Math 331 Take Home Test Fall 2015
Biomass Transfer
Consider a European forest having one or two varieties of trees. We select some of the oldest trees, those
expected to die off in the next few years, then follow the cycle of living trees into dead trees. The dead trees
eventually decay and fall from seasonal and biological events. Finally, the fallen trees become humus. Let
variables x, y, z, t be defined by
x(t) = biomass decayed into humus,
y(t) = biomass of dead trees,
z(t) = biomass of living trees,
t = time in decades (decade = 10 years).
A typical biological model ODE system model is
x’(t) = −x(t) + 3y(t),
y’(t) = −3y(t) + 5z(t),
z’(t) = −5z(t).
Suppose there are no dead trees and no humus at t = 0, with initially z(0) = 8 units of living tree biomass.
1. Solve the system, showing your work.
2. In a well-constructed final paragraph, explain what your solution is capable of telling you at t = 10 years
(after 1 decade) about the distribution of biomass in the forest.
Linear Predator-Prey Relationships
Two wolves and ten rabbits are added to an ecosystem without either species. The rate of growth of the
population of wolves is proportional to the number of wolves and the number of rabbits (food) at time t.
However the rate of growth of rabbits declines with the number of wolves but increases with the number of its
own species. The following model depicts the dynamics.
dR/dt = 2R – 2W
dW/dt = R + W
1.
2.
3.
Use the eigenvalue/eigenvector process to develop a linear survival model for these species, and show
your work.
Graph your parametric model with time until the species die out or until the model becomes no longer
viable (i.e. the population of at least one of the species goes negative. Sketch the graph as part of your
project.
Based on your graph, is the model realistic? Think hard about this, because modeling is never perfect!
Complex Linear Systems: ODEIVP
Do Number 8!