Elsevier

Ecological Complexity

Volume 21, March 2015, Pages 233-245
Ecological Complexity

Original Research Article
A food chain ecoepidemic model: Infection at the bottom trophic level

https://doi.org/10.1016/j.ecocom.2014.03.003Get rights and content

Highlights

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    First paper to our knowledge on epidemics in food chains.

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    Bistability discovered among some equilibria.

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    First paper to our knowledge to compute a separatrix surface.

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    Used a reliable approximation algorithm for the separatrix.

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    Richer dynamics than the corresponding demographic model.

Abstract

In this paper we consider a three level food web subject to a disease affecting the bottom prey. The resulting dynamics is much richer with respect to the purely demographic model, in that it contains more transcritical bifurcations, gluing together the various equilibria, as well as persistent limit cycles, which are shown to be absent in the classical case. Finally, bistability is discovered among some equilibria, leading to situations in which the computation of their basins of attraction is relevant for the system outcome in terms of its biological implications.

Introduction

Food webs play a very important role in ecology. Their study dates back to many years ago, see Fryxell and Lundberg, 1997, Gard and Hallam, 1979, Holmes and Bethel, 1972 and May (1974). The interest has not faded in time, since also recent contributions can be ascribed to this field in mathematical biology (Dobson et al., 1999) invoking the use of network theory for managing natural resources, to keep on harvesting economic resources in a viable way, without harming the ecosystems properties. This is suggested in particular in the exploitation of aquatic environments.

Mathematical epidemiological investigations turned from the classical models (Hethcote, 2000) into studies encompassing population demographic aspects about a quarter of a century ago (Busenberg and van den Driessche, 1990, Gao and Hethcote, 1992, Mena-Lorca and Hethcote, 1992). This step allowed then, on the other hand, the considerations of models of diseases spreading among interacting populations (Hadeler and Freedman, 1989). On the basic demographic structure of the Lotka–Volterra model several cases are examined in Venturino (1994), in which the disease affects either the prey or the predators. In a different context, namely the aquatic environment, diseases caused by viruses have been considered in Beltrami and Carroll (1994). More refined demographic predator–prey models encompassing diseases affecting the prey have been proposed and investigated in Venturino, 1995, Chattopadhyay and Arino, 1999 and Arino et al. (2004), while the case of infected predators has also been considered (Venturino, 2002, Haque and Venturino, 2007). In addition, other population associations such as competing and symbiotic environments could host epidemics as well (Venturino, 2001, Venturino, 2007, Haque and Venturino, 2009, Siekmann et al., 2010). It is worthy to mention that one very recent interesting paper reformulates intraguild predator–prey models into an equivalent food web, when the prey are seen to be similar from the predator's point of view (Sieber and Hilker, 2011). For a more complete introduction to this research field, see Chapter 7 of Malchow et al. (2008).

In Dobson et al. (2008), the role of parasites in ecological webs is recognized, and their critical role in shaping communities of populations is emphasized. When parasites are accounted for, the standard pyramidal structure of a web gets almost reversed, emphasizing the impact parasitic agents have on their hosts and in holding tightly together the web. The role of diseases, in general, cannot be neglected, because, quoting directly from Dobson et al. (1999), “Given that parasitism is the most ubiquitous consumer strategy, most food webs are probably grossly inadequate representations of natural communities”. A wealth of further examples in this situation is discussed in the very recent paper Selakovic et al. (2014).

Based on these considerations, then, in this paper we want to consider epidemics in a larger ecosystem, namely a food system composed of three trophic levels. We assume that the disease affects only the prey at the lowest level in the chain.

The paper is organized as follows. In the next Section we present the model, and its disease-free counterpart. Section 3 contains the analytical results on the system's equilibria. A final discussion concludes the paper.

Section snippets

The model

We investigate a three level food web, with a top predator indicated by W, the intermediate population V and the bottom prey N that is affected by an epidemic. It is subdivided into the two subpopulations of susceptibles S and infected I. The disease, spreading by contact at rate β, is confined to the bottom prey population. We assume that neither one of the other populations can become infected by interaction with the infected prey. The disease can be overcome, so that infected return to class

Boundedness

We define the global population of the system as ψ(t) = W + V + I + S. Recalling the assumptions on the parameters, for which e < c, q < n and p < h, we obtain the following inequalitiesdψdt=aS1−S+IK−cVS−βSI+γI+βIS−nIV−γI−νI−lV+eSV−hVW+qIV−mW+pVW=aS1−S+IK−(c−e)VS−(n−q)IV−(h−p)VW−lV−mW−νI<aS1−SK−lV−mW−νI.

Taking now a suitable constant 0 < η < min(ν, l, m) we can writedψdt+ηψ<(a+η)S−aS2K+(η−ν)I+(η−l)V+(η−m)W≤(a+η)S−aS2K≤K(a+η)24a=L1.From the theory of differential inequalities we obtain an upper bound on the total

Discussion

Food chain models are now classical in the literature. Here, however, we have made a step further in that we allow epidemics to affect one population in the chain.

The proposed ecoepidemic food chain presents some novel features that distinguish it from its disease-free counterpart. The purely demographic model indeed exhibits a series of transitions for which the intermediate population emanates from the situation in which only the lowest trophic level thrives when the threshold condition (4)

References (33)

  • S. Busenberg et al.

    Analysis of a disease transmission model in a population with varying size

    J. Math. Biol.

    (1990)
  • R. Cavoretto et al.

    Approximation of dynamical system's separatrix curves

    AIP Conf. Proc.

    (2011)
  • R. Cavoretto et al.

    Reliable approximation of separatrix manifolds in competition models with safety niches

    Int. J. Comput. Math

    (2014)
  • R. Cavoretto et al.

    Reconstruction of separatrix curves and surfaces in squirrels competition models with niche

  • A. Dobson et al.

    The assembly, collapse and restoration of food webs

    Nonlinear Anal.

    (1999)
  • A. Dobson et al.

    Homage to Linnaeus: How many parasites? How many hosts?

    PNAS

    (2008)

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