Abstract
We propose a game-theoretical model to describe intra-seasonal predator–prey interactions between predatory mites (Acari: Phytoseiidae) and prey mites (also 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 continuous-time dynamical model comprises predator and prey densities, along with corresponding energy levels, over the length of a season. It also includes time-dependent decision variables for the predator and the prey, representing the current portions of the predator and prey populations that are active, as opposed to diapausing (a state of physiological rest).Our aim is to find the optimal active/diapausing ratio during a season of interaction between predatory mites and prey mites: this is achieved by solving a dynamic game between predator and prey. We hereby extend our previous work that focused solely on the optimal strategy for the prey. Firstly, we analyze the optimal behavior of the prey. Secondly, we show that the optimal strategy for the predator is to stay active for the entire season. This result corresponds to biological observations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Extension of our model to multiple seasons is a subject of our future research.
References
Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. SIAM, Philadelphia, Pennsylvania
Cressman R (2003) Evolutionary dynamics and extensive form games. MIT Press, Cambridge
Danks H (1987) Insect dormancy: An ecological perspective. Biological Survey of Canada, Ottawa
Fitzgerald J, Solomon M (1991) Diapause induction and duration in the phytoseiid mite Typhlodromus pyri. Exp Appl Acarol 12:135–145
Helle W, Sabelis MW (1985a) Spider mites: their biology, natural enemies and control. World crop pests, vol 1A. Elsevier, Amsterdam
Helle W, Sabelis MW (1985b) Spider mites: their biology, natural enemies and control. World crop pests, vol 1B. Elsevier, Amsterdam, pp. 77–99
Kroon A, Veenendaal RL, Bruin J, Egas M, Sabelis MW (2004) Predation risk affects diapause induction in the spider mite Tertanychus urticae. Exp Appl Acarol 34:307–314
Kroon A, Veenendaal RL, Egas M, Bruin J, Sabelis MW (2005) Diapause incidence in the two-spotted spider mite increases due to predator presence, not due to selective predation. Exp Appl Acarol 35:73–81
Kroon A, Veenendaal RL, Bruin J, Egas M, Sabelis MW (2008) “Sleeping with the enemy”- predator-induced diapause in a mite. Naturwissenschaften 95:1195–1198
Mailleret L, Lemesle V (2009) A note on semi-discrete modelling in the life sciences. Philos Trans R Soc A 367(1908):4779–4799
Melikyan AA (1994) Necessary optimality conditions for a singular surface in the form of synthesis. J Optim Theory Appl 82(2):203–217
Melikyan AA (1998) Generalized characteristics of first order PDEs: Applications in optimal control and differential games. Birkhäuser, Boston
Melikyan AA, Olsder GJ (2010) Boundary singularities and characteristics of Hamilton-Jacobi equation. J Dyn Control Syst 16(1): 77–99
Patchepsky E, Nisbet RM, Murdoch WW (2008) Between discrete and continuous: Consumer-resource dynamics with synchronized reproduction. Ecology 89(1):280–288
Sabelis MW (1991) Life-history evolution of spider mites. In: Schuster R, Murphy PW (eds) The Acari: Reproduction, development and life-history strategies. Chapman & Hall, London, pp 23–50
Sabelis MW, Janssen A (1993) Evolution of life-history patterns in the phytoseiidae. In: Houck MA (ed) Mites: Ecological and evolutionary analyses of life-history patterns. Chapman & Hall, New York, pp 70–98
Staňková K (2009) On Stackelberg and inverse Stackelberg games & their applications in the optimal toll design problem, the energy market liberalization problem, and in the theory of incentives. PhD thesis, Delft University of Technology, Delft, the Netherlands
Staňková K, Abate A, Sabelis MW (2013) Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka-Volterra model. J Math Biol 66(4):767–794
Tauber M, Tauber C, Masaki S (1986) Seasonal adaptations of insects. Oxford University Press, New York
Veerman A (1992) Diapause in phytoseiid mites: a review. Exp Appl Acarol 14:1–60
Weibull J (1995) Evolutionary game theory. MIT Press, MA
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Staňková, K., Abate, A., Sabelis, M.W. (2013). Intra-seasonal Strategies Based on Energy Budgets in a Dynamic Predator–Prey Game. In: Křivan, V., Zaccour, G. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 13. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-02690-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-02690-9_10
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-02689-3
Online ISBN: 978-3-319-02690-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)