Coexistence of structured populations with size-based prey selection
Introduction
The ecological role of fish in aquatic communities is shaped by two characteristics: (1) individuals have a size-based preference for prey 10–1000 times smaller than themselves in terms of weight (Brose et al., 2006, Barnes et al., 2010), and (2) they have a life-history where they produce many small offspring, around 1 mg, which is several orders of magnitude smaller than their adult size ranging from 1 g to 100 kg (Duarte and Alcaraz, 1989, Winemiller and Rose, 1993, Andersen et al., 2008). Consequently this means that individuals feed on prey from successively higher trophic levels throughout ontogeny: as larvae individuals compete for shared resources of zooplankton nauplii, and as they grow they shift to adult zooplankton and later to piscivory and feeding on larger invertebrates. Individuals therefore undergo ontogenetic niche shifts (Werner and Gilliam, 1984) and specifically shifts in the trophic niche (also known as life-history omnivory; Pimm and Rice, 1987). Due to ontogenetic trophic niche shifts two individuals may interact either competitively, if they are of a similar size, or as predator and prey if the size of one individual falls into the preferred prey size range of the other. The ecological role is therefore determined by the size of an individual rather than by its species identity. This complication makes it difficult to characterise the interaction between two fish populations as “competitive” or “predator–prey” since the species-level interaction type is likely to be a complex mixture of both that depends on the size structure of the two populations (Fig. 1).
The classic way of describing species-level interactions is through unstructured models based on coupled ordinary differential equations (e.g. MacArthur and Levins, 1967; Oksanen et al., 1981; Holt and Polis, 1997). In unstructured models energy is transferred between species only through predator–prey interactions. In populations with ontogenetic trophic niche shifts energy is also transferred through somatic growth in body size. Explicit modelling of ontogenetic trophic niche shifts therefore requires the use of physiologically structured modelling approaches (Metz and Diekmann, 1986, de Roos and Persson, 2001, de Roos and Persson, 2013). The key characteristic of physiologically structured models is that they resolve food-dependent growth and reproduction, where growth can either be between stages (de Roos et al., 2008a), in a cohort model with several state variables (de Roos and Persson, 2001), or with a continuous size structure (Claessen and de Roos, 2003, Hartvig et al., 2011). The dependence of growth and reproduction on food encounter is what distinguishes physiologically structured models from the traditional age-based models used in fisheries science (Beverton and Holt, 1957). Analysis of physiologically structured models have e.g. demonstrated coexistence of several unstructured predator populations on a single structured prey population through the mechanism of emergent facilitation (de Roos et al., 2008b). It seems, however, that inclusion of ontogenetic trophic niche shifts decreases the possibility of coexistence of two size-structured species (van de Wolfshaar et al., 2006).
We explore the potential for coexistence of species experiencing ontogenetic trophic niche shifts. To simplify matters we assume that selection of food is determined solely by the differences in body size of the interacting individuals—meaning that we disregard any dietary selection based on species identity. We refer to such populations as “purely size-structured”. We apply a continuously size-structured approach where individual physiological processes, including predator–prey encounter, are parameterised by relations with the size of individuals. In this manner each individual is characterised by body size, while each species is characterised by a single trait: the size at maturation (Hartvig et al., 2011). The formulation of purely size-structured populations through a single trait allows for a general analysis of coexistence as variation of a larger set of species-specific parameters are avoided. The analysis is divided into three steps: we first characterise a resource–consumer system where one structured species feeds on a size-structured resource and potentially on itself through cannibalism. In this system we identify the resource-driven and cannibalistic population states described earlier (Claessen and de Roos, 2003) including an emergent Allee effect (de Roos et al., 2003) and bistable states (Claessen and de Roos, 2003, van Kooten et al., 2005, Guill, 2009). Next we show how two species may coexist. Depending on the relative sizes at maturation the two species can be considered part of different species-level interaction motifs. If the sizes at maturation are similar, the two species will mainly interact as competitors for the resource. If size at maturation of one species is significantly larger than the other species the interaction is a combination of competitive coexistence and predator–prey interactions (including cannibalism). Finally we explore the potential for coexistence of three or more species and show how the scope for coexistence decreases rapidly when more species are introduced into the system.
Section snippets
Model
We employ a recent size- and trait-based model framework to describe the population dynamics of purely size-structured interacting populations (Hartvig et al., 2011). The model differs from the Hartvig et al. (2011) model only by assuming that interaction strengths between species is solely due to size-dependent predation (i.e. setting the species-specific couplings strengths to unity). The model formalises the sketch in Fig. 1 by explicitly resolving the entire individual life-history from
Methods
The model (1) consists of coupled partial integro-differential equations and is solved numerically using a first order semi-implicit upwind finite-difference scheme (Press et al., 1992, Hartvig et al., 2011). The size-spectrum of all species is discretised on a mass grid with 200 logarithmically even sized mass () groups, and the time step used for integration is 0.02 years. The dynamics of the resource spectrum may be obtained analytically at each time step to save computational
Results
The analysis of a system with just one species demonstrates (Fig. 2): (1) There is an upper boundary to the size at maturation of a single species that can invade from small numbers (thick solid line). This maximum size of is determined by the resource level described by the initial feeding level ; higher resource levels leads to larger maximum size. (2) Beyond the invasion boundary species may persist but they cannot invade; these states can be reached by starting simulations from a
Discussion
Single species states
A single species with ontogenetic trophic niche shifts living on a resource is known to have the potential for several stable states (Claessen and de Roos, 2003, de Roos et al., 2003, van Kooten et al., 2005). These states are also found here and can be understood in terms of an energetic bottleneck. Through invasion in small numbers a resource-driven state may be reached where the population primarily feed on the resource spectrum. The maximum body size a species can
Acknowledgements
M.H. acknowledges support from European Marie Curie RTN FishACE through the EU’s 6th FP (MRTN-CT-2004-005578) and the Danish National Research Foundation for support to the Center for Macroecology, Evolution and Climate. K.H.A. was supported by the Villum Kann Rasmussen Centre of Excellence: Ocean Life, and the EU’s 7th framework research projects FACTS and MYFISH.
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