Predator-Prey models

Intrinsic dynamics&

Stability

Patterns and processes

Extrinsic drivers of fluctuation

• The environment can exert pressures on the organisms– Press perturbations– Pulse perturbations

• Affect growth rates or mortality rates

• The organisms lag behind

Fluctuations in biodiversity

• a,b The green and black plots show the number of known marine animal genera versus time. The trend line (blue) is a third-order polynomial fitted to the data.

• c, As b, with the trend subtracted and a 62-Myr sine wave superimposed.

• d, The detrended data after subtraction of the 62-Myr cycle and with a 140-Myr sine wave superimposed.

• Rohde & Muller 2005, Nature 434, 208-210

Intrinsic patterns in simple models

• Simple difference equation models

• Time progresses in a discrete, step-wise manner

• Births and deaths described by r

• Adjust r so that more births occur below K and more deaths above K

11 ttt rNNN

11

1 1

t

ttt N

KNrNN

Growth rate around K

• Simple linear effect on r• At K, r=0• Below K, r>0• Above K, r<0• Pushes N towards K

11

1 1

t

ttt N

KNrNN

Complex Behaviour of this equation

8

21

Chaos

Multiple equilibria

Continuous time population model

• Very similar to discrete equation

• Births occur instantaneously and N grows at all times

• N tends towards K as positive and negative growth rates around it push it back to K

KNKrN

dtdN

Can’t recreate complex dynamics

Time lag is needed

• Growth rate is now a function of the population at some point in the past (T)

• Can be hard now to reach K as growth takes time

• Lemmings populations• Solving these in the

computer a little more tricky

K

TtNKrNdtdN

Predator-prey models

• Predator intake rates• Type 3 functional

response• a = encounter rate• Th = handling time• See Chapter 11 of Ted

Case’s book An Illustrated Guide to Theoretical Ecology

2

2

1 RaTaRB

h

Prey dynamics

• a = encounter rate• Th = handling time• K = prey carrying capacity

2

2

11

RaTaRC

KRrR

dtdR

h

predatorstoduedeathsgrowthdtdR ___

• R = prey density• C = predator density

Predator dynamics

deathsbirthsdtdC

wCRaT

aRkCdtdC

h

2

2

1• a = encounter rate• Th = handling time• k = prey to predator conversion

efficiency• w = mortality rate

• R = prey density• C = predator

density

Stable populations

a=0.002, r=0.5, d=0.1, k=0.5, K=100, Th=2

Oscillatory dynamics

a=0.002, r=0.5, d=0.1, k=0.5, K=100, Th=3

Boom and Bust

a=0.002, r=0.5, d=0.1, k=0.5, K=100, Th=4

Further Reading

• An illustrated guide to theoretical ecology by Ted J Case, Oxford University Press 2000.– Chapters 5,6,11,12,13

Something to think about….

• So far we have not discussed stochasticity (random processes)

• Parameters in all these models might fluctuate either according to some seasonal pattern, or might be entirely random

• Stochasticity can be a powerful driver of non-equilibrium behaviour (or can have little influence)

Next tutorial

• Shaw et al 2004. The Shape of Red Grouse Cycles. Journal of Animal Ecology 73, 767-776 http://dx.doi.org/10.1111/j.0021-8790.2004.00853.x

• Cattadori et al. 2005. Parasites and climate synchronize red grouse populations. Nature 433, 737-741. http://dx.doi.org/10.1038/nature03276(see also the supplementary material)