## Modelling populations with a case study on orange roughy in New Zealand

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##### Date

2005##### Permanent Link

http://hdl.handle.net/10092/10106##### Degree Grantor

University of Canterbury##### Degree Name

Bachelor of Science with HonoursThe dynamics of mathematical models used for modelling populations will be investigated. Mathematical models have many applications, these will be mentioned throughout the paper. Mathematical models are particularly useful for modelling fish populations. Most fish populations are not farmed, and fish are taken from their wild environment. Models must be used to monitor population levels, and to determine the affect of harvesting. The use of simplifications is common, as models that could more accurately model a biological system are very hard to solve and require many parameters that must be measured. For example a food web that exists in nature could potentially have 50 or more species, each with a complex life cycle and each having an equally complex interaction with other species in the food web. Food webs investigated will have at most three species, so analytic solutions can be found and to reduce the number of parameters. These species will usually be prey, predator and superpredator, but alternatively can be thought of as plant, herbivore and predator. This is the beauty of using mathematical modelling, predator-prey dynamics are analysed exactly the same way as herbivore-plant dynamics, even though biologically they are different. Different topologies of food webs will be examined to determine the affect of the topology on population dynamics. A single species can be modelled with age structure in its population, by using a partial differential equation. The model examined negates interactions between species, but allows a single species to be modelled in more detail. The partial differential equation model is approximated using an individual based model, where each individual in the population ages at a constant rate, and reproduction and mortality will not be deterministic. Both of these will occur as poisson processes. The individual based model developed is used to model the population of orange roughy in New Zealand waters. Parameters concerning recruitment and mortality of orange roughy are examined. The sensitivity of these parameters are investigated, as many published parameters are best estimates or educated guesses. Orange roughy has suffered severe losses due to fishing, the estimate of current biomass is just 20% of virgin biomass. The population response to different harvesting rates is examined in detail, in particular to find if there is a sustainable harvest rate, given the population is currently so low.

##### Subjects

Field of Research::01 - Mathematical Sciences::0102 - Applied Mathematics::010202 - Biological Mathematics##### Collections

- Engineering: Reports [684]