Survival analysis and density function of stochastic stage-structured cannibalism dynamics

Cannibalism and external environmental disturbances significantly impact the survival of biological populations. To analyze their combined effects comprehensively, we introduce a novel stochastic predator-prey model incorporating stage structure and cannibalism in predators. We investigate the existence, uniqueness, and ultimate boundedness of positive solutions for the model, and conduct survival analysis in the presence or absence of cannibalism. Additionally, we derive an approximate expression for the explicit density function of the ergodic stationary distribution. In contrast to the scenario without cannibalism, the introduction of cannibalism leads to the following outcomes: (1) In the absence of environmental noise, mature predators exhibit higher population sizes, while juvenile predators and prey show lower population sizes; (2) Mild environmental noise makes both prey and mature predators more vulnerable, while juvenile predators display stronger resistance; (3) Intense environmental interference for predators results in predator population extinction, even if prey experience low levels or no environmental noise; (4) Extinction occurs across all populations when prey faces significant environmental noise. Thus, under inevitable environmental disturbances, cannibalism serves as an adaptive mechanism, enhancing the survival of mature predators and bolstering their resilience to external interference. This reduces the risk of extinction and promotes biodiversity maintenance.