Final answer:
The phenomenon of birds avoiding flying over large bodies of water due to higher energy demands can be modeled mathematically by comparing energy consumption over land and water, incorporating factors like distance, aerodynamic efficiency, and the availability of updrafts, as well as formation flight advantages.
Step-by-step explanation:
Ornithologists have observed that some species of birds avoid flying over large bodies of water during daylight hours due to the higher energy requirements compared to flying over land. To represent this phenomenon mathematically, we could propose an energy cost model for avian flight. This model could take into account factors such as the energy required to maintain altitude, aerodynamic efficiency, and the absence of thermal updrafts over water that birds use to assist with their flight over land.
For example, let E represent the energy per distance unit a bird expends while flying. We could suggest that E = c₁ * (D + c₂) over land and E = c₃ * (D + c₄) over water, where D is the distance flown, c₁ and c₃ are constants representing the base energy cost per distance over land and water, and c₂ and c₄ are constants representing additional energy factors like updraft availability and wind resistance. It's hypothesized that c₃ > c₁ and c₄ > c₂, indicating that more energy is required to cover the same distance over water.
Flying in formations can reduce energy consumption, as shown by the V formation of geese, which reduces drag. This aspect could be introduced into the model with an additional factor that decreases the energy cost when birds fly in aerodynamically favorable formations.