Calculate the time taken to forage required amount
calc_foragecapture.RdGiven the starting prey density and the competition for food, we need to estimate how long it would take a bird to gather the food it needs. Intake model: We assume that the relationship between prey quantity x and intake rate (-dx/dt) is of the form dx/dt = -ax/(h+x) for parameters a and h, the widely-used Michaelis-Menten model. This implies that the prey quantity at time t is equal to
x(t,x_0)={x:(x-x_0+at+hlog(x)-h log(x_0)) = 0}
where x_0 denotes the prey quantity at time 0. This in turn implies that the total prey consumed by foraging up to time t is equal to
y(t,x_0) = x_0 - x(t,x_0)
The solution to this cannot be written down analytically, but it can be calculated numerically e.g. using a non-linear solver. This function uses the solver, nleqslv, to create a look-up table for a given x_0, a and h for a range of values of t.