How does variation of beak type affect a birds ability to survive?

The variation in beak type enables the different individuals in a population to feed on different food sources.

The shape of the beak is key to determining the primary food source of the bird. Certain birds species have specialized beaks, but within populations of those species, there is a variation in beak shape.
This gives the population the ability to respond to changes in the environment and availability of different food sources.
For example, if because of a dry period there is a shortage of fruit, some individuals in the population have the beaks that enable them to feed on other things, survive and leave offspring.

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The rate that would change would be the  searching efficiency/attack rate. there are only two other rates in this model: the  growth rate of prey, and the  predator or consumer mortality rate.the  searching efficiency/attack rate is what indicates how effectively can a predator capture its prey whether in the searching process or actual capturing/attacking process.