In the linear model we assumed that this was proportional to nitrate concentration. However, it would be reasonable to assume that it would also be proportional to phytoplankton concentration. Thus, a better model would be J1 = k1X1X2. This is a second-order expression. When we insert it in our system, we can no longer create the matrix as we did above. We might make similar changes to the equations for other fluxes, and might even include Monod-type expressions.
These changes produce a set of nonlinear simultaneous algebraic equations. Such a system can be solved by Newton's method for systems of equations. An alternative and somewhat easier approach is to write the unsteady-state mass balances for each compartment, then simulate a sufficient period of time until steady-state is practically achieved. For example, in this case the equation for Xi would be:
The result is a system of nonlinear ordinary differential equations (ODEs), which must be solved numerically, such as by Euler's method or Runge-Kutta methods. Initial values must be specified for the X's. These can be estimates or solutions obtained from the linear case. Using the ODE model allows simulation of dynamic effects of disturbances. For example, the model could be used to determine the changes in phytoplankton populations that would result from a sudden addition of DON and ammonia, such as from a sewage spill.
Models of these kinds have been developed to simulate complete ecological systems. For example, models have been created for proposed closed ecological life support systems (CELSSs) to grow food and recycle wastes in support of human crews on long-term space missions. Global models have been used to make projections on economic resources such as food and minerals. Based on current scenarios for population growth and resource consumption, these models predict a catastrophic resource depletion some decades in the future. However, they do not take into account social responses to depletion, such as the development of new technologies, resource switching, or discovery of new resources. Nevertheless, the models make clear that our current world economy is not sustainable.
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