SLIDE 1 Last time
dN dt = [b(1 − N/k) − d]N ¯ N = 0
¯ N = k(1 − d/b) = k(1 − 1/R0) dN dt = [b − d(1 + N/k)]N ¯ N = 0
¯ N = k(b/d − 1) = k(R0 − 1) dN dt = rN(1 − N/K) ¯ N = 0
¯ N = K
Figures taken from NRC Handelsblad 11 Dec 2010 (left) and Wikipedia (right).
SLIDE 2
Theoretical Biology 2016 Chapter 3 Lotka Volterra model
SLIDE 3 Suppose measurements for the prey
(a)
mouse density R p.c. mouse birth rate b
(b)
mouse density R p.c. mouse death rate d
(c)
mouse density R mice eaten per owl dR dt = [bf(R) − d − aN]R where f(R) = 1 − R/k with maximum birth rate b, death rate d, and a predation term aRN.
y=b(1-R/k) y=aR
SLIDE 4 Suppose measurements for the prey
(a)
mouse density R p.c. mouse birth rate b
(b)
mouse density R p.c. mouse death rate d
(c)
mouse density R mice eaten per owl dR dt = [bf(R) − d − aN]R where f(R) = 1 − R/k with maximum birth rate b, death rate d, and a predation term aRN.
y=b(1-R/k) y=aR
tumor cells tumor cells tumor cells tumor cell division rate tumor cell death rate cells killed per killer cell
SLIDE 5 Measurements for the predator
where 1/δ is the expected owl life span
(d)
mice eaten per owl p.c. owl birth rate
(e)
- wl density N
- wl death rate
δ
dN dt = [caR − δ]N aR y=caR
SLIDE 6 Measurements for the predator
where 1/δ is the expected killer cell life span
(d)
mice eaten per owl p.c. owl birth rate
(e)
- wl density N
- wl death rate
δ
dN dt = [caR − δ]N aR y=caR
killer cells killer cell death rate tumor cells seen per killer cell killer cell division rate
SLIDE 7 dR dt = [bf(R) − d − aN]R where f(R) = 1 − R/k with maximum birth rate b, death rate d, and a predation term aRN. In the abscence of predators the carrying capacity is: ¯ R = k(1 − d/b) = k(1 − 1/R0) = K Number of predators: dN dt = [caR − δ]N where 1/δ is the expected life-span.
32
SLIDE 8 Steady states
Setting dR/dt = dN/dt = 0 yields R = 0 and N = 1 a [b(1 − R/k) − d] N = 0 and R = δ ca Trivial: ( ¯ R, ¯ N) = (0, 0) and ( ¯ R, ¯ N) = (K, 0). Non-trivial: ¯ N = 1 a
⇤
b
δ cak
⇥
− d
⌅
SLIDE 9 Nullclines in phase space
(a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
SLIDE 10 LV-model typically written with logistic growth
(a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
r/a
SLIDE 11 (a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
r/a What happens if we feed the prey?
Increasing K in (b):
- 1. increases the prey density
- 2. increases the predator density
- 3. increases the prey density until predators can survive
SLIDE 12 What happens if we feed the prey?
(a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
dR dt = [r(1 − R/K) − aN]R and dN dt = [caR − δ]N
r/a K
K
By feeding the prey we get predators
prey density until predators can survive
K
SLIDE 13 (a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
r/a What happens if we feed the prey?
Increasing K in (a):
- 1. increases the prey density and keeps predators the same
- 2. increases the predator density and keeps prey the same
- 3. increases both populations
SLIDE 14 What happens if we feed the prey?
(a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
dR dt = [r(1 − R/K) − aN]R and dN dt = [caR − δ]N
r/a K K
By feeding the prey we get more predators
predator density and keeps prey the same
SLIDE 15
Example: bacterial food chain
← Prey alone ← Prey with predator ← Predator
(b): The effect of nutrients on the density of prey (a): The same for prey (a: open circles) and a predator (a: closed circles). From: Kaunzinger et al. Nature 1998.
Serratia marcescens Serratia marcescens Colpidium striatium
SLIDE 16 (a)
R N
δ ca
K
b−d a (b)
R
δ ca
K dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N
dR dt = [r(1 − R/K) − aN]R and dN dt = [caR − δ]N
r/a What happens if we change a?
Prey: tumor cells Predator: killer cells a: drug changing the killing rate
SLIDE 17 1 0.5 R 1 0.5 N
Changing the predation rate a changes both
1 0.5 R 1 0.5 N
dR dt = rR(1 − R/K) − aRN , dN dt = caRN − dN .
R = d ca
N N R K K
r a
R = d ca
r a
SLIDE 18 1 0.5 R 1 0.5 N
Decreasing the predation rate increases the predator
1 0.5 b 3 1.5 N
dR dt = rR(1 − R/K) − aRN , dN dt = caRN − dN .
N N a R
R = d ca
r a
Predators with larger a have higher fitness R0
SLIDE 19 Fitness R0
For the prey R0 = b/d For the predator R0 = caR
δ
which is not a constant. Take the best possible circumstances, i.e., R = K and let R0 = caK
δ .
The prey equilibrium is at R = δ
ca or at R = K R0
This implies that a predator with an R0 = 2 is expected to halve its prey population.
SLIDE 20
Lotka Volterra model is very general
Predator-prey & host-parasite models Seals in the Waddensea infected by virus Hepatocytes infected by hepatitis Cancer cells removed by killer cells Economics: interactions industries
dR dt = [r(1 − R/K) − aN]R and dN dt = [caR − δ]N
SLIDE 21
Lotka Volterra model is very general
Many models use the Lotka Volterra equations
SLIDE 22 1 0.5 R 1 0.5 N
Increasing the killing rate decreases the killers
1 0.5 b 3 1.5 N
dR dt = rR(1 − R/K) − aRN , dN dt = caRN − dN .
N N a R
R = d ca
r a
Tumor Killer cells
SLIDE 23 For example the SARS epidemic
dI dt = [β − δ]I
dt − with R0 = 3, and β = 1.5 and δ = 0.5 per week
dS dt = rS(1 − S/K) − βSI and dI dt = βSI − δI
4 8 12 16 20 24 28 32 36 40 44 48 52
Time in weeks
10 10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
Number of people
Susceptibles Infected (Lotka Volterra) Infected (I’=(β−δ)I
SLIDE 24
Alternative: prey maintained by source
σ
dR dt = s − dN − aRN dN dt = [caR − δ]N K = s d
R N
predator remains:
δ ca
SLIDE 25 Several Lotka Volterra like models
Lotka-Volterra model (with birth and death rates): dR dt = [b(1 − R/k) − d − aN]R and dN dt = [caR − δ]N Lotka-Volterra model (with logistic growth): dR dt = [r(1 − R/K) − aN]R and dN dt = [caR − δ]N Resource maintained by a source: dR dt = s − dR − aNR and dN dt = [caR − δ]N Lotka-Volterra competition equations: dN1 dt = r1N1(1 − N1/K1 − N2/c1) and dN2 dt = r2N2(1 − N2/K2 − N1/c2)
SLIDE 26
History from Wikipedia
The Lotka–Volterra predator–prey model was proposed by Alfred J. Lotka “in the theory of autocatalytic chemical reactions” in 1910. This was effectively the logistic equation, originally derived by Pierre François Verhulst. In 1920 Lotka extended the model to "organic systems" using a plant species and a herbivorous animal species as an example, and in 1925 he utilised it to analyse predator- prey interactions in his book on biomathematics arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic independently investigated the equations in 1926.
