# Immigration-Death

The models in this section deal with the immigration-death process, which is comprised of the immigration reaction 0 → X and the death reaction X → 0.

## dsmts-002-01

The immigration and death reactions have mass-action kinetic laws with rate parameters Alpha = 1 and Mu = 0.1, respectively. Use the open button in the tool bar to import the file examples/sbml/dsmts31/dsmts-002-01.xml. The model is imported correctly and the rate laws are recognized as having mass-action kinetics. Add a method in the methods list. In the methods editor select Time Homogeneous and Time Series, Uniform with the default solver and options. Set the recording time to 50 and the number of frames to 51. To see the result of these modifications open the file examples/cain/dsmts31/dsmts-002-01.xml. Below are the plots of the mean and standard deviation of the species population calculated from 10,000 trajectories.  Note that this problem has a steady state solution. Select SteadyState in the methods list panel. In calculating the steady state solution, we let the process equilibrate for 100 time units and then record the state for 100,000 time units. We generate four trajectories and plot the steady state histogram, shown below. With the steady state output selected, click the table button and then select Estimated Error. The estimated error in the probability distribution is 0.0045.

## dsmts-002-02

For this model, the value of the Alpha is changed to 10. Below are plots of the mean and standard deviation of the species population.  ## dsmts-002-03

This model is the same as dsmts-002-02, except that the global parameter Alpha = 10 is overridden by the local parameter Alpha = 5. Cain does not support overriding global parameters. You will need to change the value of Alpha in the parameters editor.

## dsmts-002-04

This model is the same as dsmts-002-02, except that Alpha = 1,000. This increases the reaction rates and makes the simulation more expensive. Thus we only generate 1,000 trajectories in calculating the mean and standard deviation of the species populations.  ## dsmts-002-05

This model is the same as dsmts-002-02, except that there are two additional species Source and Sink, which are both boundary conditions. Cain does not support boundary conditions.

## dsmts-002-06

This model is the same as dsmts-002-05, except that Sink is not a boundary condition.

## dsmts-002-07

This model is the same as dsmts-002-05, except that Sink is a constant boundary condition.

## dsmts-002-08  