consume small amount of many different
plant species
consume a lot during life to obtain sufficient
amount of N
grazers, granivores, frugivores, herbivores
plants are not killed only reduced in biomass
top-down control – herbivore abundance is
regulated by enemies
bottom-up – herbivore abundance is
regulated by quantity and quality of plants
hfNT
fNV
K
V
r
t
V
+
−⎟
⎠
⎞
⎜
⎝
⎛
−=
1
1
d
d
V .. plant biomass
N.. herbivore density
r.. intrinsic rate of increase
K.. carrying capacity
f .. efficiency of removal
Th.. handling time
Herbivory-regrowth model
Turchin (2003)
assumptions
- continuous herbivory
- herbivore is polyphagous
- plant biomass is homogenous
- functional response Type II
- herbivore density is constant
- only small quantity of biomass is removed
hyperbolic biomass growth
time
V
0
Leishmania
microparasites: viruses, bacteria, protozoans
- reproduce rapidly in host
- level of infection depends not on the number
of agents but on the host response
macroparasites - helminths
- reproduce in a vector
- level of infection depends on the number
incidence .. number of new infections
per until time
prevalence .. proportion of population
infected =1/N
swine flu virus
cercaria
nematode
E. coli (EHEC)
predicts rates of disease spread
predicts occurrence of epidemics
predicts expected level of infection
number of deaths caused by disease exceeds that of all wars
affect also animals
- rinderpest introduced by Zebu cattle to South Africa in 1890
- 90% buffalo population was wiped out
biological control
- Cydia pomonella granulosis virus
Type I
Type II
Type III
periodic eruptions
regular pattern
iregular pattern
time
N
epidemics occur in cycles
follows 4 stages:
- establishment - pathogen increases
after invasion
- persistence - pathogen persists
within host population
- spread - spreads to other
non-infected regions, reaches peak
- epidemics terminates
rabies in Europe spread from Poland
1939
- hosts: foxes, badgers, roe-deer
spread rate of 30-60 km/year
Spread of rabies (Bacon 1985)
virus
used to simulate spread of a disease in the human
population or in the biological control
models:
- Kermack & McKendrick (1927)
- later developed by Anderson & May (1980, 1981)
3 components:
- S .. susceptible
- I .. infected
- R .. resistant/recovered and immune + dead individuals - can not
transmit disease
- latent population - infected but not infectious
- vectors (V) and pathogens (P)
- malaria is transmitted by mosquitoes, hosts become infected only when
they have contact with the vector
- the number of vectors carrying the pathogens is important
- such system is further composed of uninfected and infected vectors
β .. transmission rate - number of new infections per untit time
βSI.. density-dependent transmission function (proportional to the
number of contacts)
- mass action
- analogous to search efficiency in predator-prey model
1/β .. average time for encountering infected individual
γ.. recovery rate of infected hosts
(either die or become immune)
γ = 1/duration of disease
Assumptions:
- S0 >> I0
- ignores population change (increase of S)
- incubation period is negligible
SI
t
S
β−=
d
d
ISI
t
I
γβ −=
d
d
SI model
outbreak (epidemics) will occur if
- i.e. when density of S is high
making the population size small will halt the spread:
vaccination of S, culling or isolation of I will stop disease spread
β
γ
<0S
β
γ
>0S
Outbreaks
Assumptions:
- host population is dynamic
- newborns are susceptible
- b .. host birth rate
=1/host life-span, given exponential growth and constant population size
- m .. host mortality due to other causes
Susceptible
S
Infected
I
Resistant
R
death death death
m mm
β γ
birth
b
b
recoverytransmission
b
mSSIRISb
t
S
−−++= β)(
d
d
mIISI
t
I
−−= γβ
d
d
mRI
t
R
−= γ
d
d
SIR model
RISN ++=N .. total population of hosts per area:
R0 .. basic reproductive rate of the disease
- number of secondary cases that primary infection produces
- if R0 > 1 .. outbreak is plausible
mb
N
R
++
=
γ
β
0
fast biocontrol effect is achieved only with viruses with high β
low host population is achieved with pathogens with lower β
0
200
400
600
800
1000
1200
1400
1600
1800
1949
1951
1953
1955
1957
1959
1961
1963
1965
mothdensity
0
10
20
30
40
50
60
%infected
moth infected
Population dynamic of a moth and the associated
granulosis virus
Biological control