basque center for applied mathematics

## Modelling and Simulation in Life and Material Sciences

### Compartmental Systems

Victor Mu?Villarragut

This is a general compartmental system. Its irreducible sets are {2, 4}, {6} and {7}. It is clear that it is to those irreducible sets where all the material eventually goes. Besides, since {7} has some outflow of material toward the environment, it gets empty. On the other hand, since {6} is a singleton with no outflow of material toward the environment, the amount of material within that compartment is non-decreasing.

All the qualitative results which have been referred to here can be found in:
V. Munoz-Villarragut, S. Novo, and R. Obaya, Neutral functional differential equations with applications to compartmental systems, SIAM J. Math. Anal., 40 (2008), pp. 1003–1028.

This is an irreducible compartmental system whose compartments are set in a cyclic fashion and has some inflow and outflow of material from and toward the environment. Its compartments never get empty.

This is a special case of irreducible compartmental system with some inflow and outflow of material from and toward the environment. Its compartments never get empty. It is of interest in microbiology.

This is a compartmental system whose only irreducible set is {x3 , y3 , z3}. These compartments absorb all the material coming from the rest of the system, which in turn never gets empty due to the existence of some inflow of material from the environment. The presence of some outflow of material from the aforementioned irreducible set toward the environment is essential for the existence of bounded solutions.

This is a compartmental system whose only irreducible set is {x3 , y3 , z3}. These compartments absorb all the material coming from the rest of the system, which in turn gets empty due to the absence of any inflow of material from the environment. The presence of some outflow of material from the aforementioned irreducible set toward the environment forces compartments x3 , y3 and z3 to get empty as well.

This is a compartmental system whose only irreducible set is {x3 , y3 , z3}. These compartments absorb all the material coming from the rest of the system, which in turn gets empty due to the absence of any inflow of material from the environment. Since there is no outflow of material from the aforementioned irreducible set toward the environment,
the material remains within compartments x3 , y3 and z3.

This is an irreducible compartmental system with no inflow or outflow of material from and toward the environment. Its compartments never get empty, as expected.

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