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Lecture 3

Lecture 3. The Lotka-Volterra model. The Prey-Predator Model. In the equation: Only the logistic is controlling growth. In reality the interaction between the Prey and the Predator generates an oscillatory system. Modelo de Lotka-Volterra.

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Lecture 3

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  1. Lecture 3 The Lotka-Volterra model

  2. The Prey-Predator Model • In the equation: Only the logistic is controlling growth. In reality the interaction between the Prey and the Predator generates an oscillatory system.

  3. Modelo de Lotka-Volterra • Where Py is the concentration of the Prey, Kny is the rate of reproduction of the Prey and kmyis the rate of natural mortality of the Prey and G is the grazing rate. • Pr is the concentration of the Predator; kmris the rate of natural mortality of the Predator. Eis the losing rate (the amount of the Prey destroyed by the predator, but not used to grow). See worksheet Prey-Predator.xlsx for calculation • gzis the grazing rate, representing the amount of food per unit of mass needed by the predator. kis the semi-saturation constant.

  4. Problemas do modelo de Lotka Volterra • Não conserva a massa. A Natureza precisa de pelo menos 3 variáveis de estado: • Nota: As derivadas passaram a totais para descrever o caso de o fluido estar em movimento. • Poderá kp ser constante? Será razoável que a presa consuma detritos? Precisamos de mais variáveis...

  5. LotkaVolterramodellimitations • It does not conserve the total mass. Natureneedsatleast 3 statevariables (pneshouldbe a kindofdetritus): • Can kpbeconstant? Is itreasonablethat the prey consumes Detritus? Ifnotoneneeds more statevariables...

  6. Formof the EquationsconsideringTransport Nestas equações adicionamos o transporte difusivo.

  7. NumericalResolution • Herewe have adopted na explicitcalculationmethod. Allstatevariables are used at time “t”.

  8. Resolução Numérica • Nesta discretização admitimos que a produção e o consumo durante um intervalo de tempo são função das variáveis no início do intervalo de tempo: Modelo explícito

  9. Partiallyimplicitmethod • In this case the sourcetermisexplicitand the sinktermisexplicit. • Grazingisexplicit to assurethattehsamevalueis used in bothequations, to guaranteemassconservation.

  10. Modelo parcialmente implícito • Nesta discretização o termo de fonte é explícito e o termo de poço é implícito. • O termo de pastoreio (grazing) é explícito para ter o mesmo valor em ambas as equações.

  11. Final remarks • Lotka-Volterra model has some similitude with reality in the sense that it generates c cyclic solution but it does not conserve mass. Prey are generated from nothing and predators just vanish when they dye. • A third variable could solve the problem of mass conservation but it is too short to describe nature. Much more variables are necessary to build an ecological model. • In a realistic ecological model rates depend of the environmental conditions. They can not be constant neither in time nor in space. That is another source of complexity. • Ecological models require complex algorithms requiring complex programs to produce results. Excell worksheets can be helpful if associated to visual basic programming.

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