Modeling Metabolic State Transitions in Obesity Using a Time-Varying Lambda-Omega Framework

TL;DR

A lambda-omega model from dynamical systems theory is employed to describe metabolic regulation in response to lifestyle perturbations and introduces time-varying parameters that allow the regulatory coefficients to evolve gradually under sustained environmental and physiological stressors.

Abstract

Obesity does not emerge abruptly; rather, it develops gradually over extended periods. The gradual progression often prevents early recognition of physiological changes until excess adiposity is established. A common belief is that weight reduction can be achieved simply by "eating less and moving more". Although reductions in caloric intake and increases in physical activity are fundamental principles of weight management, this perspective oversimplifies a complex and adaptive biological system. Metabolic rate, hormonal regulation, behavioral factors, and compensatory physiological responses all influence the body's resistance to changes in weight. During weight loss, reduced metabolic rate and increased efficiency make maintaining a caloric deficit increasingly difficult. Conversely, during periods of overfeeding, resting metabolic rate, the thermic effect of food, and non-exercise activity thermogenesis increase with rising body weight, partially offsetting the caloric surplus and slowing weight gain. However, these compensatory responses are asymmetrical, with stronger and more persistent adaptations to underfeeding than to overfeeding. This asymmetry helps explain why weight gain often occurs gradually and why sustained weight loss is biologically challenging. In this work, we employ a lambda-omega model from dynamical systems theory to describe metabolic regulation in response to lifestyle perturbations. We introduce time-varying parameters that allow the regulatory coefficients to evolve gradually under sustained environmental and physiological stressors. By allowing lambda(t) and omega(t) to vary over time, the model captures progressive shifts in the metabolic set-point and deformation of the underlying dynamical landscape. This framework enables exploration of transitions between metabolic states and long-term adaptations that shape trajectories of weight gain and loss.

Figures (6)

• Figure 1: Conceptual illustration of energy balance demonstrating how the relationship between caloric intake and energy expenditure governs changes in body weight. From left to right, the panels depict a negative energy balance, in which caloric intake is less than energy expenditure, resulting in weight loss; a balanced state, in which calories consumed equal calories expended and body weight is maintained; and a positive energy balance, where caloric intake exceeds energy expenditure and leads to weight gain.
• Figure 2: Comparison between economic supply–demand equilibrium and physiological energy-balance equilibrium. Left: Classic supply–demand curves, where downward-sloping demand and upward-sloping supply intersect at a stable market equilibrium. Right: Analogous intake–expenditure curves, transformed onto a regulatory axis to mirror the economic structure. Intake (demand-like) and expenditure (supply-like) intersect at the point of energy balance, representing stable body weight. Together, the panels illustrate that both markets and body weight regulation operate as self-balancing systems in which equilibrium is determined by the intersection of opposing input and output forces.
• Figure 3: Smooth transition in nullcline geometry induces a change from unstable to stable spiral dynamics. Phase--plane snapshots of the $\lambda$--$\omega$ system illustrating how gradual deformation of the nullclines alters the stability of the fixed point. The left column (A1--A4) shows the initial configuration in which the nullclines intersect to produce an unstable spiral. As they shift smoothly from the top left to bottom right, the fixed point transitions through a bifurcation--like change in which instability is lost and stability emerges. The right column (B1–B4) depicts the subsequent strengthening of this stable state, with the fixed point evolving from weakly stable to strongly stable as the nullclines continue their smooth displacement. Red and blue shaded regions indicate the domains of outward growth and inward contraction, respectively, corresponding to instantaneous unstable and stable directions of the flow. Collectively, the panels demonstrate how continuous geometric changes in the nullclines can lead to abrupt qualitative shifts in system stability.
• Figure 4: Dynamics of a time-varying $\lambda$--$\omega$ system illustrating the transition from an unstable to a stable fixed point and the emergence of metabolic adaptation.Panel A: The phase-plane shows a smooth and continuous transition of the nullclines, which gradually changes the stability structure of the system from an unstable fixed point (supporting sustained oscillations) to a stable fixed point (attracting trajectories inward). The trajectory initially exhibits outward spiraling behavior consistent with an unstable focus, and after the nullcline transition is completed, the dynamics reorient toward an inward spiral converging to the new stable equilibrium. Panel B: The time series displays the peak amplitudes of the oscillations (gray; left axis) and their cumulative sum of these consecutive differences (black; right axis). During the early portion of the dynamics---when the fixed point is still unstable---peak amplitudes remain unchanged, indicating that the oscillatory regime persists despite the slow, ongoing structural transition. After several cycles following a perturbation, the system enters a new regime referred to as the metabolic adaptation phase (shaded region). During this phase, the system resists changes in oscillation amplitude even as the underlying stability continues to evolve. Once the fixed point becomes fully stable, the peak amplitudes begin to decrease monotonically, signaling the attenuation of the adaptive response and the gradual convergence toward a plateau that represents the new metabolic steady state. Once the fixed point becomes fully stable, the peak amplitudes begin to decrease monotonically, marking the attenuation of the adaptive response and the approach to a plateau, representing convergence to the new metabolic steady state.
• Figure 5: Dynamics of a time-varying second-order $\lambda$--$\omega$ system illustrating the smooth transition from a tiny limit cycle to the larger size limit cycle with unstable fixed point.Panel A: The phase-plane illustrates a smooth, continuous transition of the nullclines that gradually expands the size of the limit cycle. Regardless of whether the initial condition begins inside or outside the cycle, the trajectory is drawn toward it. As the nullclines shift outward and the limit cycle grows, the trajectory is correspondingly pushed in that direction, consistently guiding the system toward the evolving cycle. Panel B: This plot shows the change in peak amplitude from one cycle to the next (gray curve) and the cumulative sum of these consecutive differences, which reflects the accumulating effect of metabolic adaptation.