A degenerate cognitive model for a single species under toxicity

Abstract This paper studies a class of degenerate parabolic partial differential equation models that describe the dynamics of single-species populations with cognitive functions in toxic environments. The core innovation lies in introducing cognitive functions to simulate the ability of species to perceive toxins and adaptively adjust their behaviours, which in turn affects population density. This mechanism is characterized by a nonlinear degenerate Pratial Differential Equation. The degradation of the model is mainly manifested in the diffusion coefficient and response term approaching zero when the population density is zero or the cognitive state reaches the boundary. For this model, we have achieved the following theoretical breakthroughs: First, by using regularization techniques, we constructed a non-degenerate approximate system. Combining prior energy estimates with the fixed point theorem, we proved the existence of local classical solutions for this regularized system; subsequently, we constructed an appropriate Lyapunov function. Based on strict energy dissipation analysis and uniform prior estimates, we proved that under conditions such as bounded toxins and regular initial values, the classical solutions of the regularized system exist globally and remain bounded on the interval left bracket 0 comma normal infinity right parenthesis <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $[0, \infty )$ ultimately, based on this, through compactness theory and limit processes, the existence of the global weak solution for the original degenerate system is further derived. Numerical simulations verify the rationality of the theoretical results and visually demonstrate the dynamic regulatory effect of cognitive function on population tolerance.