I have a differential equation
diff(N(t), t) = r*N(t)*(1-N(t)/K) - p(N(t)), N(0) = N0
p(x):=piecewise(0, x<0, S/(N_crit)*x, 0<= x<N_crit, S)
This equation represents the population growth model where the predator is present.
The predator is unspecialized, so its population is not important (=not present in the model).
N(t)...the size of population in time t
r, K, S, N_crit ... model parameters
p ... the function of "how many preys (modeled population) the predator kills"
The problem is that the predator kills proportionally many preys ("N(t)") only to some critical
value "N_crit". That's why there is a piecewise function.
Is there a way how to "easily" solve this in Maple?
I can separate the DE into two cases - solve them according to the value of "N(t)",
but "N(t)" can be above "N_critical" some time, some time below and it is pretty
complicated to find the solution for the given time range in this way.
I hope you understand what I mean.