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2007, 4(3): 431-456. doi: 10.3934/mbe.2007.4.431

Effects of predation efficiencies on the dynamics of a tritrophic food chain

Maria Paola Cassinari 1, , Maria Groppi 2, and  Claudio Tebaldi 3,

1. 

Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy
2. 

Dipartimento di Matematica, Università di Parma, V.le G.P. Usberti 53/A, 43100 Parma, Italy
3. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received  July 2006 Revised  January 2007 Published  May 2007

In this paper the dynamics of a tritrophic food chain (resource, consumer, top predator) is investigated, with particular attention not only to equilibrium states but also to cyclic behaviours that the system may exhibit. The analysis is performed in terms of two bifurcation parameters, denoted by $p$ and $q$, which measure the efficiencies of the interaction processes. The persistence of the system is discussed, characterizing in the $(p,q)$ plane the regions of existence and stability of biologically significant steady states and those of existence of limit cycles. The bifurcations occurring are discussed, and their implications with reference to biological control problems are considered. Examples of the rich dynamics exhibited by the model, including a chaotic regime, are described.
Keywords: stability, tritrophic system, bifurcations, limit cycles..
Mathematics Subject Classification: 92D25, 34C2.
Citation: Maria Paola Cassinari, Maria Groppi, Claudio Tebaldi. Effects of predation efficiencies on the dynamics of a tritrophic food chain. Mathematical Biosciences & Engineering, 2007, 4 (3) : 431-456. doi: 10.3934/mbe.2007.4.431
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