The eruption of a supervolcano that casts a thick layer of ash into the atmosphere, dimming the sun by 50% over a valley, would have significant cascading effects on the plant, rabbit, and fox populations. Let's examine the immediate, medium-term, and long-term effects step-by-step:

### Immediate Effects

1. **Plants:**
   - **Reduced Photosynthesis:** With sunlight diminished by 50%, photosynthesis in plants will be severely impacted. Plants rely on light to convert carbon dioxide and water into glucose and oxygen. Less light means less energy for growth.
   - **Stunted Growth and Reduced Reproduction:** Plant growth will slow down, and reproductive processes (such as flowering) may be delayed or reduced.

2. **Rabbits:**
   - **Decreased Food Availability:** As plants struggle to grow and reproduce, there will be less vegetation available for the rabbits. This could lead to malnutrition and weakened health among the rabbit population.
   - **Increased Competition:** Rabbits may compete more intensely for the limited resources, leading to higher stress levels and potential conflicts within the population.

3. **Foxes:**
   - **Reduced Prey Availability:** With fewer healthy rabbits due to decreased plant availability, foxes will face a shortage of their primary food source. This could result in reduced energy intake and weaker health for the foxes.

### Medium-Term Effects (1-2 Years)

1. **Plants:**
   - **Survival of the Fittest:** Some plant species may adapt better than others to the lower light conditions, potentially outcompeting less resilient species. However, overall plant diversity might decrease.
   - **Shrinking Canopy and Leaf Area:** Many plants will have smaller canopies and fewer leaves, reducing their ability to capture what little light is available.

2. **Rabbits:**
   - **Population Decline:** The rabbit population is likely to decline due to insufficient food and increased competition. Weaker individuals may die off, leaving behind only those with better survival strategies.
   - **Behavioral Changes:** Rabbits may expand their search areas for food, potentially leading to increased encounters with predators, including foxes.

3. **Foxes:**
   - **Shift in Diet:** Foxes may need to diversify their diet, possibly preying on other animals or scavenging for alternative food sources like insects or carrion.
   - **Territorial Disputes:** With fewer rabbits, fox territories may become more contested, leading to increased aggression and territorial fights.

### Long-Term Effects (Several Years to Decades)

1. **Plants:**
   - **Evolutionary Adaptation:** Over time, plant species may evolve to better tolerate low-light conditions. Some may develop larger leaves or more efficient photosynthetic mechanisms.
   - **Recovery:** Once the volcanic ash settles and sunlight returns to normal levels, plants that have adapted or survived will begin to recover and flourish again.

2. **Rabbits:**
   - **Repopulation:** As plants recover and provide more food, the rabbit population will gradually increase. However, it may take several generations to return to pre-eruption numbers.
   - **New Survival Strategies:** Rabbits may have developed new behaviors or physical traits that enhance their ability to find food in challenging conditions.

3. **Foxes:**
   - **Recovery and Adaptation:** Foxes that managed to survive by diversifying their diets will thrive once the rabbit population rebounds. Those that did not adapt may face extinction.
   - **Population Dynamics:** The fox population will stabilize at a level that the ecosystem can support, given the new balance between plant, rabbit, and fox populations.

### Summary of Causal Chain:
- **Diminished Sunlight** → **Reduced Photosynthesis** in plants → **Less Vegetation** → **Decreased Rabbit Population** → **Reduced Prey for Foxes** → **Fox Population Decline**.
- Over time, **adaptation and recovery** in all populations will occur, but the exact outcomes depend on the resilience and adaptability of each species involved.