MULTISPECIES MODELS

10.2 MULTISPECIES MODELS

The first formal representation of interactions be- tween predator and prey were published by Lotka (1925) and Volterra (1926) and the system of equa- tions they proposed to describe such interactions, now known as Lotka–Volterra equations, are still used – often in strongly modified forms – for ex- ploring mathematically the consequences of cer- tain features of systems of interacting organisms (see Smith, Chapter 4, this volume). The basic form of this system is:

(10.1a) (10.1b)

where r is the intrinsic rate of population growth of the prey, g is the coefficient of negative growth (de-

cline) of the predators (N 2 ) in the absence of prey (N 1 ) and c 1 ,c 2 are interaction coefficients. Given certain sets of values for these parameters, the predator and the prey will oscillate, sometimes violently: the prey grows, the predator follows and over-consumes the prey which declines, fol- lowed by the predator with the prey then picking up again. Volterra, after modifying these equations such as to cover any number of species, teamed up with his son-in-law, the fisheries scientist Umber- to D’Ancona, to apply the new formalism to the in- terpretation of observed cycles in the landings of small prey and large predatory fishes from the Adri- atic. The details of this story are in Tort 1996 and the historical setting is given by Smith (Chapter 4, this volume). We note in passing that such cycles, nowadays, would invariably be (if often unjustifi- ably) attributed to environmental fluctuations.

The exploration of systems of equations such as (10.1a and b) has led to a number of insights about interacting species, notably, that their combined

2 gcNN 2 1 =-+ 2 ( ) ,

d N d t rcNN 1 1 2 = 1 - ( )

yield to a fishery depends on the character and strength of these interactions (Kirkwood 1982). Thus, the combined yield of a prey and predator is always less than these organisms considered sepa- rately. On the other hand, species with mutually beneficial interactions increase the combined yield. These results seem trivial; however, we must realize that single-species fisheries models still largely assume that predation effects can be neglected, with optima being proposed separately for each species, as if they did not interact. On the other hand, the familiarity that marine scientists and ecosystem modellers now have with the Lotka–Volterra approach has had numerous nega- tive impacts. Notably, modellers have uncritically taken over the mass-action assumption implied in the Lotka–Volterra model, i.e. that the consump- tion of prey by predators is a function of, and only of, their respective numbers or biomass in the ecosystem, just as is the case with reactants in a chemical vat. As we shall see, this assumption ignores the use of spatial structure by potential prey to enable part of their population to hide from, or otherwise render itself invulnerable to, their predator(s) (Krause et al., Chapter 13, Volume 1). This dampens the oscillation otherwise im- plied by the system. Spatial structure also enables the coexistence of two very similar competing species, whereas the equation systems such as equation (10.1) would predict that one would have to go extinct (Gause 1934).

Another more practical approach to analysis of multispecies situations is to add up yield-per- recruit (i.e. the results of yield-per-recruit analy- ses, see Shepherd and Pope, Chapter 8, this volume) for a number of species into an overall yield-per-recruit for a given area. Multispecies yield-per-recruit analysis, while tedious to imple- ment, is conceptually straightforward (Beverton and Holt 1957; Munro 1983; Murawski et al. 1991). However, the results suffer from the same defect that plagues single-species yield-per-recruit analy- ses, that is, strong assumptions must be made as to the level of recruitment for each of the species con- sidered. Usually, this level of recruitment is as- sumed constant, and the solutions, which then rely on equilibrium assumptions, ignore the vio-

Ecosystem Models

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Chapter 10

lent transient effects often resulting from fishing other, each subjected to a different stage-specific mortality changes (Shepherd and Pope, Chapter 8, mortality, and the vagaries of a varying physi- this volume). However, when the signal from the cal environment. Still, the aggregate behaviour fishery being investigated is very strong, such as in of the model was credible; notably, it generated fisheries in which a wide range of species is being

a particle-size distribution such as observed in- growth overfished, this approach can provide esti- dependently (see Section 10.8 below), and natural mates of the gain that can be obtained, by, say, re- mortalities for juvenile fishes well in excess of the ducing fishing mortality on a wide range of species. values then assumed, for example about M = 2 In such cases, indeed, the method is particularly year -1 instead of about 0.1 year -1 for juvenile her- powerful in that it allows one to distinguish be- rings. This result was later confirmed by multi- tween fisheries with different selection patterns. species virtual population analysis, MSVPA (see It also allows one to identify, again under the Pope and Shepherd, Chapter 7, this volume). assumptions of constant recruitment and of no

However, it is clear that few of Andersen and biological interactions, the mix of fishing effort Ursin’s colleagues were convinced of the practical- by size or age which optimizes overall yield-per- ity of this tool for fisheries management. Rather, recruit. Thus, in spite of its clear drawbacks, these what it established was that simulating a large techniques are likely to continue being used, spe- marine ecosystem was not an intractable task. cially since they are not very demanding in terms Moreover, several of the elements of the Andersen of input parameters. These are growth parameters, and Ursin model – notably its size-selection sub- estimates of natural mortality, and others, most of model (Ursin 1973) – were found to be extremely which, at least for commercial species, can be ob- useful on their own, and have seen use in tained online; see www.fishbase.org. One software later modelling approaches, e.g. the length-based system that can be used to implement this MSVPA described by Christensen (1995a) and techniqueisFiSAT(Gayaniloetal.1996),also avail- Shepherd and Pope, Chapter 7, this volume. able online (www.fao.org/fi/statist/fisoft/fisat/

The simulation of the Northeastern Pacific project.htm) .

ecosystem developed by Laevastu and collabora- tors, though similarly ambitious, differed from