The topic of the SLIP network is one of the central problems in ecology (Levin 1992, Levin et al., 1997) and receives increasing attention in fisheries research (e.g. Pelletier and Magal 1996, Mason and Brandt 1999, Walters and Bonfil 1999, Sibert et al. 1999). The aim of the network is to provide a link between the basic research at Danish universities in single fish behaviour and physiology and the applied research in population biology and dynamics undertaken at public and semi-private research institutions. It offers opportunities for research across the boundaries of mathematics, statistics and biology.
The importance of spatial and temporal scaling in fisheries and marine ecology
One of the challenges in contemporary ecology is to understand how the dynamics of a population arise from the behaviour of its members. This is a challenge because the individuals differ due to their previous life histories and current states. However, even if the members of a population were all alike, non-linear interactions between the individuals and a heterogeneous environment makes it unlikely that a simple relationship would exist between the life history of the average individual and the dynamics of the population.
Marine ecosystems exhibit spatial and temporal heterogeneity over a range of scales. This heterogeneity is important for populations whose long-term persistence depends on the physical (e.g. temperature and salinity) and biological (e.g. prey and predator density) characteristics of their surroundings. On the smallest spatio-temporal scale, interactions between individual plankton organisms, and between plankton organisms and their environment, are governed by small-scale fluid motion and diffusion. For example, zooplankton organisms perceive their environment (predator, prey, mate) by hydro-mechanical and chemical cues. The mediation of such cues may be modified by ambient fluid motion that, in turn, is governed by processes occurring at much larger scales (e.g. tides, wind).
At the larger scale, hydrographic fronts, where water masses of different physical characteristics meet, are known to influence the distribution and abundance of plankton (Franks 1992) as well as different life stages of fish. It is generally believed that a major part of the energy available for higher trophic levels is generated in relation to the frontal areas (Kiørboe 1991) and it has been demonstrated that these areas provide favourable habitats for larval and juvenile fish. We know relatively little about how behavioural patterns, such as diurnal migrations and frontal circulation, jointly generate the distribution of primary and secondary producers in the front. We know even less about how temporal changes in the production in frontal areas affect the movement and behaviour of planktivorous and piscivorous fish.
Fish undertake small-scale horizontal and vertical movements on a daily basis. Fish stocks perform large-scale seasonal migrations between feeding and spawning grounds. Spatial structure is observable at the level of the population, but is generated by the behaviour of the individuals. Interactions between predators and prey are likely to depend on habitat structures and behavioural processes at the scale of meters and minutes to hours. Field studies can detect patterns at a given time and place; otolith microstructure, fatty acid compositions and biomarkers contain information about the previous life history of an individual; and laboratory experiments can reveal the individuals response to a limited range of environmental cues. Models are necessary to integrate the response of the individuals to explain the patterns observed in the field.
The problem of how to integrate individual behaviour to the population level has not yet been solved. However, a number of models are available and can be used as points of departure (Tyler and Rose 1994, Giske et al. 1998). Individual-based models (IBM's) describe the fate of each individual and how it is determined by its history and present conditions, e.g. its satiation level, the local environment and the abundance of prey and predators. Reaction-advection-diffusion models describe spatial heterogeneity by assigning different diffusion parameters and/or life history parameters to sub populations in different areas depending on local conditions. Meta-population models describe a population as consisting of a set of spatially separated sub-populations between which the individuals move.
Moving around in a spatially heterogeneous environment involves changes in growth and survival generated by gradients in food availability, temperature, salinity and predation risk. How the animal responds to these gradients will determine its survival success and how much genetic material it will pass on to the following generation. For instance, the trade off between predation risk and growth opportunities makes certain types of aggregative behaviour such as schooling advantageous in some situations, in other situations the individuals are better off on their own. While the immediate response to environmental gradients is triggered by the information received by the sense organs, the motivation to respond can best be understood in an evolutionary context. How the individuals behave is likely to be determined by their expected long-term reproductive success.
It has recently become apparent that even highly migratory marine species are not genetically homogenous but are subdivided into more localized populations (Ruzzante et al. 1998). Some species exhibit a meta-population structure, where sub-populations are linked by migration and where individual sub-populations sometimes go extinct and other new sub-populations are founded (Hanski and Gilpin 1997). It is important to know the time scale at which individual sub-populations exist in order to assess the extent to which they contain specific adaptations and are likely to follow separate evolutionary trajectories. On a long (evolutionary) time scale the phylogeny of alleles contains information about the history of populations, such as bottlenecks and population expansions (Avise 1994). On a shorter (ecological) time scale the temporal stability of populations at selected and neutral loci can be studied by analysis of DNA from contemporary and archived (scales, otoliths) samples (Nielsen et al. 1999). A deeper knowledge of the genetic structure of marine fish populations is important for their conservation and sustainable exploitation. Conservation of marine fish populations poses a number of new challenges compared to conservation of freshwater and terrestrial organisms. For instance, it is a major concern that changes in selective regimes due to high (and selective) fishing pressure may lead to changes in life-history traits in exploited populations. Also, it has generally been assumed that collapses of fish populations due to overfishing would have no or limited consequences at the level of genetic diversity, as census and effective population sizes even in "collapsed" populations would never drop to a level (hundreds or fewer) assumed to be critical in traditional conservation biology. However, Ryman et al. (1995) have demonstrated that drastically decreased population sizes, as seen in fisheries induced population "collapses", inevitably will lead to the loss of numerous rare alleles that might have been important in the future evolution of the population or species. Thus, in the future management of marine fish populations it will be important to look at the interactions between individuals and populations from a conservation biology angle.
Spatial heterogeneity has long been regarded as important in ecology. It is well known that incorporation of spatial structure in population models can change population and ecosystem dynamics by, for instance, increasing the stability of interacting populations and facilitating co-existence of species (Levin 1992, Kareiva 1994). Despite this, fisheries researchers often ignore individual differences and spatial heterogeneity when they model fish growth, predator-prey interactions and the sustainability of fisheries. Most single and multispecies fish stock assessment models assume that populations consist of identical individuals whose spatial distribution remains constant with time. They also suppose that individuals from a given location are able to mix instantly with individuals from anywhere within the distribution area of the population. There are cases where this caricature of population dynamics has had serious implications for management. The collapse of the Grand Banks cod stock was for example caused by overfishing and a failure to account for temperature related changes in spatial distribution and catchability (Atkinson et al. 1997).
It is likely that the lack of spatial structure will bias results from multispecies fisheries models. Studies of food selection suggest that predators are likely to exhibit negative switching, i.e. to maintain a more conservative diet than expected. Rindorf et al.(submitted) thus found that the relative proportion of a given item in the diet of cod and whiting in the North Sea changed less than in the environment. Such behaviour makes current multispecies population models, such as MSVPA, unstable, perhaps because dynamic changes in fine scale distribution cannot be accounted for. In a spatially resolved model it is possible that changes in the distribution of predators and prey will counteract the destabilizing influence of a conservative diet.
Spatial and temporal heterogeneity receives increasing attention in fisheries management. The wish to impose management regulations at a finer spatial scale than previously, e.g. the closing of small areas to protect breeding colonies of seabirds from competition from sandeel fishermen or vulnerable benthos from beamtrawling, necessitates spatially resolved models in which migration and advection of animals across the boundaries can be accounted for. In order to design protected marine areas and refuges in an optimal way it is necessary to know more about the importance of spatial scale and heterogeneity for population and ecosystem dynamics.
Mathematical models and scaling
Since the network will address the relationship between phenomena at different scales, and different mathematical descriptions are natural at the different scales, the mathematical toolbox will be large and diverse.
The smallest scale in space and time addresses individual behaviour. Here, it is natural to model each individual as a stochastic dynamic system with a state representation. In simple mechanistic models the state is simply the position and orientation of the individual (e.g. Flierl et al. 1999); in other models the state will contain information such as age, maturity, hunger, and behavioural patterns. One way to model these behavioural patterns is to use a neural network to map stimuli to response (e.g. Huse 1998).
At this level, the task is to construct suitable mathematical descriptions of individual behaviour, by combining knowledge in the fields of biological and stochastic processes. This involves simulation models, analysis of stochastic processes, and estimation of parameters in stochastic models.
The largest spatial scale covers an entire ocean. At this spatial scale and at the time scale of population dynamics, it is natural to represent spatial distribution in terms of population densities, the dynamics of which can be described by advection-diffusion type equations. In a single species setting, it is possible to formulate such models so that they allow parameter estimation, and so that they have a reasonable level of explanatory power (Sibert et al. 1999). However, much work is still needed regarding interacting populations, where product densities appear, and regarding the non-linearities in the equations that describe density-dependent behaviour. Also, these models typically contain a very large number of parameters, so the choice of suitable estimation techniques is critical and largely unexplored.
The coupling between these levels is an important, and largely open, problem. It is promising to make use of the theory of spatial point processes, and spatio-temporal point processes, in which notions of densities and product densities are given precise meaning, and which allows statistical analysis of observed patterns. Some steps in this direction have already been taken (e.g. Orensanz 1999). Also techniques from statistical physics may be useful to move from individual dynamics to macroscopic descriptions (e.g. Flierl et al. 1999). Finally model reduction techniques from system theory may be used to simplify large dynamic models of interacting individuals.
At an intermediate scale, groups (schools or patches) of individuals become important. While there exist several mathematical models of individual behaviour that lead to formation of groups, and while the advantage of groups in an evolutionary perspective is at least partly understood (e.g. Parrish and Hamner 1997), it remains unclear how to represent groups in population dynamic models. One option is to use a finite state representation of each group; a very simple example is describing the group by the number of individuals, their mean position and mean velocity, and the diameter of the group. Techniques from the geometry of random closed sets (e.g. Stoyan et al. 1995) and the field of multi-phase fluid flows may be helpful.
References
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