# Calculating Foraging Energy Balance: Solving for Net Change in Energy Level

• ufs
In summary, the conversation discusses the net change in the energy level of a forging animal over a period of time and the factors that affect it, such as energy intake and expenditure. The time at which the net change is zero is determined by the formula E/j = t. For the answer to be biologically meaningful, certain conditions must be satisfied by the constants f and j, such as them being greater than zero. The conversation also explores the implications of these conditions on the foraging behavior of the animal.

## Homework Statement

"1. The net change, C, in the energy level of a forging animal over a period of time t is equal to the energy intake, I, minus the energy expended, E: C= I-E
Suppose that the energy intake (per unit time, j>0, is a constant. Then, over a time interval t, the total energy intake is I=jt.
Further, suppose that B is the basal metabolic rate, and f is the energy required for foraging activity per unit time t, (where B, f>0 are constants). Then the total energy required to forage for time t is
E= ft+B.

a) Determine the time at which energy intake I balances energy spent, E i.e the time at which the net change in the energy level is zero.

b) What conditions must be satisfied by the constants f and j (other than f,j>0) defined above for your answer to be biologically meaningful?

## The Attempt at a Solution

a) C= I-E
0=I-E
0=(jt)-E
E/j=t

b) I am not sure what it wants but this is what I did randomly.

E-B/t=f and I/t=j

c) Not sure on what to say due to (b)

The wording is certainly confusing.
I think there are two time variables implied. One is for a period of existence, while the other is for a period of foraging within that. The first mention of 't' is for existence, while the rest are for foraging (the period of existence now being taken as 1). Thus, although B is described as a rate, it appears in the equation as a quantity of energy (having been multiplied by a unit period of existence).
To untangle this, let's introduce T as a period of existence, and let h be the fraction of time spent foraging. Thus t = hT. Presumably the energy intake rate, j, only applies while foraging.
We now have that the average rate of expending energy is B+hf, while the average rate of obtaining energy is hj. To put it another way, in time T, energy intake is jt = hjT, while energy expenditure is BT+ft = BT + hfT.
Does that help?