding extinction.
Extinction Times at Different Harvesting Rates
We calculate neₓtiₙct for each species at harvesting rates f = 0.3, 0.5, 0.7, and 1.0 using the formula:
neₓtiₙct = ⌈ (ln P₀)⁄(-ln G) ⌉where
G = (1 - f) + f × rFor species with r ≥ 1, such as Glowy Rose, G ≥ 1, meaning the population does not decline, and extinction never occurs. Below are the P₀ and r values for each species and skill level:
- Dark Wizard: P₀ = 112, r = 0.338181818 (Skill 0); P₀ = 112, r = 0.72 (Skill 3)
- Brimmed Bonnet: P₀ = 109, r = 0.330275229 (Skill 0); P₀ = 109, r = 0.597014925 (Skill 3)
- Witch's Ringlet: P₀ = 228, r = 0.350877193 (Skill 0); P₀ = 228, r = 0.732919255 (Skill 3)
- Glowy Rose: P₀ = 206, r = 1.0 (Skill 0); P₀ = 206, r = 2.0 (Skill 3)
Dark Wizard
Skill 0 (P₀ = 112, r = 0.338181818)
- 30% Harvesting:
G = 0.7 + 0.3 × 0.338181818 ≈ 0.8015 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.8015) ⌉ ≈ 22 days- 50% Harvesting:
G = 0.5 + 0.5 × 0.338181818 ≈ 0.6691 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.6691) ⌉ ≈ 12 days- 70% Harvesting:
G = 0.3 + 0.7 × 0.338181818 ≈ 0.5367 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.5367) ⌉ ≈ 8 days- 100% Harvesting:
G = 0.338181818 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.338181818) ⌉ ≈ 5 daysSkill 3 (P₀ = 112, r = 0.72)
- 30% Harvesting:
G = 0.7 + 0.3 × 0.72 = 0.916 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.916) ⌉ ≈ 54 days- 50% Harvesting:
G = 0.5 + 0.5 × 0.72 = 0.86 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.86) ⌉ ≈ 32 days- 70% Harvesting:
G = 0.3 + 0.7 × 0.72 = 0.804 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.804) ⌉ ≈ 22 days- 100% Harvesting:
G = 0.72 → neₓtiₙct = ⌈ (ln 112)⁄(-ln 0.72) ⌉ ≈ 15 daysBrimmed Bonnet
Skill 0 (P₀ = 109, r = 0.330275229)
- 30% Harvesting:
G = 0.7 + 0.3 × 0.330275229 ≈ 0.7991 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.7991) ⌉ ≈ 21 days- 50% Harvesting:
G = 0.5 + 0.5 × 0.330275229 ≈ 0.6651 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.6651) ⌉ ≈ 12 days- 70% Harvesting:
G = 0.3 + 0.7 × 0.330275229 ≈ 0.5312 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.5312) ⌉ ≈ 8 days- 100% Harvesting:
G = 0.330275229 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.330275229) ⌉ ≈ 5 daysSkill 3 (P₀ = 109, r = 0.597014925)
- 30% Harvesting:
G = 0.7 + 0.3 × 0.597014925 ≈ 0.8791 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.8791) ⌉ ≈ 37 days- 50% Harvesting:
G = 0.5 + 0.5 × 0.597014925 ≈ 0.7985 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.7985) ⌉ ≈ 21 days- 70% Harvesting:
G = 0.3 + 0.7 × 0.597014925 ≈ 0.7179 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.7179) ⌉ ≈ 15 days- 100% Harvesting:
G = 0.597014925 → neₓtiₙct = ⌈ (ln 109)⁄(-ln 0.597014925) ⌉ ≈ 10 daysWitch's Ringlet
Skill 0 (P₀ = 228, r = 0.350877193)
- 30% Harvesting:
G = 0.7 + 0.3 × 0.350877193 ≈ 0.8053 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.8053) ⌉ ≈ 26 days- 50% Harvesting:
G = 0.5 + 0.5 × 0.350877193 ≈ 0.6754 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.6754) ⌉ ≈ 14 days- 70% Harvesting:
G = 0.3 + 0.7 × 0.350877193 ≈ 0.5456 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.5456) ⌉ ≈ 9 days- 100% Harvesting:
G = 0.350877193 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.350877193) ⌉ ≈ 6 daysSkill 3 (P₀ = 228, r = 0.732919255)
- 30% Harvesting:
G = 0.7 + 0.3 × 0.732919255 ≈ 0.9199 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.9199) ⌉ ≈ 65 days- 50% Harvesting:
G = 0.5 + 0.5 × 0.732919255 ≈ 0.8665 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.8665) ⌉ ≈ 38 days- 70% Harvesting:
G = 0.3 + 0.7 × 0.732919255 ≈ 0.8130 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.8130) ⌉ ≈ 27 days- 100% Harvesting:
G = 0.732919255 → neₓtiₙct = ⌈ (ln 228)⁄(-ln 0.732919255) ⌉ ≈ 18 daysGlowy Rose
Skill 0 (P₀ = 206, r = 1.0)
- For any $f$:
G = (1 - f) + f × 1.0 = 1.0- Since G = 1, the population remains stable: never extinct.
Skill 3 (P₀ = 206, r = 2.0)
- For any $f$:
G = (1 - f) + f × 2.0 = 1 + f > 1- Since G > 1, the population grows: never extinct.
Summary Table: Days to Extinction
| Specimen | Skill | 30% Harvesting | 50% Harvesting | 70% Harvesting | 100% Harvesting |
|---|---|---|---|---|---|
| Dark Wizard | 0 | 22 days | 12 days | 8 days | 5 days |
| Dark Wizard | 3 | 54 days | 32 days | 22 days | 15 days |
| Brimmed Bonnet | 0 | 21 days | 12 days | 8 days | 5 days |
| Brimmed Bonnet | 3 | 37 days | 21 days | 15 days | 10 days |
| Witch's Ringlet | 0 | 26 days | 14 days | 9 days | 6 days |
| Witch's Ringlet | 3 | 65 days | 38 days | 27 days | 18 days |
| Glowy Rose | 0 | Never | Never | Never | Never |
| Glowy Rose | 3 | Never | Never | Never | Never |
Observations
- Lower Harvesting Rates: Reducing the harvesting rate significantly extends extinction times for species with r < 1. For example, Witch's Ringlet at Skill 3 lasts 65 days at 30% harvesting compared to 18 days at 100%—a 3.6-fold increase.
- Skill Level Impact: Higher skill levels (Skill 3) increase r, leading to longer extinction times due to improved seed yield ratios. For instance, Dark Wizard at Skill 3 lasts 54 days at 30% harvesting versus 22 days at Skill 0.
- Glowy Rose: With r ≥ 1, this species is sustainable under all harvesting conditions, never facing extinction. At Skill 3 (r = 2.0), the population even grows regardless of harvesting rate.
Ideal Harvesting Rate
For species with r < 1, any harvesting rate f > 0 leads to eventual extinction without natural propagation. Since an "ideal" rate that prevents extinction indefinitely is impossible, we define "ideal" as the maximum harvesting rate that sustains the population for at least 100 days. We solve for the maximum f where neₓtiₙct ≥ 100:
(ln P₀)⁄(-ln [(1 - f) + f r]) ≥ 99Rearranging to solve for f:
f ≤ frac{1 - e⁻⁽lⁿ P_⁰⁾⁄⁽⁹⁹⁾}{1 - r}Calculations for Each Species at Skill 3
- Dark Wizard ($P_0 = 112$, $r = 0.72$):
f ≤ frac{1 - e⁻⁽lⁿ ¹¹²⁾⁄⁽⁹⁹⁾}{1 - 0.72} ≈ 0.1764 (17.64\%)- Verification: With f = 0.1764, G = (1 - f) + f r ≈ 0.941, and neₓtiₙct = (ln 112)⁄(-ln 0.941) ≈ 100 days.
- Brimmed Bonnet ($P_0 = 109$, $r = 0.597014925$):
f ≤ frac{1 - e⁻⁽lⁿ ¹⁰⁹⁾⁄⁽⁹⁹⁾}{1 - 0.597014925} ≈ 0.1163 (11.63\%)- Verification: With f = 0.1163, G ≈ 0.936, and neₓtiₙct ≈ 100 days.
- Witch’s Ringlet ($P_0 = 228$, $r = 0.732919255$):
f ≤ frac{1 - e⁻⁽lⁿ ²²⁸⁾⁄⁽⁹⁹⁾}{1 - 0.732919255} ≈ 0.2035 (20.35\%)- Verification: With f = 0.2035, G ≈ 0.944, and neₓtiₙct ≈ 100 days.
Harvest Yield at Ideal Rates
The initial daily harvests at these ideal rates are:
- Dark Wizard (Skill 3):
0.1764 × 112 ≈ 19.76 specimens, seeds = 19.76 × 0.72 ≈ 14.23- Brimmed Bonnet (Skill 3):
0.1163 × 109 ≈ 12.68 specimens, seeds = 12.68 × 0.597014925 ≈ 7.57- Witch’s Ringlet (Skill 3):
0.2035 × 228 ≈ 46.40 specimens, seeds = 46.40 × 0.732919255 ≈ 34.01Since the population declines over time for species with r < 1, daily harvests decrease accordingly. For Glowy Rose, with r ≥ 1 (1.0 at Skill 0 and 2.0 at Skill 3), no extinction occurs, allowing harvesting to be optimized for yield without risking depletion. For example:
- Glowy Rose (Skill 3, $r = 2.0$): At f = 1.0 (100%), harvest = 206 specimens daily, seeds = 412, and the population remains stable or grows.
Conclusion
For species with r < 1 (Dark Wizard, Brimmed Bonnet, Witch’s Ringlet), the "ideal" harvesting rate is the maximum f that sustains the population for at least 100 days, ranging from 11.63% to 20.35% at Skill 3. Initial harvests are modest (12.68 to 46.40 specimens daily), reflecting the need to balance yield with longevity. Glowy Rose, with r ≥ 1, supports higher harvesting rates without depletion, making it uniquely sustainable among the species studied.
The population declines as Pₙ = P₀ Gⁿ, reducing daily harvests over time. Total cumulative harvest over 100 days approximates (P₀)⁄(1 - r):
- Dark Wizard: (112)⁄(0.28) ≈ 314 specimens
- Witch’s Ringlet: (228)⁄(0.267081) ≈ 603 specimens
- Brimmed Bonnet: (109)⁄(0.402985) ≈ 166 specimens
Analysis
- Unskilled Harvesting (Skill 0) Unsustainability:
- At Skill 0, specimens with r < 1 (Dark Wizard, Brimmed Bonnet, Witch's Ringlet) face rapid extinction due to low seed yield ratios. For example, Brimmed Bonnet lasts only 5 days at 100% harvesting.
- Higher harvesting rates accelerate collapse (e.g., Dark Wizard drops from 26 days at 30% to 6 days at 100%).
- Skilled Harvesting (Skill 3) Benefits:
- Skill 3 increases r, significantly extending survival times. For instance, Witch's Ringlet at 30% harvesting lasts 61 days at Skill 3 vs. 26 days at Skill 0 (a 2.35-fold increase).
- The benefit is most notable at lower harvesting rates, where higher r keeps G closer to 1, slowing population decline.
- Specimen Variation:
- Brimmed Bonnet is the most vulnerable at Skill 0 due to its low P₀ (109) and r (0.330275229), resulting in the shortest survival times.
- Witch's Ringlet benefits most from Skill 3, with the highest r (0.732919255) among declining specimens, achieving the longest survival (61 days at 30%).
- Glowy Rose is immune to extinction: with r = 1 at Skill 0, it’s stable; with r = 2 at Skill 3, it grows.
- Conclusion:
Unskilled harvesting (Skill 0) leads to rapid population collapse due to insufficient seed yields, especially at high harvesting rates. Skilled harvesting (Skill 3) mitigates this by improving r, prolonging survival and underscoring the importance of sustainable practices.
Analysis and Illustration
The calculations below show that unskilled harvesting (Skill 0) leads to extinction much faster than skilled harvesting (Skill 3) for species with seed ratios (r) less than 1. Glowy Rose, with r ≥ 1, remains sustainable. Below is the detailed analysis:
Seed Yield Impact
- Skill 0: Lower seed ratios (e.g., Dark Wizard Mushroom: r = 0.338181818, Brimmed Bonnet: r = 0.330275229) result in fewer seeds per harvested specimen, leading to a smaller growth factor G = r under full harvesting, which accelerates population decline.
- Skill 3: Higher ratios (e.g., Dark Wizard Mushroom: r = 0.72, Witch’s Ringlet: r = 0.732919255) slow the decline, though extinction still occurs when r < 1.
Extinction Timeline (100% Harvesting)
The number of days to extinction when r < 1 is calculated using:
neₓtiₙct = ⌈ (ln P₀)⁄(-ln r) ⌉- Dark Wizard Mushroom:
- Skill 0: P₀ = 112, r = 0.338181818
- ln 112 ≈ 4.7185, -ln 0.338181818 ≈ 1.0846
- (4.7185)⁄(1.0846) ≈ 4.35, neₓtiₙct = lceil 4.35 rceil = 5 days
- Skill 3: P₀ = 112, r = 0.72
- ln 112 ≈ 4.7185, -ln 0.72 ≈ 0.3285
- (4.7185)⁄(0.3285) ≈ 14.36, neₓtiₙct = lceil 14.36 rceil = 15 days
- Brimmed Bonnet:
- Skill 0: P₀ = 109, r = 0.330275229
- ln 109 ≈ 4.6913, -ln 0.330275229 ≈ 1.108
- (4.6913)⁄(1.108) ≈ 4.23, neₓtiₙct = lceil 4.23 rceil = 5 days
- Skill 3: P₀ = 109, r = 0.597014925
- ln 109 ≈ 4.6913, -ln 0.597014925 ≈ 0.515
- (4.6913)⁄(0.515) ≈ 9.10, neₓtiₙct = lceil 9.10 rceil = 10 days
- Witch’s Ringlet:
- Skill 0: P₀ = 228, r = 0.350877193
- ln 228 ≈ 5.4293, -ln 0.350877193 ≈ 1.0477
- (5.4293)⁄(1.0477) ≈ 5.18, neₓtiₙct = lceil 5.18 rceil = 6 days
- Skill 3: P₀ = 228, r = 0.732919255
- ln 228 ≈ 5.4293, -ln 0.732919255 ≈ 0.3106
- (5.4293)⁄(0.3106) ≈ 17.48, neₓtiₙct = lceil 17.48 rceil = 18 days
- Glowy Rose:
- Skill 0: r = 1.0, population stable (no extinction).
- Skill 3: r = 2.0, population grows exponentially (no extinction).
Survival Multiplier
The survival time ratio (Skill 3 / Skill 0) highlights the benefit of skilled harvesting:
- Dark Wizard Mushroom: (15)⁄(5) = 3.0
- Brimmed Bonnet: (10)⁄(5) = 2.0
- Witch’s Ringlet: (18)⁄(6) = 3.0
This demonstrates that Skill 3 extends survival by 2.0 to 3.0 times compared to Skill 0, based on the exponential decay model Pₙ₊₁ = Pₙ × r at 100% harvesting.
Additional Statistical Insights
Rate of Decline
- Skill 0: Seed ratios range from 0.330275229 (Brimmed Bonnet) to 0.350877193 (Witch’s Ringlet), ensuring rapid population decline under full harvesting.
- Skill 3: Seed ratios range from 0.597014925 (Brimmed Bonnet) to 0.732919255 (Witch’s Ringlet), slowing the decline but still leading to extinction when r < 1.
Consistency Across Species
- Species with r < 1 (all except Glowy Rose) face inevitable extinction under 100% harvesting, with Skill 3 delaying the outcome.
- Glowy Rose’s r ≥ 1 ensures sustainability or growth, underscoring the importance of seed ratios.
Understanding the Propagation Rate and Percentage Increase
The minimum propagation rate lambdamiₙ is the growth factor required for the unharvested population to prevent decline, given a harvest fraction f and seed ratio r:
lambdamiₙ = (1 - f r)⁄(1 - f)The percentage increase is:
Percentage Increase = (lambdamiₙ - 1) × 100\%- Positive: Unharvested population must grow by this percentage.
- Zero: Stability without additional growth.
- Negative: Excess sustainability (set to 0% practically).
Table 5: Minimum Propagation Required to Prevent Extinction
| Species | Skill | $r$ | 30% Harvesting ($f = 0.3$) | 50% Harvesting ($f = 0.5$) | 70% Harvesting ($f = 0.7$) | 100% Harvesting ($f = 1.0$) |
|---|---|---|---|---|---|---|
| Dark Wizard Mushroom | 0 | 0.338 | 1.284 (28.4%) | 1.662 (66.2%) | 2.544 (154.4%) | Impossible |
| Dark Wizard Mushroom | 3 | 0.72 | 1.120 (12.0%) | 1.280 (28.0%) | 1.653 (65.3%) | Impossible |
| Brimmed Bonnet | 0 | 0.330 | 1.283 (28.3%) | 1.660 (66.0%) | 2.521 (152.1%) | Impossible |
| Brimmed Bonnet | 3 | 0.597 | 1.173 (17.3%) | 1.403 (40.3%) | 1.940 (94.0%) | Impossible |
| Witch’s Ringlet | 0 | 0.351 | 1.286 (28.6%) | 1.665 (66.5%) | 2.532 (153.2%) | Impossible |
| Witch’s Ringlet | 3 | 0.733 | 1.114 (11.4%) | 1.267 (26.7%) | 1.623 (62.3%) | Impossible |
| Glowy Rose | 0 | 1.0 | 1.0 (0%) | 1.0 (0%) | 1.0 (0%) | 1.0 (0%) |
| Glowy Rose | 3 | 2.0 | 0.571 (-42.9%) | 0 (0%) | 0 (0%) | 0 (0%) |
Notes:
- lambdamiₙ is calculated as (1 - f r)⁄(1 - f) and rounded to three decimal places. $r$ is rounded to three decimal places.
- Percentage increase is (lambdamiₙ - 1) × 100\%, rounded to one decimal place.
- For f = 1.0, if r < 1, extinction is inevitable ("Impossible"); if r ≥ 1, the population sustains or grows ("Never extinct").
- For Glowy Rose at Skill 3, negative lambdamiₙ values indicate excess sustainability, but 0% is used practically.
Conclusion
Unskilled harvesting (Skill 0) is unsustainable for species with r < 1, leading to extinction in 5–6 days, while skilled harvesting (Skill 3) extends survival to 10–18 days. Glowy Rose, with r ≥ 1, thrives under both skill levels, highlighting the pivotal role of seed ratios in ecological sustainability. Prohibiting unskilled harvesting could significantly delay extinction, supporting conservation efforts.
Recommendations
A harvesting rate of 10-15% offers a practical balance for sustainability:
- 10% Harvesting: For Dark Wizard Mushroom at Skill 3, G = 0.972, n ≈ 166 days, initial harvest ≈ 11.2 specimens.
- 15% Harvesting: G ≈ 0.958, n ≈ 110 days, initial harvest ≈ 16.8 specimens.
The "ideal" rate depends on daily harvesting needs. For example, if 10 specimens are required daily for Dark Wizard Mushroom with an initial population P₀ = 112 at Skill 3, the harvesting fraction is f = (10)⁄(112) ≈ 0.0893 or 8.93%, yielding approximately n = (ln 112)⁄(-ln (1 - 0.0893 + 0.0893 × 0.72)) ≈ 189 days of survival. However, adjusting to f ≈ 11.4\% (f = 0.114) gives G = (1 - 0.114) + 0.114 × 0.72 = 0.968, and n ≈ (ln 112)⁄(-ln 0.968) ≈ 139 days. Generally, rates between 10-20% optimize both longevity and yield, adjustable based on specific requirements.
To sustain the species at Skill 3, a propagation rate of at least 17.3%–40.3% is required to offset harvesting losses. If specimens are harvested at only 30% capacity daily, the propagation rate need only be between 11.4%–17.3% to prevent extinction. This necessitates strict harvesting rules and additional measures:
- Prohibit Unskilled Harvesting of Vulnerable Species
- Immediate Enforcement:
- Ban Skill Level 0 harvesting for Dark Wizard Mushroom, Witch’s Ringlet, and Brimmed Bonnet within 90 days.
- Deploy enforcement teams to monitor compliance.
- Deploy GUNC Expert Botanists:
- Ensure proper harvesting techniques are applied and provide seeds to those in need.
- Confiscate illegally harvested specimens and genetically engineer these species to restore them to the natural environment, simulating propagation lost to excessive harvesting.
- Mandatory Harvester Training Programs
- Certification Requirements:
- Mandate Skill Level 3 certification via exam within 6 months.
- Issue 3-year renewable licenses with refresher courses.
- Educational Initiatives:
- Launch Pyromycis Harvester Academy with subsidized training.
- Partner with universities to train 1,000 harvesters annually.
- Establishment of Conservation Protocols
- Harvesting Quotas:
- Limit harvesting to a fraction f where the population next day satisfies:
Pₙ₊₁ = Pₙ (1 - f + f × ratio) ≥ PₙFor\text{ratio} < 1, incorporate the natural reproduction rater, requiring:f ≤ (r)⁄(1 - ratio)
- Example: For Dark Wizard Mushroom (Skill 3, ratio = 0.72), assuming r = 0.3, the sustainable harvesting fraction is f ≤ (0.3)⁄(1 - 0.72) ≈ 1.07, indicating sustainability if f is kept below this threshold.
- Monitoring Systems:
- Use drones or other video monitoring equipment to track harvesting rates.
- Public Awareness Campaigns
- Community Engagement:
- Launch multimedia campaigns to reach as many people in "The Garden" as possible.
- Host annual “Fungi Festivals” for education and stewardship.
- Stakeholder Collaboration:
- Form a Botany Conservation Alliance meeting quarterly.
- Offer tax breaks for sustainable harvesting investments.
Conclusion
Conclusion This analysis reveals the dire consequences of current harvesting practices on Pyromycis’ flora, with unskilled harvesting accelerating extinction for species with sub-unitary seed yield ratios (e.g., Dark Wizard Mushroom, Witch’s Ringlet, Brimmed Bonnet). While Skill Level 3 delays depletion—for instance, Dark Wizard Mushroom at Skill 0 lasts 5 days (n = ⌈ (ln 112)⁄(-ln 0.338181818) ⌉ = 5n = ⌈ (ln 112)⁄(-ln 0.338181818) ⌉ = 5) versus 15 days at Skill 3 (n = ⌈ (ln 112)⁄(-ln 0.72) ⌉ = 15n = ⌈ (ln 112)⁄(-ln 0.72) ⌉ = 15)—only comprehensive interventions—strict regulations, advanced training, and conservation technologies—can ensure survival. The Glowy Rose’s resilience (Skill 0: r=1.0r = 1.0r = 1.0 , Skill 3: r=2.0r = 2.0r = 2.0 , no extinction) offers hope, underscoring sustainable management potential. Immediate action, supported by research, is essential to safeguard Pyromycis’ ecological legacy. Without significantly higher propagation rates and proper harvesting controls, extinction is inevitable for these mushroom species. It is GUNC’s duty to raise awareness and act decisively to prevent the loss of this critical flora.
Acknowledgments
Thanks to the Galactic Union of Natural Conservation for funding, XenoLife Research Institute and Pyromycis field researchers for their efforts under challenging conditions.

