nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

6.1 Fixed Points of the IM Model 59
unknown, and it also reduces the number of parameters to keep track of. Per automation
we also nondimensionalized the IM (including crowding terms) – cf. appendix A.6
– but since we are in the rare situation of having estimates of all the parameters, we
have chosen not to work with the nondimensional version.
A brief note on linearizing
Some newcomers to the field of mathematical modelling may think that we oversimplified
things by linearizing the system when we analyzed the stability of the fixed points
because we casually threw away the higher order nonlinear terms without making any
remarks about these. So what gave us the right to do so?
Hartman-Grobman’s theorem 8 informally states that the flow of a nonlinear system
is topologically equivalent to that of a linear system in the neighborhood of a steady
state solution, provided that the eigenvalues of the Jacobian of the linear system are
not purely imaginary or take on zero value. 9 The equivalence here being a homeomorphism,
i.e. a continuous deformation with a continuous inverse, that preserves the
sense of time (Guckenheimer and Holmes (2002)). More so the eigenspaces attributable
to the eigenvectors, of the linearized system, where the eigenvalues are subject to the
behavior given above, are tangent to the invariant manifolds of the fixed points of the
nonlinear system (Guckenheimer and Holmes (2002)) – provided that the fixed points
are hyperbolic. This last statement is the essence of the theorem known as The Stable
Manifold Theorem for a Fixed Point; see e.g. Guckenheimer and Holmes (2002) page
13. When we are in three or less dimensions this fact should lend a helping hand to the
intuitive understanding of the situation.
Let us recapitulate in (pseudo) laymen terms: when the eigenvalues of the Jacobian
does not take on zero or purely imaginary values, the behavior obtained from the linear
system is in qualitative agreement with that of the nonlinear system.
One could hope that as long as we knew all the eigenvalues we could precisely determine
what kind of behavior is in play, and in fact it is so when we are in R 2 . But when
we enter R 3 and higher dimensions there are no theorems such as e.g. the Poincaré-
Bendixson theorem, that yield guarantees about the dynamical behavior, when we are
dealing with systems that evolve in more than two dimensions (Strogatz (2000)) – the
Poincaré-Bendixson theorem states that given a trajectory confined to a compact subset
of the plane, R 2 , that does not contain any fixed points, then said trajectory will either
approach a closed orbit, or it is itself a closed orbit (Strogatz, 2000, p.203). When
dealing with systems that evolve in R 3 and above a plethora of dynamics, including
chaos, can arise. 10
Now that we have elaborated a little on the relationship between a nonlinear system
and the linearized version thereof we will return to the analysis of the intermediate
model for that story is not completely over.
8 For a formal statement of the Theorem cf. appendix A.3.
9 Note that we are not given any promises regarding the size of said neighborhood.
10 Actually we would find it boring if there was no more to the study of systems of differential equations
than determining the eigenvalues.