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.

maxmax0 / ccckk

kcdtdc n

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

Modelo de Lotka-Volterra

• Where Py is the concentration of the Prey, Kny is the rate of reproduction of the Prey and kmy is the rate of natural mortality of the Prey and G is the grazing rate.

GPkPkdtdP

ymyynyy

rmrr PkGEdtdP

.

kPP

PEg

Gy

yr

z

• Pr is the concentration of the Predator; kmr 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).

• gz 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 kp ser constante? Será razoável que a presa consuma detritos? Precisamos de mais variáveis...

zmzzpggppD

zmzzpggz

zpgppp

ckcckeckdtdc

ckcckedtdc

cckckdtdc

1

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 kp be constant? Is it reasonable that the prey consumes Detritus? If not one needs more state variables...

zmzzpggppD

zmzzpggz

zpgppp

ckcckeckdtdc

ckcckedtdc

cckckdtdc

1

Form of the Equations considering Transport

zmzzpggppj

D

jj

Dj

DD

zmzzpggj

z

jj

zj

zz

zpgppj

p

jj

pj

pp

ckcckeckxc

xxc

vtc

dtdc

ckcckexc

xxc

vtc

dtdc

cckckxc

xxc

vtc

dtdc

1

Nestas equações adicionamos o transporte difusivo.

Numerical Resolution

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

GPkPkdtdP

ymyynyy dtGkkdtPP myny

ty

dtty 1

rmrr PkGEdtdP

. mrtr

dttr kGEdtPP .1

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

GPkPkdtdP

ymyynyy dtGkkdtPP myny

ty

dtty 1

rmrr PkGEdtdP

. mrtr

dttr kGEdtPP .1

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.

mynyty

dtty kdtdtGkdtPP 1/*1

kPP

PEg

Gy

ytr

z

mrtr

dttr kdtGEdtPP *1/**

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.

mynyty

dtty kdtdtGkdtPP 1/*1

kPP

PEg

Gy

ytr

z

mrtr

dttr kdtGEdtPP *1/**

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.