Relating behavior to population dynamics: a predator–prey metaphysiological model emphasizing zooplankton diel vertical migration as an inducible response

Abstract

We present a continuous-time deterministic model, for the population dynamics of a predator-prey system. The model which includes a function of refuge-use by the prey is developed in order to account for the interaction between lake zooplankton and fish populations, assuming the ‘predator avoidance hypothesis’ as the ultimate explanation for the behavior of zooplankton diel vertical migration. Nevertheless, this model is flexible enough as to be utilized in other qualitatively similar scenarios. The model is based on the metaphysiological approach to population modeling proposed by Getz [Getz, W.M., 1994. A metaphysiological approach to modeling ecological populations and communities. In: Levin, S.A. (Ed.), Frontiers in Mathematical Biology. Lecture Notes in Biomathematics 100. Springer-Verlag, New York, pp. 411–442]. Thermal stratification of the environment is explicitly incorporated in the model and, consequently, a temperature-dependence function for physiological processes. A Holling type II sub-model is used as the extraction function of prey-biomass by predators, and an abruptness function is used for the induction of the zooplankton vertical migration by fish density. The model is reparametrized, resulting in a ten-parameter equivalent system, where we analytically identify the presence of three equilibrium points. By means of computer simulation, some dynamic properties of the system are studied, evidencing the occurrence of limit-cycles, and trivial and non-trivial locally stable equilibrium points, depending on the parameter values. Stability of dynamic properties of the model are studied on selected bidimensional parameter-spaces, and the ecological significance of results is discussed.