Non-algebraic limit cycles in Holling type III zooplankton-phytoplankton models

Abstract

We prove that for certain polynomial differential equations in the plane arising from predator-prey type III models with generalized rational functional response, any algebraic solution should be a rational function. As a consequence, limit cycles, which are unique for these dynamical systems, are necessarily trascendental ovals. We exemplify these findings by showing a numerical simulation within a system arising from zooplankton-phytoplankton dynamics.

Keywords

• predator-prey models
• functional-response
• puiseux series
• newton polygon
• limit cycles
• invariant algebraic curve