Question: Solving DE with a piecewise function.

Hi everyone,

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.



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