The Lotka-Volterra predator-prey model

The Lotka-Volterra predator-prey model
Vlastimil Krivan
Biology Center
Academy of Sciences
and
Faculty of Sciences
University of South Bohemia
Ceske Budejovice
Czech Republic
vlastimil.krivan@gmail.com
www.entu.cas.cz/krivan

Talk outline:
1. Some historical background on foundations of mathematical ecology (U.
Ancona, V. Volterra, etc.)
2. The Gause predator-prey model with discontinuous functional response
3. Some models of animal behavior.
4. Effects of animal behavior on the Lotka-Volterra population dynamics.

“Why did the percentage of predatory fish
increase during the WWI when fishing effort
ceased?
I cannot find any reasonable way how to explain
this trend… but...I will meet Luisa and Vito tonight
…”
Table: % of predatory fish in total catch in the port of Fiume
Umberto D'Ancona
1896-1964 Fiume, Italy
Percent of
predatory fish
40
30
20
10
0
1912 1914 1916 1918 1920 1922 1924
Years

Later on that day … Vito Volterra is reading Ancona’s notes …
“Hmmm, Umberto raises interesting questions
that make me think really hard…maybe biology
will be more fun than integral equations...”
Vito Volterra
(1860 – 1940)
Nature, October 1926

…and the (Lotka-)Volterra predator-prey model was
conceived…
dR
dt
= R(r−λC)
dC
dt
= (eλR−m)C
Equilibrium=Average number:
R ∗ = m eλ and C∗ = r λ .

This had several consequences:
1. Mathematical (theoretical) ecology was born
2. Umberto married Luisa Volterra and he continued his life-long productive
collaboration with V. Volterra
INTERNATIONALE REVUE DER GESAMTEN
HYDROBIOLOGIE UND HYDROGRAPHIE
Volume 18, Issue 3-4, 1927, Pages: 261–295
Ulteriori osservazioni sulla Daphnia cucullata del
Lago di Nemi.
Per
la Dott. Luisa Volterre D’Ancona.
Con 12 Figure.
Le osservazioni da me eseguite sulla Daphnia c u c u l l a t a , originaria dal Lago Danese di
* * *

…soon after, a young student at Moscow University is reading the
Volterra’s article …
That Calculus course was really useful. The idea
about predator-prey oscillations is so co...ool.
This looks like a real lab project for me!
…and that work could finally convince the
people at the Rockefeller Foundation to give me
the fellowship to work with Raymond Pearl …’’
Georgii Frantsevich Gause
(1910–1986)

…and Gause experimented with protists…

…and with mites…
Outcome of experiments:
1. Disappearance of predators at insufficient prey densities
2. Complete destruction of prey by predators
3. Disappearance of predators in a dense population of prey

…and with protists and yeast…
Paramecium bursaria
Yeast
Observation: Periodic fluctuations,
but not of the LV conservation type

…but he never observed the Lotka-Volterra type oscillations in his experiments …
...but...
he was smart and tried to understand what assumptions of the Lotka-Volterra
model were not met in these experiments …

Instead, he postulated two new mechanisms for predator-prey coexistence:
1. Consumption of prey by predators is not a linear function of prey density
(i.e. λR in the LotkaVolterra model), but a non-linear saturating function f(R)
2. At low concentrations, prey are in a refuge (yeast in the sediment was not
accessible to predators)

…and this lead him to the Gause model…
dR
dt
= rR−Cf(R)
dC
dt
= (ef(R)−m)C
Gause‘s stability criterion: Let (R ∗ , C ∗ ) be an equilibrium of the above model.
Provided
df
dR (R∗ ) > f(R∗ )
R ∗
then the equilibrium is locally asymptotically stable.
Destabilizing f
Potentially stabilizing f

Functional response (Holling 1959)
Question: When prey density is R, how many prey a predator can capture in
time T ?
f(R) =
λR
1 + hλ R

Functional response
Population dynamics

Effect of a refuge on prey-predator population dynamics
(Gause et al. 1936)
dR
dt
dC
dt
= rR − Cf(R)
= (ef(R) − m)C
0.6
0.5
f
f
f
0.4
0.3
0.2
0.1
R c
R
0.0
0.0 r 0.5 1.0 1.5 2.0 2.5 3.0
R c
R

Using this functional response, Gause predicted existence of a predator-prey limit cycle

dR
dt
dC
dt
For R > R c :
= rR − Cf(R)
= (ef(R) − m)C
dR
dt
dC
dt
{ λR
1+λhR
if R > R c
f
f =
0 if R < R c
1.0
= rR−C λR
0.8
1 + hλR
eλR
= (
1 + hλR − m)C 0.6
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
R
0.4
For R < R c :
dR
dt
dC
dt
= rR
= −mC
R c
0.2
0.0
0.0 0.5 1.0 1.5 2.0
C
But Gause did not realize that this model is not well posed!

C⊣⊓⌋〈†√∇≀⌊↕⌉⇕⇐∞⇒
C⊣⊓⌋〈†P⌉⊣\≀⊑⊣⊔〈⌉≀∇⌉⇕¬C≀\⊔〉\⊓〉⊔†≀{fI↽⌉§〉∫⊔⌉\⌋⌉≀{∫≀↕⊓⊔〉≀\∫≀{⊔〈⌉
C⊣⊓⌋〈†P⌉⊣\≀⊑⊣⊔〈⌉≀∇⌉⇕¬C≀\⊔〉\⊓〉⊔†≀{fI↽⌉§〉∫⊔⌉\⌋〉⌉\∇≀⊑\〉⌋⌉⇐∞⇒
Existence of solutions for ODEs
dx
dt
x(0) = x 0
= f(t, x(t))
(1)
x = (x 1 , . . . , x n ) a f : I × D ⊂ R × R n → R n .
Cauchy-Peano theorem: Continuity of f =⇒ existence of solutions to the Cauchy
problem (1)

LetR n = G 1 ∪ G 2 ∪ M
Filippov definition of a solution
(Filippov 1960, 1985)
{
dx f
dt = 1 (x) x ∈ G 1
f 2 (x) x ∈ G 2
(DR)
F (x)≡
⎧
⎪⎨
⎪⎩
{f 1 (x)} x ∈ G 1
conv{lim y∈G 1
y→x
f 1 (y), lim y∈G 2
y→x
f 2 (y)}
x ∈ M
{f 2 (x)} x ∈ G 2
dx
dt ∈ F(x)
(DI)
Definition: Solutions of differential equation (DR) in the Filippov sense are
solutions of differential inclusion (DI)

Prey isocline
R
1.5
1.0
Limit cycle
R c
0.5
0.0
0.0 0.5 1.0 1.5 2.0
C
R
C

Small refuge size, small handling time
Small refuge size, larger handling time
Large refuge size, small handling time
Large refuge size, larger handling time

Gause’s work resulted in publication of his book in 1934 when he was 24 years old…

How a functional response can become discontinuous?

Behavioral response of prey to predation risk
Dragonfly larvae
(Peacor and Werner 2001)
λ

Behavioral and morphological responses of prey to
predation risk
Leoicephalus carinaturs
Predatory lizard
(Losos et al. 2006)
Anolis sagrei
Native lizard at Bahmas

Lotka-Volterra
dR
dt
= (r − λP ) R
dP
dt
= (eλR − m) P
Model with predator/prey activities
dR
dt
= (r 1 u + r 2 − (λ 1 u + λ 2 v)P ) R
dP
dt
= (e (λ 1 u + λ 2 v)R−(m 1 + m 2 v)) P
Parameter
prey per capita
growth rate
interaction strength
conversion of prey
to new predators
predator mortality
rate
Lotka-
Volterra
r constant
λ constant
e constant
m constant
Model with predator/prey activities
r 1 u+r 2
λ 1 u+ λ 2 v increases with prey & predator
activity
e constant
increases with prey activity
m 1 + m 2 v increases with predator
activity

What are the prey and predator optimal activity levels?
Herbert Spencer
Charles Darwin
“Survival of the fittest”

What are the prey and predator optimal activity levels?
Prey fitness ∼ per capita prey population growth rate =
W R = r 1 u + r 2 − (λ 1 u + λ 2 v)P
Predator fitness ∼ per capita predator population growth rate
W P = e(λ 1 u + λ 2 v)R − (m 1 + m 2 v)
0 ≤ u ≤ 1 = prey activity
0 ≤ v ≤ 1 = predator activity

Optimal prey activity
Optimal prey activity:
u(P ) =
{
1 if P < Ps = r 1
λ 1
0 if P > P s
Prey should be active if predator density is below the critical threshold. When above
they should be inactive.
Optimal predator activity:
v(R) =
{
1 if R > Rs = m 2
eλ 2
0 if R < R s
Predators should be active if the benefit from activity overweights the increased
mortality due to predator activity. Otherwise they should be inactive.

Ps
Rs
Both prey and predators are adaptive
Prey inactive
Predator inactive
Prey inactive
Predator active
Predator abundance
P s
u = 0
v = 0
Prey active
Predator inactive
u = 0
v = 1
Prey active
Predator active
u = 1
v = 0
u = 1
v = 1
R s
Prey abundance

Prey functional response
Prey functional response: f(R, P ) = (λ 1 u(P ) + λ 2 v(R))R
4
f
3
2
f
1
0
0.0 0.5 1.0 1.5 2.0
R
R
P

Lotka-Volterra
Lotka-Volterra with activity
The activity level at the population equilibrium
u = m 1λ 2
m 2 λ 1
, v = r 2λ 1
r 1 λ 2

Prey switching
Acanthina
λ 1
λ 2
Barnacles
Mussels

Prey switching
Guppies
Proportion of Drosophila
eaten
Proportion of Drosophila
In aquarium
Tubificid worm
Drosphila

A Lotka-Volterra model with two prey types
(Holt 1977, Krivan 1997)
C
dR 1
dt
= R 1 (r 1 − u 1 λ 1 C)
u 1
u 2
R 1
R 2
dR 2
dt
= R 2 (r 2 − u 2 λ 2 C)
dC
dt
= C(e 1 u 1 R 1 + e 2 u 2 R 2 − m)
u 2 = 1 − u 2
5
4
R 1
3
2
1
0
0 20 40 60 80 100
time
C
R 2
Proposition 1 (Gause Competitive exclusion principle) When preferences
u 1 and u 2 are fixed, the resource with the lower ratio of r i /(u i λ i ) will be outcompeted.

Optimal prey switching
Consumer fitness ~ per capita population growth rate=
1
C
dC
dt = e 1u 1 R 1 + e 2 u 2 R 2 − m → max
(u 1 ,u 2 )
N(x) = {(u 1 , u 2 ) | u i ≥ 0, u 1 + u 2 = 1, e 1 u 1 R 1 + e 2 u 2 R 2 − m → max}
Consumer optimal strategy:
u 1 (R 1 , R 2 ) =
{
1 for e1 R 1 > e 2 R 2
0 for e 1 R 1 < e 2 R 2
u 2 (R 1 , R 2 ) = 1−u 1 (R 1 , R 2 )

The Lotka-Volterra functional response
for prey switching
f 1 (R 1 , R 2 ) =
{ 0 e1 R 1 < e 2 R 2
λ 1 R 1 e 1 R 1 > e 2 R 2
2.0
1.5
1.0
0.5
0.0
f 1
R 1
0.0 0.5 1.0 1.5 2.0

No adaptive predation
Adaptive predation
5
4
3
2
1
0
0 20 40 60 80 100
time
5
R 1
R 1
4
R 2
3
C
C
2
1
R 2
0 20 40 60 80 100
time
Conclusion: Adaptive consumer foraging behavior relaxes apparent competition
between resources and leads to their persistence.

Proposition 1 (Boukal and Krivan 1999, Krivan and Vrkoc 2007) When
predators behave optimally, all trajectories with positive initial conditions converge
to a global atractor (shown in blue). The atractor is in the plane e 1 R 1 =
e 2 R 2 (shown in cyan) and it is formed by the solutions of differential equation
(
dR 1
r1 λ 2 + r 2 λ 1
= R 1 − λ )
1λ 2
C
dt
λ 1 + λ 2 λ 1 + λ 2
dC
= C(e 1 λ 1 R 1 − m)
dt
that satisfy C(t)≥ r 1−r 2
λ 1
(i. e., they are above the dashed line). Preferences of
the predator on the attractor satisfy
u 1 = r 1− r 2 + λ 2 C
(λ 1 + λ 2 )C
C
R 1
R 2

Population dynamics
dx i
dt = x if i (x, u),
i = 1, . . . , n
Controls (animal strategies, or phenotypic plastic traits):
u = (u 1 , . . . , u k ) ∈ U=U 1 × · · · × U k
Fitness of the i-th individuals:
G i (u i ; u, x) = f i (x, u)
Strategies that maximize animal fitness are the Nash equilibria (or Evolutionary Stable
Strategies) at current population numbers:
N(x) = { u ∈ U | G i (u i ; u, x) ≥ G i (v; u, x) for any v ∈ U i , i = 1, . . . , k } .
Feedback control:
u ∈ N(x)
This approach assumes time scale separtion: Behavioral processes operate on a much
faster time scale than do population dynamics

Question: Are these predictions falsifiable?

Population growth of bacteria on two substrates
Diauxie (J. Monod):
microbial cells consume two or more
substrates in a sequential pattern, resulting in two separate growth
phases (phase I and II). During the first phase, cells preferentially
metabolize the sugar on which it can grow faster (often glucosebut
not always). Only after the first sugar has been exhausted do the
cells switch to the second. At the time of the ”diauxic shift”, there
is often a lag period during which cells produce theenzymesneeded
to metabolize the second sugar.
Lac operon: Molecular mechanism that regulates diauxic growth (F. Jacob
and J. Monod, Nobel prize 1965)
Adaptation: Evolution should result in optimal timing of the diauxic switch
so that bacterial fitness maximizes
Question: Is the lac operon evolutionarily optimized?

Michaelis-Menten batch population kinetics
C
u 1
u 2
S 1 S 2
dS 1
dt
dS 2
dt
dC
dt
= − 1 Y 1
S 1
K 1 + S 1
u 1 C
= − 1 S 2
u 2 C
Y 2 K 2 + S
( 2
µ1 S 1
=
u 1 + µ )
2S 2
u 2 C
K 1 + S 1 K 2 + S 2
S 1 , S 2 - sugar concentration (eg. glucose and lactose)
C - bacterial population
u i -bacterial preference for the i−th (u 1 + u 2 = 1, i = 1, 2) sugar

Optimal strategy maximizing bacterial fitness
Fitness= per capita bacterial population growth rate i.e.,
1
C
dC
dt = µ 1S 1
u 1 + µ 2S 2
u 2 ↦→ max
K 1 + S 1 K 2 + S 2 u i
The optimal strategy:
µ 1 S 1
K 1 + S 1
> µ 2S 2
K 2 + S 2
=⇒ u 1 = 1
µ 1 S 1
K 1 + S 1
< µ 2S 2
K 2 + S 2
=⇒ u 1 = 0

dS 1
dt
dS 2
dt
dC
dt
= − 1 Y 1
S 1
K 1 + S 1
u 1 C
= − 1 S 2
u 2 C
Y 2 K 2 + S
( 2
µ1 S 1
=
u 1 + µ )
2S 2
u 2 C
K 1 + S 1 K 2 + S 2
u 1 =
{ 1 when
µ 1 S 1
K 1 +S 1
> µ 2S 2
K 2 +S 2
0 when
µ 1 S 1
K 1 +S 1
< µ 2S 2
K 2 +S 2
µ 1 S 1
K 1 +S 1
= µ 2S 2
K 2 +S 2

Model predictions
1. First, bacteria utilize the resource on which their population growth rate
µ
is highest, i.e., the resource with highest i S i
K i +S i
. At high concentrations
this ratio is approximately eqal to µ i .
2. Concentration of the preferred resource will decrease and at certain time
bacterial fitness on both susbstrates will be the same. Since that time
bacteria will utilize both substrates.
3. The switching time can be estimated from the model. The parameters
we need to know are only those that describe bacterial growth on a single
substrate.

Growth rate parameters of Klebsiela oxytoca on single
substrates
(Kompala et. al. 1986)
glucose
µ K YY
1.08 0.01 0.52
arabinose
1.00
0.05
0.5
fruktose
0.94
0.01
0.52
xylose
0.82
0.2
0.5
lactose
0.95
4.5
0.45

Conclusion: There is no significant difference between observed times of
switching and predicted times of switching. Thus, bacteria switch between
different sugars at times at which their fitness is maximized. This shows that
the lac operon is evolutionarily optimized.

What do these models predict?
1. Animal short term behavior has a potential to influence population dynamics
(i.e., behavioral effects do not necessarily attenuate at the longer population
time scale).
2. Adaptivity in animal behaviors promotes species coexistence, without
necessarily promoting species stability
3. It pays off to read biology literature as a source of interesting math problems

[1] Volterra V. 1926. Fluctuations in the abundance of a species considered mathematically,
Nature 118:558-560.
[2] Gause, G. F. 1934. The struggle for existence,Williams and Wilkins, Baltimore.
[3] Gause, G. F., Smaragdova, N. P. , Witt, A. A. 1936. Further studies of interaction between
predators and prey. Journal of Animal Ecology 5:1-18.
[4] V. Krivan, Dynamic ideal free distribution: Effects of optimal patch choice on predator-prey
population dynamics, American Naturalist 149 (1997) 164-178
[5] Boukal, D., Krivan, V. 1999. Lyapunov functions for Lotka-Volterra predator-prey models
with optimal foraging behavior. Journal of Mathematical Biology 39:493-517.
[6] Krivan, V. 2011. On the Gause predator-prey model with a refuge: A fresh look at the history.
Journal of Theoretical Biology 274:67-73.