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I am studying a 3D system of ODEs derived from a biological model (tumor growth). All parameters are defined to be positive, as negative values would lack biological meaning (e.g., negative loss rates of tumor cells).

A text I read identifies a Zero-Hopf bifurcation (a fold-Hopf bifurcation) in this system. This bifurcation is proven to imply the emergence of periodic solutions through averaging theory. However, the analytical condition for this bifurcation requires that some parameters become negative.

This seems to create a contradiction: the model is only biologically significant for positive parameters, but the interesting complex dynamics (periodic orbits) are detected at a point in parameter space that is biologically inaccessible.

My question is: What is the mathematical and practical value of such a result? Why would one analyze a bifurcation that occurs outside the physically relevant domain of the model?

If, from a purely mathematical point of view this is interesting, then so is any other system that comes from real-world models and applications? One can just extrapolate parameters (an other possible restrictions) and call it ``interesting to mathematicians''?

Does this analysis still provide insight into the behavior of the system within the positive parameter cone? Are there many published papers that goes this way?

Thanks in advance.

Mand
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    I don't see the contradiction. Why must the system of equations be considered from a biological point of view at all? The presence of the bifurcation is interesting to mathematicians (maybe), not the fact that parameters were extrapolated past an arbitrary bound – whpowell96 Sep 06 '25 at 20:05
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    Additionally, there are places where extrapolating to physically inaccessible parameter regimes provides insight, like taking limits as positive parameters go to zero, e.g., inviscid fluid flow, many problems in perturbation theory – whpowell96 Sep 06 '25 at 20:09
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    Here is an example from a different domain that is amusing: prices are generally considered to always be positive and this was a problem in some of the earliest models for prices that assumed Gaussian movements (Bachelier iirc)--the model predicted negative prices with non-zero chance. Eventually, Geometric Brownian motion were used to exclude this: if $S_t$ solves $dS=\mu S dt+\sigma S dB$ then $S_t>0$ for all $t$ almost surely. However, West Texas Intermediate oil prices briefly went negative for the first time in history in April 2020. So, who is right? Bachelier or Black-Scholes? – Nap D. Lover Sep 06 '25 at 20:09
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    It is possible that the bifurcation (here fold-Hopf) at negative parameter values leads to a solution branch that continues to exist for positive values of these parameters. So the bifurcation analysis could give an explanation for the existence of solutions that would not be available if one were to consider only biologically plausible parameters. – Hans Engler Sep 06 '25 at 20:09
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    In a similar vein, sometimes the existence of "nonphysical" solutions is important to understand as they may appear in numerical approximations and steps must be taken to eliminate them and enforce physicality in approximations. Expansion shocks in computational fluid dynamics are a classic example – whpowell96 Sep 06 '25 at 20:12

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