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. A interacção entre a presa e o predador dá origem a um sistema oscilatório.

Modelo de Lotka-Volterra Where P y is the concentration of the Prey, K ny is the rate of reproduction of the Prey and k my is the rate of natural mortality of the Prey and G is the grazing rate. Pr is the concentration of the Predator; k mr is the rate of natural mortality of the Predator. E is the losing rate (the amount of the Prey destroyed by the predator, but not used to grow). g z is the grazing rate, representing the amount of food per unit of mass needed by the predator. k is the semi-saturation constant. See worksheet Prey- Predator.xlsx for calculation

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 k p ser constante? Será razoável que a presa consuma detritos? Precisamos de mais variáveis...

Lotka Volterra model limitations It does not conserve the total mass. Nature needs at least 3 state variables (pne should be a kind of detritus): Can k p be constant? Is it reasonable that the prey consumes Detritus? If not one needs more state variables...

Form of the Equations considering Transport Nestas equações adicionamos o transporte difusivo.

Numerical Resolution Here we have adopted na explicit calculation method. All state variables are used at time “t”.

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

Partially implicit method In this case the source term is explicit and the sink term is explicit. Grazing is explicit to assure that teh same value is used in both equations, to guarantee mass conservation.

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.

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.