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Fig. 3. Exponential (exp; broken lines) and pure polynomial (poly; solid lines)
parts of Eq. 5 plotted separately as functions of 
. The crossing
points between the functions are the solutions to the characteristic
polynomial of the system. values are shown in the negative real axis
portion, thus, crossing points correspond to pure real negative and,
therefore, asymptotically stable nodes in the corresponding eigenvector.
Notice that at =0, the polynomial part is zero and the exponential
part is negative (-bk5). (A) A putative large homeotherm
(>1000 g) is represented. Notice the existence of three crossing points.
(B) Small (
100 g) homeotherm. (C) 50 g homeotherm, represents a putative
limiting condition. The functions touch each other just twice. Any further
decrease in body mass would make the functions fall apart and two eigenvalues
would have imaginary parts, rendering the system a focus. The focus is
asymptotically stable while the real part of each complex conjugate
belongs to the negative real axis. The focus would become unstable when the
complex conjugate has positive real part. This potentially occurs
when the functions are `far away' from each other, as depicted in (D),
representing a 10 g homeotherm.
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