## Applications

Development of mathematical ecology is tightly connected to applications. First a subject of study is most often taken from observations of a natural phenomenon; hence it arises from a need for application. For example, a prey-predator interaction has been analyzed as a possible cause of observed fluctuations in prey and predator population dynamics. Second, the model which has been constructed may be applied to more or less similar ecosystems. Third, the model may be an inspiration to construct a similar model which may serve to explain another phenomenon. For example, competition of firms in the same market as mentioned earlier has been studied by using a competition model of two populations. Last but not least, the type of analysis carried on a model may be applied to some extent, or be an inspiration, for a similar analysis on another model. In this way, applications of mathematical ecology have been made to other disciplines, and methods and models from other disciplines have been applied to problems in mathematical ecology.

Chronologically perhaps the first grand application was to find out how many people can this planet support, that is, to find the parameter K for the human population. Another related problem was to find out when in the history did the human population begin. This is done by extending solutions of equations ([1], [2], or [3], or other more realistic models) backward in time.

We have already mentioned the application of the Lotka-Volterra prey-predator model to fisheries. More realistic model variants have been applied to find out the optimum fishing on predators or prey or both. Of course, applications presuppose that one defines what is meant by an optimum fishery. Is it keeping the population of prey or predator or both at a certain safe level, or is it to fish the maximum sustainable number of predatory fish? Naturally, depending on the definition of the optimum, one will get different solutions including a possibility that the solution does not exist. This last possibility exists because what is optimal for fishermen may be in contradiction to the persistence of some species in the ecosystem.

Application to eutrophication problems include models of marine and freshwater ecosystems. These models are composed of a number of equations representing species or groups of species and in general include food webs with cycles of matter. Models are run with an increase in nutrient inflow and analyzed for consequences in ecosystem dynamics. Of interest is occurrence and timing of massive phytoplankton blooms, extinction of species in a studied area and hypoxia near the bottom.

Models in mathematical ecology have been applied to global change problems. The interest has been to find out how marine and forest ecosystems will react to the expected temperature rise, and whether this will threaten the survival of some species and therefore change functioning of considered ecosystems.

Successful applications have also been sought in predicting and controlling spread of epidemics. This includes occurrence and spread of diseases in humans and species in marine, freshwater, and forest ecosystems.

See also: Age-Class Models; Biological Control Models; Competition and Coexistence in Model Populations; Competition and Competition Models; Fisheries Management; Fundamental Laws in Ecology; Limits to Growth; Metapopulation Models; Prey-Predator Models; r-Strategist/K-Strategists.

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