Okay, let's think through this step-by-step:

Immediate effects:
1. The thick layer of volcanic ash in the atmosphere would reduce the amount of sunlight reaching the valley by 50%.
2. With less sunlight, the rate of photosynthesis in the plants would decrease significantly.
3. Reduced photosynthesis would lead to slower plant growth and less overall plant biomass in the valley.

Medium-term effects:
4. As the plant population declines due to reduced sunlight, the rabbits' primary food source would become scarce.
5. With less food available, the rabbit population would begin to decline due to starvation and reduced reproduction rates.
6. As the rabbit population decreases, the foxes' main food source would become less abundant.

Long-term effects:
7. The fox population would start to decline due to the reduced availability of rabbits, their primary prey.
8. Foxes might begin to search for alternative food sources, such as small rodents or birds, but these may not be sufficient to sustain the original fox population size.
9. Over time, the fox population would likely stabilize at a lower level, in balance with the reduced carrying capacity of the environment (i.e., the number of foxes the valley can support based on the available food resources).
10. If the supervolcano eruption is a short-term event (a few years), plant, rabbit, and fox populations might slowly recover to pre-eruption levels as sunlight levels return to normal. However, if the dimmed sunlight persists for an extended period, the valley ecosystem may undergo a more permanent shift, with lower populations of plants, rabbits, and foxes.

The reasoning behind each step:
- Steps 1-3 are based on the understanding that sunlight is the primary energy source for plants through photosynthesis.
- Steps 4-6 follow from the knowledge that rabbits are herbivores and rely on plants for food, while foxes are predators that primarily feed on rabbits.
- Steps 7-9 are based on the concept of carrying capacity and the idea that predator populations are limited by the availability of their prey.
- Step 10 considers the potential for ecosystem recovery or permanent change based on the duration of the volcanic ash's effect on sunlight levels.

In summary, a supervolcano eruption dimming sunlight by 50\% would likely cause a cascading effect through the valley ecosystem, with reduced plant growth leading to a decline in rabbit and fox populations over time, until a new balance is reached in response to the altered environmental conditions.