Temperature-dependent development rates fundamentally shape insect phenology, population dynamics, and geographical distribution. However, the mathematical models commonly used to describe these relationships often provide good statistical fits without directly interpretable biological parameters. Here, we developed and evaluated three alternative parameterizations of the Gaussian thermal performance curve that explicitly incorporate entomologically meaningful thermal thresholds: base temperature (Tbase), optimum temperature (Topt), and maximum temperature (Tmax), along with dimensionless width parameters (ω). These reparameterizations transform abstract curve descriptors into parameters that directly correspond to the thermal limits governing insect life cycles, thereby enhancing both biological interpretation and practical application in phenology prediction. We developed three distinct model variants by adjusting the parameterization of the Gaussian core: (1) a lower-threshold variant using Tbase, suitable for cold-limited systems; (2) an upper-threshold variant using Tmax, appropriate for heat-limited scenarios; and (3) an asymmetric dual-threshold variant incorporating both limits. Each of these three model variants was implemented separately, fitted independently to development rate data, and produces its own thermal performance curve and set of parameter estimates. They were implemented in R using multi-start nonlinear regression and applied to development rate data for four life stages (eggs, larvae, pupae, and egg-to-adult) of the cotton bollworm, Helicoverpa armigera (Hübner, 1808), a major agricultural pest. The lower- a nd u pper-threshold v ariants performed nearly i dentically t o the traditional Taylor-Lamb formulation (R² > 0.98) while yielding directly interpretable thermal parameters. Tbase estimates ranged from 11.0°C to 16.4°C across life stages, and the dimensionless width factor ω (0.63-0.90) provided a scale-independent measure of thermal specialization. In contrast, the dual-threshold variant showed clear overparameterization (adjusted R² as low as 0.60), with biologically implausible estimates (e.g., Tbase = -0. 97°C for larvae) and boundary-constrained width factors. This demonstrates that asymmetric formulations require richer datasets than typically available from constant-temperature experiments. The lower- and upper-threshold variants offer parsimonious alternatives that bridge the gap between mathematical description and biological meaning, facilitating more robust phenology predictions and improved communication between modelers and applied entomologists. We recommend the lower- or upper-threshold variant for most insect phenology applications, reserving the dual-threshold formulation for comprehensive datasets spanning both lower and upper thermal limits.

ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
    1. Development of modified gaussian thermal responsemodels
    2. Goodness-of-fit assessment
    3. Helicoverpa armigera biological data
RESULTS
    1. Visual comparison of fitted thermal performancecurves
    2. Parameter estimates and goodness-of-fit assessment
DISCUSSION
CONCLUSIONS
ACKNOWLEDGMENTS
REFERENCES

• Thameur Bouslama(IRESA/LR21AGR03/2PHD, Regional Research Centre on Horticulture and Organic Agriculture (CRRHAB), PO Box 57 Chott-Mariem 4042 Sousse, Tunisia) Corresponding author
• Asma Laarif(IRESA/LR21AGR03/2PHD, Regional Research Centre on Horticulture and Organic Agriculture (CRRHAB), PO Box 57 Chott-Mariem 4042 Sousse, Tunisia)