Below, I'll trace the most likely chain of events following the supervolcano eruption, which dims sunlight over the valley by 50% for several years. This dimming reduces the energy available for photosynthesis, disrupting the food chain: sunlight → plants → rabbits → foxes. I'll break it down into immediate (hours to days), medium-term (weeks to months), and long-term (years) effects on each population. My reasoning is based on ecological principles, including energy flow in trophic levels (each level depends on the one below it), population dynamics (e.g., carrying capacity, starvation, and reproduction rates), and the fact that the ash-induced dimming is temporary but prolonged (several years). I'll explain the causal chain step by step, noting that while the dimming is uniform, effects propagate up the food chain with delays due to stored energy and population inertia.

### Immediate Effects (Hours to Days)
The eruption casts ash into the atmosphere, immediately reducing sunlight by 50%. This is the initial trigger, directly impacting photosynthesis but with minimal immediate changes to animal populations, which rely on existing food stores.

- **Plants**: Photosynthesis rates drop by approximately 50% (assuming a linear relationship between light intensity and photosynthetic output, which holds for many plants under moderate light reduction). Plant growth slows immediately, as they produce less glucose for energy and biomass. However, existing plants don't die right away—they draw on stored carbohydrates. No significant population decline yet, but new growth (e.g., seedlings) is stunted.
  
- **Rabbits**: No immediate population change. Rabbits continue foraging on existing plants, which are still abundant from pre-eruption growth. Their numbers remain stable, as they have short-term food reserves in the environment and their bodies.

- **Foxes**: No immediate change. Foxes prey on rabbits, which are still plentiful. Their population remains stable, supported by existing rabbit numbers.

**Reasoning for this step in the causal chain**: The dimming directly affects plants (the base of the food chain) because they rely on sunlight for energy. Herbivores (rabbits) and carnivores (foxes) experience a lag, as they depend on biomass already produced. This is akin to a "bottom-up" control in ecosystems, where primary production limits higher trophic levels.

### Medium-Term Effects (Weeks to Months)
As reduced sunlight persists, plants can't replenish biomass quickly, leading to food shortages that cascade up the chain. Populations begin to decline, starting with plants and then affecting consumers.

- **Plants**: Population declines moderately (e.g., 20-40% reduction in density and biomass). With halved photosynthetic output, plants grow slower, produce fewer seeds/offsprings, and some weaker individuals die from energy deficits. The valley's plant cover thins, but resilient species (e.g., those with deep roots or shade tolerance) survive better.

- **Rabbits**: Population starts declining (e.g., 30-50% drop). With less plant biomass available, rabbits face food scarcity, leading to malnutrition, reduced reproduction, and higher starvation rates. Competition among rabbits intensifies, accelerating the decline. However, rabbits reproduce quickly (short generation time), so some persist if any plants remain.

- **Foxes**: Population begins to decline slightly (e.g., 10-30% drop). Fewer rabbits mean less prey, causing foxes to expend more energy hunting with lower success rates. This leads to initial starvation, especially among juveniles or weaker adults. Foxes might switch to alternative prey if available (though the problem specifies rabbits as their main food), but overall, their numbers start falling due to the rabbit shortage.

**Reasoning for this step in the causal chain**: Reduced plant productivity lowers the ecosystem's carrying capacity for herbivores. Rabbits, as direct consumers of plants, feel the impact next—starvation reduces their survival and birth rates, following Lotka-Volterra predator-prey dynamics where prey decline precedes predator decline. Foxes lag behind because they can survive short-term on fewer rabbits, but sustained shortages erode their population. This is a classic trophic cascade, amplified by the ongoing dimming.

### Long-Term Effects (Years, Including Post-Dimming Recovery)
The dimming lasts several years, allowing effects to compound: populations crash severely due to sustained energy reduction. After the ash clears (beyond "several years"), sunlight returns, enabling gradual recovery. However, long-term outcomes depend on how close populations approach extinction and external factors like migration or seed banks.

- **Plants**: Severe population crash (e.g., 50-80% decline) during the dimming years, with potential local die-offs in sun-dependent species. Biomass remains low, leading to soil erosion and reduced biodiversity. Once sunlight returns, plants recover relatively quickly (e.g., within 1-2 years to pre-eruption levels) via seed banks, wind-dispersed seeds, or regrowth from roots. Resilient species dominate the recovery, potentially altering the plant community composition.

- **Rabbits**: Major population collapse (e.g., 70-90% decline or near-extirpation). Prolonged plant scarcity causes widespread starvation and failed reproduction cycles. Rabbits have high reproductive potential, but if numbers drop too low, Allee effects (e.g., difficulty finding mates) could hinder recovery. Post-dimming, with plant recovery, rabbit numbers rebound (e.g., exponential growth within 1-3 years), potentially overshooting due to abundant food and low predation initially.

- **Foxes**: Significant decline or possible local extinction (e.g., 80-100% drop). With rabbits scarce for years, foxes face chronic food shortages, leading to emigration (if possible), starvation, and reproductive failure. As tertiary consumers, they are most vulnerable in long disruptions. Recovery lags behind others—post-dimming, foxes depend on rabbit rebound, so their numbers might take 3-5 years to stabilize, possibly requiring immigration from outside the valley. If foxes go extinct locally, the ecosystem could shift (e.g., rabbit overpopulation until predators recolonize).

**Reasoning for this step in the causal chain**: Over years, the cumulative energy deficit scales up: plants can't sustain herbivores, leading to rabbit crashes, which in turn decimate foxes (a "bottom-up" collapse). This follows the principle that higher trophic levels are more sensitive to prolonged disruptions due to energy loss at each level (only ~10% transfers per trophic level per the 10% rule). Recovery post-dimming is "top-down" delayed—plants recover first (as autotrophs), enabling rabbit resurgence, which then supports fox recovery. However, if any population hits a critical low (e.g., via stochastic events like disease), extinction risks increase, potentially leading to a new ecosystem equilibrium.

### Overall Chain of Events and Key Considerations
1. **Trigger**: Ash dims sunlight by 50%, halving photosynthetic energy input.
2. **Immediate**: Plants slow growth; animals unaffected.
3. **Medium-term**: Plant decline → rabbit food shortage → rabbit decline → fox food shortage → fox decline.
4. **Long-term**: Amplified crashes across all populations → post-dimming recovery starting with plants, then rabbits, then foxes.

This chain assumes no significant external interventions (e.g., no human aid) or complicating factors (e.g., ash toxicity directly killing organisms, though the problem emphasizes dimming). In reality, variations could occur—e.g., if foxes have alternative food sources, their decline might be milder. Empirical examples (e.g., post-Mt. St. Helens eruption in 1980) show similar patterns: initial plant suppression, herbivore declines, and predator lags, followed by recovery. If the dimming were permanent, the ecosystem might collapse entirely, but the temporary nature allows resilience.