A Temperature-Structured PDE Model for Population Dynamics

Abstract

This paper proposes a population model structured by temperature, in which thermal variation is incorporated directly as a continuous internal variable rather than treated as an external input. The resulting formulation leads to a non-linear parabolic partial differential equation that includes diffusion along the temperature axis to represent environmental variability and redistribution effects. Population growth and decay are governed by temperature-dependent nonlinear terms of logistic type. The analytical study is conducted in an infinite-dimensional functional setting. The construction of solutions is achieved by approximating the system within finite-dimensional subspaces and then passing to the limit using suitable compactness properties together with uniform estimates. In addition, uniqueness is obtained by exploiting continuity conditions imposed on the nonlinear reaction component and applying integral inequality techniques. These results collectively guarantee the well-posedness of the model. From a biological standpoint, the formulation captures how populations adjust their distribution across heterogeneous thermal environments, offering a more realistic description than classical ordinary differential or stage-based approaches. The framework is particularly relevant in situations where temperature strongly regulates biological processes, such as in ectothermic species and vector-borne disease dynamics under climate variability. Overall, this study provides a mathematically consistent foundation for temperature-structured population models and supports further theoretical development as well as numerical exploration of environmentally structured systems.