A Predator-Prey ModelA Predator-Prey Model

G. F. Fussmann, S. P. G. F. Fussmann, S. P. Ellner, K. W. Shertzer and G. Ellner, K. W. Shertzer and G.

Hairston Jr.Hairston Jr.

(2000) (2000) ScienceScience, 290, 1358-, 290, 1358-13601360

BackgroundBackground

Models for the interaction of prey Models for the interaction of prey and predators date back to the and predators date back to the beginning of the 19beginning of the 19thth century century (Volterra-Lotke equations)(Volterra-Lotke equations)

In some conditions, both predator In some conditions, both predator and prey populations oscillate.and prey populations oscillate.

The famous Canadian lynx data show The famous Canadian lynx data show this, for example.this, for example.

The ExperimentThe Experiment

A tank contains a species of algae, A tank contains a species of algae, Chlorella vulgarisChlorella vulgaris. .

Nitrogen is the resource that limits Nitrogen is the resource that limits algae growth, and this is controlled in algae growth, and this is controlled in the experiment.the experiment.

The predator is a planktonic rotifer, The predator is a planktonic rotifer, Brachionis calyciflorusBrachionis calyciflorus, that feeds on , that feeds on the algae.the algae.

VariablesVariables

NN: Nitrogen concentration: prey : Nitrogen concentration: prey nutrientnutrient

CC: Prey concentration - : Prey concentration - ChlorellaChlorella RR: Predator concentration - : Predator concentration -

Reproducing Reproducing BrachionusBrachionus BB: Total : Total BrachionusBrachionus concentration concentration

[ ( )]

[

( )

( )

[ )]

/

] ( )

(

i c

c B

B

B

N F N C

F C B

DN N

DC F N C

DB m B

DR m

R

C

C

R

F

F

Predator-Prey differential Predator-Prey differential equationsequations

where the two F functions are thresholding functions that vary between zero and an upper asymptote:

( ) /( )

( ) /( )C C C

B B B

F N b N K N

F C b C K C

Experimental ConstantsExperimental Constants

bbcc = 3.30, = 3.30, bbBB = 2.25 = 2.25 KKcc = 4.3, = 4.3, KKBB = 15.0 = 15.0 εε = 0.25 = 0.25 MM = 0.055 = 0.055 λλ = 0.400 = 0.400 NNii = 80 = 80

Model parameter Model parameter δδ (dilution (dilution level)level)

This parameter controls how rapidly This parameter controls how rapidly the tank nitrogen level responds to a the tank nitrogen level responds to a change in input nitrogen, and also change in input nitrogen, and also how rapidly the prey population how rapidly the prey population responds to this change.responds to this change.

• At low dilution level At low dilution level δδ, both , both predator predator and prey populations and prey populations decay to zero.decay to zero.• At medium rates, the two At medium rates, the two populations populations are stable.are stable.• At higher rates, the two At higher rates, the two populations populations oscillate.oscillate.• At even higher rates, the two At even higher rates, the two populations again decay.populations again decay.

Fig. 1. Population dynamics predicted Fig. 1. Population dynamics predicted by the original model (left panels) by the original model (left panels) and observed in the chemostat and observed in the chemostat experiments (right panels). experiments (right panels).

What we would like to doWhat we would like to do

Use the data to estimate dilution rate Use the data to estimate dilution rate δδ,,

compute the fit to the data based on compute the fit to the data based on the differential equation,the differential equation,

allow for some unexplained residual allow for some unexplained residual variation,variation,

and deliver a reasonable confidence and deliver a reasonable confidence interval for interval for δδ..