Application of invasion thresholds
for the analysis of competition
in a water column
Alexey Ryabov, Bernd Blasius
University of Oldenburg
Can these plants survive if their growth rate (seed production)
is just slightly greater than mortality?
Unfavorable
Favorable
In general they cannot, because some seeds will drop
into unfavorable area and will be lost. The GLOBAL
growth rate can become smaller than mortality
To avoid this we need to isolate the environment
So all seeds will settle within the favorable area
Mathematically it’s a complex
eigenvalue problem
But it is quite clear
intuitively
Biomass, P
Survival
mmax-m
-m
0
L
growth
patch size
mortality
dispersal
Extinction
Ryabov & Blasius MMNP 2008
“Population growth and persistence…”
What if a wall does prevent diffusion?
The closer to an impenetrable boundary the
favorable patch is located the lower are the
losses into unfavorable area
0
L
0
L
0
L/2
Biomass losses decrease
Cantrell and Cosner 1998
Ryabov & Blasius MMNP 2008
“Population growth and persistence…”
Ryabov & Blaisus JTB . 2011
We consider competition for two resources in a water
column
Unfavorable
Favorable
Unfavorable
Light (I)
Biomass (P)
Nutrient (N)
For growth phytoplankton need light and a nutrient.
Production occurs in the layer where both resources
are available. So-called production layer
Ryabov & Blaisus JTB . 2011
Different species can shape environment differently
Biomass (P)
Nutrient (N)
Light (I)
Depth
Depth
Light (I)
Biomass (P)
Nutrient (N)
• How can we describe the effect of species on the resource
distribution?
• What can we say about possibility of species coexistence if we
know the effect of each species in the environment?
• How can we predict changes in community structure?
Q: What shall we do if the system were uniform?
Tilman 1980
Resource 2
A: We can represent the concentrations of the two resources as
a point in the resource space.
Supply point
Consumption
vector
𝑹∗𝟐
System state point
Zero net growth isocline
(ZNGI)
𝑹∗𝟏 Resource 1
The competitor is characterized by its minimal resource requirements (ZNGI)
Supply point - concentrations in the absence of the competitor
System state point – concentrations in the presence of the competitor
The vector from the supply point to the system state point shows the species
effect on the environment – consumption vector
For a patchy environment
Tilman 1980
Resource 2
If we would have many patches we could represent them as
separate supply points, in the case of small exchange rate
between them the patches.
Supply points
𝑹∗𝟐
System state point
Zero net growth isocline
(ZNGI)
𝑹∗𝟏 Resource 1
Different supply points results in different systems state points
For a patchy environment
Tilman 1980
Resource 2
If we would have many patches we could represent them as
separate supply points, in the case of small exchange rate
between them the patches.
Supply points
𝑹∗𝟐
System state points
Zero net growth isocline
(ZNGI)
𝑹∗𝟏 Resource 1
Different supply points results in different systems state points
But this assumes we have no unfavourable environments, and
therefore, no source – sink dynamics
Ryabov et al. JTB 2010
How can we represent resource
distribution in a water column?
Light (I)
Resource plane
Resource 2, Light
Depth
Spatial distribution
Unfavorable
Favorable
ZNGI
I*
Unfavorable
Nutrient (N)
N*
Resource 1 (Nutrient)
The system state curve shows the effect of consumer on the resource distribution
Because of the source-sink effect resource concentrations within the favorable layer
should greater than species minimal resource requirements
Ryabov et al. JTB 2010
Can we now determine if a new
species can invade or not?
Gray species cannot invade in the presence
of the black species but the black species can
invade in the presence of gray species.
Dominance of black species
Neither gray nor black species can invade in
the presence of the competitor. For gray
specie the resource levelsa are too low, for
black species the production layer is too
narrow
From this analysis we can only that a new species cannot invade if there is
no favorable layer. But the presence of a favorable layer, does not guaranty
that the new species can survive. This layer, might be still too small.
Q: How to select successful invaders?
N*resident
Depth
Growth rate
Depth
I*resident
Growth rate
Depth
A: At least
numerically we can
find a threshold
between successful
and unsuccessful
invaders in space of
resource
requirements.
The green line is
an invasion threshold.
Ryabov & Blaisus Ecol. Lett. 2011
Growth rate
Q: Slope of the invasion threshold?
It turned out that the
invasion thresholds
under quite general
conditions has the
same slope as the
system state curve
Thus is we know this
solution, we can find
which species can
invade
Ryabov & Blaisus Ecol. Lett. 2011
I*resident
N*resident
Q: How does slope depend on resource distribution?
A: We have shown that 𝜸 =
Ryabov & Blaisus Ecol. Lett. 2011
𝒄𝑰 𝒅 ln 𝐼 𝒅 ln 𝑵
=
/
𝒄𝑵
𝒅𝒛
𝒅𝒛
log 𝐼
Large g =
large gradient of
light or
small gradient of
nutrients
favors
good nutrient
competitors
Depth
log I*
log 𝐼
log 𝑵∗
Small g =
small gradient of
light or
large gradient of
nutrients
favors
good light
competitors
Depth
log I*
log 𝑵
log 𝐼
log 𝑵
log 𝑵∗
Ryabov & Blaisus Ecol. Lett. 2011
The invasion threhosl in the presenece of anothre
competitor will also ahave another slope
I*
I*
N*
N*
Red species CAN invade
Red species CANNOT invade
Green species CAN invade
Green species CAN invade
Coexistence
Green species dominance
Ryabov & Blaisus Ecol. Lett. 2011
Competition outcomes
I*
I*
N*
N*
Coexistence
I*
Green species dominance
I*
N*
Red species dominance
N*
Bistability
Ryabov & Blaisus Ecol. Lett. 2011
Q: How does the slope of invasion threshold depend on parameters?
𝜸𝒓𝒆𝒅
𝜸𝒓𝒆𝒅
gcr
Good nutrient competitor
can invade. gred>gcr
𝜸𝒈𝒓𝒆𝒆𝒏
𝜸𝒈𝒓𝒆𝒆𝒏
gcr
Good light competitor
can invade. ggreen<gcr
Nutrient supply
Now if we know how the slope changes with some parameter and how this change
depends on the species characteristics we can predict changes in species composition
Ryabov & Blaisus Ecol. Lett. 2011
Q: How does species composition will change with the parameter?
𝜸𝒓𝒆𝒅
𝜸𝒓𝒆𝒅
gcr
Good nutrient competitor
can invade. gred>gcr
𝜸𝒈𝒓𝒆𝒆𝒏
𝜸𝒈𝒓𝒆𝒆𝒏
Good light competitor
can invade. ggreen<gcr
gcr
Sp 1
Sp 2
1 or 2
Nutrient supply
Now if we know how the slope changes with some parameter and how this change
depends on the species characteristics we can predict changes in species composition
Ryabov Theor. Ecol. 2012
Q: What is the effect of environmental factors?
log I*
log I*
𝜸=
log 𝑵∗
log 𝑵∗
Good light competitors win
Good nutrient competitors win
Good light
competitor
Good nutrient
competitor
Increasing both light intensity or
the nutrient supply decreases g
Species composition depends on the absolute
level of resource supplies rather than on the
resource ratio.
𝒄𝑰
𝑳𝒊𝒈𝒉𝒕 𝒈𝒓𝒂𝒅𝒊𝒆𝒏𝒕
=
𝒄𝑵 𝑁𝒖𝒕𝒓𝒊𝒆𝒏𝒕 𝒈𝒓𝒂𝒅𝒊𝒆𝒏𝒕
Ryabov & Blaisus Am Nat 2014
Q: What does happen if the dependence of the slope g is non-monotonic?
A: There will be multiple areas of coexistence and bistability
Slope g
gcr
Good nutrient competitor
can invade. gred>gcr
Good light competitor can
invade. ggreen<gcr
Slope g
Sp 1
Sp 2
1 or 2
Sp 1
1&2
Sp 2
1 or 2
Ryabov & Blaisus Am Nat 2014
For a pair of two species we observe transitions between all possible competition outcomes
Light
supply
Nutrient supply
Nutrient supply
Ryabov & Blaisus Am Nat 2014
Q: How does the water surface effect species survival?
Deep chlorophyll
maximum
Sub-surface layer
Surface layer
Depth
0
Zbot
Species have
comparable losses
The species at shallower
depth has smaller losses
Species have comparable
losses
No spatial segregation
Ryabov & Blaisus Am Nat 2014
Q: Who does win?
Deep chlorophyll
maximum
Sub-surface layer
Surface layer
Depth
0
Zbot
Any species can win.
This depends on the
resource distributions.
Good nutrient competitor
is in more favorable
conditions
(closer to the surface )
Good light competitor
wins, because light
limitation is crucial
Ryabov & Blaisus Am Nat 2014
Species composition as a function of the average biomass depth
Descy et al. 2005, 2010
Temperature
Temperature
Temperature
22° 24° 26° 28° 30°
22° 24° 26° 28° 30°
22° 24° 26° 28° 30°
Depth , m
0
20
Temperature
40
Chlorophytes
Cyanobateria
60
80
100
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
– nutrient competitor
– light competitor
Ryabov & Blaisus Am Nat 2014
Species composition as a function of the average biomass depth
Surface
layer
Sub-surface
layer
Deep chlorophyll
maximum
Temperature
Temperature
Temperature
22° 24° 26° 28° 30°
22° 24° 26° 28° 30°
22° 24° 26° 28° 30°
Depth , m
0
20
Temperature
40
Chlorophytes
Cyanobateria
60
80
100
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
– nutrient competitor
– light competitor
Ryabov & Blaisus Am Nat 2014
Species composition as a function of the average biomass depth
0
100
80
20
(b)
40
60
60
40
(c)
20
80
0
100
5
10
15
20
25
30
35
40
45
Chlorophytes, %
good nutrient competitor
Cyanobacteria, %
good light competitor
(d)
50
Biomass average depth, m
(b)
(c) Temperature
Temperature
22° 24° 26° 28° 30°
(d)
22° 24° 26° 28° 30°
Temperature
22° 24° 26° 28° 30°
0
Temperature
Chlorophytes – nutrient competitor
Cyanobateria – light competitor
Depth, m
20
40
60
80
100
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
Ryabov & Blaisus Am Nat 2014
Species composition as a function of the average biomass depth
0
100
80
20
(b)
40
60
40
60
2
D= 4 cm /s
2
D= 5 cm /s
2
D= 6 cm /s
(c)
20
80
100
0
5
10
15
20
25
30
35
40
45
Chlorophytes, %
good nutrient competitor
Cyanobacteria, %
good light competitor
(d)
50
Biomass average depth, m
(b)
(c) Temperature
Temperature
22° 24° 26° 28° 30°
(d)
22° 24° 26° 28° 30°
Temperature
22° 24° 26° 28° 30°
0
Temperature
Chlorophytes – nutrient competitor
Cyanobateria – light competitor
Depth, m
20
40
60
80
100
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
0.0
0.5
Chl-a, mg/l
1.0
The species composition correlates with the depth of the biomass maximum
Surface
layer
Good light competitor
dominates
Light limitation
Sub-surface
layer
A refuge for a good
nutrient competitor
Boundary effect
(the surface affects
biomass losses)
Deep chlorophyll
maximum
Good light competitor
Ratio of resource
gradients
Good nutrient
competitor
Ryabov et al.
Ryabov &Blasius
Ryabov
Ryabov &Blasius
J Theor . Biol. 2010
Ecol . Lett. 2011
Theor. Ecol. 2012
Am.Nat. 2014
Many thanks for your attention!
Surface
layer
Good light competitor
dominates
Light limitation
Sub-surface
layer
A refuge for a good
nutrient competitor
Boundary effect
(the surface affects
biomass losses)
Deep chlorophyll
maximum
Good light competitor
Ratio of resource
gradients
Good nutrient
competitor
Ryabov et al.
Ryabov &Blasius
Ryabov
Ryabov &Blasius
J Theor . Biol. 2010
Ecol . Lett. 2011
Theor. Ecol. 2012
Am.Nat. 2014