ix southern-summer school on mathematical biology · 2020. 1. 14. · references j.d.murray:...
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IX Southern-Summer School on Mathematical Biology
Roberto André Kraenkel, IFT
http://www.ift.unesp.br/users/kraenkel
Lecture III
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 1 / 31
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Outline
1 Competition
2 Mathematical Model
3 Interpretation!
4 Protozoa, ants and plankton!
5 References
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 2 / 31
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Outline
1 Competition
2 Mathematical Model
3 Interpretation!
4 Protozoa, ants and plankton!
5 References
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 2 / 31
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Outline
1 Competition
2 Mathematical Model
3 Interpretation!
4 Protozoa, ants and plankton!
5 References
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 2 / 31
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Outline
1 Competition
2 Mathematical Model
3 Interpretation!
4 Protozoa, ants and plankton!
5 References
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 2 / 31
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Outline
1 Competition
2 Mathematical Model
3 Interpretation!
4 Protozoa, ants and plankton!
5 References
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 2 / 31
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Competition
Consider competition betwenn two species.
We say that two species compete if the presence of one of them is detrimental forthe other, and vice versa.
The underlying biological mechanisms can be of two kinds;I exploitative competition: both species compete for a limited resource.
F Its strength depends also on the resource .I Interference competition: one of the species actively interferes in the acess to
resources of the sother .I Both types of competition may coexist.
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Models for species in competition
We are speaking of inter-specific competitionIntra-specific competition gives rise to the models like the logisticthat we studied in the first lecture.In a broad sense we can distinguish two kinds of models forcompetition:
I implicit: that do not take into account the dynamics of the resources.I explicit where this dynamics is included.
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Mathematical Model
Let us begin with the simplest case:I Two species,I Implicit completion model,I intra-specific competition taken into account.
We proceed using the same rationale that was used for thepredator-prey system.
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Lotka-Volterra model for competition
Let N1 and N2 be the two species in question.
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Lotka-Volterra model for competition
Each of them increases logistically in the absence of the other:
dN1
dt= r1N1
[1−
N1
K1
]
dN2
dt= r2N2
[1−
N2
K2
]
where r1 and r2 are the intrinsic growth rates and K1 and K2 are thecarrying capacities of both species in the absence of the other..
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Lotka-Volterra model for competition
We introduce the mutual detrimental influence of one species on the other:
dN1
dt= r1N1
[1−
N1
K1− aN2
]
dN2
dt= r2N2
[1−
N2
K2− bN1
]
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Lotka-Volterra model for competition
Or, in the more usual way :
dN1
dt= r1N1
[1−
N1
K1− b12
N2
K1
]
dN2
dt= r2N2
[1−
N2
K2− b21
N1
K2
]
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Lotka-Volterra model for competition
Or, in the more usual way:
dN1
dt= r1N1
1− N1
K1−
↓︷︸︸︷b12
N2
K1
dN2
dt= r2N2
1− N2
K2−
↓︷︸︸︷b21
N1
K2
where b12 and b21 are the coefficients that measure the strength of the
competition between the populations.
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Lotka-Volterra model for competition
This is a Lotka-Volterra type model for competing species. Pay attentionto the fact that both interaction terms come in with negative signs. All the
constants r1, r2,K1,K2, b12and b21 are positive.
dN1
dt= r1N1
[1−
N1
K1− b12
N2
K1
]
dN2
dt= r2N2
[1−
N2
K2− b21
N1
K2
]
Let’s now try to analyze this system of two differential equations .
Roberto A. Kraenkel (IFT-UNESP) IX SSSMB January 2020 6 / 31
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Analyzing the model I
dN1
dt= r1N1
[1 −
N1
K1− b12
N2
K1
]
dN2
dt= r2N2
[1 −
N2
K2− b21
N1
K2
]
We will first make a change of variables, bysimple re-scalings.
Define:
u1 =N1
K1, u2 =
N2
K2, τ = r1t
In other words,we are measuring populationsin units of their carrying capacities and thetime in units of 1/r1.
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Analyzing the model II
du1
dτ= u1
[1− u1 − b12
K2
K1u2
]
du2
dτ=
r2r1
u2
[1− u2 − b21
K1
K2u1
]
The equations in
the new variables.
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Analyzing the model III
du1
dτ= u1 [1− u1 − a12u2]
du2
dτ= ρu2 [1− u2 − a21u1]
Defining:
a12 = b12K2
K1,
a21 = b21K1
K2
ρ =r2r1
we get these equations.It’s a system of nonlinear ordinarydifferential equations.
We need to study the behavior of their solutions
.
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Analyzing the model IV
du1
dτ= u1 [1− u1 − a12u2]
du2
dτ= ρu2 [1− u2 − a21u1]
No explicit solutions!.
We will develop a qualitative analysis of these equations.
Begin by finding the points in the (u1 × u2) plane such that:
du1
dτ=
du2
dτ= 0,
the fixed points.
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Analyzing the model V
du1dτ
= 0⇒ u1 [1− u1 − a12u2] = 0
du2dτ
= 0⇒ u2 [1− u2 − a21u1] = 0
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Analyzing the model V
u1 [1− u1 − a12u2] = 0
u2 [1− u2 − a21u1] = 0
These are two algebraic equations for ( u1 e u2).We FOUR solutions. Four fixed points.
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Fixed points
u∗1 = 0
u∗2 = 0
u∗1 = 1
u∗2 = 0
u∗1 = 0
u∗2 = 1
u∗1 =
1− a121− a12a21
u∗2 =
1− a211− a12a21
The relevance of those fixed points depends on their stability. Which, in turn, depend on thevalues of the parameters a12 e a21. We have to proceed by a phase-space analysis, calculatingcommunity matrixes and finding eigenvalues......take a look at J.D. Murray ( MathematicalBiology).
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Stability
If a12 < 1 and a21 < 1
u∗1 =
1− a121− a12a21
u∗2 =
1− a211− a12a21
is stable.
If a12 > 1 and a21 > 1
u∗1 = 1 e u∗
2 = 0
u∗1 = 0 e u∗
2 = 1
are both stable.
If a12 < 1 and a21 > 1
u∗1 = 1 e u∗
2 = 0
is stable.
If a12 > 1 and a21 < 1
u∗1 = 0 e u∗
2 = 1
is stable.
The stability of the fixed points depends on the values of a12 and a21.
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Phase space
To have a more intuitive understanding of the dynamics it is useful toconsider the trajectories in the phase spaceFor every particular combination of a12 and a21 – but actuallydepending if they are smaller or greater than 1 – ,we will have aqualitatively different phase portrait.
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Phase Space II
Figura: The four cases. The four different possibilities for the phase portraits.
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Coexistence
Figura: a12 < 1 and a21 < 1. The fixed point u∗1 and u∗
2 is stable and represents thecoexistence of both species. It is a global attractor.
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Exclusion
Figura: a12 > 1 and a21 > 1. The fixed point u∗1 and u∗
2 is unstable. The points (1.0) and (0, 1) are stable buthave finite basins of attraction, separated by a separatrix. The stable fixed points represent exclusionof one of thespecies.
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Exclusion
Figura: a12 < 1 and a21 > 1. The only stable fixed is (u1 = 1, u2 = 0).A global attractor. Species (2) isexcluded.
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Exclusion
Figura: This case is symmetric to the previous. a12 > 1 and a21 < 1. The only stablefixed point is (u1 = 1, u2 = 0). A global attractor. Species (1) is excluded
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Interpretation of the results
What is the meaning of these results?Let us recall the meaning of a12 and a21:
du1
dτ= u1 [1− u1 − a12u2]
du2
dτ= ρu2 [1− u2 − a21u1]
I a12 is a measure of the influence of species 2 on species 1. How detrimental 2 is to1.
I a21 measures the influence of species 1on species 2. How detrimental 1 is to 2.
So, we may translate the results as:I a12 > 1⇒ 2 competes strongly with 1 for resources.I a21 > 1⇒ 1 competes strongly with 2 for resources.
This leads us to the following rephrasing of the results :
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If a12 < 1 and a21 < 1The competition is weak and both can coexist.
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If a12 > 1 and a21 > 1The competition is mutually strong . One species always excludes the
other. Which one "wins"depends on initial conditions.
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If a12 < 1 e a21 > 1Species 1 is not strongly affected by species 2. But species 2 is affectedstrongly be species 1. Species 2 is eliminated, and species 1 attains it
carrying capacity.
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Se a12 > 1 e a21 < 1This is symmetric to the previous case. Species 1 is eliminated and
Species 2 attains its carrying capacity
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Competitive exclusion
In summary: the mathematical model predicts patterns of exclusion.Strong competition always leads to the exclusion of a speciesCoexistence is only possible with weak competition.The fact the a stronger competitor eliminates the weaker one is knownas the competitive exclusion principle.
Georgiy F. Gause (1910-1986), Russian biolo-
gist, was the first to state the principle of com-
petitive exclusion (1932).
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Paramecium
The experiences of G.F. Gause where performed with a protozoa groupcalled Paramecia.
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Paramecium
The experiences of G.F. Gause where performed with a protozoa groupcalled Paramecia .Gause considered two of them: Paramecium aurelia e ParameciumCaudatum.
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Paramecium
The experiences of G.F. Gause where performed with a protozoa groupcalled Paramecia .Gause considered two of them: Paramecium aurelia e Parameciumcaudatum. They where allowed to grow initially separated, with a logisticlike growth .
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Paramecium
The experiences of G.F. Gause where performed with a protozoa groupcalled Paramecia .Gause considered two of them: Paramecium aurelia e ParameciumCaudatum. They where allowed to grow initially separated, with a logisticlike growth .When they grow in the same culture, P. aurelia survives and P. caudatum iseliminated.
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Paramecium
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Paramecium
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Paramecium
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Ants
Figura: The Argentinean ant (Linepithema humile) and the Californian one(Pogonomyrmex californicus)
The introduction of the Argentinean ant in California had the effect toexclude Pogonomyrmex californicus.Here is a plot with data....
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Ants II
Figura: The introduction of the Argentinean ant in California had the effect ofexcluding Pogonomyrmex californicus
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Plankton
In view of the principle of competitive exclusion, consider the situation of phytoplankton.
Phytoplankton are organisms that live in seas andlakes, in the region where there is light.
You won’t see a phytoplankton with naked eye..
You can see only the visual effect of a large numberof them.
It needs light + inorganic molecules.
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The Plankton Paradox
The plankton paradox consists of the following:There are many species of phytoplankton. It used a very limitednumber of different resources. Why is there no competitive exclusion?
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One paradox, many possible solutions
Competitive exclusion is aproperty of the fixed points. Butif the environment changes, theequilibria might not be attained.We are always in transientdynamics.
We have considered no spatialstructure. Different regions couldbe associated with differentlimiting factors, and thus couldpromote diversity.
Effects of trophic webs.
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References
J.D. Murray: Mathematical Biology I (Springer, 2002)F. Brauer e C. Castillo-Chavez: Mathematical Models in PopulationBiology and Epidemiology (Springer, 2001).N.F. Britton: Essential Mathematical Biology ( Springer, 2003).R. May e A. McLean: Theoretical Ecology, (Oxford, 2007).N.J. Gotelli: A Primer of Ecology ( Sinauer, 2001).G.E. Hutchinson: An Introduction to Population Ecology ( Yale,1978).
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Online Resources
http://ecologia.ib.usp.br/ssmb/
Thank you for your attention
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