Are there any animals that gradually approaches the predator to search for food?
The behaviour that you describe is common in most animal species, as part of the natural trade-off between access to food, minimizing risk, habituation and hunger. Animals usually choose to forage in high-quality habitats that has a low risk of predation, but if food sources are depleted (or competition and/or territoriality is high) they will move to other areas with lower food quality or higher risk (since they are desperate for food).

In general, the effect of predators are both direct (killing of prey) and indirect (changes in prey behavior to avoid predation). The indirect effects means that prey will often choose to feed in habitats that are suboptimal from the perspective of forage quality, which will have negative effects on their foraging efficiency. These types of indirect effects of predators are often labelled as 'The landscape of fear' or the 'ecology of fear' (see e.g. Brown et al, 1999). However, as food resources run out this will trade-off against the fear of predators (along with possible habituation effects), which will lead to the behaviour you are asking for. You might also be interested in Gamelius et al (2014), as an example of how prey behaviour can be influenced by the presence of predators (here, roe deer and lynx), but there is a large number of studies looking into these issues. The review "Fear in animals: a meta-analysis and review of risk assessment" by Stankowich &amp; Blumstein (2005) will also give you a good background and many examples.

The use of scarecrows is a simple way to understand the 'landscape of fear'. By placing a scarecrow in a field you strengthen the "landscape of fear" (by adding an artificial 'predator'), which will modify the behaviour of perceived pest species (prey), so that they avoid the areas around the scarecrow. However, if the animals you want to scare away ("pests") run out of food they will take larger risks and venture closer to the scarecrow to forage. Over time, they will also become habituated to the scarecrow, and will discover that it doesn't seem to pose a big threat, which will decrease the effectiveness of the scarecrow.

References


Brown et al. 1999. The Ecology of Fear: Optimal Foraging, Game Theory, and Trophic Interactions. Journal of Mammalogy  
Gamelius et al. 2014. Habitat Selection and Risk of Predation: Re-colonization by Lynx had Limited Impact on Habitat Selection by Roe Deer. Plos One
Stankowich &amp; Blumstein. 2005. Fear in animals: a meta-analysis and review of risk assessment. Proceedings B