Existence of equilibrium solutions to a size-structured predator-prey system with functional response

Abstract

In this paper, we consider a nonlinear size-structured population model with functional response, which describes the dynamics of a predator-prey system living in a common habitat. We present a kind of functional response for the prey being a plant or algae, and explain its biological meanings. When the vital rates depend both on the individual's size and on the total population or only depend on the former, we obtain the existence of equilibrium solutions.Structured population models are the most important mathematical models in Population Ecology. The aim of the study is to know how the difference of individuals indicated by structural variables has an effect on the dynamics of the whole population. As usual, structural variables include size, age, rank, length and so on. According to structural variables, structured population models can be classified as age-structured models, size-structured models and so on.Since the first popular age-structured population model was presented in 1911, many scholars [1][2][3][4] have focused all their attention on finding a solution to the problem during the last several decades. However, there is less literature on size-structured population models, linear or nonlinear (for an account of the various treatments see the monograph [5] and the papers of Ackleh et al [6][7][8][9] , Calsina et al [10,11] , Cushing [12] , Kraev [13] and Kato et al [14,15] and the references cited therein).Functional responses in traditional predator-prey models were notably introduced by Holling in 1959 [16] . Cosner et al [17] gave a unified mechanism approach for the derivation of various forms of functional responses in predator-prey models. As far as we know, such a functional response is seldom considered for size-structured population dynamics.In this paper, a size-structured population model with functional responses is presented and we give a specific form of functional responses for the prey being a plant or algae and explain the biological meanings of the