Journal of Mathematical Biology

, Volume 66, Issue 4–5, pp 767–794

Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka–Volterra model

  • Kateřina Staňková
  • Alessandro Abate
  • Maurice W. Sabelis


We propose an optimal control framework to describe intra-seasonal predator–prey interactions, which are characterized by a continuous-time dynamical model comprising predator and prey density, as well as the energy budget of the prey over the length of a season. The model includes a time-dependent decision variable for the prey, representing the portion of the prey population in time that is active, as opposed to diapausing (a state of physiological rest). The predator follows autonomous dynamics and accordingly it remains active during the season. The proposed model is a generalization of the classical Lotka–Volterra predator–prey model towards non-autonomous dynamics that furthermore includes the effect of an energy variable. The model has been inspired by a specific biological system of predatory mites (Acari: Phytoseiidae) and prey mites (so-called fruit-tree red spider mites) (Acari: Tetranychidae) that feed on leaves of apple trees—its parameters have been instantiated based on laboratory and field studies. The goal of the work is to understand the decisions of the prey mites to enter diapause (a state of physiological rest) given the dynamics of the predatory mites: this is achieved by solving an optimization problem hinging on the maximization of the prey population contribution to the next season. The main features of the optimal strategy for the prey are shown to be that (1) once in diapause, the prey does not become active again within the same season and hence diapause is an irreversible process; (2) for the vast majority of parameter space, the portion of prey individuals entering diapause within the season does not decrease in time; (3) with an increased number of predators, the optimal population strategy for the prey is to start diapause earlier and to enter diapause more gradually. This optimal population strategy will be studied for its ESS properties in a sequel to the work presented in this article.


Predator–prey problems Fruit-tree red spider mites  Game theory Optimal control Singular characteristics 

Mathematics Subject Classification (2000)

49L20 92B05 93C15 

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kateřina Staňková
    • 1
  • Alessandro Abate
    • 2
  • Maurice W. Sabelis
    • 3
  1. 1.Department of Knowledge EngineeringMaastricht UniversityMaastrichtThe Netherlands
  2. 2.Delft Center for Systems and ControlDelft University of TechnologyDelftThe Netherlands
  3. 3.Institute for Biodiversity and Ecosystem DynamicsUniversity of AmsterdamAmsterdamThe Netherlands

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