Construction of population growth equations in the presence of viability constraints
AbstractA mathematical method based on the G-projection of differential inclusions is used to construct dynamical models of population biology. We suppose that the system under study, not being limited by resources, may be described by a control system 
$$\dot x\left( t \right) = f\left( {x\left( t \right),u\left( t \right)} \right)$$
\] where u is a control describing the choice of resources. Then considering the constraints that the system must satisfy we define a viability set K. Since there may not exist a control u(·) such that the corresponding solution satisfies x(t) ε K, we have to change the dynamics of the control system to get a viable solution. Using the G-projection we introduce so-called “projected” control system 
$$\dot x\left( t \right) = \Pi _{T_K }^G f\left( {x\left( t \right),u\left( t \right)} \right)$$
 that has a viable solution. The projected system has usually simpler dynamics than traditional models used in population biology.