How do we calculate the probability that a system will continue to survive?

01 / Starting from a reconstructed state

One possible present can lead to many modeled futures.

Lesson 14 gave us a conditional reconstruction law over possible January 2020 forest states. For this lesson, take one such state, call it X₀, and fix a set of model parameters and one scenario. The stochastic model can now generate many future histories from that same starting point.

Some stems die. Others grow. New stems may enter the measured population. Different sampled outcomes produce different trajectories, even when the initial reconstruction and scenario are held fixed.

The model evaluates horizons of 1, 5, 10, and 20 years. These are simulated outcomes under a declared demographic law, not observed future forests.

02 / What changes during a modeled year?

Mortality, growth, and candidate entry update the cohort state.

If a hypothetical modeled forest begins with 1,000 stems, loses 50, and gains 30, its next count is 980. The change in total count alone does not settle whether the system has satisfied its declared continuation requirements.

03 / Which future maintains the defined organization?

The specification checks structural support and how long an interruption lasts.

Paper V asks about two quantities at the annual checkpoints: basal area, BA(X_t), and juvenile support, J(X_t). The juvenile measure is a declared count proxy for woody stems below 10 cm diameter. The reference quantities come from the earlier census.

A prospective history must also satisfy the declared terminal structural conditions and a bound on its longest consecutive interruption. The parameter τ specifies the tolerated number of consecutive annual checkpoints below the structural criteria. For example, τ = 2 permits a qualifying interruption of at most two consecutive checkpoints.

These values are declared demonstration requirements. The study does not establish them as scientifically validated thresholds for the health of Harvard Forest.

04 / Candidate, lawful, viable

A sampled history must pass two different checks before it counts as viable.

First, the model's lawfulness check, Lawful_G, assesses valid cohort counts and diameters, chronology, and population accounting. This verifies consistency with the declared computational rules; it does not establish biological correctness.

Second, a lawful history must pass the continuation criteria just described. A lawfully simulated future may lose the specified organization, or remain structurally insufficient for longer than allowed.

This is the candidate → lawful → viable architecture from Papers I and IV, now evaluated on a finite sample bank. Histories that fail either test retain their original probability weight in the accounting that follows.

05 / A bank of 64 possible futures

Here is a simple example we can calculate exactly.

Imagine a single conditional bank of 64 equally weighted future histories. The following counts are invented to explain the mathematics and are separate from the study's reported numerical results.

The viable-mass estimate for this equally weighted bank is 48/64 = 0.75. A quarter of the original conditional sample weight remains attached to histories that do not qualify as viable.

This estimate is a statement about the declared simulation bank. It is not an independently calibrated 75% ecological survival probability.

06 / Keep the denominator

Discarding failures would quietly change the probability question.

The second calculation is true as a conditional statement after we deliberately select successful futures. It cannot answer the original question about how much mass the model assigns to possible failure.

The viable subprobability measure retains the original weights of viable histories and restricts attention to them. Its total mass can be less than one. The failed histories no longer contribute to viable response capacity, while their excluded weight remains explicitly accounted for under the original law.

07 / The first question · Q₁

How much of the original probability mass supports a viable future?

Introduce a declared minimum viable probability θ (theta). The first adequacy question asks whether viable mass reaches this level.

If the starting forest also presently realizes the declared organization, the same Q₁ adequacy answer becomes its full prospective Health answer for the specified scenario and horizon. That present-realization condition remains essential.

08 / The second question · Q₂

A viable future may still finish with a depleted juvenile reserve.

Paper V adds an additional terminal reserve requirement. For a simulated history γ, compare the terminal juvenile-support count with its initial value.

Call the original probability mass of histories that are both viable and reserve-qualified m_R(ρ). The second adequacy question is:

Suppose 40 of the 48 viable histories in the invented bank also qualify for juvenile reserve. Then m_R = 40/64 = 0.625. At θ = 0.75, Q₁ passes and Q₂ fails. The additional question detects something that the first does not require.

10 / What the actual finite design computes

Different declared scenarios yield sharply different continuation masses.

We can now return to the reported experiment. The primary design uses 1,024 combinations of reconstructed starting states, sampled predictive parameters, and growth-model variants. Each combination produces 64 future histories per scenario. Health is evaluated separately for each conditional unit.

The following 20-year results use α = 0.75, β = 0.4, τ = 2, θ = 0.75, and ρ = 0.5.

These figures are conditional numerical outputs from the declared simulation design, not calibrated probabilities for the real Harvard Forest and not an operational health classification.

The reference scenario assigns all sampled histories viable and reserve-qualified mass in the displayed design. The combined-stress scenario assigns none. Those are properties of the assumed dynamics, forcing, reconstruction, and finite sample, not claims about actual ecological outcomes.

11 / One subtle averaging distinction

The average viable mass is not the fraction of units that pass the Health requirement.

Look at D1. Its mean viable mass is 0.2099, or 20.99%. The fraction of conditional units satisfying Q₁ is 0.1973, or 19.73%. This difference has a precise source: the study applies the declared threshold to each unit's mass and then averages the true-or-false judgments.

For example, if a model ensemble contains many units with very low mass but a smaller set with mass above 75%, the average viable mass and the fraction of units passing the 75% threshold need not be numerically similar.

The fractions in Table 3 refer to computational model/reconstruction units, not proportions of independently sampled real forests.

12 / The scope of the calculation

We now have an auditable numerical interpretation of prospective capacity.

Paper V makes the inference chain explicit: census evidence enters a reconstruction procedure; a conditional state and parameter sample enter a stochastic future model; simulated histories undergo model-lawfulness and continuation tests; the original candidate probability law supplies viable and reserve-qualified mass; declared adequacy predicates yield conditional Health judgments.

The calculation is inspectable inside its declared model. Its empirical reliability depends on reconstruction calibration, mortality and entry assumptions, growth dynamics, the probability law over histories, and the ecological justification for the chosen requirements.

A mathematically correct model-conditional result is not automatically a reliable probability for the real forest. This study does not claim to establish an operational Health status for Harvard Forest.