The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure

Abstract

In this paper, fear effect and stage structure are introduced in a free boundary problem of a prey-predator model. This system simulates the spread of an invasive or newly introduced predator species, taking into account the presence of both immature and mature stages of prey that are affected by fear of the predator. The predator's predation behavior on adult prey induces fear in the prey, which in turn causes the prey to seek out safer habitats. While this short-term survival strategy may be effective, it ultimately leads to a decrease in the prey's long-term survival fitness, including reduced reproductive ability. Consequently, the overall population of prey is expected to decline over the long term. The existence and uniqueness of the solution is given, and the comparison principle is used to discuss the long-term behavior of the solution by constructing a sequence of upper and lower solutions. We obtain a spreading–vanishing dichotomy for this model, in other words, when the predator can only spread in a limited area, the predator will eventually become extinct, the population density of the two stages of prays will tend to two positive constants, and when the predator can spread to infinity, the predator ultimately survives, and their population density, defined as (<i>u</i>, <i>v</i>, <i>w</i>) will eventually tend to (<i>u^*</i>, <i>v^*</i>, <i>w^*</i>) which we defined blow.